advances in nuclear science and technology: volume 25 (advances in nuclear science & technology)

Advances inNuclear Scienceand TechnologyVOLUME 25

Advances inNuclear Scienceand TechnologySeries Editors

Jeffery LewinsCambridge University, Cambridge, England

Martin BeckerOregon Graduate Institute of Science and TechnologyPortland, Oregon

Editorial Board

R. W. AlbrechtErnest J. HenleyJohn D. McKeanK. OshimaA. SesonskeH. B. SmetsC. P. L. Zaleski

A Continuation Order Plan is available for this series. A continuation order will bring deliveryof each new volume immediately upon publication. Volumes are billed only upon actualshipment. For further information please contact the publisher.


Edited by

Jeffery LewinsCambridge UniversityCambridge, England

and

Martin BeckerOregon Graduate Institute of Science and TechnologyPortland, Oregon

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PREFACE

The present review volume not only covers a wide range of topics pertinentto nuclear science and technology, but has attracted a distinguished internationalauthorship, for which the editors are grateful. The opening review by Drs. JanetTawn and Richard Wakeford addresses the difficult matter of questioning scien-tific hypotheses in a court of law. The United Kingdom experienced a substantialnuclear accident in the 1950s in the form of the Windscale Pile fire. This in itselfhad both good and bad consequences; the setting up of a licensing authority toensure nuclear safety was one, the understandable public sentiment concerningnuclear power (despite the fire occurring in a weapons pile) the other. Windscaletoday is subsumed in the reprocessing plant at Sellafield operated by BritishNuclear Fuels plc and it was inevitable perhaps that when an excess cluster ofchildhood leukaemia was observed in the nearby village of Seascale that publicconcern should be promoted by the media, leading to the hearing of a claim ofcompensation brought on behalf of two of the families of BNFLs workers whohad suffered that loss. The review article demonstrates the complexity of under-standing such a claim against the statistical fluctuations inherent and shows howthe courts were persuaded of the need to propose a biological mechanism ifresponsibility were to be held. The Company were undoubtedly relieved by thefinding. At the same time, the concerns raised led to a deeper understanding ofsuch matters which our authors summarise admirably.

An analogous technique involving stochastic modelling, better known toour readers perhaps as the Monte Carlo method, is shown in the next chapter tobe usefully applied to the problem of determining parameters for elementaryreactor dynamics equations. From the United States, Tim Valentine, however,shows that this view is unnecessarily limited; the Monte Carlo simulation can beused more directly to evaluate time-dependent development in nuclear reactorsof considerable (and therefore realistic) complexity. It makes particular sense inextracting cross-spectral data for noise analysis, as the author shows.

Chapter 4 is also from a United Kingdom author. It may have a further linkto the first through the word “gene” but Jonathan Carter actually takes us throughthe biological analogy of the genetic optimisation algorithm that has provedsuccessful in optimising reactor cores. It might be recollected that despite thecharacterisation of nuclear power as capital intensive, running cost cheap,nevertheless as much money is committed in the plant lifetime to supply the fuelas is spent on the original construction. Improvements in the specification ofreload cores can then provide dramatic returns. The genetic algorithm has proveda powerful way to optimise the highly constrained and very large, non-linearproblem that emerges. The techniques, including new developments reviewedhere, have application to a wide range of problems, such as oil-well diagnostics.Perhaps the most exciting is the “tabu” algorithm that might with advantage bemade known to a wider audience therefore than the nuclear community.

v

Professor Marseguerra, our Italian expert, furthers the dynamic under-standing of reactor systems by reviewing the foundations of wavelet analysis.Fourier analysis and Fourier transformation have not only great practical strengthin applications but have led to a deep understanding of functional analy-sis—readers may recollect that Fourier himself was denied the Gold Medal inthree successive competitions before the deep significance of his work wasappreciated, so remarkable a development it was. There can be little doubt thatwavelet analysis provides a powerful technique which will be of value to theprofessional nuclear engineer; our Italian author can be thanked for offering justsuch an interpretation.

From Belgium in our next article, but by an author again of internationalstanding, Professor Devooght brings us a further stochastic study, the applicationof Monte Carlo simulation to the study of system reliability. The review showsclearly why the Monte Carlo method is efficient in solving complex problemsand takes us further into the problem of employing variational-adjoints and othermethods to promote accuracy in the face of rare events. Our author provides asignpost to modern studies of dynamic reliability that can be expected to providemore realistic estimates of the hazards of a real and changing world.

The TMI accident and the Chernobyl accident emphasised the centralsignificance of both the plant operators and the control room. Modern computersshould make it easier for the operators to function, routinely, in accident condi-tions—and not overlooking the need for improved production efficiency. But todo this requires a blend of hardware, software, and knowledge of humanbehaviour. A jointly authored review of the problems in computerising controlrooms is provided by Dr. Sun, from the US, and his colleague Dr. Kossilov(formerly of the IAEA), both acknowledged experts in this area combiningelectronic-digital technology with ergonomics.

The clew of this volume, however, returns to matters arising from Cher-nobyl itself, the nuclear accident of epic proportions that has had such wideimplications. But although much of the world may be concerned with the publicimage of nuclear power in the aftermath of the 1986 accident, in Russia, Ukraine,and Byelorussia (Belarus), there is an immediate, substantial, and very practicalproblem. Our Russian authors from the Kurchatov Institute provide an accountof the consequences of Chernobyl that many will find profoundly moving in thewholesale involvement of so many workers in the clean-up process and thespecification of the difficulties yet to be overcome. To compare these problemswith those facing colleagues in the rest of the world is sobering indeed.

As the Christian millennium looms larger, it is right to seek a balancedview of nuclear power. Clearly France and many Far Eastern countries hold to acontinuing dependence, in some cases enlargement, on nuclear power to produceelectricity. The stations must be operated safely; the processing industry must beoperated efficiently; and above all politicians must come to grips with questionsof disposal versus waste storage. But at the same time, burgeoning concerns overthermal warming and the need to contain carbon-dioxide release and the finitesupply of fossil fuels, are just examples of the reassessment that may yet see the

vi PREFACE

next century accept a balanced programme of nuclear power. Much depends ona demonstration of renewed confidence by the United States of America. Wehope to keep our readers not only abreast of the changes but to contribute to theirability to bring them about. For this, as always, we thank our distinguishedauthors for their timely contribution.

Jeffery LewinsMartin Becker

vii

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Childhood Leukaemia and Radiation: The Sellafield Judgment

E. Janet Tawn and Richard Wakeford

1.2.3.4.5.6.7.8.9.

10.11.12.13.14.

Reactor Dynamics from Monte Carlo Calculations

Timothy E. Valentine

1.2.3.4.5.6.7.

Notes on a Simplified Tour: From the Fourier to the Wavelet Transform

Marzio Marseguerra

1.2.3.4.5.6.

Introduction

ix

Childhood Leukaemia around Nuclear InstallationsThe Gardner ReportThe Legal CasesThe Nature of EpidemiologyBackground to the Gardner Hypothesis and Subsequent FindingsRadiation Genetic RiskGenetic Risk from Sellafield Occupational Radiation ExposureHeritability of LeukaemiaEtiology and Heterogeneity of CasesAnimal StudiesDose Rate EffectsUnconventional Genetic MechanismsThe Legal Judgment and Aftermath

References

IntroductionReview of Statistics of Stochastic ProcessesReactor Transfer FunctionsTime Delay EstimationMonte Carlo SimulationApplication to the Advanced Neutron Source ReactorSummary

References

IntroductionPreliminariesThe Continuous Windowed Fourier TransformFramesThe Continuous Wavelet TransformThe Discrete Windowed Fourier TransformThe Discrete Wavelet Transform

CONTENTS

1314151718181920

22

124579

31323438394351

51

53545865697478

x CONTENTS

7.8.9.

The Multiresolution AnalysisSub-Band FilteringConclusions

References

Genetic Algorithms for Incore Fuel Management and Other RecentDevelopments in Optimisation

Jonathan N. Carter

1.2.3.4.5.

IntroductionOptimisation ProblemsOptimisation Algorithms for Combinatorial OptimisationOptimisation Methods for Continuum ProblemsConclusions

References

Appendix: Gray Coding

The Computerization of Nuclear Power Plant Control Rooms

Bill K.H. Sun and Andrei N. Kossilov

1.2.3.4.5.6.7.

IntroductionHuman-Machine InterfaceComputerization in Nuclear Power Plant Control RoomsSafety and LicensingImplementation and Maintenance IssuesFuture TrendsConclusions

References

Consequences of Chernobyl: A View Ten Years on

A. Borovoi and S. Bogatov

1.2.3.4.5.6.7.8.9.

10.

IntroductionThe AccidentArea PollutionCreation of the Sarcophagus: Its Advantages and ShortcomingsResearch Activities Associated with the SarcophagusWhat is the Threat from the Sarcophagus?Necessity and Strategy for the Transformation of the SarcophagusRemediation of Contaminated AreasMedical Consequences: Residence in the Contaminated AreasConclusion

References

81103111

112

113113121144147

149

153

155156158163165167168

169

171172182190191192199200203211

212

CONTENTS xi

Dynamic Reliability

Jacques Devooght

1.2.3.4.5.6.7.8.9.

10.11.

12.

INDEX

IntroductionPhysical SettingThe Chapman-Kolmogorov EquationsReduced FormsExit ProblemsSemi-Markovian GeneralizationSubdynamicsApplication to Event TreesSemi Numerical MethodsThe Monte Carlo MethodExamples of Numerical Treatment of Problems of DynamicReliabilityConclusions

References

215216218222227230233236242245

261272

274

279

CONTENTS OF EARLIER VOLUMES1

CONTENTS OF VOLUME 10

Optimal Control Applications in Nuclear Reactor Design and Operations,W. B. Terney and D. C. Wade

Extrapolation Lengths in Pulsed Neutron Diffusion Measurements,N. J. Sjsötrand

Thermodynamic Developments, R. V. Hesketh

Kinetics of Nuclear Systems: Solution Methods for the Space-TimeDependent Neutron Diffusion Equation, W. Werner

Review of Existing Codes for Loss-of-Coolant Accident Analysis,Stanislav Fabic


Nuclear Physics Data for Reactor Kinetics, J. Walker and D. R. Weaver

The Analysis of Reactor Noise: Measuring Statistical Fluctuations in NuclearSystems, N. Pacilio, A. Colombina, R. Mosiello, F. Morelli and V. M. Jorio

On-Line Computers in Nuclear Power Plants—A Review, M. W. Jervis

Fuel for the SGHWR, D. O. Pickman, J. H. Gittus and K. M. Rose

The Nuclear Safety Research Reactor (NSSR) in Japan, M. Ishikawa andT. Inabe

Practical Usage of Plutonium in Power Reactor Systems, K. H. Peuchl

Computer Assisted Learning in Nuclear Engineering, P. R. Smith

Nuclear Energy Center, M. J. McKelly


Characteristic Ray Solutions of the Transport Equation, H. D. Brough andC. T. Chandler

Heterogeneous Core Design for Liquid Metal Fast Breeder Reactors,P. W. Dickson and R. A. Doncals

Liner Insulation for Gas Cooled Reactors, B. N. Furber and J. Davidson

Outage Trends in Light Water Reactors, E. T. Burns, R. R. Pullwood andR. C. Erdman

Synergetic Nuclear Energy Systems Concepts, A. A. Harms

Volumes 1–9 of the series were published by Academic Press.

xiii

1

Vapor Explosion Phenomena with Respect to Nuclear Reactor SafetyAssessment, A. W. Cronenberg and R. Benz


Radioactive Waste Disposal, Horst Böhm and Klaus Kühn

Response Matrix Methods, Sten-Oran Linkahe and Z. J. Weiss

Finite Approximations to the Even-Parity Transport Equation, E. E. Lewis

Advances in two-Phase Flow Instrumentation, R. T. Lahey and S. Benerjee

Bayesian Methods in Risk Assessment, George Apostolakis


Introduction: Sensitivity and Uncertainty Analysis of Reactor PerformanceParameters, C. R. Weisben

Uncertainty in the Nuclear Data used for Reactor Calculations, R. W. Peeble

Calculational Methodology and Associated Uncertainties, E. Kujawski andC. R. Weisben

Integral Experiment Information for Fast Reactors, P. J. Collins

Sensitivity Functions for Uncertainty Analysis, Ehud Greenspan

Combination of Differential and Integral Data, J. H. Marable, C. P. Weisbinand G. de Saussure

New Developments in Sensitivity Theory, Ehud Greenspan


Eigenvalue Problems for the Boltzmann Operator, V. Protopopescu

The Definition and Computation of Average Neutron Lifetimes, Allen F. Henry

Non-Linear Stochastic Theory, K. Saito

Fusion Reactor Development: A Review, Weston M. Stacey, Jr.

Streaming in Lattices, Ely M. Gelbard


Electrical Insulation and Fusion Reactors, H. M. Bamford

Human Factors of CRT Displays for Nuclear Power Plant Control,M. M. Danchak

Nuclear Pumped Lasers, R. T. Schneider and F. Hohl

Fusion-Fission Hybrid Reactors, E. Greenspan

Radiation Protection Standards: Their Development and Current Status,G. C. Roberts and G. N. Kelly

xiv CONTENTS OF EARLIER VOLUMES


A Methodology for the Design of Plant Analysers, T. H. E. Chambers andM. J. Whitmarsh- Everies

Models and Simulation in Nuclear Power Station Design and Operation,M. W. Jervis

Psychological Aspects of Simulation Design and Use, R. B. Stammers

The Development of Full-Scope AGR Training Simulators within theC. E. G. B., C. R. Budd

Parallel Processing for Nuclear Safety Simulation, A. Y. Allidina, M. C. Singhand B. Daniels

Developments in Full-Scope, Real-Time Nuclear Plant Simulators, J. Wiltshire


Realistic Assessment of Postulated Accidents at Light Water Reactor NuclearPower Plants, E. A. Warman

Radioactive Source Term for Light Water Reactors, J. P. Hosemann andK. Hassman

Multidimensional Two-Phase Flow Modelling and Simulation, M. Arai andN. Hirata

Fast Breeder Reactors—The Point of View of French Safety Authorities,M. Laverie and M. Avenas

Light Water Reactor Space-Dependent Core Dynamics Computer Programs,D. J. Diamond and M. Todosow


Festschrift to Eugene Wigner

Eugene Wigner and Nuclear Energy, A. M. Weinberg

The PIUS Principle and the SECURE Reactor Concepts, Kåre Hannerz

PRISM: An Innovative Inherently Safe Modular Sodium Cooled BreederReactor, P. H. Pluta, R. E. Tippets, R. E. Murata, C. E. Boardman,

C. S. Schatmeier, A. E. Dubberley, D. M. Switick and W. Ewant

Generalized Perturbation Theory (GPT) Methods; A Heuristic Approach,Augusto Gandini

Some Recent Developments in Finite Element Methods for Neutron Transport,R. T. Ackroyd, J. K. Fletcher, A. J. H. Goddard, J. Issa, N. Riyait,M. M. R. Williams and J. Wood

CONTENTS OF EARLIER VOLUMES xv


The Three-Dimensional Time and Volume Averaged Conservation Equationsof Two-Phase Flow, R. T. Lahey, Jr. , and D. A. Drew

Light Water Reactor Fuel Cycle Optimisation: Theory versus Practice,Thomas J. Downar and Alexander Sesonske

The Integral Fast Reactor, Charles E. Till and Yoon I. Chang

Indoor Radon, Maurice A. Robkin and David Bodansky


Nodal Methods in Transport Theory, Ahmed Badruzzaman

Expert Systems and Their Use in Nuclear Power Plants, Robert E. Uhrig

Health Effects of Low Level Radiation, Richard Doll and Sarah Darby

Advances in Optimization and Their Applicability to Problems in the Field ofNuclear Science and Technology, Geoffrey T. Parks

Radioactive Waste Storage and Disposal in the U. K., A. D. Johnson,P. R. Maul and F. H. Pasant


High Energy Electron Beam Irradiation of Water, Wastewater and Sludge,Charles N. Kurucz, Thomas D. Waite, William J. Cooper andMichael J. Nickelsen

Photon Spectroscopy Calculations, Jorge F. Fernández andVincenzo G. Molinari

Monte Carlo Methods on Advanced Computer Architecture, William R. Martin

The Wiener-Hermite Functional Method of Representing Random Noise andits Application to Point Reactor Kinetics Driven by Random ReactivityFluctuations, K. Behringer


Contraction of Information and Its Inverse Problems in Reactor SystemIdentification and Stochastic Diagnosis, K. Kishida

Stochastic Perturbation Analysis Applied to Neutral Particle Transfers,Herbert Rieff

Radionuclide Transport in Fractured Rock: An Analogy with NeutronTransport, M. M. R. Williams


Chernobyl and Bhopal Ten Years on, Malcolm C. Grimston

xvi CONTENTS OF EARLIER VOLUMES

Transport Theory in Discrete Stochastic Mixtures, G.C. Pomraning

The Role of Neural Networks in Reactor Diagnostics and Control, Imre Pázsitand Masaharu Kitamura

Data Testing of ENDF/B-VI with MCNP: Critical Experiments,Thermal-Reactor Lattices and Time-of-Flight Measurements,

Russell D. Mosteller, Stephanie C. Frankle, and Phillip G. Young

System Dynamics: An Introduction and Applications to the Nuclear Industry,K.F. Hansen and M.W. Golay

Theory: Advances and New Models for Neutron Leakage Calculations,Ivan Petrovic and Pierre Benoist

Current Status of Core Degradation and Melt Progression in Severe LWRAccidents, Robert R. Wright

CONTENTS OF EARLIER VOLUMES xvii

CHILDHOOD LEUKAEMIA AND RADIATION: THE SELLAFIELDJUDGMENT

E. Janet Tawn1 and Richard Wakeford2

1Westlakes Research InstituteMoor RowCumbria CA24 3JZUK

2British Nuclear Fuels plcRisleyWarrington WA3 6ASUK

INTRODUCTION

In October 1993, a year from the commencement of court proceedings, Judgmentwas given in the High Court of Justice, London, for the Defendants, British Nuclear Fuelsplc (BNFL) in cases brought by two individuals claiming compensation for the causationof leukaemia/lymphoma. This review will examine the events leading up to the trial andthe scientific issues which were raised then, and subsequently, in addressing the questionof a possible role for ionising radiation in excesses of childhood leukaemia occurringaround nuclear installations.

Ten years earlier a television documentary “Windscale - the Nuclear Laundry” haddrawn attention to an apparent excess of childhood leukaemia in the coastal village ofSeascale 3 km from the Sellafield nuclear installation in West Cumbria. The “cluster”consisted of 6 cases which had occurred since 1950, when nuclear operations commencedat Sellafield. This number of observed cases was about 10 times the number expected onthe basis of national rates. The documentary suggested that exposure to radiation fromradioactive discharges from Sellafield might be responsible for this excess of childhoodleukaemia. The concern generated by this programme prompted the UK Government toset up an Independent Advisory Group, chaired by Sir Douglas Black, to enquire intothese claims, and in 1984, the Group confirmed the excess of childhood leukaemia.1

However a radiological assessment carried out by the National Radiological ProtectionBoard (NRPB) found that radiation doses to the children of Seascale from Sellafielddischarges were several hundred times too small to account for the excess.2 This led tospeculation that the risk of radiation-induced childhood leukaemia in Seascale had beengreatly underestimated3 and a programme of scientific work was recommended to clarify

Advances in Nuclear Science and Technology, Volume 25Edited by Lewins and Becker, Plenum Press, New York, 1997 1

the risk of childhood leukaemia in Seascale, including a case-control study of leukaemiaand lymphoma in West Cumbria.

The report of the Black Advisory Group1 also recommended that an expert groupwith “significant health representation” should be set up to examine the effects ofenvironmental radioactivity and the Committee on Medical Aspects of Radiation in theEnvironment (COMARE) was established in 1985. COMARE’s first task was to reassessdischarges from Sellafield following suggestions that certain emissions of radioactivematerial had been underestimated, in particular the atmospheric releases of irradiateduranium oxide particles from the chimneys of the early air-cooled military reactors atSellafield (the “Windscale Piles”) during the mid-1950s. A reassessment of discharges4

did indicate that particular releases had been underestimated in the original assessmentcarried out for the Black Advisory Group, but the revision of the discharge record hadrelatively little impact upon the radiological reassessment carried out by the NRPBbecause the original assessment had been based, wherever possible, upon environmentalmeasurements rather than discharge records. The First Report of COMARE5 found thatthe conclusions of the Black Advisory Group,1 that Sellafield radioactive discharges couldnot account for the excess of childhood leukaemia in Seascale, remained unchanged by theadditional discharge information, but the Report expressed dissatisfaction over the waythis information had come to light. The COMARE First Report5 reinforced therecommendations of the Report of the Black Advisory Group.1

Two of the recommendations of the Black Advisory Group1 concerned cohort studiesof children born in Seascale and children attending schools in Seascale. These cohortstudies would provide more secure information than the geographical correlation studieswhich had identified the excess of childhood leukaemia in Seascale. The results of thesecohort studies were published in 1987.6,7 The excess of childhood leukaemia wasconfirmed although the results suggested that the excess was concentrated among thoseborn in the village. The authors considered that this finding might indicate that one ormore risk factors might be acting on a “locality specific basis before birth or early in life”.

CHILDHOOD LEUKAEMIA AROUND NUCLEAR INSTALLATIONS

Meanwhile, several other reports of raised levels of childhood leukaemia aroundcertain other nuclear installations in Britain were being made. In particular, researchersfrom the Scottish Health Service reported that a tenfold excess of leukaemia had occurredamong young persons living within 12½ km of the Dounreay nuclear establishment innorthern Scotland during 1979-84.8 This finding was based upon 5 cases. In addition, anexcess of childhood leukaemia cases was found around the nuclear weapons facilities atAldermaston and Burghfield in West Berkshire.9 COMARE investigated both of thesereports and they became the subject of the COMARE Second Report10 and Third Report.11

The COMARE Second Report10 was published in 1988 and examined the incidenceof leukaemia and non-Hodgkin’s lymphoma among young people under 25 years of ageliving in the vicinity of Dounreay. The Committee confirmed the excess of childhoodleukaemia around Dounreay, particularly in the coastal town of Thurso about 13 km eastof the site, but noted that the excess was confined to the period 1979-84 out of the entireperiod 1968-84 available for study. Also, the Thurso cases were restricted to the westernpart of the town, within 12½ km of Dounreay. There was no apparent reason for thisspace-time pattern, and COMARE were cautious in their interpretation. Nevertheless, theobservation of an excess of childhood leukaemia around the only other nuclear fuelreprocessing plant in Britain led COMARE to suggest that some “feature” of Sellafieldand Dounreay might be leading to an excess risk of childhood leukaemia around the sites.

2 E. JANET TAWN AND RICHARD WAKEFORD

Again, a detailed radiological assessment carried out by the NRPB demonstrated thatradiation doses due to radioactive discharges were much too low (by a factor of around1000) to account for the excess of childhood leukaemia in Thurso.12,13 COMARErecognised that conventional radiation risk assessment could not account for the raisedlevel of childhood leukaemia around Dounreay, but the Committee suggested that researchbe undertaken to investigate whether there might exist unrecognised routes wherebyradiation exposure could materially raise the risk of childhood leukaemia. The COMARESecond Report10 made a number of recommendations for scientific work to be carried outto investigate this possibility.

The COMARE Third Report11 was published in 1989 and dealt with the observationof a raised level of childhood leukaemia around Aldermaston and Burghfield.9 TheCommittee confirmed an excess of leukaemia in young children living in the area, andnoted that the excess also extended to other childhood cancers, a phenomenon which hadnot been observed in other studies. A study of childhood cancer around the nearbynuclear establishment at Harwell which was carried out for the COMARE Third Report11

found no evidence of an excess risk of childhood leukaemia around this site. Anotherdetailed radiological assessment carried out by the NRPB14 demonstrated yet again thatradiation doses to children from discharges of radioactivity from Harwell, Aldermastonand Burghfield were far too low to account for the excess of childhood leukaemia.Indeed, the doses from the minute quantities of radioactive material released from theBurghfield facility, around which the excess of childhood leukaemia appeared to beconcentrated, were about a million times too low to be capable of accounting for theadditional cases. Near Aldermaston, doses from the discharge of naturally occurringradionuclides from the coal-fired boilers were greater than the doses received from theemission of nuclear material.15 COMARE recognised that there was no value incontinuing to investigate individual reports of raised levels of childhood leukaemia nearparticular nuclear sites. They suggested that no further detailed investigations should beconducted until the background pattern of childhood leukaemia, and how this mightinfluence childhood leukaemia near nuclear sites, was better understood. The Committeerecommended a continuing programme of fundamental scientific research to betterunderstand childhood leukaemia and the role of radiation in its induction.

Other studies of childhood leukaemia around nuclear installations in Britain had alsobeen carried out. Of particular importance was a study undertaken by the Office ofPopulation Censuses and Surveys and the Imperial Cancer Research Fund whichexamined cancer incidence and mortality in local authority areas around all the majornuclear facilities in England and Wales.16,17 This study was of particular importancebecause sites were not selected on the basis of prior knowledge of the data, a source ofuncertainty which had affected several other studies of childhood leukaemia aroundnuclear installations.18,19 The investigation found no evidence for a generally raised riskof cancer among those living in areas around nuclear sites, but it did find some evidenceto suggest a raised risk of childhood leukaemia near nuclear establishments whichcommenced operations before 1955. However, interpretation of this finding wascomplicated by the level of childhood leukaemia in control areas (with which the areasaround nuclear sites were being compared) being especially low, rather than the levelaround installations being especially high. Subsequent work which was carried out toaddress this problem of control areas confirmed the excess of childhood leukaemia inlocal authority districts around nuclear sites, but the authors noted that this excessoccurred in a large area around the facilities and that there was no evidence of a trend ofincrease of childhood leukaemia with nearness of a district to a facility.20 A further studywhich examined cancer in districts around potential sites of nuclear power stations found apattern which was very similar to that found around existing sites.21 The implication was

3CHILDHOOD LEUKAEMIA AND RADIATION

that the raised level of childhood leukaemia found in districts near existing nuclearinstallations was more to do with the nature of the area in which a nuclear facility wassited, rather than the facility itself.

By the end of the 1980s, it was recognised that the risk of leukaemia among childrenliving around certain nuclear installations in Britain appeared to be raised, although theinterpretation of this finding was far from straightforward.18,22 It was broadly agreed thatdirect exposure to ionising radiation was most unlikely to be the cause of the raised levelsof childhood leukaemia, and research had not revealed any gross underestimates of therisk of radiation-induced childhood leukaemia which could account for the observeddiscrepancy between actual numbers of childhood leukaemia cases and the numberspredicted by radiological assessments.23,24 For example, studies of childhood leukaemiaafter the highest levels of atmospheric nuclear weapons testing in the early 1960s25,26 havenot detected any unexpected rise in rates due to doses from radionuclides present in thefallout, and these are very similar to the radionuclides released in the effluent of nuclearfuel reprocessing plants. Therefore, these studies provide evidence against suchradionuclides deposited within the body making an unexpectedly high contribution to therisk of childhood leukaemia.

The finding concerning raised levels of childhood leukaemia around certain nuclearinstallations in Britain led to a number of studies being carried out in other countries. Noconvincing evidence of a raised risk of childhood leukaemia around nuclear facilities hasbeen found in the USA,27 France,28,29 Germany,30 Canada31 and Sweden.32 Recently,however, evidence has been published of an excess of childhood leukaemia near the LaHague nuclear reprocessing plant in Normandy,33 although the results of a detailedinvestigation of these cases have yet to be published.

Following the trial, Bithell et al. 34 reported the results of the most detailedinvestigation of the distribution of childhood leukaemia and non-Hodgkin’s lymphoma inelectoral wards near nuclear installations in England and Wales during 1966-87. Sitesconsidered were the 15 major installations, 8 minor installations and 6 potential sites ofpower stations. In no instance was a statistically significant excess of cases found in areaswithin 25 km of an installation. Tests for a trend of increasing incidence with nearness ofa ward to an installation were performed. The only significant results were for Sellafield(which was entirely accounted for by the 6 cases in Seascale) and for the minor facility atBurghfield. One potential site also gave a significant trend. The authors concluded thatthere was “virtually no convincing evidence for a geographical association of childhoodleukaemia and non-Hodgkin’s lymphoma with nuclear installations in general”. Thisstudy must be regarded as the definitive geographical correlation study of childhoodleukaemia and nuclear installations in England and Wales.

THE GARDNER REPORT

As a result of a recommendation made in the Report of the Black Advisory Group,1

Gardner and his colleagues carried out a case-control study of leukaemia and lymphomain those born in West Cumbria and diagnosed during 1950-85 while under 25 years of ageand resident in the district. This was published in February 1990 and became known asthe Gardner Report.35 Many factors potentially influencing the risk of childhoodleukaemia in the district were examined in this study, but the most striking finding was astatistically significant association between doses of radiation as recorded by film badgesworn by men employed at Sellafield before the conception of their children and leukaemiain their children. An association was found with two measures of preconceptional dose: arecorded external dose of accumulated by a father before conception and a dose

E. JANET TAWN AND RICHARD WAKEFORD4

of received in the 6 months immediately preceding conception. These doseswere calculated from annual summaries of film badge records, and doses for part-yearswere obtained pro rata from the annual doses. Relative risks of around 6 to 8 werereported for these highest dose categories. Similar associations were also found forleukaemia and non-Hodgkin’s lymphoma combined, although the results were driven bythe leukaemia cases. The statistically significant associations were based on just 4 cases ofleukaemia and a similarly small number of controls, and the same 4 cases wereresponsible for all the significant associations. Consequently, lower 95% confidencelimits for the significantly raised relative risks were all between 1 and 2.

Examination of the preconceptional doses received by the fathers of the 5 Seascale-born leukaemia cases in the study led the authors to suggest that the association couldeffectively explain the excess of cases in the village. In the absence of any other factorswhich could account for the excess, particularly direct exposure to radiation fromradioactive effluent, the Gardner hypothesis, as it became known, was attractive and thecase-control study had apparently achieved its objective. The causal hypothesis putforward to explain the association between paternal preconceptional radiation exposureand childhood leukaemia suggested that the excess cases were the result of radiation-induced sperm mutations which manifested themselves in first generation progeny i.e. adominantly inherited effect. The implications of this hypothesis were commented on atthe time, particularly by scientists in the fields of human genetics and radiobiology.36-41

Two issues were of prime concern: the discrepancy between the genetic risks implied bythe Gardner hypothesis and those generated by the International Commission onRadiological Protection (ICRP),42 and the lack of evidence to suggest that leukaemia has astrong heritable component. Nevertheless the Gardner Report received considerablemedia attention and the statistical association was translated into a causal link.

THE LEGAL CASES

It was on the publication of the COMARE Second Report10 that the possibility oflegal action on behalf of individuals (or their families) who had developed leukaemia ornon-Hodgkin’s lymphoma while living near nuclear installations began to be seriouslyconsidered. The London firm of solicitors, Leighs (later Leigh, Day & Co) took theunusual step of advertising in a local Cumbrian newspaper for clients wishing to pursueclaims against BNFL. Writs were issued in 1989 on behalf of children who haddeveloped leukaemia or non-Hodgkin’s lymphoma while living near Sellafield. Thesewrits claimed personal injury resulting from exposure to radiation from Sellafield. Writswere actually served in 1990 within weeks of publication of the Gardner Report and thefocus of the claims shifted from environmental exposure to paternal preconceptionalirradiation.

The two cases of Reay BNFL and Hope BNFL were heard concurrently before asingle judge between the period October 1992 and June 1993. The Judge arrived at hisdecision on the balance of probabilities, based on evidence presented by expert witnesseswho prepared one or more written reports and gave oral evidence in Court. In both casesthe father had received cumulative film badge doses prior to the child’sconception, although only Mr Reay had received in the 6 months precedingconception. Dorothy Reay was born in the town of Whitehaven, 13 km north of Sellafieldand died of leukaemia there in 1962 at the age of 10 months. Vivien Hope was born inthe village of Drigg, 3 km south of Seascale, in 1965 and moved to Seascale at the age of6 years. In 1988 while still living in Seascale she was diagnosed as having non-Hodgkin’slymphoma and to date she is in remission.

CHILDHOOD LEUKAEMIA AND RADIATION 5

Because of the heavy reliance placed upon the Gardner Report, the Court ordered theUK Medical Research Council (MRC) and BNFL to make available to both Plaintiffs andDefendants, on a confidential basis, all relevant base data and associated dosimetry data ofthe West Cumbria case-control study. This enabled experts in dosimetry, statistics andepidemiology to examine and reanalyse the data used in the Gardner Report for thepurposes of preparing expert evidence. In addition the legal process of Discovery obligedboth sides to make available to each other any document held by them that was ofrelevance to the issues in the legal action. BNFL and its predecessor, the United KingdomAtomic Energy Authority (UKAEA) gave copies of many thousands of documents to thePlaintiffs relating to occupational radiation dose and discharges to the environment. As aconsequence the Plaintiffs suggested that the use by Gardner et al.35 of annual summariesof film badge readings had underestimated, perhaps significantly, the actual dose receivedby, for example, not including doses to the testes from internally deposited radionuclides.In addition, individual monitoring for neutrons had not been carried out in the early yearsof operations at Sellafield, personal protection being achieved through area monitoring.As a consequence neutron doses had to be individually assessed for a number of fathers inthe West Cumbria case-control study. Further, the accuracy of early film badges undercertain conditions was questioned and individual dose records were examined todetermine what affect this might have upon preconceptional doses. Considerable debatetook place outside the Court but in the end photon doses, assessed neutron doses andinternal doses were agreed for the purposes of the litigation for Mr Reay and Mr Hope andfor the relevant case and control fathers in the Gardner Report. These doses were used inthe Gardner Report reanalyses. Mr Reay’s total preconceptional dose was agreed to be530 mSv and that for Mr Hope 233 mSv.

The Plaintiffs received an enormous amount of information during the Discoveryprocess relating to discharges of radioactive material from Sellafield but were faced withsome difficulty in claiming that environmental exposure to radiation was the most likelycause of the malignant diseases since this possibility had been examined in considerabledetail in the period from the broadcast of the original television documentary in 1983.Those who had examined this possibility, including the NRPB,2,4 had concluded thatradiation doses were a factor of several hundred times too low to account for the excess ofchildhood leukaemia in Seascale, and the various pathways and mechanisms that had beensuggested as possibly leading to an underestimation of risk had effectively beendiscounted.23 It was partly because of this that the Gardner hypothesis looked attractive asan explanation for the Seascale “cluster”. In order to pursue environmental radiationcausation it was necessary, therefore, either to identify a gross underestimate ofradioactivity discharged, or to demonstrate that dose to target tissue was considerablygreater than had been thought, or to postulate that the risk of childhood leukaemia due tosomatic exposure to radiation had been substantially underestimated. Moreover, theselarge discrepancies would have to have been missed by the detailed examinations that hadbeen carried out. This was a formidable task. In the end, the Plaintiffs did not put expertevidence concerning environmental doses before the Court, and contented themselves withcross-examining the Defendants’ experts in this field. The probability that the twocancers had been caused by radioactive material discharged from Sellafield was calculatedto be very low: 0.16% for Dorothy Reay and 1.4% for Vivien Hope. During the trial, thePlaintiffs conceded that radionuclides discharged from Sellafield could not alone be thecause of the two cases or of the excess of childhood leukaemia in Seascale.


THE NATURE OF EPIDEMIOLOGY

Epidemiology is the study of patterns of disease in groups of humans with theobjective of identifying those factors influencing the risk of disease. Results are obtainedthrough the statistical analysis of data concerning potential risk factors in such groups.Epidemiology is an observational, that is a non-experimental, science which uses datagathered under the scientifically uncontrolled conditions of everyday life. Since data havenot been generated under experimental conditions, particular care has to be taken in theinterpretation of the results of epidemiological studies. An epidemiological associationcould directly reflect an underlying cause-and-effect relationship, but other interpretationsmust always be considered. One such alternative explanation is that a statisticalassociation has occurred through the play of chance, and does not represent a genuineeffect. By definition, statistically significant results will occur at a low frequency bychance alone. Special care has to be taken in the interpretation of statistically significantresults in a study in which many statistical tests have been performed. This was the casein the West Cumbria case-control study35 which was essentially exploring data in a searchfor an indication of what might be the cause of the Seascale childhood leukaemia“cluster”. Chance can act to produce statistically significant results in many insidiousways, particularly in exploratory studies, and especially when the structure of an analysishas been influenced, however unintentionally, by some prior knowledge of the data.

A further explanation of an epidemiological association which is not causal is thatbiases or systematic effects have entered into the study, such that the association isartificial. Because epidemiological data are generated under scientifically uncontrolledconditions, biases can be introduced into studies if sufficient care is not taken. Thus,systematic errors can occur through, for example, an unrepresentative selection of cases orcontrols, or through biased collection of information. Careful design and execution of anepidemiological study is required to avoid the introduction of bias.

Another way in which an association found in an epidemiological study does notrepresent a direct cause-and-effect relationship is through confounding. In this case,unlike with chance or bias, a causal relationship is responsible for the epidemiologicalassociation, but only indirectly. The factor identified in the study is, in fact, related to agenuine causal factor and is not, in itself, the cause of the disease.

In an observational study, the problems normally associated with experimentalresearch are magnified, and this requires considerable caution in the interpretation offindings. Inferential frameworks have been proposed to assist in the scientific assessmentof epidemiological associations. Probably the most famous of these was proposed by theeminent British epidemiologist, Sir Austin Bradford Hill in 1965.43 He suggested thefollowing nine criteria of causality as a guide to the interpretation of epidemiologicalfindings:Temporality

This is a necessary condition for a causal relationship and requires that a putativecause must precede the effect.Consistency

Owing to the difficulties encountered in epidemiological research, an associationshould be found under different conditions if a cause-and-effect interpretation is viable.Thus the association should be found with different study designs, in different groups ofpeople, under as wide a variety of circumstances as possible.Strength of Association

An association which is large and has high statistical significance is less likely tohave been produced by chance or bias alone.


Biological GradientAn epidemiological association should demonstrate an appropriate dose response

relationship, that is the association should tend to increase as the dose, or level ofexposure to the supposed cause, increases.Coherence

An association found in a limited group of individuals should be coherent with awider body of epidemiological data. Therefore, if cigarette smoking is found to beassociated with lung cancer in a study group of individuals, then national rates of lungcancer would be expected to increase (after an appropriate time lag to account for thelatent period) with the increased consumption of cigarettes at a national level. They havein fact done so.Biological Plausibility

An epidemiological association should not run counter to the established body ofbiological knowledge. This includes results from animal experiments. The criterion ofbiological plausibility is growing increasingly important in the interpretation ofepidemiological associations as this knowledge expands.Human “Experiments”

If circumstances of exposure change, mimicking experimental conditions, then theproposed consequent effect should change accordingly. Therefore, if cigaretteconsumption decreases at the national level as a result of public health campaigns, thenlung cancer rates should decrease after an appropriate time lag, which they have done.Analogy

It may be that a particular epidemiological association is analogous to anotherassociation for which the evidence for causality is stronger. Such an analogy wouldprovide additional assurance that the association represented a cause-and-effectrelationship.Specificity

An association can be more likely to be causal if a specific effect is caused by aspecific exposure, rather than a more general association with a spectrum of diseases withdifferent causes.

Sir Austin Bradford Hill said of these nine criteria of causality: “None of my nineviewpoints can bring indisputable evidence for or against the cause-and-effect hypothesisand none can be required as a sine qua non. What they can do, with greater or lessstrength, is to help us to make up our minds on the fundamental question - is there anyother way of explaining the set of facts before us, is there any other answer equally, ormore, likely than cause and effect?” “No formal test of [statistical] significance cananswer those questions. Such tests can, and should, remind us of the effects that the playof chance can create, and they will instruct us in the likely magnitude of those effects.Beyond that they contribute nothing to the ‘proof’ of our hypothesis.”

It will be seen that the interpretation of an epidemiological association is a complexprocess which has to be carried out with considerable care if erroneous inferences are notto be made. There have been many instances in epidemiology where apparentlyconvincing associations have turned out not to be causal. Epidemiology could be seen asa statistical stopgap which is used in the absence of detailed biological mechanisticknowledge. Epidemiology would not be necessary if such mechanistic knowledge wereavailable, but in most cases this ideal situation is far from being reality. Epidemiologydoes have the considerable strength of using direct observations on groups of humans; inother words it is a direct measure of health under particular circumstances. Butepidemiological studies do need to be conducted, and results interpreted, with theappropriate caution.


BACKGROUND TO THE GARDNER HYPOTHESIS AND SUBSEQUENTFINDINGS

From the discussion above, it is clear that the initial interpretation of the associationbetween paternal preconceptional radiation dose and childhood leukaemia found in theWest Cumbria case-control study35 had to be made in the context of the body of scientificknowledge existing at that time. The epidemiological evidence supporting a causalinterpretation was, at best, weak and there was evidence against a direct cause-and-effectrelationship. Leukaemia among 7387 offspring of the Japanese atomic bomb survivorshad been studied.44 The results provided no support for irradiation of fathers beforeconception of their children materially increasing the risk of leukaemia in these children,the observed number of cases being 5 against an expected number of 5.2, even though thedoses received during the bombings were, on average, much higher than those received bythe Sellafield fathers. A review of pre-1989 studies concerning the health effects of low-level preconceptional radiation by Rose for the then Department of Health and SocialSecurity concluded that there was no reliable evidence of a causal association betweenchildhood malignancies and low level doses of preconception paternal or maternalirradiation.45 Although a Chinese study initially reported an association of leukaemia withpaternal diagnostic X-ray dose,46 a follow-up study failed to confirm these findingssuggesting that the original association was probably due to recall bias.47 (The samegroup has recently reported a study in the USA which found a positive association ofpaternal preconceptional X-rays with infant leukaemia48 but again there are difficultieswith self-reporting of data which could lead to recall bias). The COMARE Second andThird Reports10,11 had examined possible mechanisms by which occupational exposure,including a preconceptional effect, could be involved in the induction of childhood cancerand had concluded that such mechanisms were “highly speculative”. The Committee did,however, advise that these needed to be explored if only to be dismissed.

COMARE, in a Statement of Advice to Government (Hansard 2 April 1990) issuedtwo months after the publication of the West Cumbria case-control study35 noted thestatistical association between recorded external radiation dose and the incidence ofchildhood leukaemia but were cautious in their interpretation since the conclusions of thestudy were based on very small numbers and the novel findings had not previously beenrecorded. The Committee further noted that “this type of study cannot provide evidenceof causal relationship” and that additional evidence was required before firmer inferencescould be drawn.

Epidemiological studies of relevance to the scientific interpretation of the associationbetween recorded external radiation dose and childhood leukaemia found in the GardnerReport soon followed. Yoshimoto et al.49 updated the study of cancer among the liveborn offspring of one or both parents who had been irradiated during the atomic bombingsof Hiroshima and Nagasaki. Fourteen cases of leukaemia were found in children ofparents who could be assigned the latest gonadal dose estimates against 11.2 expectedfrom an unirradiated control group, a non-significant excess. Little compared the resultsof Gardner et al.35 with those from the Japanese study. He demonstrated that the excessrelative risk coefficient (the excess risk per unit dose) derived from the Japanese data,whether considering parental or paternal doses, was statistically incompatible with thatderived from the West Cumbrian data, the coefficient obtained from the Gardner Reportbeing about 50 to 80 times higher than those obtained from the offspring of the Japanesebomb survivors.50 Using data obtained from the Radiation Effects Research Foundation inJapan, Little51 also showed that the risk of childhood leukaemia among those Japanesechildren born to fathers irradiated during the bombings and conceived within


approximately half a year of this exposure was also statistically incompatible with thefindings of Gardner et al.35

In the COMARE Second Report10 the recommendation was made that a leukaemiaand lymphoma case-control study be carried out around Dounreay and the results of thisstudy were reported in 1991.52 After the publication of the Gardner Report the primaryinterest in the Caithness case-control study concerned the findings of relevance to paternalpreconceptional irradiation. Urquhart et al.52 found that of 8 cases of childhoodleukaemia and non-Hodgkin’s lymphoma resident within 25 km of Dounreay at diagnosisonly 2 had fathers who had received a dose of radiation in the nuclear industry prior to thechild’s conception and both of the preconceptional doses were <50 mSv. The authorsobserved that paternal preconceptional irradiation could not explain the excess ofchildhood leukaemia and non-Hodgkin’s lymphoma in the vicinity of Dounreay andtherefore any “feature” common to both Sellafield and Dounreay which was raising therisk of childhood leukaemia in the vicinity of these installations (as suggested byCOMARE in 198810) could not be paternal exposure to radiation prior to conception.However the number of cases considered by the Caithness case-control study was so smallthat the findings concerning paternal radiation dose were statistically compatible with boththose of the Gardner Report and those of the Japanese study. A further case-controlstudy53 of childhood leukaemia and non-Hodgkin’s lymphoma in three areas whereexcesses of childhood leukaemia had been reported (including West Cumbria) did findevidence of a raised incidence of childhood leukaemia and non-Hodgkin’s lymphomaassociated with paternal preconceptional irradiation but there was considerable overlapwith the study by Gardner et al.35 and therefore this study did not provide independentsupport for the Gardner hypothesis.

The results of the first large childhood leukaemia case-control study designed to testthe Gardner hypothesis became available just before the commencement of the trial in199254 and were published subsequently. McLaughlan et al.54,55 examined 112 childhoodleukaemia cases and 890 matched controls born to mothers living around operatingnuclear facilities in Ontario, Canada and diagnosed during 1950 - 1988. No evidence wasfound for an increased risk associated with total cumulative paternal preconceptional doseor with paternal exposure to radiation in the 6 months immediately prior to conception.Looking specifically at the associations found in the study of Gardner et al.35 it is notablethat no case but 5 controls were associated with cumulative paternal preconceptional doses

and no case but 7 controls were associated with 6 month preconceptional dosesPaternal exposure to tritium (principally associated with operation of CANDU

reactors) was also assessed. No case, but 14 controls were associated with paternalexposure to tritium before conception. The Ontario case-control study, therefore,provided no support for the Gardner hypothesis. However, even a study of this size couldnot produce results which were statistically incompatible with those of the Gardner Reportas demonstrated in expert evidence given by G.R. Howe during the trial and later byLittle.56

During the trial Kinlen et al.57 published the results of a further large case-controlstudy which examined childhood leukaemia and non-Hodgkin’s lymphoma throughoutScotland in relation to radiation doses received in the nuclear industry. This study of 1369cases and 4107 controls covering the period 1958-1990 found no association with paternalpreconceptional irradiation whether this be total cumulative dose or the dose received 6months prior to conception. The Scottish case-control study does not support the findingsof Gardner et al.35 but again the results are not incompatible statistically with those of theWest Cumbria study. However, when Little56 later combined the results of the Ontarioand Scottish studies and compared these with the Gardner Report the difference was ofmarginal statistical significance (p = 0.10-0.15).


One feature of the West Cumbria study which seemed anomalous was theconcentration of leukaemia cases associated with relatively high doses of paternalpreconceptional irradiation in Seascale with 3 of the 4 cases in the highest dose groupsbeing born in the village. This appeared to run contrary to the general knowledge that themajority of Sellafield workers resided in West Cumbria outside Seascale. Parker et al.58

investigated this peculiarity by placing onto a computer database records from the birthcertificates of just over 250 000 livebirths registered within Cumbria during 1950-89.Using these data from birth certificates, and data on employees held at Sellafield, just over10 000 Cumbrian born children were linked to fathers who were employed at Sellafieldbefore the child’s conception. Of these, 9256 were associated with paternalpreconceptional radiation exposure. Only 8% of these children were born in Seascale andonly 7% of the collective dose of paternal preconceptional irradiation (whether cumulativeor 6 months) was associated with these Seascale births. These proportions are highlyinconsistent with paternal preconceptional irradiation providing the explanation for theSeascale leukaemia cluster since on the basis of the association seen in Seascale manymore cases of childhood leukaemia would be expected to have been found in the rest ofWest Cumbria. The results of this study were presented in expert evidence by one of theauthors (RW) during the trial, and provided strong grounds for believing that paternalpreconceptional irradiation could not be the sole explanation for the Seascale cluster.

Again during the trial, Kinlen59 published the results of a study of childhoodleukaemia and non-Hodgkin’s lymphoma incidence in children living in Seascale. Hefound a significant excess not only among those born in the village but also among thoseborn elsewhere. The significant excess in those born outside the village was based on 5cases, 4 of which were associated with no (or in one case a trivial) dose of paternalpreconceptional irradiation. Kinlen concluded that the dose of radiation received by afather before the conception of his child could not explain the excess of childhoodleukaemia and non-Hodgkin’s lymphoma in Seascale because it could not account for thesignificant excess of cases in those born outside Seascale, but diagnosed while resident inthe village.

G.R. Howe, in expert evidence presented during the trial, showed that the raisedrelative risk associated with a paternal preconceptional irradiation dose of waseffectively confined to those children born in Seascale. This observation was confirmedby the comprehensive study by the UK Health and Safety Executive (HSE)60,61 which waspublished after Judgment in the Reay and Hope BNFL cases had been delivered. TheHSE study found that a statistically significant association between childhood leukaemiaand non-Hodgkin’s lymphoma and paternal preconceptional irradiation was apparentamong those born in Seascale, but that the raised relative risk in Seascale was statisticallyincompatible with the absence of a significantly raised relative risk among those bornoutside the village. No significant association between childhood leukaemia and non-Hodgkin’s lymphoma and the paternal dose of radiation received shortly beforeconception (based upon original dose records) was found for either children born inSeascale or children born in the rest of West Cumbria, the association originally reportedby Gardner et al.35 being heavily influenced by the proportioning of annual dosesummaries, which led to erroneous dose estimates. The HSE study found that the positiveassociation for childhood leukaemia and non-Hodgkin’s lymphoma did not extend to otherchildhood cancers. Indeed a negative association was found between these cancers andcumulative paternal preconceptional dose.

One matter which received attention during the trial was whether the film badgedoses producing the paternal preconceptional irradiation association in the Gardner Reportmight be acting as a surrogate for some other occupational exposure, in particularexposure to neutrons or internally incorporated radionuclides. G.R. Howe, in expert


evidence, demonstrated that although photon and neutron doses were correlated, theGardner association appeared to be driven principally by photon doses. This finding wassupported by the HSE study which, in a qualitative assessment of neutron exposures,found no association with assessed potential for exposure to neutrons. G.R. Howe alsofound no evidence for an independent association with internal dose. Again, this wasconfirmed by the HSE study and by the absence of an association with tritium in theOntario case-control study54,55 and with the high alpha particle doses from thorium and itsdecay products in Danish patients injected with the contrast medium, Thorotrast, in thelater study of Andersson et al.62 These results make internal irradiation of the testes anunlikely explanation for the Gardner association. Indeed, no occupational factorexamined in the HSE study could explain the restriction of the significant excess ofchildhood leukaemia and non-Hodgkin’s lymphoma among offspring of Sellafieldworkers to that small fraction who were born in the village of Seascale. The HSE studyprovided strong confirmation of the validity of the evidence upon which Judgment in theReay and Hope cases was based. Little et al.63,64 subsequently showed that the Seascaleassociation was statistically incompatible not only with the lack of an association in therest of West Cumbria but also with the negative results of all other substantive studiesusing objective measures of radiation dose.

One recommendation of the COMARE Third Report11 was that a case-control studyof childhood leukaemia should be carried out around Aldermaston and Burghfield and theresults of this study were published during the trial.65 Although an association ofborderline significance was found with being monitored for exposure to radiation beforeconception, unlike the Gardner study there was no convincing evidence of an associationwith actual recorded external dose. The doses received were trivial in all instances, thecumulative doses being <5 mSv. The authors speculated that some unmonitored exposureto radioactive substances or chemicals could be responsible for the observed associationbut it was clear that the association could not account for the excess of childhoodleukaemia in the area around Aldermaston and Burghfield, only 4 fathers of affectedchildren being employed in the nuclear industry before conception.

The results of two relevant geographical correlation studies were published during thecourse of the trial. Draper et al.66 examined cancer incidence in the vicinity of Sellafieldbetween 1963 and 1990. They confirmed the excess of childhood leukaemia andnon-Hodgkin’s lymphoma in Seascale over the period studied by the Black AdvisoryGroup and found that this excess appeared to persist into the period after the Seascalecluster was originally reported in the television documentary in 1983. This excess ofcases did not extend to those over 25 years of age, nor did it extend generally to theremaining area of the two local authority districts around Sellafield or to the rest of thecounty of Cumbria. Craft et al.67 examined cancer incidence among young people livingin over 1200 electoral wards in the north of England during 1968-85. They found thatSeascale had the most significant excess of childhood leukaemia and non-Hodgkin’slymphoma. Interestingly, they also identified another electoral ward, (Egremont North),situated 7 km north of Sellafield with an excess of acute lymphoblastic leukaemia whichfell within the top five most significant excesses in wards in the north of England.However, these two geographical correlation studies by themselves could not identify thecauses of these excesses.

The demonstration in evidence put before the Court that the association betweenpaternal preconceptional irradiation and childhood leukaemia was effectively restricted tothe village of Seascale, and did not extend to the great majority (>90%) of the offspring ofthe Sellafield workforce who were born outside this village, led to the Plaintiffssuggesting that the explanation for this phenomenon was an interaction between paternalradiation exposure and some factor (“Factor X”) which was essentially confined to


Seascale. This explanation proposed that the irradiation of testes predisposed offspring tothe action of some factor which affected the individual after conception, and that “FactorX” operated predominantly, but not exclusively, in Seascale. “Factor X” was suggested tobe an infectious agent such as a virus, or environmental radioactivity. Although it is notunusual in epidemiology to find two risk factors which act together more than additively(for example, radon and cigarette smoke as risk factors for lung cancer), it is most unusualto find factors acting together more than multiplicatively. G.R. Howe, in expert evidence,demonstrated that the restriction of the paternal dose association to Seascale was soextreme that any interaction between paternal preconceptional irradiation and “Factor X”would have to be substantially greater than multiplicative, and that such an interaction wasimplausible. The need for the interaction to be appreciably supramultiplicative in order toexplain the Seascale cluster was later confirmed by Little et al. 63

Further evidence against an interactive process being capable of explaining therestriction of the paternal preconceptional irradiation association to Seascale was providedby the Egremont North childhood leukaemia excess.67 One of the authors (RW) presentedevidence to show that none of the four cases which occurred in Egremont North wasassociated with paternal preconceptional irradiation, even though the collective dose ofsuch irradiation in this electoral ward was greater than in Seascale. If some factor inEgremont North is increasing the risk of childhood leukaemia, then it would be mostremarkable that this factor did not affect those children who are supposedly most at risk,that is the children whose fathers were exposed to radiation prior to conception. Thatpaternal preconceptional irradiation should take part in an exceptionally strong interactionwith “Factor X” in Seascale, but not play any role in an excess some 10 km away seemshighly unlikely. These findings were later reported by Wakeford and Parker.68

RADIATION GENETIC RISK

The causal hypothesis to account for the statistical association between paternalpreconceptional irradiation and childhood leukaemia proposed that irradiation of fathershad induced mutations in sperm which had been transmitted to their offspring andsubsequently resulted in the development of leukaemia. The biological plausibility of thisproposed mechanism came under considerable scrutiny. Although this was only one ofthe criteria proposed by Sir Austin Bradford Hill in 1965 for establishing causationepidemiologically, considerable advances have been made in biological knowledge sincethat time and the necessity of there being an acceptable causal mechanism to explain theassociation was considered by both epidemiologists and geneticists to be of primeimportance.

Despite extensive studies on the offspring of the A-bomb survivors, no evidence hasemerged of any adverse health effects attributable to inherited radiation-inducedmutations. Indeed, no studies of exposed human populations have indicatedunequivocally a radiation-induced heritable effect. In discussing their findings Gardneret al.35 noted that their report was “the first of its kind with human data” and this pointwas also made in the accompanying editorial in the British Medical Journal.69 Because ofthis lack of human information, genetic risk estimates have had to rely heavily on mousedata derived in controlled experiments using relatively large numbers of animals. The 7specific-locus test developed by Russell70 has been used extensively for studying thequalitative and quantitative effects of radiation. Such experiments suggest a geneticdoubling dose for low LET acute irradiation of 0.4 Sv.71 A significant finding to emergefrom these early studies was that the genetic damage was greater by a factor of 3 if a givendose is delivered acutely rather than chronically.72 Applying a dose rate reduction factor


of 3 to the mouse data for acute exposures gives a doubling dose of 1.2 Sv for chronic lowLET radiation. The United Nations Scientific Committee on the Effects of AtomicRadiation (UNSCEAR) have extrapolated this estimate to humans and use a doubling doseof 1 Sv, which is applied to incidence data on human genetic conditions to obtain riskestimates.73,74 The International Commission on Radiological Protection (ICRP) apply therisk estimates generated by UNSCEAR to derive appropriate protection practices.42 Whenconsidering first generation effects in a reproductive population, the risk for Mendelianplus chromosomal disorders from 10 mSv of low LET radiation exposure is estimated tobe 18 per 106 live births. A further 12 cases of disease with a multifactorial origin isexpected of which >90% will be of adult onset.

Although no adverse effects were seen in Japan, a recent review by Neel et al. 75 hascombined the analyses of a range of endpoints, each weighted for its heritable component,and shown that this combined data is consistent with a minimal doubling dose of between1.7 and 2.2 Sv. Following examination of the Japanese parental exposures the authorshave opted for a dose rate reduction factor of 2 as being most appropriate for theseconditions and thus derive a doubling dose for chronic ionizing radiation of 3.4 to 4.4 Sv.Disturbed by the discrepancy between this estimate and the 1 Sv used for risk estimation,Neel and Lewis undertook a reanalysis of the mouse data for 8 different genetic endpointsand derived a doubling dose of 1.35 Gy.76 When a dose rate reduction factor of 3 isapplied this gives an estimate for the doubling dose for chronic radiation of 4.05 Gy. Thisstudy also noted that there are good reasons to believe that the 7 specific recessive locioriginally chosen by Russell are particularly mutable and more recent comparative studieshave shown that they are 10 times more sensitive per locus than dominant cataractmutations.77 ICRP have noted the work of Neel and Lewis and acknowledge that the useof a doubling dose of 1 Sv for determining genetic risk from low dose rate low LETradiation in man is conservative.78

GENETIC RISK FROM SELLAFIELD OCCUPATIONAL RADIATIONEXPOSURE

The genetic risk imposed by occupational radiation exposure on workers at Sellafieldwas the subject of expert evidence for the trial and has since been published. 79,80,81 Thestudy by Parker et al.58 revealed that during 1950-1989 a total of 9256 children were bornin Cumbria to fathers who had been occupationally exposed to radiation at Sellafield.This group of men had a collective preconceptional dose of 539 person Sv giving a meandose of 58 mSv. If the ICRP risk estimate42,78 is applied, the expectation is ofapproximately 1 excess case of Mendelian plus chromosomal disorders in this populationof children. Since the background frequency of such disorders is 16 300 per 10 livebirths(10 000 dominant, 2500 recessive, 3800 chromosomal73,74) this one extra case would beoccurring in addition to approximately 150 spontaneous cases. A further contribution of<1 case of severe childhood disease with multifactorial origin would occur against anestimated background of about 185 births (2%) suffering from major congenitalabnormalities of multifactorial etiology. Clearly any genetic effect in this population of9256 children will not be discernible against statistical fluctuations in the background rate.Furthermore no genetic effect is going to be detectable in the subgroup of 774 childrenborn to fathers with paternal preconceptional irradiation from Seascale in whom the doseprofile is similar.

The collective paternal preconceptional dose for the Seascale children is 38 person Svand if the 5 leukaemia cases in this group are attributed to this dose the implied mutationrate, assuming involvement of only one locus, is per locus per Sv. A


considerable amount of experimental data is available on the induction of mutations byradiation both in vitro and in vivo. These indicate induced rates which range from1 to per locus per Sv.82,83,84 The rate deduced from the Seascale data is therefore3 to 5 orders of magnitude greater. Postulating the involvement of a number of differentgenes with typical radiosensitivity will not explain this difference, sincethis would necessitate that the majority of the functional genes in man (50 000 - 100 000)could mutate in the germline and result in the single endpoint of leukaemia in thesubsequent offspring.

To circumvent the difficulty of having to confine a radiation-induced heritablemechanism of leukaemia induction to Seascale, it has been suggested, as discussed earlier,that preconceptional irradiation could be interacting with a postconceptional co-factor(“Factor X”) specific to the village of Seascale. However, since the proposed initiating orpredisposing event is an inherited genetic change, the incidence of leukaemia, at the veryleast, represents the mutation rate of a dominant gene with complete penetrance. If,however, such a mutation only confers a modest degree of predisposition and not all thosewho inherit a defective gene contract leukaemia, then an even greater underlying mutationrate has to be implied.

HERITABILITY OF LEUKAEMIA

The Gardner hypothesis postulates an inherited etiology for the excess of childhoodleukaemia in Seascale. However, unlike such childhood malignancies as retinoblastomaand Wilm’s tumour, where inheritance of a defective tumour suppressor gene provides ahigh risk of malignancy, evidence has not emerged for a similar mechanism forleukaemia. Although 10 times more common than retinoblastoma, which exhibits wellestablished familial patterns, trawling of registers has revealed only a few family clustersof childhood leukaemia with only one apparent parent to offspring transmission.85 Whenconsidered against background rates in large populations this could be a chance finding.An increase in leukaemia in the offspring of patients who have received treatment forcancer could be indicative of genetic transmission, either as a result of an already pre-existing familial predisposing gene or resulting from a new germline mutation induced byradiotherapy and/or chemotherapy. In fact, no such increase has been observed either forchildren of cancer patients generally or for children of survivors of leukaemia or non-Hodgkin’s lymphoma.86,87 Further evidence arguing against an inherited predisposinggene for leukaemia comes from studies of childhood cancers in different immigrantgroups. Racial incidence figures are maintained for such cancers as retinoblastoma andWilm’s tumour whereas the incidence of leukaemia is associated with country ofresidence rather than ethnic origin, suggesting an environmental rather than heritableetiology.88

Data which, when first evaluated, was thought to point to a heritable component isthe high rate of concordance for leukaemia in monozygotic twins. This, in the main, isconfined to leukaemias in the first year of life and is attributable, not to an inheritedmutation but to initiating events occurring in one twin in utero with subsequenttransplacental passage to the other.89 The molecular nature of the genetic change in theMLL (or HRX) gene which characterises infant null acute lymphoblastic leukaemia hasbeen analysed in three pairs of twins with the disorder. The rearrangement, which is notconstitutional, was found to be unique to each twin pair thus providing evidence for themutational event occurring during pregnancy.90 The high rate of concordance inmonozygotic twins for infant leukaemia is the result of all steps towards malignancyoccurring in utero. For leukaemia in older children, although the initiating genetic change


may occur prenatally and become established in both twins, different postnatal eventsresult in a much lower rate of concordance.

Nevertheless leukaemia is a genetic disease and mutational changes associated withleukaemia are readily identified by the presence of chromosomal rearrangements.91 Thesechanges are acquired somatic mutations, many of which are associated with the activationof proto-oncogenes. Chromosome rearrangements move such genes from their normalregulatory control sequences placing them next to promoting regions thereby causingincreased activity which results in a breakdown of proliferative constraints in the cell.Activated oncogenes invariably act in a dominant manner and thus only one of a pair ofproto-oncogenes in a cell needs to be altered for such an effect to occur. There is noevidence from human studies to suggest that chromosome rearrangements resulting inproto-oncogene activation can be transmitted through the germ line. Genetic engineeringhas produced viable transgenic mice with constructs of oncogenes plus certain specificpromotors incorporated into the constitutional genome but these have been produced forthe study of the sequential steps of tumour development and the effect of variouscarcinogenic agents on the progression from the predisposed cell to malignancy and notfor the study of heritability.92 Indeed attempts to introduce the bcr-gene with its naturalpromotor failed to produce viable offspring93 and activated oncogenes in the germ cellwill most likely disrupt normal fetal development.94,95

This considerable evidence which refutes the central tenet of the Gardner hypothesis,i.e. that leukaemia can be caused by the inheritance of a mutated dominant gene, wasraised by those questioning the biological plausibility of the causative mechanismsuggested by Gardner et al.35 and was addressed in detail by H.J.Evans and J.V.Neelduring the trial. There are however a number of rare well defined recessively inheritedsyndromes, such as ataxia telangiectasia and Fanconi’s anaemia, in which leukaemiaoccurs as one of a range of clinical endpoints.96 Such diseases are characterised by DNArepair deficiency, chromosome instability, and reduced immunocompetence, all factorswhich are thought to contribute to the enhanced risk of the somatic induction andmaintenance of leukaemia initiating events. Leukaemia is also one of a range ofmalignancies seen in the Li Fraumeni syndrome, a familial syndrome in which enhancedcancer predisposition is associated with inheritance of a defective tumour suppressorgene.97 However, consideration of the Seascale in the Gardner Report and also the twocases brought before the Court indicate nothing to suggest that any are part of a widersyndrome and they appear to be indistinguishable clinically from other sporadic cases ofleukaemia/lymphoma. This would seem to rule out any possibility that the effect reportedby Gardner et al.35 was due to a recessive mutation in individuals already heterozygous fora cancer predisposing syndrome. In any event it would seem unlikely that the populationof Seascale should contain a disproportionate number of such individuals.

In the event, a review of the evidence led to the acceptance in the trial that theheritable component of childhood leukaemia is likely to be small. Neel has suggested thatit is probably no more than 5%,98 and since the incidence of childhood leukaemia in theUnited Kingdom is about 1 in 1500 the background incidence of inherited leukaemia istherefore unlikely to be greater than 1 in 30 000. However if the 5 cases of leukaemiaamongst the 774 births in Seascale associated with paternal preconceptional irradiation areattributable to an inherited etiology, the incidence in this group is 1 in 150. This implies a200-fold increase in a dominantly inherited disorder in these Seascale children. Such aradiation-induced effect would not be expected to be confined to one outcome and if thesame increase was operating for other gene mutations then an epidemic of genetic diseasewould be expected. If the same effect was being induced in the rest of West Cumbriawhich has 13 times the Seascale collective paternal preconceptional dose and 11 times thenumber of births associated with paternal irradiation exposure 58 such an increase is not


likely to have gone unnoticed. Although no large scale study has yet been reported, Jonesand Wheater99 found no increase in abnormal obstetric outcomes in Seascale. It is alsoinconceivable that such an effect would not have been observed in the offspring of theA-bomb survivors and indeed should also be detectable in areas of high backgroundradiation.

ETIOLOGY AND HETEROGENEITY OF CASES

Diagnostic information for Reay and Hope was examined for the purposes of the trialand the diagnoses agreed. For Dorothy Reay, who died in 1962, details were sparse andthere were only a few poor quality haematological slides available for examination.Nevertheless it was agreed that this was a case of infant null acute lymphoblasticleukaemia. Evidence emerging around the time of the trial, and subsequently confirmed,identified a specific alteration of chromosome 11q23 involving the MLL (or HRX) genein the vast majority of such cases.100 The mutational origin of this leukaemic eventprobably occurs in utero and is not therefore an inherited genetic change of the typepostulated by the Gardner hypothesis. Furthermore Dorothy Reay was the one case ofleukaemia amongst 1588 children born in West Cumbria outside Seascale to fathers whohad received a preconceptional dose >100 mSv, 79 not an unexpected finding.

More diagnostic information was available for Vivien Hope, including pathologicalmaterial. During the course of the disease there had been no bone marrow involvementand it was agreed that this was a case of non-endemic (sporadic) Burkitt’s lymphoma.The etiology of Burkitt’s lymphoma has been well characterised.101 An important event isthe translocation between chromosome 8 and chromosome 14 which results in myconcogene activation. Unfortunately chromosome analysis was not undertaken in tumourtissue from Hope but in view of the diagnosis the expectation was that this event wouldhave occurred. A hallmark of endemic Burkitt’s lymphoma is the association with earlyinfection by Epstein-Barr virus (EBV), The polyclonal proliferation of EBV infected Bcells remains unchecked particularly in areas associated with malaria. A transformedclone arises as a consequence of myc gene activation in one cell of this increased cellpopulation. This somatic chromosome rearrangement is thought to arise as an accident ofthe normal developmental process of immunoglobulin gene rearrangement and is crucialto the parthenogenesis of Burkitt’s lymphoma. Although the role of EBV in the etiologyof endemic Burkitt’s lymphoma is well established, the co-factor fulfilling a similar rolein the etiology of non-endemic or spontaneous Burkitt’s lymphoma is unknown. Burkitt’slymphoma has, however, been reported in patients who are immuno-suppressed as a resultof HIV infection.102 Whatever the agent is that mimics the growth-promoting action ofEpstein-Barr virus in non-endemic (sporadic) Burkitt’s lymphoma its action will likewisebe somatic. Familial cases of Burkitt’s lymphoma have been reported in males with X-linked immunodeficiency syndrome,103 a disorder characterised by a predisposition toEBV related disease. There was, however, no evidence that Hope suffered from this orany other inherited immunodeficiency disorder.

Examination of the diagnostic details of the Seascale born leukaemia cases withpaternal preconceptional irradiation in the Gardner Report indicates further diversity:59

two young children with an unknown subtype of ALL, one child with null AL, one youngchild with AML and a young adult with CML. This heterogeneity must raise questionswhen searching for a common etiology and distinct causative mechanism for the Seascalecases and the two cases which were the subject of the legal action.


ANIMAL STUDIES

Support for a plausible mechanism to support the Gardner hypothesis had originallybeen drawn from the experimental work by Nomura on radiation-inducedtransgenerational carcinogenesis in mice. Nomura has recently reviewed his studies whichdate back to the 1970’s.104 The bulk of this work concerns the induction of lungadenomas in offspring of irradiated male mice. Unfortunately the experimental detailsprovided give rise to queries with respect to the use of concurrent controls and the numberof independent matings used for the heritability experiments. Examination of the data105

reveals small numbers in each dose and germ cell stage group and when lung adenomasare considered separately from total tumours the significance of the results is in doubt, atleast for spermatogonial exposure. Furthermore Nomura had previously noted thattumour incidence was related to body size106 and yet there appears to have been norecognition of the possibility raised by P.B. Selby in evidence in the trial that theinduction of dominant lethal mutations could have influenced litter size and hence size ofthe offspring. To date no studies have confirmed Nomura’s findings. A recent reanalysisof data from a lifespan study undertaken at Oak Ridge showed no increase in tumourincidence in offspring of irradiated mice107 and an attempt to repeat Nomura’s work byCattanach et al., published after the trial,108 found that lung tumour incidence was notrelated to paternal radiation dose nor were there significant differences between germ cellstages irradiated. This latter work did, however, find that tumour rates in experimentaland control groups varied in a cyclic way thus emphasising the necessity of concurrentcontrols for this type of study.

In his more recent publications Nomura has reported data on leukaemia incidence inthree different mouse strains.104,109 Numbers of leukaemias were small and in only onestrain was there a significant increase following spermatagonial exposure. Even if takenat face value this increase, from 1 in 244 in the control offspring to 9 in 229 following5.04 Gy spermatogonial exposure109 implies an induced mutation rate of per Svwhich is still orders of magnitude lower than that deduced from the Seascale leukaemiacases. Furthermore the dose in these experiments was acute and a dose rate factor of 3should be applied for comparison with the chronic exposure received by the Seascalefathers.

Transgenerational carcinogenesis has been observed following chemical exposure110

but the interpretation of these experiments is difficult111,112 and positive effects followingpreconceptional maternal exposure may in some cases be attributable to transplacentaltransfer of the carcinogen to the fetus rather than to induction of a germ cell mutation.

DOSE RATE EFFECTS

The biological effect of a given dose can be greatly influenced by the dose rate. Forlow LET radiation, as the dose rate is lowered and the exposure time becomes moreprotracted, the effect of a given dose is reduced. A dose and dose rate effectiveness factoris therefore commonly applied when deriving risk estimates for both somatic and genetic(i.e. heritable) risks for low dose low dose rate low LET radiation from high dose acuteexposure data. For high LET radiation little or no dose rate effect is observed, this beingconsistent with the damage caused by high LET radiation resulting from a single denselyionising track. In contrast damage caused by low LET radiation can be the result of theinteraction of more than one initial lesion and protraction of dose can allow the separationof these in time, thus allowing repair to occur (for review see Hall).113


In recent years evidence has emerged that for high LET radiations protracting theexposure may lead to an increase in biological effect. This led to speculation that theassociation between paternal exposure and leukaemia observed in Seascale could be due tosuch an inverse dose rate effect. However the majority of these studies have involvedoncogenic transformation assays on somatic cells in vitro and in general the enhancedeffect seldom exceeds a factor of 2. Brenner and Hall114 have developed a model which isconsistent with this experimental data and indicates that the results are heavily influencedby cell cycle kinetics. When this model is applied to the agreed neutron doses of the twofathers of the legal cases i.e. 11 mGy (220 mSv) and 3.5 mGy (70 mSv) dose rateeffectiveness factors are derived of 3.5 and 1.5 respectively. Furthermore, examination ofthe doses of the Seascale case fathers indicates no neutron dose greater than 5 mSv and onthe Brenner-Hall model no inverse dose rate effect would be expected. There is thus noevidence that the large discrepancy between the Seascale findings and the A-bomb datacan be reconciled in this way. Perhaps more importantly in view of the postulated germ-line effect, in vivo experiments on neutron irradiation of mice spermatogonia115

demonstrated an inverse dose rate effect only at high doses and this can be explained by agreater incidence of cell killing.

UNCONVENTIONAL GENETIC MECHANISMS

In the absence of evidence pointing to conventional mutation as an explanation forthe Gardner hypothesis, consideration has been given to the possibility of other radiationgenetic or epigenetic events which could be responsible.116 These were examined duringthe trial. One such phenomenon is transposon mediated mutation. Transposable geneticelements can change their positions within the genome and have been shown to beimplicated in mutations of the haemophilia A gene.117 However, in man, such events areextremely rare and there is no evidence, to date, that the transposition of DNA elements iseffected by radiation.

Further speculation has centred round the possible involvement of sequence elementscharacterised by tandem repeats. These minisatellite regions occur throughout the genomeand are highly polymorphic making them ideal markers for personal identification. Thehigh rate of spontaneous mutation resulting in a change in repeat copy number has led tosuggestions that minisatellite mutations could be used for population monitoring ofgermline mutations. Minisatellite mutations have been shown to occur at high frequencyin the offspring of male mice following spermatid irradiation but the evidence forspermatogonial irradiation is less persuasive.118,119 Following the Judgment, Dubrovaet al. 120 have found the frequency of minisatellite mutations to be twice as high in familiesfrom Belarus exposed to contamination from the Chernobyl accident compared to a UKcontrol group. Mutation rates correlated with levels of thus giving support to acausal link although it was recognised that other environmental contaminants could alsobe involved. The authors acknowledge that the limited target size would rule out thepossibility that the increase in mutations is due to direct damage to minisatellite DNA, amore likely explanation being that non-targeted effects stimulate minisatellite instability.A similar study in families of the Japanese A-bomb survivors has, however, failed to showincreased mutation frequencies.121 It has yet to be established if minisatellite mutationsplay a role in the causation of human disease. Krontiris122 has suggested that minisatellitemutations could affect gene expression, and thus cell transformation, but the role ofminisatellite DNA in normal cellular function remains unclear. In any event, in thecontext of the Seascale cases, any proposed role in transgenerational radiation-inducedleukaemia is unlikely to have been confined to one village.


The role of fragile sites in malignancy has also come under scrutiny. These regionsof chromosomal instability are revealed under certain in vitro cell culture conditions andalthough an association with the sites of somatic chromosome exchanges seen inleukaemia has been observed, no functional relationship has been demonstrated.123 Thesequencing of the fragile site associated with X-linked mental retardation has shown thedisease to be associated with an amplification of repeat DNA sequences.124 Expansion oftrinucleotide repeat sequences has also been observed in other inherited diseases, e.g.Huntingdon’s chorea and myotonic dystrophy124 but to date there is no evidence of anassociation with the haemopoietic disorders associated with leukaemia. Nor is there anysuggestion that radiation can induce fragile sites in germ cells or preferentially damagepre-existing fragile sites.

A recently recognised epigenetic phenomenon is genomic imprinting. This acts at thegamete level and is the result of parental origin causing differential expression of the twoalleles of a gene. Imprinting has been implicated in a number of human diseases, the bestdocumented being deletion of chromosome 15q11-3 which results in Prader-Willisyndrome when the origin of the deleted chromosome is paternal and Angelman syndromewhen the deleted chromosome originated maternally.125 Parent-of-origin effects havebeen reported for the Philadelphia chromosome,126 a somatic exchange which occurs inchronic myeloid leukaemia, but this interpretation has now been refuted127 and whetherimprinting plays a role in leukaemia initiation is still unknown. In any event, to date,imprinting has not been shown to be either modified or induced by radiation.

Susceptibility to the induction of spontaneous and induced somatic mutations is likelyto be influenced by a range of mechanisms affecting, for example, the fidelity of DNAreplication and repair, genomic surveillance and chromatin configuration. Similarly theinduction of genomic instability or a mutator phenotype which then enhances thelikelihood of further progression to malignancy will be affected by factors involved inapoptosis (programmed cell death) and the maintenance of genomic integrity. There is noevidence to suggest, however, that inherited radiation-induced genetic changes affectingany of these processes will occur at the orders of magnitude needed to explain the excessof leukaemia cases in Seascale nor that the results of such events would result only inhaemopoietic malignancies.

THE LEGAL JUDGMENT AND AFTERMATH

Judgment was given in the cases of Hope and Reay BNFL in October 1993.128 TheJudge concluded “In my judgment, on the evidence before me, the scales tilt decisively infavour of the Defendants and the Plaintiffs, therefore, have failed to satisfy me on thebalance of probabilities that ppi [paternal preconceptional irradiation] was a materialcontributory cause of the Seascale excess, or, it must follow, of (a) the leukaemia ofDorothy Reay or (b) the non-Hodgkin’s lymphoma of Vivien Hope” “In the result, theremust be judgment for the Defendants”. A review of the evidence has been published.129

Although arguably the Court is not the best place to settle scientific controversy thelegal cases focused the minds of a number of highly reputable scientists and acceleratedthe scientific investigation of the Gardner hypothesis. Subsequent events have notchallenged the Judge’s decision but rather reinforced the view that paternalpreconceptional irradiation cannot be responsible for the excess of childhood leukaemia inSeascale. In a comprehensive review, Doll et al.89 stated “In our opinion, the hypothesisthat irradiation of the testes causes detectable risk of leukaemia in subsequent offspringcannot be sustained”. “We conclude that the association between paternal irradiation andleukaemia is largely or wholly a chance finding”. In contrast, epidemiological evidence


for a role of infection in childhood leukaemia has grown. In particular, the hypothesis putforward by Kinlen130,131 that unusual population mixing induces unusual patterns ofinfection which increase the risk of childhood leukaemia has obtained considerablesupport. That such a process might apply to Seascale, an extremely unusual community interms of social class composition, isolation and population mixing, is highly attractive.The 1993 report from UNSCEAR74 which considered hereditary effects of radiation,reviewed radiation genetic risks in the context of the Gardner Report and concluded thatsince no epidemic of genetic diseases has been reported around Sellafield or other nuclearsites it is highly unlikely that the conclusions of Gardner et al.35 are correct. In 1994UNSCEAR132 reviewed epidemiological studies concerning the incidence of leukaemiaaround nuclear sites, thus adding to the assessment of the genetic implications of theGardner hypothesis and concluded “A tentative explanation based on an association ofchildhood leukaemia and paternal exposure has largely been discounted followingextensive investigations of the Sellafield area and elsewhere and because there is no soundgenetic basis for this effect”.

The COMARE Fourth Report133 has re-examined the Seascale leukaemia cluster inthe light of epidemiological and dosimetric data and the advances in radiobiology andradiation genetics that have emerged since the report of the Black Advisory Group.1 TheCommittee confirmed again the significantly raised level of malignancies among youngpeople in Seascale reported in the Black Report for the period 1963-83 and note that thishas continued in the later period of study from 1984-92. This is primarily due to anexcess of lymphoid leukaemia and non-Hodgkin’s lymphoma. Once more NRPB hascarried out a detailed radiological assessment for the Seascale population134 leadingCOMARE to confirm that discharges from Sellafield could not account for the excess. Inreviewing the hypothesis put forward by Gardner et al.,35 that paternal occupationalradiation exposure had resulted in germ cell mutations leading to a material increase in therisk of childhood leukaemia, the Committee have stated “We have not found anyepidemiological study elsewhere to support Gardner’s findings in Seascale in relation topreconception radiation effects” and “the level of risk implied by this explanation isinconsistent with the radiation doses actually received via occupational exposure andcurrent estimates of genetic risk”. “We consider that ppi [paternal preconceptionalirradiation] cannot account for the Seascale childhood leukaemia excess.” Whilstrecognising that questions still remain on microdosimetry and mechanisms of biologicalresponse, particularly in relation to radionuclides incorporated in the germ cells, andrecommending further research in these areas, the Committee acknowledge that it isunclear how these uncertainties could apply only to the village of Seascale. TheCOMARE Fourth Report133 also examined and dismissed a role for environmentalexposure to chemicals. The Committee recognised that Kinlen’s hypothesis of populationmixing130 which facilitates an infectious mechanism could play a role in the excess ofchildhood leukaemia in Seascale, but was not convinced that this could be the sole cause.The possibility that a combination of factors might account for the increase in cancer inSeascale was also examined but, while unable to rule this out, it was difficult to envisage asituation whereby such an interaction would be unique to Seascale.

The cause of the Seascale leukaemia cluster remains unresolved. The YorkshireTelevision programme in 1983 focused attention on the nearby Sellafield installation andalthough radioactive discharges were discounted as a causal factor, when Gardner et al.35

reported the association with paternal preconceptional irradiation this appeared tosubstantiate the link. The emotive nature of the media coverage caused considerableanxiety to the Sellafield workforce who were suddenly informed that in doing their dailywork, and being exposed to radiation at levels within the limits recommended by ICRPand NRPB, apparently they were materially putting their children’s health at risk.


Unfortunately the media sensationalism also influenced the scientific literature and thestatistical association between paternal preconceptional irradiation and childhoodleukaemia was treated by some as being causal when a more rational appraisal would havedemonstrated the implausible implications of this view. As a consequence there was aconsiderable amount of personal anxiety, and an expensive and lengthy Court action. Itcould also be argued that concentration on Seascale has deflected resources fromaddressing the wider issues of leukaemia etiology and causation. For the sake of scienceand humanity it is to be hoped that in future when a study yields a most unusual result itwill be interpreted with the caution it deserves until the full implications have beenevaluated and it receives independent confirmation.

ACKNOWLEDGEMENTS

We thank Lynda Buckland, Anne Poyner and Chris Beresford for preparation of themanuscript.

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REACTOR DYNAMICS FROM MONTE CARLO CALCULATIONS

Timothy E. Valentine

Instrumentation and Controls DivisionOak Ridge National LaboratoryP. O. Box 2008Oak Ridge, TN 37831-2008

I. INTRODUCTION

Point kinetics models are commonly used to evaluate certain accident scenarios fornuclear reactors which, along with other safety analyses, provide the basis for the design ofthe reactor control and safety systems. However, the validity of the point kinetics models isa relevant issue if the results of these safety analyses are to be credible. Rod oscillatorexperiments1 have been performed in the past to determine reactor kinetics parameters.Other techniques such as varying coolant temperature or flow rate have been employed tochange the reactivity of a reactor to investigate the kinetic behavior of a reactor core.2

Although these methods provide a direct measure of the reactor transfer function, theyrequire a reactor perturbation. Noise techniques have been applied for over 40 years todetermine reactor parameters. Analyses of the measured noise statistics (auto-or cross-power spectral densities) provide quantities that are directly related to the reactor transferfunction. The early work by de Hoffman3 and Courant and Wallace4 analyzed neutronfluctuations for reactors operating at zero power. The number of neutrons from fissionfluctuates about an average value that results in local variations in the neutron flux. Thedistance a neutron travels between collisions is random and results in stochasticfluctuations of the neutron population. The type of neutron event (fission, absorption, etc.,)is also random and results in momentary fluctuations in the neutron flux. In 1958, Moore5

suggested that measurements of the fission rate autocorrelation of a reactor operating atsteady state could be used to determine the reactor transfer function. An application of thistechnique was performed by Conn6 in 1959 at Argonne National Laboratory. Cohnmeasured the autocorrelation of an ionization chamber output to obtain the reactor transferfunction and performed a least squares fit to the transfer function to obtain a value ofSimilar analyses have also been performed at other facilities. The use of these techniquesto estimate reactor parameters led to the development of stochastic models to study thefluctuations in the neutron populations.7,8 These theories modeled the probabilities ofvarious neutron events in the reactor and obtained the moments of these probability


distribution functions. In 1968, Zolotar9 performed the first Monte Carlo analysis usingstochastic models. Zolotar developed a one-speed Monte Carlo code to simulatefluctuations in the neutron populations for a bare reactor and to estimate parameters of theprobability distributions. In doing so, Zolotar demonstrated that the Monte Carlotechniques could estimate parameters of the probability distributions that could becompared with point reactor models.

Since Zolotar’s analyses, little had been done to use Monte Carlo techniques fornoise analysis calculations until the development of the codes KENO-NR10 and MCNP-DSP11. This work highlights a unique synthesis of noise analysis techniques with theMonte Carlo method. The time dependent nature of the Monte Carlo simulation is ideallysuited for performing noise and/or correlation calculations. Explicit modeling of detectorsin the Monte Carlo simulations allow for the direct calculation of time dependent detectorresponses. With the development of the Monte Carlo codes KENO-NR and MCNP-DSP,noise analysis measurements using neutron and/or gamma ray detectors can be simulated.Although these codes were originally developed to calculate source-driven noise analysismeasurements,12 they can also be applied to a number of noise analysis simulations, and inparticular, the determination of reactor transfer functions for steady-state reactorconfigurations.

This article describes how noise analysis techniques incorporated into a MonteCarlo neutron transport code can be used to either investigate the applicability of pointreactor kinetics transfer functions or, alternatively, to develop a more sophisticated modelfor the reactor transfer function. This article also demonstrates how noise analysistechniques can be used to determine the time delay between core power production andexternal detector response, an important parameter for design of the reactor control andsafety systems. This manuscript describes the reactivity and source transfer functionsobtained from the point reactor kinetics models and how the source transfer function isestimated using noise analysis techniques; it describes how noise analysis parameters maybe estimated using Monte Carlo calculational models; and it presents an application ofthese techniques to a conceptual design of the Advanced Neutron Source (ANS) reactor.13

Finally, some conclusions are drawn about the applicability and limitations of thesemethods.

II. REVIEW OF STATISTICS OF STOCHASTIC PROCESSES

Before proceeding to a detailed discussion of estimation of noise statistics it isnecessary to briefly review the statistics of stochastic processes. A random physicalphenomenon cannot be explicitly described by mathematical relationships because eachobservation of the phenomenon will be unique and any given observation will onlyrepresent one of the many possible states of the process.14 The collection of all possiblestates of the random or stochastic process is called an ensemble. Properties of a stochasticprocess are estimated by computing averages over the ensemble that is characterized by aprobability density function, The probability density function describes theprobability that the process will assume a value within some defined range at any instant oftime.14 To determine the statistics of a stochastic process, knowledge of the probabilitydensity is required. For example, the mean value of an ensemble of random variablesx(t) is the expectation value

32 TIMOTHY E. VALENTINE

REACTOR DYNAMICS FROM MONTE CARLO CALCULATIONS 33

The autocorrelation of x(t) is the expectation of

where is a joint probability density function. The joint probability densityfunction describes the probability that both random processes will assume a value withinsome defined pair of ranges at their respective times and 14 Likewise, the cross-correlation between two random variables x(t) and y(t) is

where is a joint probability density function for and 14 A stochasticprocess is stationary if the statistics are independent of time. Therefore, the mean value is aconstant and the correlation functions depend only on the difference in time, Thecomplication of estimating the statistics of a stochastic process is greatly simplified if theprocess is ergodic. A stochastic process is ergodic if its ensemble averages equalappropriate time averages. In practice, random data representing stationary physicalphenomena are generally assumed to be ergodic. For a stationary ergodic process, the meanvalue can be estimated as

the autocorrelation of the random variable x(t) is

and the cross-correlation between two random variables x(t) and y(t) is

For stationary ergodic processes, Eqs. 2.4 through 2.6 are equivalent to Eqs. 2.1 through2.3. References 14 and 15 provide a detailed treatment of the statistics of randomprocesses. In practice, the mean and the correlation functions are calculated from timeaverages of finite length and are estimated from discrete time signatures. The mean valueis estimated from

where N is the number of points and is a discrete sample of x(t) at time Theautocorrelation is estimated as

By taking the expectation of Eq. 2.7, the expected value of the estimate of mean value of xcan be shown to be the same as the true mean value. Likewise, the expected values of theestimates of the correlation functions can be shown to be the same as the true values of thecorrelation functions. These estimates are unbiased estimates of the statistics of the randomprocess. Although the estimates are unbiased, there is a variance associated with theestimates that depends on the number of discrete samples of the signal, N. The variance ofthe estimates approaches zero as N approaches infinity.

The statistics of random processes are also described in terms of frequency domainquantities. Spectral density functions are the frequency domain equivalent of thecorrelation functions. Like the cross-correlation function the cross-power spectral density(CPSD) between x and y represents the amount of common information between x and y.In general, the CPSD is a complex quantity because the cross-correlation function is not aneven function. Likewise, the auto-power spectral density (APSD) is the Fourier transformof the auto-correlation. The CPSD may defined as the discrete Fourier transform of thecross-correlation function

where k is the frequency index, n is the lag index, and N is the number of discrete samplesof the correlation function.

III. REACTOR TRANSFER FUNCTIONS

3.1 Introduction

Classic reactor transfer function measurements attempt to relate fluctuations inreactivity to fluctuations in the neutron population. These fluctuations in reactivity areinduced by oscillating control rods or otherwise perturbing the reactor to measure reactor-dynamic parameters. The point reactor kinetics equations are commonly used to describethe reactor transfer function for small fluctuations about an equilibrium condition althoughthe use of space-dependent neutron dynamics has grown rapidly for analysis of reactortransients. However, there are other reactor transfer functions that can be measured and donot require a perturbation of the reactor and that may be compared to theoretical models forvalidation. The source transfer function can be obtained for a steady-state reactorconfiguration with time-dependent sources. In a subcritical reactor, a neutron source isneeded to maintain the neutron flux at a quasi-static level. If the nuclear parameters of asubcritical reactor are assumed to remain invariant with respect to time and inherent noise


where n is the lag index. The cross-correlation is estimated as

contributions are negligible, the fluctuations in the neutron production are mainly due tofluctuations in the neutron source. Uhrig16 has described how noise techniques can be usedto obtain the source transfer function or impulse response of the subcritical reactor for asteady-state configuration. The nuclear parameters are then obtained by fitting themeasured transfer function with an appropriate model. The general idea of this work is toincorporate noise analysis techniques in Monte Carlo calculations to obtain the sourcetransfer function. However, because Monte Carlo calculations represent the full space,time, and energy dependence of the neutron populations, Monte-Carlo-estimated transferfunctions can not only be compared to the point reactor kinetics transfer function but canalso provide the basis for a more sophisticated kinetics model. This section reviews thepoint kinetics representation of the source transfer function. This section also describeshow the Monte-Carlo-calculated transfer function is related to the point reactor kineticsequations and how the Monte Carlo transfer function is estimated from the noise spectra.Some frequency dependent features of the noise spectra are also discussed and their use inanalyzing the applicability of point kinetics to describe the source transfer function isdescribed.

3.2 Point Kinetics Transfer Function

The point kinetics equations used to describe the dynamic behavior of a subcriticalfissile assembly are17

and

where n(t) is the neutron density, is the i’th precursor density, is the total delayedneutron fraction, is the reactivity, is the neutron generation time, is the decayconstant of the i’th precursor, is the delayed neutron fraction of the i’th precursor, ands(t) is the neutron source density. For reactivity oscillation analyses, the neutron density,precursor density, and the reactivity are assumed to fluctuate about their average values,

and

where the average values are assumed to be equal to equilibrium values. By substitutingthese expressions into Eqs. 3.1 and 3.2 and assuming the oscillations are sufficiently smallsuch that the product of is negligible, one obtains the following linearized equationsfor the fluctuating components,


A similar derivation may be performed to obtain the source transfer function. For asteady-state subcritical reactor at a constant reactivity, the neutron density will fluctuateabout its quasi-static level due to fluctuations in the neutron source. As before, the neutron,precursor, and source densities can be assumed to be composed of an average value and afluctuating component while the reactivity is treated as a constant. After performingsimilar substitutions and algebraic manipulations as those required to obtain the reactivitytransfer function, one obtains the following expression for the source transfer function

where is the Laplace transform of Comparing Eqs. 3.7 and 3.8 the reactivitytransfer function can be shown to be related to the source transfer function by

Because current Monte Carlo calculations do not include delayed neutrons, they are notincluded in comparisons of the point reactor kinetics models and the Monte Carlocalculated models for the source transfer function. The introduction of delayed neutrons isnot necessary because this is a steady-state analysis and the delayed neutron contribution ismost significant for transient analyses. For transient analyses, the delayed neutroncontribution could be calculated analytically and the prompt transfer function could beestimated from the Monte Carlo simulations. The source transfer function without delayedneutrons is


and

where is the Laplace transform of and is the Laplace transform ofTherefore, the reactivity transfer function is simply

Taking the Laplace transform of Eqs. 3.4 and 3.5 and combining the resulting algebraicequations yields

In some measurements and in the Monte Carlo calculations, fission detectors are used toanalyze the fluctuations in the neutron population; therefore, the fluctuations in the neutronpopulation must be converted to fluctuations in the neutron production rate. The fission rateis related to the neutron population by

where l is the neutron life time, is the macroscopic fission cross-section, and is themacroscopic absorption cross-section. The neutron production rate is Bysubstitution of the standard definitions for k and the neutron production rate is expressedas

Therefore, the transfer function relating the neutron production to the neutron source is

Rearrangement of integration and expectation yields

where


This transfer function may be compared with the Monte Carlo calculated transfer function.

3.3 Reactor Transfer Functions from Noise Analysis Techniques

As previously mentioned, the idea of obtaining the reactor transfer function fromsteady-state operation of a reactor is not new; this idea was first proposed by Moore anddemonstrated experimentally by Cohn. The early work employed correlation techniques todetermine the reactor transfer function. These techniques and their frequency domainequivalent are discussed in this section.

A subcritical reactor with a source driving function and a fission detector can beconsidered as a linear single-input single-output system. The output, o(t), is related to theinput, i(t), via the convolution integral

where h(t) is the impulse response of the system. The transfer function can be obtainedfrom the cross-correlation between the input and output. The cross-correlationfunction is the expectation value

The cross-power spectral density (CPSD) between the input and the output is

where is the reactor transfer function and is the auto-power spectral density(APSD) of the input. The APSD of a Poisson-distributed white noise source is a constantfunction of frequency. The CPSD between the source and a fission detector is given by

where A is the value of the APSD of the source and is proportional to the sourcedisintegration rate, and we have substituted in the expression for the transferfunction, Eq. 3.13. If the point reactor kinetics models are applicable, the real part of theCPSD is positive because the reactivity is less than zero for a subcritical reactor. Thisproperty of the CPSD is useful for determining the applicability of the point reactor kineticsmodel.

If two identical fission chambers are used to measure the fluctuations in the neutronproduction, the CPSD between the two detectors can be used to ascertain if the source anddetectors can be considered a single-input dual-output system. For a single-input dual-output system, the CPSD between two detectors with identical transfer functions,can be expressed as

where is the APSD of the input. Note that because all the terms in the CPSD are realthe phase of the CPSD between the two detectors is zero. In experiments, zero phase in themeasured CPSD between the two detectors verifies the assumption that the transferfunction between the source and a detector is the same for both detectors. This is useful fordetermining if all symmetric modes are measured by all detectors.

IV. TIME DELAY ESTIMATION

The time delay between core power production and external detector response isone of the most important parameters in the design of control and safety systems of reactorswith external detectors. In the event of a rapid core power change, the control and safetysystems must respond quickly to mitigate the possibility of a severe excursion. Noiseanalysis algorithms incorporated into Monte Carlo calculations can also be used to estimatetime delay between core power production and external detector response. The externaldetector response can be considered as the output of a single-input single-output systemwhose input is the neutron production in the reactor core. The time delay can be estimatedfrom the phase of the CPSD between the input and the output. Estimation of the time delayin this manner is equivalent to pulsed neutron measurements as will be shown. Aspreviously mentioned, the output of the system is related to the input via the convolutionintegral


where D is the delay time, and is the Dirac delta function. Therefore, the output issimply The cross-correlation between the input and the output is

where is the input autocorrelation. The CPSD between the input and the output is

Consequently, the phase of the CPSD is Therefore, the delay is

where f is the frequency in hertz. In this manner, the phase of the CPSD between corepower production and external detectors may be used to estimate the time required forneutrons produced in the core to reach the external detectors.

V. MONTE CARLO SIMULATION

5.1 Introduction

The Monte Carlo method can be formulated in terms of an integral form of theBoltzmann transport equation or from a physical viewpoint. One integral form of theBoltzmann transport equation commonly used for describing the estimation of the averageneutron behavior in an assembly is the emergent particle density equation,18

where P denotes the phase space, is the emergent particle density (density ofparticles emerging from a collision or a source), is the kernel, and S(P) is thesource term. This equation is a Volterra-type integral equation that may be solved using aNeumann series approach. The random walk procedure is implemented by representing theemergent particle density by a Neumann series,

where is the density of particles emerging from the nth collision and is determinedfrom


If the output is a fraction, g , of the delayed input, the impulse response is

Using this formulation, the transport equation is solved by first tracking the sourceparticles, to their first collision site to estimate The surviving first collisionparticles are then transported throughout the system to estimate the next collision sites.This process is repeated for a sufficient number of particle histories to obtain an estimate of

which is then used to estimate some desired quantity such as a neutron capture in adetector.

Although the integral form of the Boltzmann equation is useful for improving ormodifying the standard random walk procedure, the random walk procedure can besimulated without even referring to the transport equation. Estimates of desired quantitiesare obtained by observing the behavior of a large number of individual radiation particles.Because actual particle transport is a stochastic process, the simulation should also bestochastic. All that is required for simulating particle transport is a probabilistic descriptionof the particle emissions and interactions. With a description of the geometricalboundaries and the materials of the system, the particle distance to collision and interactionevents can be randomly determined from the nuclear cross section data for the material.The Monte Carlo calculation is ideally suited for performing calculations of the highermoments like CPSDs of the neutron transport process in a fissile assembly because thevariations of the neutron events are probabilistic in nature. Because typical biasingtechniques are employed to reduce the variance of estimates of first moment quantities,they do not preserve the higher moments; therefore, analog Monte Carlo calculations mustbe performed when analyzing parameters which are directly related to the higher momentsof the neutron populations. Because the use of average quantities reduces the statisticalfluctuation of the neutron population, average quantities such as the average number ofneutrons from fission are not used, instead, appropriate probability distribution functionsare sampled.

The Monte Carlo codes used for this type of analysis were developed to simulate afrequency analysis measurement that uses a source, which may be treated as aPoisson-distributed white noise source, and a variety of particle detectors to characterizefissile assemblies. The source is contained in an ionization chamber to detect eachspontaneous fission. The source and detector responses are accumulated and the resultingsequences segmented into data blocks. A data block is a sample of the detector response fora specified period of time. The auto-and cross-power spectral densities are accumulatedand averaged over many blocks of data. This measurement technique is used to characterizefissile materials and could be used to measure the transfer function of a reactor bymeasuring the CPSD between the source and radiation detectors placed near the core.There are two codes available to simulate the frequency analysis measurement. The firstcode is the Monte Carlo code KENO-NR which is a modified version of KENO-Va19 anduses group averaged neutron cross sections. This code calculates the auto- and cross-powerspectral densities between a source and neutron detectors. The other code is MCNP-DSPwhich is a modified version of MCNP4a.20,TM MCNP-DSP is a continuous energy MonteCarlo code which calculates the auto- and cross-power spectral densities and auto-andcross-correlation functions between a source and radiation detectors for both neutrons andgamma rays. The Monte Carlo calculation does not impose limitations on the spatialdependence of the simulation except for the accuracy of representing physical systems. Theonly limitation of the energy dependence is that imposed by the cross section data files

MCNP is a trademark of the Regents of the University of California, Los Alamos National Laboratory.


™


whether continuous or group averaged and that imposed by the representation of the energyof neutrons and/or gamma rays from fission.

5.2 Monte Carlo Simulation

This is a brief review of the basic random walk of particles in the Monte Carlosimulation. The reader is referred to Lux and Koblinger21 for a complete description of theMonte Carlo method. The Monte Carlo calculation typically proceeds by tracking aspecified number of source particles and their progeny. However, in the KENO-NR andMCNP-DSP Monte Carlo calculations the tracking procedure consists of an additionalouter loop over data blocks because the time and frequency statistics are averaged overmany data blocks. The outer loop sets the current data block, and the inner loop tracksevents following source fissions in the current data block. This is illustrated schematicallyin Fig. 5.1. The outer loop starts with nps=1. If nps is greater than the number of specifiedblocks to be accumulated (bks), then the final outputs are obtained. Otherwise, the numberof source events per data block (nsdpb) is sampled from a Poisson distribution. The sourceparticles and their progeny are then tracked until the particles are either absorbed or escapefrom the system. After all source particles for a given data block have been tracked, theblock counter is incremented and the process repeats until all blocks have beenaccumulated.

A block diagram of the inner loop structure is given in Fig. 5.2. The inner loopbegins by obtaining information about the source event. The source is treated as a pointsource whose directional distribution is either isotropic or determined from an appropriatedistribution function. The times of the source fission events are uniformly distributedwithin the data block. The energy of the neutrons are sampled from a correctedMaxwellian distribution. The number of neutrons from the spontaneous fission of aresampled from Spencer’s distribution.22 This information is stored in the bank for all but one

of the neutrons for the given source spontaneous fission. Next, the distance to the cellboundary is determined from the geometry description of the region and the direction of thesource neutron, and the distance to collision is determined probabilistically from the totalmacroscopic cross-section of the material in which the source is located. If the distance tocollision is shorter than the distance to the cell boundary, then the collision type and timeare determined from the direction and velocity of the neutron; otherwise, the particle istransported to the boundary of the region and the process is repeated. The type of collisionis probabilistically determined from the nuclear cross-section data. For example, theprobability of fission event is determined from the ratio of the fission cross-section to thetotal cross-section. If a fission event occurs, the number of neutrons from fission aresampled from the appropriate probability distributions, the fission neutron directions aresampled isotropically, and their energies are sampled from the fission spectrum of the targetnucleus. The birth time of the secondary particles is the sum of the source particle birthtime and the transit time to the collision site. These progeny would then be stored in thebank to be tracked latter. If the collision event occurred in the detector material, thedetector response at the time of collision is incremented. If the particle survives thecollision, the distance to the cell boundary is recalculated along with the distance tocollision and the process is repeated. If the particle is absorbed or causes fission, then thenext particle from the bank is retrieved and the distance to the boundary and the distance tocollision are determined for this particle. If there are no particles in the bank the inner loopcounter is incremented. All particles are tracked until they are either absorbed or escapefrom the system. After all source and secondary particles for a given data block have beentracked, the detector responses are accumulated. This procedure is repeated for thespecified number of data blocks to obtain average estimates of the detector responses. TheAPSDs and CPSDs are estimated from the detector responses by complex multiplication ofthe Fourier transform of the data blocks and averaged over blocks.

5.3 Detector Simulation

In these Monte Carlo calculations, the detector is specified by defining the detectormaterial, type, and any energy thresholds. The detector response is segmented into timebins for each data block whose width is specified. There are three types of detectorsavailable in these calculations: capture, scatter, and fission detectors. The detector responseof capture detectors is due to neutron absorption followed by emission of secondarycharged particles that ionize the detection media and produce an electrical pulseproportional to the kinetic energy of the secondary charged particle. In the Monte Carlosimulation, neutron absorption in the detector material results in a count at the appropriatetime in the data block. In KENO-NR, all neutron absorptions in the detector media resultin a count. In MCNP-DSP, a specified fraction of the neutron absorptions in the detectormedia lead to a count to allow for detector thresholds typically set in the measurements.Scattering detectors are those in which the detector response is due to neutron scattering inthe detection media. These detectors are typically used to simulate liquid and/or plasticscintillators. To observe a count in a scattering detector, the neutron must deposit enoughenergy to the recoil nucleus to excite electrons in the scintillation material which producelight that is converted into an electrical pulse by a photomultiplier tube. In KENO-NR, theresponse of the scattering detectors is determined by the energy of the incident neutron. InMCNP-DSP, the response of the scattering detector is determined by the neutron energydeposition in the detection media and multiple scattering events of particles in the detectorsare taken into account. The MCNP-DSP treatment is more realistic than the KENO-NRtreatment of scattering detectors. In fission detectors, the fission fragments travel throughthe detection media ionizing the atoms in the detector. The large energy release per fission



allows for easy discrimination of other events that may also produce ionized atoms in thedetector. In the calculations, a count is registered each time a fission occurs in thedetection media, and the fission neutrons are stored for tracking. In MCNP-DSP, gammarays can also contribute to the detector response for capture and scatter detectors if thecalculation is a coupled neutron-photon calculation.

VI. APPLICATION TO THE ADVANCED NEUTRON SOURCE REACTOR

6.1 Advanced Neutron Source (ANS) Reactor

The ANS reactor was to be designed as an experimental facility for neutronresearch. Conceptual designs of the reactor included numerous facilities for neutronscattering analyses, materials irradiation, isotope production, and nuclear science studies.The reactor was designed to achieve a high flux in the reflector tank in order to performunprecedented neutron scattering studies. To achieve the necessary flux, the conceptualdesigns consisted of a small compact reactor and a heavy water reflector. Figure 6.1 is aschematic of one ANS conceptual designs. In this design, two annular concentric fuelelements comprised the reactor core. The lower fuel element has a 200 mm id. and a 340mm od., and the upper element has a 350 mm id. and a 470 mm od. Each fuel elementconsisted of involute fuel plates that were approximately 1.2 mm thick and spaced 1.2 mmapart. The plates were to be constructed of a highly enriched uranium silicide composite inan aluminum matrix. The silicide thickness varied radially. Uranium silicide was chosenbecause of its high density. In the following analysis, the two different diameter annularfuel elements were vertically displaced. Located in the inner region of the annular fuel

elements were three 71 mm od. hafnium control rods. The control rods were to be drivenupward from the bottom of the reactor to the top of the reflector vessel to facilitate the fuelloading and removal. In-core irradiation facilities were located between the inner radius ofthe upper fuel element and control rods. A core pressure boundary tube separated the corecoolant flow from the heavy water reflector. This would allow the large heavy waterreflector region to be maintained at a low pressure relative to the core region pressure.Eight hafnium shutdown rods were located outside the core pressure boundary tube in theheavy water reflector. The shutdown rods would move down from the top of the reactorvessel, where they would be located during reactor operation, to the midplane of the reactorto affect shutdown of the reactor. The reflector vessel had a 3.5 m diameter and wasapproximately 4.3m high. A 1.3 m thick pool of light water surrounded the reflector vesselon the sides and was surrounded by a biological shield within the reactor building. In thisdesign, eight beam tubes, two cold neutron sources, and one hot neutron source werelocated in the reflector vessel.

6.2 ANS Monte Carlo Model

The KENO-NR Monte Carlo model of this conceptual design of the ANS reactorwas simplified significantly. The model used in these analyses consisted of the two annularfuel elements, three hafnium control rods, the heavy water reflector, and the light waterpool. The tip of the inner control rods was positioned at the reactor midplane. Theinvolute fuel plates of the annular fuel elements were not explicitly modeled. Instead, theannular fuel element models were comprised of 13 radial zones and 25 axial zones torepresent the variations of the fuel density in the annuli. The beam tubes and cold and hotneutron sources were not modeled in some calculations. The source was positioned atthe midplane on the axis of the reactor core to initiate the fission process. Part of the upperfuel element and part of the lower fuel element were treated as fission detectors.Calculations were performed at various radial positions of the external detectors (fromapproximately 400 mm to 2500 mm, the latter being the location of the external fissiondetectors). To increase the precision of the estimates of the noise statistics, the externaldetector was modeled as an annular ring of the relevant moderator for agiven radial position that was 10 mm wide and 1080 mm high. Neutron scattering was theevent scored as a detection. Modeling the detectors in this fashion decreased thecalculation time by increasing the detector efficiency relative to that of a point detector.The presence of the structural components in the heavy water reflector reduce the corereactivity. To account for this negative reactivity, the neutron emission probabilities werearbitrarily reduced by 2% to produce a subcritical configuration rather than changing thecontrol rod position. Variations to this model were also made for some calculations. Insome calculations, the light water pool was replaced by a heavy water pool to determine theeffects of the pool material on the calculated time delays. To account for the presence ofthe experimental facilities in the heavy water reflector, an amount of aluminum equivalentto the mass of aluminum in these experimental facilities was added to the heavy waterreflector for some of the calculations. In the calculations which included the effect of theexperimental facilities in the heavy water reflector, the neutron emission probabilities werenot reduced.

6.3 ANS Reactor Transfer Function

The ANS reactor transfer function was estimated by dividing the CPSD between thesource and fuel element fission detectors by the APSD of the source. This analysis wasperformed using reactor models both with and without the experimental facilities in theheavy water reflector modeled. The reactivity was obtained from the Monte Carlo



calculation and the neutron generation time was estimated by other calculations; theseparameters were then used in point kinetics transfer function compared against the MonteCarlo calculated transfer function. For calculations which did not model the experimentalfacilities in the heavy water reflector, the reactivity was -0.0142, and the neutrongeneration time was 1.6 ms. This unusually large generation time is caused by the heavywater reflector. A comparison of the point reactor kinetics transfer function and the MonteCarlo calculated transfer function is shown in Fig. 6.2. There is excellent agreementbetween the Monte Carlo calculated transfer function and the prompt point kinetics transferfunction for each annular fuel element. The presence of the experimental facilities in theheavy water reflector greatly reduces the reactivity. For the calculation which modeled thealuminum in the heavy water reflector, the reactivity was -0.0534, and the neutrongeneration time was 1.0 ms. In this calculation, the neutron emission probabilities were notreduced because the aluminum in the heavy water reflector significantly reduced thereactivity of the reactor. As expected, the neutron generation time was decreased when thealuminum was added to the heavy water reflector. A comparison of the Monte Carlocalculated transfer function with the point kinetics transfer function for the reactor withaluminum in the heavy water reflector is shown in Fig. 6.3. Once again, there is excellent

agreement between the Monte Carlo calculation and the point reactor kinetics model.Consequently, the preceding calculations demonstrate that point kinetics was applicable forthese geometrical configurations of the ANS reactor.

The applicability of point kinetics can also be verified by examining knownproperties of the CPSDs. For point kinetics to be applicable the real part of the CPSDbetween the source and the fuel element fission detector must be positive. Figure 6.4illustrates that the CPSD real part between the source and the fuel element fission detectorare indeed positive for all frequencies. The CPSD for the upper element has a greater valuethan the CPSD for the lower element because the upper fuel element has a higherconcentration of than the lower fuel element. If the transfer function between thesource and fuel element fission detector is the same for both fuel elements, the phase of theCPSD between the fuel element fission detectors must be zero. Furthermore, if the phase iszero, then the detected neutrons represent those of the fundamental mode, and the two fuelelements behave as one element. The phase of the CPSD between the two fuel elements isapproximately zero as shown in Fig. 6.5. This was to be expected because the transferfunctions between the source and the fuel element fission detectors are essentially the same.These properties of the power spectral densities have shown that point kinetics is applicablefor this geometrical configuration of the ANS reactor and that the two vertically displacedannular fuel elements behave as one core.

6.4 ANS Calculated Time Delays

The time delay between core power production and the external detectors wasestimated from the phase of the CPSD between the fuel elements and the external scatteringdetectors. These calculations were performed with and without aluminum in the heavywater reflector. The time delay was evaluated at several positions in the heavy waterreflector and in the light water pool. The time delay increased as a function of distancefrom the core until it reached a saturation value in the light water pool. The results of thecalculations are presented in Table 6.1 and are shown in Fig. 6.6. The time delay had amaximum (~ 18 ms) in the reflector and then decreased to 15 ms in the pool forthe calculations that did not include the aluminum in the heavy water reflector. Thedecrease in the time delay is due to the fact that once a slowed-down neutron reached thelight water pool it was absorbed locally due to hydrogen capture. The flight path of aneutron to a radial point is not a direct path but consists of many scattering paths in alldirections while “diffusing” to the radial point. Therefore, the calculated time delay is theaverage flight time of thermal neutrons “diffusing” to each radial point. Because moreneutron scattering occurred in the the average flight path was longer in the heavywater reflector; hence, the average flight time will be longer. Neutrons that have numerousscattering collisions in the may be scattered back into the core and would notcontribute to the detector response in the light water pool. To further enhanceunderstanding of this decrease in the time delay, a calculation was performed with the lightwater replaced by heavy water. For the case, the time delay increased with distanceas shown in Fig. 6.7, thus confirming the effects of neutron absorption in the light water.Because the experimental facilities in the heavy water reflector were not modeled, the timedelays were overestimated. Additional calculations were performed that included aluminumin the heavy water reflector. The aluminum in the heavy water reflector significantlyreduced the time delays because neutrons that stayed in the heavy water reflector forrelatively long periods could be absorbed by the aluminum. The results from thesecalculations for the three detector positions in the light water pool are given in Table 6.2.Including the aluminum in the heavy water reflector decreased the time delay to theexternal detectors by 5 ms. The time delay at the position of the external fission detectorsis 10 ms.


VII. SUMMARY

Noise analysis and Monte Carlo methods are widely used in characterizing nuclearreactors. The combination of these two methods in the Monte Carlo codes MCNP-DSPand KENO-NR is unique and provides a useful tool for design of reactor safety and controlsystems. This article has demonstrated how Monte Carlo methods can be applied toinvestigate the applicability of point kinetics for steady-state reactor configurations or,alternatively, to provide a more detailed model for the reactor transfer function bycomputing the reactor transfer function from power spectral densities. The relationshipbetween the point reactor kinetics transfer function and the Monte Carlo calculated transferfunction was defined to compare the transfer functions obtained from the two models.Properties of the Monte Carlo calculated power spectral densities also provide additionalmeans to investigate the applicability of point reactor kinetics. The phase of the CPSDbetween neutron detectors can be used to determine the time delay between core powerproduction and external detector response. The time delay between core power productionand the external detector response is essential to the design of reactor safety systems of areactor. These Monte Carlo analyses apply to a specific steady-state configuration of thereactor and cannot be used to asses the applicability of point kinetics for otherconfigurations or during reactor transients. Additional Monte Carlo calculations could beperformed to investigate other reactor configurations.

The Monte Carlo analysis was applied to a conceptual design of the ANS reactor todetermine the applicability of point kinetics for the reactor operating in a steady-stateconfiguration. This analysis has shown for the slightly subcritical configuration of thereactor that point kinetics is applicable and that the two vertically displaced annular fuelelements behave as one fuel element. If the point kinetics model for the reactor transferfunction had not been applicable, the Monte Carlo calculated transfer function could haveserved as a model for the reactor transfer function. The reactor transfer function could bemeasured using source in an ionization chamber placed at the center of the core and afission detector adjacent to the core. The Monte Carlo calculations also provided estimatesof the time delay between core power production and the external detector response. Forthis conceptual design of the ANS reactor, the time delay between the core powerproduction and the external detector response is approximately 10 ms if the aluminumequivalent of the heavy water reflector components is included in the Monte Carlo model.The estimate of the time delay without aluminum in the heavy water reflector isconservative and provided a limit for the design of the reactor safety systems. The timedelay could also be obtained by measuring the CPSD between a detector near the core andexternal detectors.

Acknowledgment. The author is very grateful to his colleague J. T. Mihalczo for manydiscussions and advice regarding this work, and is indebted to J. K. Mattingly, R. B. Perez,and J. A. March-Leuba for their detailed review of this manuscript.

Keepin, G. R., Physics of Nuclear Kinetics, Addison-Wesley Publishing Co. Inc.,Reading, Massachusetts, (1965).Thie, J. A., Reactor Noise, Rowand and Littlefield, Inc., New York, New York,(1963).

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de Hoffman, F., Intensity Fluctuations of a Neutron Chain Reactor, MDDC-382,LADC-256 (1946).Courant, E. D. and Wallace, P. R., “Fluctuations in the Number of Neutrons in aPile,” Phy. Rev., 72, 1038 (1947).Moore, M. N., “The Determination of Reactor Transfer Functions fromMeasurements at Steady Operation,” Nucl. Sci. Eng., 3, 387 (1958).Cohn, C. R., “Determination of Reactor Kinetic Parameters by Pile NoiseAnalysis,” Nucl. Sci. Eng., 5, 331 (1959).Saito, K. and Otsuka, M., “Theory of Statistical Fluctuations in NeutronDistributions,” J. Nucl. Sci. Technol., 2, 304 (1965).Harris, D. R., “Neutron Fluctuations in a Reactor of Finite Size,” Nucl. Sci. Eng.,21, 369 (1965).Zolotar, B. A., “Monte Carlo Analysis of Nuclear Reactor Fluctuation Models,” Nucl.Sci. Eng., 31, 282 (1968).E. P. Ficaro and D. K. Wehe,"Monte Carlo Simulations of theNoise Analysis Measurements for Determining Subcriticality," ProceedingsInternational Topical Meeting Advances in Mathematics, Computations and ReactorPhysics, Pittsburgh, Pennsylvania, April 28-May 2, 1991, Vol. 1, p. 5.22.1,American Nuclear Society, 1991.Valentine, T. E. and Mihalczo, J. T., “MCNP-DSP: A Neutron and Gamma RayMonte Carlo Calculation of Source Driven Noise-Measured Parameters,” Ann.Nucl. Energy, 23, 1271 (1996).Mihalczo, J. T., Pare, V. K., Ragen, G. L., Mathis, M. V., and Tillet, G. C.,“Determination of Reactivity from Power Spectral Density Measurements withCalifornium-252,” Nucl. Sci. Eng., 66, 29 (1978).D. L. Selby, R. M. Harrington, and P. B. Thompson, The Advanced Neutron SourceProject Progress Report, FY 1991, ORNL-6695, Oak Ridge National Laboratory,Oak Ridge, TN, January 1992.Bendat, J. S. and Piersol, A. G., Random Data Analysis and MeasurementProcedures, John Wiley & Sons, New York, New York, (1986).Papoulis, A., Probability, Random Variables, and Stochastic Processes, McGraw-HillInc., New York, New York, (1984).Uhrig, R. E., Random Noise Techniques in Nuclear Reactor Systems, Ronald PressCo., New York, New York (1970).Henry, A. F., Nuclear Reactor Analysis, Massachusetts Institute of Technology,Cambridge, Massachusetts (1975).Carter, L. L. and Cashwell, E. D., Particle-Transport Simulation with the MonteCarlo Method, Energy Research and Development Administration, Oak Ridge,Tennessee, (1977).Petrie, L. M. and Landers, N. F. ORNL/NUREG/CSD-2, Oak Ridge NationalLaboratory (1984).Briesmeister, J. F., Ed., “MCNP4A-A Monte Carlo N-Particle Transport Code,”LA-12625-M, Los Alamos National Laboratory (1993).Lux, I. and Koblinger, L., Monte Carlo Particle Transport Methods: Neutron andPhoton Calculations, CRC Press, Inc., Boca Raton, Florida (1991).Spencer, R. R., Gwin, R., and Ingle, R., “A Measurement of the Average Number ofPrompt Neutrons for Spontaneous Fission of Californium-252,” Nucl. Sci. Eng., 80,(1982).


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NOTES ON A SIMPLIFIED TOUR:

FROM THE FOURIER TO THE WAVELET TRANSFORM

Marzio Marseguerra

Dept. of Nuclear EngineeringPolytechnic of MilanItaly

INTRODUCTION

Any signal may, in principle, be viewed as resulting from a linear superposition(possibly an integral) of assigned elementary functions. If the set of these functionsis sufficiently large the coefficients in the superposition constitute the transform ofthe signal. A classic example is given by Fourier analysis in which the elementaryfunctions are sinusoids of different frequencies which may be viewed as generatedby varying the frequency of an initial mother sinusoid. The wavelet transform maybe similarly viewed: here the elementary functions are generated by the dilatationsand translations of a mother function which may be selected with some degree offreedom provided it has reasonably good time and frequency localization features. Itturns out that this mother function should also have zero mean so that it resemblesa small wave: hence the name mother wavelet. An example of a mother wavelet isthe derivative of a bell–shaped curve such as a Gaussian function. With respect tothe Fourier analysis, a mother wavelet with a reasonably good localization both intime and frequency gives to the wavelet transform the so called zoom in and zoomout capability – i.e. the capability of resolving both the details and the trend of asignal – which represents the basis for a list of several applications.

In recent years, particularly since the second half of the 1980s, a tremendousinterest has emerged towards the application of wavelet analysis to different researchareas which include both theoretical and applied aspects. Examples may be foundin a variety of fields, ranging from functional analysis and digital multidimensionalsignal processing to image and sound analysis or to the biomedical domain. Impor-tant results have been achieved in specific applications such as digital video datacompression, a very important subject in the telecommunication field, or turbulentflow, of interest to hydraulic, naval and aerospace engineers and to meteorologists.Correspondingly, a great deal of papers and books have been published in which


the theoretical aspects of the wavelet methodology have been deeply investigatedwith the aid of the whole arsenal of tools at the disposal of mathematicians. How-ever, this scientific literature is generally written by specialists for specialists so thatthe mathematical demands of these sophisticated tools scale up considerably. Non–mathematicians hardly fully understand the matter and are often lost since in generalthey do not always have the clew of thread to get out of the labyrinth of theorems,propositions, lemmas, applicability conditions etc., required by rigorous mathematicalreasoning.

The present paper has been written mainly for an audience with a mathematicalbackground at the engineering degree level; since it is devoted to people who work inapplied areas, the so called theorem–structured way has been deliberately avoided,on account of the possibility that it might obscure the matter and discourage non–mathematicians from entering this new and fascinating field. A plain expository stylehas therefore been preferred, sometimes at the expenses of mathematical rigour, andthe wavelets are presented as a succession of easily understandable analytical ex-pressions. The present exposition is simply my understanding of a possible way forcrossing the pillars of Hercules towards the wavelet sea. On the whole I followed, withsome heuristic simplifications, the line of reasoning presented by G. Kaiser (1994) inthe first part of his beautiful book A Friendly Guide to Wavelets which I system-atically refer to for detailed explanations. Much of what is reported here I learnedfrom Kaiser and an enormous intellectual debt is owed to him. Some parts are alsoborrowed from two other outstanding books, namely Ten Lectures on Wavelets by I.Daubechies (1992) (a book difficult for non–mathematicians, even though it receivedan AMS prize for exposition) and An Introduction to Wavelets by C. Chui (1992)(less difficult for non–mathematicians).

I hope, the present paper will be of some help to professionals in engineeringscience for understanding specialized papers or up–to–date surveys such as those con-tained in the special issue on wavelets (1996) edited by and I. Daubechies.In essence, the purpose of this paper is to encourage engineers, and particularly nuclearengineers, to add the wavelet technique to the widely adopted Fourier methodology.

The plan of the paper is as follows. Initially some preliminary mathematicalnotions are recalled; Section 2 then reviews the well established windowed Fouriertransform; in Section 3 the concept of a frame is introduced; this concept is usedin Section 4 where the continuous wavelet transform is introduced in a way whichparallels that of the windowed Fourier transform; Sections 5 and 6 address the discreteversions of the above transforms; the multi–resolution analysis is presented in Section7 and, finally, the link between the wavelets and the sub–band filtering is given inSection 8.

At the end of the paper a list of essential references is given: comprehensive listsof up to date references on specific issues may be found, for example, in IEEE (1996).

1. PRELIMINARIES

1.1 Basis of a vector space

Let us consider an M–dimensional vector space A standard basis in isa sequence of M column vectors such that all the elements ofthe generic vector are zeros except the element which is one, viz.,

where is the usual Kronecker–delta.

54 MARZIO MARSEGUERRA

A sequence of M linearly independent column vectors is a basis in ifany column vector can be written as where is thecomponent of along the vector When considering any two bases andin the vectors of one basis may be expressed as linear combinations of those ofthe other basis, viz., where is the component of along

1.2 Dual spaces and dual bases

Consider the linear operator F which maps column vectors in to columnvectors in viz. also written as Let and beany bases in and respectively. Then the result of applying F to a columnvector is the column vector soexpressed

NOTES ON A SIMPLIFIED TOUR 55

where is the component of the vector along the vector andThen, in the given bases, F is represented by a matrix of order (N, M)

with elements and the application of F to is computed as a matrix product.Particular cases:

In words: when the N–dimensional (column) vector (regarded as an operator)operates on the 1–dimensional vector the result is the product of

now viewed as a scalar in C, timesN = 1, i.e. or In the following we change symbolsand write N instead of M. Thus the mapping here considered will be writtenas or The operator F, called a linear functional, isrepresented by a matrix of order (1, N), i.e. by a row vector of order N. The N–dimensional space spanned by these row vectors will be denoted by Sincethe matrix product of the (row) vector times the (column) vector

is the 1–dimensional vector (a scalar) we may say that any (row)vector may be viewed as a linear operator mapping

Given any basis (column vectors) of we may construct a basis(row vectors) of such that when operates on a vector the result is

the component of with respect to the basis viz., Thusthe row vector may be viewed as a linear functional. In particular,the vector composition rule (parallelogram rule) yields so thatwe have the so called duality relationCorrespondingly, is called the dual space of and is the dual basisof In practice the basis vectors are given by assigning their components

in the standard basis. In this basis the componentsof are the solutions to the system

ii.

i. M = 1, i.e. or The operator F is represented by a matrixof order (N, 1), i.e. by a column vector of order N. Since the matrix productof the (column) vector times the 1–dimensional vector (a scalar) isthe (column) vector we may say that any (column) vectormay be viewed as a linear operator mapping Formally:

Thus, the N algebraic systems(1.2) of N equations each, must be solved in orderto get the whole sequence

1.3 Inner product

Given any inner product must satisfy the following axioms:

where the overbar denotes complex conjugation. If are the components ofand with respect to the standard basis, the standard inner product between andand the norm of are defined as and If the innerproduct vanishes we say that the two vectors are orthogonal. Clearly the concepts oforthogonality and length (square root of the norm) are relative to the choice of theinner product. More generally, a weighted inner product is defined as

1.4 Adjoints

Given any inner products in and consider the operator F, representedby a matrix of order (N, M), which maps column vectors from to viz.

The adjoint of F is defined as the operator F* which performs theinverse mapping

It can be shown that the adjoint F* of F exists and it is unique.The determination of the matrix representative of the operator F* is simple in

case of orthonormal bases and in and respectively. In this casewe have:

Multiplying the first equation by and the second one by yields

Application of the adjoint condition to one of the above equations, for example tothe second equation, yields Thus the matrix representing F* is theHermitian conjugate of the one representing F.

A particularly interesting case of adjointness occurs when M = 1. Thenbecomes C. Let be any two 1–dimensional vectors in C. We choose as the innerproduct in We consider any as the operator (see eq.(1.1))

The adjoint of is the operator which performs the inverse mapping, i.e. thefunctional (row vector) For any and we getsuch that We also have so that

such that


Thus the inner product (a scalar) of any two (column) vectors in maybe split in the (matrix) product of the adjoint (a row vector) of times This isa result of paramount importance and its generalization in the Hilbert space will beextensively used in the sequel.

1.5 Reciprocal basis

Let us consider the space with a basis and let be the dual basisof We have seen that each row vector may be viewed as an operatorperforming the mapping The adjoint of is a column vectorwhich may be viewed as an operator performing the inverse mapping, i.e.

It can be shown that the sequence forms a new basis of this basis iscalled the reciprocal basis of The importance of the reciprocal basis stemsfrom the fact that it allows us to compute the components of any vectorin the given basis by taking an inner product, just as in the orthogonal case.Indeed from eq.(1.5) we get In the particular case of

the above equation becomes But, since thenand, finally

1.6 The resolution of unity

The dual bases and linked by the duality relation (1.2), have theimportant property of resolution of unity. Since every can be written inthe basis and since by definition of we have

Regarding as an operator from and as anoperator from then is an operator from i.e. from

1 The expression which follows is valid provided G is not singular. To this aim,given any define Then

andThus G is positive definite, and has, therefore, an inverse.

The above represents a very important result: any given basis and its reciprocalbasis are (mutually) orthogonal.

Given any basis in we have seen that the vectors of the reciprocalbasis are the adjoints of the vectors of the dual basis and that these maybe computed from the by solving N systems (1.2). However, we may directlycompute the from the of the original basis by resorting to the metric operator,defined as

and represented, e.g. in the standard basis, by a square matrix of order N. Indeed,from eqs.(1.5) and (1.6), we have

Thus 1


and we can write so that the operator in brackets equals

the identity operator viz. or Theabove expressions are an example of a resolution of unity, also called resolution ofidentity. We would like to stress that in order to have a resolution of unity, we needeither a pair of dual bases in and in or a pair of reciprocal bases

and both in

2. THE CONTINUOUS WINDOWED FOURIER TRANSFORM

2.1 Introduction

We are interested in analyzing a non–stationary, square integrable functionof time aiming at resolving its frequency content along time. For example, withreference to a musical piece or to a speech, the problem amounts to identifying thetime occurrence of a single note or phoneme. In nuclear engineering this problemoccurs for example in the field of early detection of failures. This is an importantissue since many major failures are preceded by the random occurrence of sudden,high frequency disturbances of short duration which should be detected and alsodetected as soon as possible.

At first sight it might appear that a solution to these problems could be foundwithin the classic Fourier approach by looking at the frequency content of theHowever, this approach does not generally solve the specific problem here considered.Indeed, since the Fourier Transform (FT) of a signal is usually computed after thesignal has been measured during a time interval of suitable length, the frequencycontributions due to single notes or to phonemes or to small disturbances of shortduration are hidden in the bulk of the transform; moreover, in the Fourier approach,the signal is decomposed in the sum (actually, integral) of sinusoids having all possiblefrequencies whose amplitudes are fixed and related to the integral of the signal overthe whole measurement time. In these integrals the time occurrence of a single note orphoneme or of a single short disturbance is lost, In this view, for example, a musicalpiece is considered as resulting from the superposition (constructive or destructive) ofinfinitely many pure notes, each generated by a musician who plays always the samenote with same intensity (amplitude). In the mathematical language we say that theanalyzing functions of the Fourier approach, namely the complex exponentials, sufferfrom lack of time localization.

A possible way to circumvent this drawback of the Fourier approach was pro-posed by D. Gabor (1946) who introduced the so called Windowed Fourier Transform(WFT), also named short time FT, which is a sort of moving FT. In principle, foreach time we consider only a suitably weighted portion of the signal nearfor example contained within a (possibly small) interval T preceding and performthe usual FT. By so doing, for each we obtain a function of the frequency inwhich the contributions of the sinusoids due to the rest of the signal outside T aremissing and in which the various harmonics thereby computed are well localized sincewe know that they occurred within the time interval With reference tothe above mentioned musical piece, each musician still plays the same note, but nowchanges the amplitude with time. The limits of the WFT approach will be discussedin Section 2.7.


2.2 Notations

In the following we shall proceed heuristically, thus skipping most of the subtletiesof a rigorous mathematical approach. The main assumption is that the results sofar obtained with reference to the N–dimensional vector space can be sensiblygeneralized to the case of an infinite dimensional Hilbert space 2. In this space weconsider the set of measurable functions of time e.g. whereand C are the sets of all real and complex numbers, respectively. The inner productbetween and and the norm of are (in the following, unless explicitly stated, allintegrals run from to ):

2 Recall that a Hilbert space is a linear vector space which is normed, completeand equipped with an inner product. The space is said to be complete if any Cauchysequence converges to an when A sequence is called aCauchy sequence if when both

In analogy with the finite dimensional case (see eq.(1.5)), the inner product will beoften written in vectorial form. In these cases we see every as a bounded linearoperator acting on in such a way that i.e. correspondingly,we define a bounded linear operator performing the inverse mappingsuch that

The linear functional represents a particular case of an adjoint operator, definedsimilarly to eq.(1.4): given any two Hilbert spaces in correspondence of anybounded operator there exists a unique operator performing theinverse mapping such that

We will also generalize the basis of a vector space by introducing the concept of frameof a vector space. For the moment it is sufficient to say that a frame of vectorsin a space represents a redundant reference system with more vectors than strictlynecessary to form a basis: for example, in the bidimensional plane, three non collinearvectors form a frame. The rigorous definition of a frame will be given in the nextsection, when we rephrase in the frame language the results here obtained for thewindowed Fourier transform.

Concerning the FT, we consider the space of square integrable functionsof one continuous variable which will be called time but which could equally be space,voltage, etc. Given a function of time its analysis is performed by theFT, viz.,

where is the frequency (Hz).The synthesis of is performed by the Inverse Fourier Transform (IFT) of

viz.,


Both the above integrals are assumed to exist.

i.e. if it has a compact support of length it issufficient to know only in correspondence of the discrete sequence of frequencies

( where denotes the set of all integers) where Then theeqs.(2.2) and (2.1) become (in the following, if no limits are indicated in a summationsign, then is understood)

i.e. if for it is sufficient to knowonly in correspondence of the discrete sequence of times where

This last condition follows from the requirement that the Nyquist frequencyshould be larger than or equal to the maximum frequency in Then

the eqs.(2.1) and (2.2) become

2.3 The analysis of

The idea underlying the WFT may be formalized by introducing a suitable win-dow function which vanishes, or is negligible, outside an interval oflength T near for example outside (–T , 0), and whose norm isTo visualize the situation we may think of as a bell shaped curve or a rectanglestarting at and ending at For completeness, we assume that maybe a complex valued function, although in practice it is real. This window, suitablytranslated in time, will be used to localize in time 3. Then, for every insteadof we consider the new function

3 Gabor (1946) selected a Gaussian curve for

Moreover, a little algebra shows that

Moreover, a little algebra shows that

and define the WFT of as the usual FT of viz.,


In words, as a function of may be interpreted as a pure note located nearwhich oscillates at the frequency within the envelope constituted by the carriertranslated at More formally, eq.(2.7) shows that the family is constitutedby a continuous sequence of vectors in doubly indexed by the time–frequencypair and eq.(2.6) shows that the WFT maps an defined in toan defined in the plane of the indexes. Since the familyis obtained by translating and modulating with a complex exponential the window

this window may be properly called the mother of all the notesand these may be written as

where is the adjoint of Another expression for may be obtained byresorting to Parseval’s theorem, viz.,


where

whose FT is

Since also and eq.(2.6) may be written as an innerproduct

The norm of is

where we have introduced the operator

This operator will be soon recognized as being the metric operator (see eq.(1.7)) ofthe WFT transform. Application of G to any and freely interchangingthe integration orders, yields

so that

Thus the operator G reduces to the norm of the window which is a positiveconstant, possibly equal to one if the window is normalized in energy. Then eq.(2.11)reads

The above expression tells us that the univariate energy density per unit time of thesignal namely is resolved in the bivariate energy density per unit timeand frequency, namely

MARZIO MARSEGUERRA62

2.4 Time and frequency localization

Assume that in addition to being well localized in time, is also well localizedin frequency. In other words, assume that is a window function which vanishesoutside a possibly small interval near Then the eqs.(2.6) and (2.10) allowunderstanding the time/frequency localization features of Indeed eq.(2.6)tells us that in correspondence of a given value, is related to the functionof frequency resulting from Fourier transforming a portion of near Morespecifically, the frequency content of is limited to the frequency content of thatportion of (weighted by translated at ) which belongs to the time lengthT of the window. The frequency content of this portion of not immediatelyevident from eq.(2.6), is better clarified by the companion eq.(2.10). This equationtells us that in correspondence of a given value, is related to the functionof time resulting from an inverse Fourier transform of the portion of (weightedby translated at ) which belongs to the frequency interval of the window. Inother words, the two equations (2.6) and (2.10), taken together, tell us thatis related not to the whole frequency content of the portion of near but onlyto those frequency components of this portion portion which are near

In conclusion, it can be said that if vanishes (or is very small) outside a smallinterval T and its Fourier transform vanishes (or is very small) outside a smallfrequency interval the WFT of namely provides good informationon both the time and the frequency behaviour of In this respect it shouldbe remembered that any window obeys an uncertainty principle which states that

constant. This result agrees with our intuition according to which we mustobserve a signal during a finite time interval in order to estimate its frequencycontent during that interval. If that is if the window supportthen it is impossible to infer the frequency value of thesignal at from the knowledge of the single value Analogously, if (inthis case that is if the window support tends to the whole real axis,then independent of the WFT becomes the usual FT and anytime localization is lost.

2.5 Reconstruction formula

The WFT of a function taken with respect to a window is given byeq.(2.6), reported below for convenience

or, in vectorial form

Inverse Fourier transforming, we get

Since the window vanishes everywhere except in the (small) time interval Tnear we cannot recover from the above expression. Instead we multiplyeq.(2.17) by and integrate

NOTES ON A SIMPLIFIED TOUR

Solving with respect to we obtain the inverse windowed Fourier transform(IWFT):

where

with G the operator defined by eqs.(2.12) or (2.13). Note that from the proportionalitybetween and it follows that is also the mother of the family

The two equations (2.15) and (2.18) represent the pair of windowed Fourier directand inverse transforms of a function with respect to the window Substitu-tion of eq.(2.16) in eq.(2.18), yields the vectorial form

which originates the continuous resolution of unity, viz.,

where I is the identity operator and where the second equation is obtained by takingthe adjoint of the first one. Note that, from the definition (2.12) of G, eq.(2.21) maybe written as Recall that in the discrete case described in §1.6, we haveseen that a resolution of identity requires a pair of reciprocal bases, namely and

Therefore, in analogy with the discrete case, we may say that andconstitute a pair of reciprocal families. Moreover we also recall from §1.5 that thereciprocal vectors can be directly obtained by applying the inverse of the metricoperator to the vectors (see eq.(l.8)). In the present case, eq.(2.19) indicatesthat the operator G, defined by eq.(2.13), does exactly the same job, so that actuallyit can be recognized as being the metric operator of the WFT.

The resolution of unity is a useful expression for obtaining important relations.As a first example, eq.(2.14) may be easily obtained as follows

Since the integral at is the norm of in the space the equationmay be written as We shall see in §3.2 that this

63

expression means that the vectors form a continuous tight frame with framebound C.

A second, more important, example of application of the resolution of unityconcerns the linear dependency among the vectors. Indeed multiplication ofthe second of eqs.(2.21) by shows that each may be expressed in termsof a linear integral operator acting on the others


where

is the so–called reproducing kernel.The same situation occurs for the Taking the adjoint of eq.(2.22) and

multiplying by from eq.(2.9) we have

From eq.(2.22) it appears that the vectors are not independent so that theyconstitute a frame and not a basis. Correspondingly, the family may besaid to constitute the reciprocal frame.

2.6 Consistency condition

We have seen that the WFT of a function is the functionWe are now interested in investigating the inverse situation, namely in de-

termining the conditions – called consistency conditions – under which a functionis the WFT of a function Clearly not any such

function can satisfy the consistency conditions since in that case we could select afunction with arbitrary time–frequency localization features, thus violating the uncer-tainty principle. Let us call the proper subspace of such that any function

is the WFT of a function It can be easily proved that afunction is also if it satisfies the following consistencycondition [write and substitute (2.20) for ]:

where is the reproducing kernel defined by eq.(2.23).In many instances we deal with a function which we only know to belong

to Given a window we may equally use eq.(2.18) to perform its IWFT,viz.,

In general, if we take the WFT of we do not return to indeed, sinceis the proper subspace of (usually a small portion of the whole spacein general We arrive instead to a function

The above situation is of interest for example when we try to reduce the noise in agiven function To this aim we firstly perform the WFT of thus arrivingat and then we modify the transform in correspondence ofthe pairs in a suitable domain (e.g. setting to zero the values below a certainthreshold). By so doing becomes an generally however its IWFT,

represents the least square approximation to the initial in the sense thattheir transforms are as close as possible in

3. FRAMES

3.1 Introduction

In the present section we introduce the concept of a frame and show that thescheme thereby derived lends itself to easily obtaining the expressions relevant totransforms like that described in the preceding section. This approach is quite general,as it will be seen in the next sections where it is applied to the wavelet transform ina straightforward manner.


which represents the least square approximation to in This fol-lows from a theorem which states that, among the WFT of all functions

is the closest to in the sense that

2.7 Drawbacks of the WFT

A distinct feature of the WFT is that the window has always the samesize, independently of the local behaviour of the function to be analyzed. Whenwe are interested in resolving the frequency content of along time, this featurerepresents a drawback of the procedure since it limits the usefulness of the approach incase of a wide–band, non–stationary signal. In this case the short duration details ofthe signal, i.e. its high frequency content, could be better investigated by means of anarrow window, whilst the analysis of the long term trends, that is the low frequencycontent, calls for a window with a wide support. Actually both kinds of windowsshould be simultaneously available since a narrow window looses the general trendwhile a wide one smears out the details of the signal. Thus the two kinds of windowscomplement each other and a collection of windows with a complete distribution ofwidths would be necessary.

3.2 Continuous frames: definitions and main features

In the framework of the WFT it has been shown (see eqs.(2.15) and (2.18))that the analysis and the synthesis of a function are carried out byresorting to two families of vectors in – or, more generally, in – namely thefamily for the analysis and the reciprocal family for the synthesis, bothindexed by the pair The analysis of (see eq.(2.15)) may be viewedas a continuous linear combination of the vectors with coefficients whichgives rise to a function However, eq.(2.22) shows that the vectors

are not independent and, correspondingly, from eq.(2.24) it appears that alsothe are not independent. Thus the family does not represent a basis

The inner product representing a mapping from the spaceto the space may be viewed as performed by a frame operator

(where is the range of T) so defined

4 The generalization to a multi–dimensional case is straightforward.5 We shall always consider honest, i.e. measurable, functions.6 Application of T* to implies returning to but not necessarily to in other

words, in general,


so that

We now introduce the adjoint T* of the operator T, which performs the inversemapping 6, i.e. To obtain a formal expression for let us

The above expression is called the frame condition or also the stability conditionsince it compels within fixed limits, the ratio B/A representing a stabilityindicator; the closer to 1 is B/A, the more stable is If A = B we say that theframe is tight and the above condition becomes

A family of indexed vectors is called a frame in if there existtwo constants A > 0 and called frame bounds, such that for anythe inner product satisfies the inequality constraint

With the new notations, a frame may be so defined:

Then is also a Hilbert space, called the auxiliary Hilbert space ofThe link between the new notations and the old ones relating to the WFT is

in Intuitively speaking, we may say that, with respect to a basis, this familycontains a redundant number of vectors which are said to represent a frame in

In order to define the frame in a slightly more general scheme, in the followingwe shall consider two families of vectors in namely the family andits reciprocal family both indexed by where represents anypair of continuous indexes belonging to a bidimensional set 4. Generalizing theweighted inner product (1.3), different weights may be given to different byintroducing a measure so that the integral of a function5 isWe also denote by the set of square–summable functions with measure

for which

i.

At first sight the above definition of frame might appear a mathematical subtlety;that definition is instead the key for a deep and general understanding of the matter.Indeed the following consequences may be drawn from the above definition:

is the so called metric operator, which plays a role analogous to that definedin §1.5 for the discrete case. From the above definition it appears that G isHermitian, self–adjoint and positive definite, so that it possesses an inverseThe formal expression of G in terms of the frame vectors may be obtained bysubstituting eqs.(3.2) and (3.3) in the definition (3.5),viz.,

7 In case of a tight frame with A = B = 1 the metric operator reduces to theidentity operator, i.e. G = I.


consider any two vectors and Then

and therefore, in a weak sense,

ii. In terms of the operators T and its adjoint T*, the norm of is

The frame condition (3.1) now plays its fundamental role: it guarantees that Gand are both bounded. To see this, (3.4) is substituted in (3.1), viz.,

so that and and, finally

which indicate that G and are both bounded 7. Then, in correspondenceof any the vectors belong to and inner products involvingthem are admissible.We now define the reciprocal vectors asiii.

Then, from eq.(3.6) we get

where

and

The above item iii. illuminates the fundamental role played by the inverse metricoperator for obtaining the reciprocal vectors. Unfortunately, in many instances,we are not able to find an explicit expression for the inverse of a given G. Never-theless we may obtain good estimates for by resorting to the first inequalityin (3.8). This inequality tells us that G may be expressed as the mean of the twobounds AI and BI, plus a remainder. A suitable expression for G is

8 Proof: any vector may be decomposed in the sum of a vector

plus a component in the orthogonal complement of

The operator TS projects in itself and gives rise to a zero vector in

when applied to Indeed To show thatconsider any and let Then the inner product of times

vanishes, viz. Since is any vector in this

expression implies and, finally,


which is the resolution of unity for the case at hand. In vectorial form, theresolution of unity and its adjoint are

iv.

Then and Substitution of(3.11) in (3.8) yields Since the series convergesin norm to and then In practice A and B arereasonably close to each other so that the above series converges rapidly to itslimit and thus it may be well approximated by a few terms.In case of a tight frame, namely A = B, the situation is quite simple: fromeq.(3.8) we get G = AI and then In this case the reciprocalvectors are proportional to the frame vectors and both families of vectors havethe same mother.We are now in a position to derive the reconstruction formula for Indeed,defining the operator

v.

from eq.(3.5) one has so that S is the left inverse of T.Then the reconstruction formula reads

or, written in length (see eqs.(3.3) and (3.9)),

In summary, in order to reconstruct from its transform we resort tothe reciprocal vectors which, in turn, are immediately obtained from the

once is computed.The expression (3.12) shows that, actually, the operator S performs the inversemapping of any vector to a vector However, thisdoes not mean that if we apply the direct mapping T to the so obtained weshould necessarily return to In other words, not necessarily equalssince the operator TS performs the orthogonal projection from to therange of the operator T 8 . If then TS

3.3 Discrete frames

We shall now consider the case in which the set of continuous indexesis replaced by a discrete set constituted by a grid of points

taken from Then the integral of a function over is replaced bythe sum

4. THE CONTINUOUS WAVELET TRANSFORM

4.1 Introduction

The drawbacks concerning the time and frequency localization capability of theWFT mentioned in Section 2.7 are substantially reduced by the use of the wavelettransform technique. As in the Fourier approach, one starts with a window function

called the mother wavelet, which takes the place of the mother theword wavelet is added since, besides the usual requirements of a finite normand of a good time and frequency localization, the function should also have zeromean. These requirements imply that must exhibit an oscillatory behaviouralong the small time interval in which it is localized (where it is sensibly differentfrom zero); then it must resemble a short wave or a wavelet. In general maybe a complex valued function, but in practice it is a real function. Analogously tothe pure (musical) notes considered in the Fourier case, we construct a set offunctions called wavelets, which are versions of the mother wavelet scaled by

and translated by t, viz., 9

9 Since the whole family is obtained from a single function the basis is saidto be structured.


whose Fourier transform is

In eq.(4.1), the scale factor in the argument of represents a dilatation or a compres-sion of the mother wavelet according to whether respectively;negative values of indicate also a reflection of the wavelet around the ordinate axis.The scale in bears a strong similarity to the inverse of the frequency ap-pearing in the pure notes of the WFT: an increase of represents a dilatationof and, analogously, a decrease in represents a dilatation of the sinusoidswithin The difference between the two is that when is varied, the supportof remains constant, whilst that of varies with being larger at larger

and the norm of F is where is the weight of thesingle point possibly equal to unity. The frame condition (3.1) becomes

scales and viceversa. As we shall see, this feature represents the basis for the zoomcapability of the wavelet approach. Moreover, from eq.(2.8) we see that the FT of

is related to the FT of at a frequency translated by from eq.(4.2) wesee that the FT of is instead related to the FT of at a frequency multi-plied by This difference is in favour of the wavelet approach since the filtering inthe frequency domain is generally performed in terms of octaves, so that frequencymultiplications by a fixed factor are preferable to frequency translations by a fixedamount.


4.2 The direct and inverse transforms

The wavelet transforms will now be obtained as described in the general proce-dure given in Section 3 with the following change of notations

Firstly we find the condition under which the family constitutes a frame.Given any we define the Continuous Wavelet Transform (CWT) as the

inner product (this product exists since so that

where is the frame operator (see eq.(3.2)). The CWT (or the frame operatorT) performs a mapping from the space to the space having measure

ds dt where is a weight function to be suitably defined later. The linkbetween the norm of and that of is given by eq.(3.4), viz.,

where G is given by eq.(3.6), which now reads

The wavelet family constitutes a frame provided G is a bounded operator:in that case G is the metric operator of the wavelet approach (see eq.(3.9)). To seethat G is bounded, we apply G to any Freely exchanging the integrationorders yields

Substituting eq.(4.1), the factor in brackets becomes

The second integral on the does not depend on provided we defineIn this case we have

and eq.(4.5) reads and, finally, since is any function in weget G = C. Thus G is actually bounded provided the mother wavelet is chosenin such a way that it satisfies the admissibility condition:

which is the above mentioned zero–mean requirement for the mother wavelet.The conclusion is that the family generated by any admissible mother

wavelet actually constitutes a frame in Therefore, from eq.(3.9), the recip-rocal wavelets are

Moreover, since the frame operator G is a constant, the frame is tight (see item iv.of §3.2) and the vectors of the reciprocal wavelet family, required for the synthesis of

are proportional to the frame vectors, viz.,

where

is the mother of the reciprocal wavelets, also generated by Note that thereciprocal wavelets are generated by their mother exactly as the frame vectorsare generated from From eq.(3.12) we finally have the reconstructionformula

It might appear that there is something wrong in the above equation; indeed if weintegrate both members over the is zero since, from eq.(4.8), the integral over

is zero, whereas the integral of the i.e. the area under may wellhave a finite value. The explanation of this apparent incongruence is that eq.(4.9)was obtained by resorting to eq.(3.12), which makes use of the operator T*, definedby (3.3) in a weak sense. Therefore, eq.(4.9) also holds true in weak sense, so thatit can only be used to compute inner products It may be shown(Daubechies, 1992, p.25) that the integral on the of (4.9) converges to in

but not in In words, if we perform the double integration overa finite area when and the difference between the two


so that

In passing, note that the above condition also implies that, for


members of eq.(4.9) tends to a very flat, stretched, function which has same area asthat of but vanishing norm in indeed this norm is an inner product and,coherently with the mentioned weak sense convergence, (4.9) holds true not per sebut when used to compute inner products.In the case at hand the resolution of identity (3.10) is written

In general we consider positive scales only and in this case from the expression (4.6),we get

where the upper sign refers to and viceversa. The condition required for theadmissibility condition of reads

The metric operator G is now piecewise constant, being

As before, the reciprocal wavelets are obtained by resorting to eq.(3.9)

where

This time we have to distinguish between negative and positive frequencies, as requiredby eq.(4.13); to this aim we first Fourier transform the above equation (4.15) and thenwe take its inverse transform, separating the two frequency ranges, viz.,

where

and

In case of a real mother wavelet we have so thatand from (4.11) we get Then eqs.(4.15) and

(4.14) become

and

In this case from eq.(4.3) we have

Finally, eqs.(4.10) and (4.3) yield

so that

in which the norm of is taken with respect to the measureThis expressions means that the vectors

form a tight frame with frame bound C/2.

Moreover is required to decay at infinity fast enough so that

Then it turns out that

Proof: The condition (4.16) clearly implies that so that we maywrite


4.3 A feature of the mother wavelet in transient analysis

As pointed out in §4.1, the CWT of a signal lends itself to localize and analyzetransient phenomena such as singularities, peaks, etc. Obviously, the efficiency ofthis methodology strongly depends on the choice of the mother wavelet which,to a large extent, is up to the analyst. In this respect we have seen that, besidesa somewhat vague requirement of good time and scale localization properties, themain requirement imposed to is represented by the admissibility condition (4.7)which implies the vanishing of its zero–th order moment. We now show that thevanishing of as much higher moments as reasonably achievable is a further featurehighly desirable in signal transient analysis. To this aim let us consider (Daubechies ,1992, p.49; Cohen, 1996) a function continuous, bounded and square integrabletogether with its derivatives up to the order, with In order to investigatethe regularity of at some point the approximating polynomial of degree isintroduced, viz.,

Further, assume that is regular at i.e. assume that the followinginequality holds true

where and is a strictly positive constant. Now, select a motherwavelet having vanishing moments up to at least the order, viz.,

Finally, eq.(4.18) follows from the condition (4.17) provided it is further assumed that

The expression (4.18) tells us that, as the scale (which implies that alsoby the condition (4.19)) then:

decays very fast to zero if is an ”honest” or a ”regular” function ati.e. if at it is with a large

slowly decays to zero if is ”irregular” at i.e. if its indexis small.

Obviously these statements are valid provided is such that its moments are zeroup to the order, with In that case, the conclusion is that wavelettransforming an and looking at small scales, we may observe peaks in thein correspondence to values of where is irregular.

A possible application of methodologies analogous to the described one may befound in image analysis and compression. Indeed most of the information in animage is contained in the discontinuities of the graph. A deep and comprehensiveinvestigation of this problem may be found in Mallat’s works (Mallat, 1992).

where the last equality follows from eq.(2.15). Having discretized the frequency insteps of size we proceed by discretizing the time in steps of length By sodoing the set of continuous indexes is replaced by a discrete set constitutedby the grid of points where Eq.(5.l) reads

10 Actually we are free to chose any frequency step smaller than 1/T sincesuitably padded with zeros, may be thought of as having a support larger than T.


5. THE DISCRETE WINDOWED FOURIER TRANSFORM

5.1 Time–limited window

We now deal with a time localized (or time limited) window, that is a windowwith compact support having a length as described in §2.2.

Therefore also has the same support (translated by ), whateverthe support of may be, and it can be expanded in a discrete Fourier series(eq.(2.3))

where is the frequency step 10, and

where

To obtain the reconstruction formula for we follow a procedure completely anal-ogous to that of the continuum case (eq. (4.4)), viz.,

where The family constitutes a frame provided Gis bounded; in this case G would then be the metric operator of the discrete windowFourier approach. To see that G is actually bounded we apply G to anyviz.,

Substitution of eq.(5.2) in which and are replaced by and yields

so that G is

11 apart from an inessential zero measure set in case of and

and

In conclusion, provided provided the time step is smaller than thewindow support T, the do constitute a frame; the operators G and areboth bounded so that G actually is the metric operator. We may now utilize eq.(3.9)to get the reciprocal vectors required for the synthesis of viz.,

Remembering that is that is we get

where


which is a function which approximates Unlike the continuum case,the operator G is not a constant; however we may say that, since has a compactsupport, the of (5.3) contains only a finite number of nonvanishing terms, so that

Moreover, for so that, ifthat is, if then in

and within the intervals Then is notbounded and the do not constitute a frame. In the opposite case, namelyfor is always positive 11. Then is confined between a greatestlower bound and a lowest upper bound so that, almost everywhere

is the mother of the reciprocal vectors The above expressions indicatethat also the vectors can be obtained by means of translations and modulationsof a single function, namely the which differs from the window as much as thefunction differs from the constant

The reconstruction formula for i.e. the synthesis of is finally obtainedfrom eq.(3.12) which in the present case reads

To obtain the reconstruction formula for we again proceed as in the continuumcase (see eq.(4.4)), viz.,

12 Actually we are free to choose any positive time step smaller than sincesuitably padded with zeros, can be thought of as having a band width larger than

which is the discrete counterpart of eq.(2.18).

5.2 Band limited window

The present analysis closely follows that of the preceding section, the differencebeing essentially that the roles of time and frequency are interchanged. A window

is said to be band limited if its FT is zero outside a specified frequency band,viz., for The starting point is represented by the eq.(2.10), herewritten as

Since is band limited, also has the same band width (translatedby ), whatever the band of may be, and then it can be expanded in a discreteFourier series

where is the discrete time step 12. From the second of eqs.(2.4) it is seen thatthe coefficients are given by the of eq.(5.4) written for Then

Having discretized the time in steps of length we proceed by discretizing also thefrequency in steps of size As for the case of a time localized window, theset of continuous indexes is then replaced by the discrete set constitutedby the grid of points where and From eqs.(5.4) and(2.8) we get


is a non negative function which plays a role analogous to that ofof the time limited window case, the difference being that now the inner product istaken in the frequency domain so that we have to come back to the time domain.Moreover, from eq.(5.8) it appears that a possible metric operator would act as afilter on Proceeding similarly, it can be shown that, providedis confined within the interval so that also

then

and finally, since by Parseval’s theorem

Thus the family does actually constitute a frame. Having ascertained thatis bounded, the reconstruction formula for immediately follows from eq.(5.7)

viz.,


From eqs.(2.8) and (5.5) we get

Substituting in (5.6) yields

where

where

Inverse Fourier transforming (5.10),

Thus the sequence whose members are the inverse FT of the isthe required reciprocal family of

6. THE DISCRETE WAVELET TRANSFORM

6.1 Introduction

Given a signal let us consider its dilated version having same totalenergy, viz.

where is a scale factor. A feature of the CWT is that the transformof computed at the scale–time coincides with that of computed incorrespondence with the scale–time

Indeed, we have

so that

In the discrete wavelet approach, the continuum of scales and times is substitutedwith a regular grid. When sampled with a time step the initial continuous signalmay be thought as taken at the initial scale so thatwhere The condition (6.1) implies that the CWT of this signal dilatedby i.e. at the scale and taken with a larger time step at times

coincides with If now the dilated signal takes theplace of the initial one, an additional dilatation leads to a scale andto a time step so that Proceeding in this way, at the

step the scale is the time step is and thediscrete time sequence is In conclusion, in the discrete approach, thecondition (6.1) is satisfied provided that geometrical sequences of scales and times areselected.

13 This condition follows from the well known requirement that the Nyquist fre-quency be larger than or equal to

6.2 Band–limited mother wavelet

We now consider the case of a band–limited mother wavelet, i.e. a mother waveletfor In a discrete approach, the maximum time step at the

scale is then 13. Application of Parseval’s theorem to eq.(4.3) andsubstitution of expression (4.2) for gives


As in the preceding cases, we must now establish whether the wavelets so discretizedconstitute a frame (actually, a subframe of the continuous wavelet frame). In otherwords, we should see whether also in the present case of the discrete windowed trans-form we can arrive to a frame condition similar to that of eq.(5.9). To this aim weestablish the link between the norm of and that of From the generalexpression (4.3) and from Parseval’s theorem we get

where


Since is band–limited, also is band– limited, viz. forand we may expand in a Fourier series the quantity

where is the scale–dependent time step at scale The coefficients aregiven by the inverse Fourier transform

Having discretized the time, we now discretize the scale according to the values of thegeometrical sequence, viz. Eq.(6.2) is then written

Correspondingly, instead of the continuous wavelets we shall consider theirdiscretized versions. To simplify the notations we write

Then

From eqs.(6.4) and (6.3) we get

so that

which is a real function everywhere positive, apart from a possible inessential setof measure zero. Eq.(6.6) is analogous to eq.(5.8) of the discrete windowed Fourierapproach; again, it turns out that a possible metric operator would act as a filter on

The next step consists in investigating whether is constrained within fixedlimits; in this case the of eq.(6.6) would give rise to a constraint for andthen, from Parseval’s theorem, also for To this aim we note that


so possible bounds for may be determined with reference to one period

The mean value of over a period isii.

where are the same constants of the continuous wavelet transform in case ofpositive scales (see eq.(4.11));It may be easily shown that where is defined byeq.(4.11).

iii.

From the above remarks it follows that in each period andare essentially positive and oscillate around their mean values and

respectively, which obey the admissibility condition (4.12). Therefore in each period– and then everywhere – the function is constrained between a greatest lowerbound which is positive everywhere except at where and alowest upper bound viz.,

If we define and we end up with therequired constraint condition for viz.,

which plays the role of the admissibility condition (4.12). Therefore

Since by Parseval’s theorem, eq.(6.6) then becomes

which is the required condition for the to be a frame. Having ascertainedthat the discrete family actually constitutes a frame, we may determine the

i. is scale–periodic, viz.,

reciprocal family required for reconstructing To this aim we writeeq.(6.5) as follows

where the i.e. the IFT of are the required reciprocal wavelets.The above analysis shows that, unlike the case of the continuous transform, the

discrete reciprocal wavelets are not proportional to the wavelets Inthe present case, starting from an admissible mother wavelet we must proceedalong the following steps. Firstly, we compute the discrete wavelets (eq.(6.4))and their Fourier transforms successively we filter them with thefilter, thus obtaining (eq.(6.9)); finally we get the required by inversetransforming

Concerning the filtering action, we note that in the limit the functiontends to which has the constant values in this case the above filtering

action is missing and the proportionality between wavelets and reciprocal wavelets isrestored.

7. THE MULTIRESOLUTION ANALYSIS

7.1 Introduction

The multiresolution analysis (MRA) represents an important improvement inthe field of the wavelet transform, essentially because it allows obtaining recursiveexpressions well suited for the computations

7.2 The nested spaces

Given a function a scale factor and a time step weintroduce the following quantities:


The factor in brackets on the is and we know from eq.(6.7) that its inverseexists. Then

where we have introduced the Fourier transform of the reciprocal wavelets, viz.,

Inverse Fourier transforming eq.(6.8) yields

Within the present context of the MRA, is generally a real, bell–shaped functioncentered near like the window of the WFT case. If is a suitably definedwidth of then is centered near and has a width of Itfollows that is a sample of weighted over an interval of lengthnear Clearly, the greater the interval over which is averaged, the lessthe details of the profile of contribute to the computation of the sample value. Inparticular, the above interpretation of the inner product must be valid also forwhatever the translation of may be: therefore we require that should obey thefollowing first condition

14 Proof: in correspondence of an arbitrary consider the quantity

Since and then we have


and

From the definitions it follows thatand are unitary operators, viz.,

Thus and analogously forb - application of the adjointness condition (1.4) to eq.(7.1) yields

c - the two operators do not commute; they enjoy the following properties

The FT of is

which implies continuity of the Fourier transform for all 14. Howevernote that this condition is convenient but not essential for the MRA.

a -


A second condition imposed on the scaling function is the orthonormality betweentranslated versions, viz.,

which implies the same kind of orthonormality between translations at anyscale:

For any define the vector space as the space whose orthonormalbasis is the family In particular, is the space whose orthonormalbasis is the family of translated scaling functions; thus by definition.Then any vector may be written as

where we have introduced the translation operator

where

In particular, from eq.(7.8) it follows that any may be written as

and we may identify with the set of all square summable sequenceswith The link between and is (see eq.(7.3)):

Taken together, the two above equations read

Since its norm is finite, so that

This condition guarantees that the operator is bounded even if the sum ineq.(7.9) contains an infinite number of terms. In practice we approximate the vectors

in by considering a finite number of terms in the sum appearing in (7.8): corre-spondingly becomes a polynomial in T and (a Laurent polynomial) andtherefore it is bounded in In the following we shall assume that any opera-tor like is a Laurent polynomial operator. These operators will be also calledfunctions of T.

Eq.(7.8) states that the vector is obtained by applying the operatorto the vector so that is the image (the range) of under theoperator

In addition to the mapping performed by we also introduce the orthogonalprojection the relevant operator is

15 Proof: we have


Indeed, given any we get a vector inThus is a projection operator. If is already in from the orthonormality

condition (7.7) it follows that the application of leaves it unchanged. Finally,since is self–adjoint and idempotent 15, viz. and then

which means that the projectionis orthogonal. Since and theoperator may be written as

We are now in a position to fix the third condition of the MRA. We have seen thatwithin i.e. at scale the time shift between successive basis vectors and

is then successive samples and may be intuitivelyviewed as representing sampled with a time step Going to the next

scale the corresponding samples are taken with a time step times larger,so that the family constituted by vectors in contains more information on

than the family of the vectors in From this intuitive consideration,the third condition of the MRA follows, viz.,

From this condition and from eq.(7.13) it follows that so thatmust satisfy the fundamental dilatation equation


where the operator dependent on the choice of the scaling function is

The dilatation equation tells us that the dilated scaling function is a weightedaverage of translated scaling functions the coefficients being the weights.The polynomial operator acting on vectors in is called averaging operator andthe weights are called filter coefficients. From eq.(7.16) we get

We shall see that this equation represents the basis for obtaining eq.(7.59) which givesan explicit expression for

Two additional conditions are imposed on the MRA. The requirement (7.15) maybe written as

so that the family constitutes a hierarchical structure of nested spaces, eachcontaining those relating to scales with higher index Then one is lead to intuitivelythink that the space relating to the smallest possible index should coincide withand that relating to the highest possible index should consist only of a constant,whose value can only be zero, due to the requirement of belonging to Theseare exactly the fourth condition and the fifth condition imposed to MRA, viz.,

which, with some abuse of notations, may also be written as

Let us now consider the orthogonal complements of the spaces Eq.(7.15) tells usthat the space contains therefore can be split in the sum of plusthe orthogonal complement viz.,

Correspondingly, in addition to the orthogonal projection operatorwe introduce the orthogonal projection operator such that, forany we get

The operator is related to in the same way as is related to (see eq.(7.14)),viz.,

Since we have

16 More generally, H and G may be defined in Here and in the following weconsider their restrictions as indicated.


where, for example means that for any thenFrom eq.(7.20), after successive substitutions, we getThen the orthogonal projection of any over may be written

where the last equality follows from the idempotent character of Whenwe getSince is contained in then which is contained in iscontained in as well. Therefore is orthogonal not only to

but also to and the family is constituted by disjoint,mutually orthogonal subspaces; the additional condition (7.18) also implies

We complete this section by considering the action of the operator on a functionIn other words, we would like to know on which space does the function

live. The orthogonal projection of over is

since when The orthogonal projection of over is

since when Thus maps ontoThis result, together with eq.(7.13), indicates that the operator performs boththe following mappings

7.2 Useful operators and their relationships

Define the operators and 16

H and G will be identified as the low-pass and high-pass filters, respectively. Fromthe above definition we get

17 Note that and areisometric operators: e.g. for the case i.e. for so that


Analogously 17

These operators enjoy the following properties:

The products and sums of the two operators are

and

From the above definitions and from eqs.(7.21) it follows that

The above relations are pictorially described in Figure 1 in which the andspaces are drawn as segments and the vectors in a given space are represented by pointsin the corresponding segment. A space is contained in another one if the downwardvertical projection of the segment representative of the former falls within the segmentrepresentative of the latter, e.g. The null space of an operator isrepresented as {0}. However note that the above scheme of vertical projections mustnot be applied to points representative of vectors; indeed the vertical from a point, e.g.in falls either within the segment representative of or within that representativeof and one might erroneously infer that the vector corresponding to thatpoint belongs either to or to whilst it is clear that can be split in a componentin and another in


We further introduce the up–sampling and down–sampling linear operators. Theup–sampling operator is defined as follows

and derives its name from the fact that its application to any vector yields

Note that is obtained from by multiplying the time shifts of all thecomponents by Then, represents an enlarged version ofin the sense that, like it is made up by the same linear combination of basisvectors, but these are now separated by a larger distance instead of In particular

Let us now apply to any From eqs.(7.3) and (7.16)we get

so that

Note that so that actually maps onto the restriction of

The adjoint of is immediately obtained from (7.36), viz.,

we get


where we have introduced the down–sampling operator defined as the adjoint of theup–sampling operator, namely and the adjoint of (see eq.(7.2))

Up to now the scaling factor was allowed to take any real number greater thanone; to further proceed with the MRA we need some relationships which can only beobtained if we restrict our considerations to a scaling factor

Henceforth we shall assume that takes the above value and, correspondingly, wewill have a so called dyadic sequence of scales and the operators and will besubstituted by and respectively. First of all, we obtain the key relationshipbetween and Given any two vectors in correspondence ofany we have

Since

and therefore

The above expressions, together with eqs.(7.33) and allow one to obtain thefollowing relations (recall eq.(7.35)) i.e. )

Note that is obtained from by first deleting the odd componentsand then by halving the time shifts of the remaining even components, i.e.

In this process half of the information is lost and the support ofis half of that of

7.4 The MRA approach

In the preceding sections we have seen that, given a suitable scaling functionand a scale factor we may construct a ladder of nested spaces

such that any may be splitin the sequence of its orthogonal projections onto the spaces (eq.(7.23)).We shall now see that this sequence actually represents the WT of More precisely,

18 The orthogonal projections of any to andare zero since is orthogonal to both and


assume that we are given a signal which we view as the orthogonal projectionof an unknown signal to We successively consider the orthogonal projections

of to and respectively, and then the orthogonal projectionsof to and respectively, etc. 18. In general we get

where

By so doing, after M steps we arrive at the expression (7.22) here reported for con-venience

The vectors belong to the sequence of disjoint, mutually orthogo-nal spaces and represent the details of at the various scales

the vector represents the residue of at the M–th scale. In thelimit and the sequence is the complete WT of

The main feature of the MRA approach is that, for any it allows one toobtain the pairs by means of a recursive procedure,numerically easy to implement, based on the introduction of a new sequence of vectors

the wavelets, such that represents an orthonormal structuredbasis in All these wavelets may be obtained from a single function (dependenton the scaling function ), the mother wavelet, exactly as the sequence isobtained from Moreover, a similar recursive procedure allows one to go backwards,namely to obtain given the residue and the sequence

7.5 The wavelets

We have seen that the space is made up by the sum of plus the orthogonalcomplement of namely (eq.(7.19)). Any vector may then besplit in the sum of its orthogonal projections on and viz.,

Obviously, if its projection on vanishes; analogously, ifits projection on vanishes. We are now interested in finding explicit

expressions for these two projections in terms of the operators H and H*.

Projection From eq.(7.30) or from eq.(7.36) it appears that is theimage of H* acting on any vector in therefore any vector in – and therefore also

– may be viewed as resulting from the application of H* to someThus

The problem of getting will be considered below.


Projection From eq.(7.32) it appears that is the null space of H.Therefore any vector in – and therefore also – obeys the condition

Substitution of eq.(7.37) for H yields

Since and are both Laurent polynomials, their product is also a Laurentpolynomial and, from the first of eqs.(7.39) with and from (7.11)we get

This equation is satisfied if is odd, i.e. if it is a polynomial containing oddpowers of T only. A possible solution consists in factorizing in the product of asuitably assigned function times an even function to be determined, viz.,

An appropriate function is

whose adjoint is

so that

Substituting the above expressions (7.43) and (7.42) in eq.(7.41) yields

We now deal with the problem of determining the translation operators and

Consider the operator applied to eq.(7.47).The contribution of the second term on is zero since belongs tothe null space of H. From eq.(7.36) we get

Substitution of the second of (7.25) yieldswhere the last equality follows from the fact that The required expressionfor is then

Consider the operator applied to eq.(7.47). Thecontribution of the first term on is zero since belongs to which,

19 Proof: Any vector may be obtained by applying the dilatation operator Dto a corresponding vector (see eq.(7.8)). Then, from eqs.(7.3) and (7.16)we get

Application of to yields (see eq.(7.45))which is zero since is an odd function of T (see eq.(7.38)).


in addition to being the null space of G, is also the null space of 19. Then

Eqs.(7.44) and (7.45) yield

Since the sum on is an odd function of T, we get from eq.(7.38) and fromeqs.(7.36), (7.37)

Substituting in (7.48) and also from the second of (7.25), yields

where the last equality follows from the fact that belongs to so that itsorthogonal projection onto leaves it unchanged. In conclusion, the required ex-pression for is

We are now in a position to introduce the mother wavelet and all the wavelets therebygenerated.

We have seen that and are the orthogonal projectionsof a generic vector onto and respectively. Both these projectionsmay also be obtained with the aid of the dilatation operator D. The first projectionmay be obtained by applying D onto the vector

where the last equality has been obtained making use of the dilatation equation (7.17).The second projection may be analogously obtained by applying D onto the vector


where the vector is so defined 20

In §7.8 we shall see that obeys the zero–mean condition (4.8): assuming that it isalso localized in time, it may be called a wavelet. Figure 2, drawn with same symbolsas the preceding one, depicts this situation.

Thus, we have been able to obtain the orthogonal projections of a generic vectoronto the subspaces and its orthogonal complement in by apply-

ing the dilatation operator D to the vector or to the vectorrespectively. Both these vectors are generated by the translation operator or

applied the former directly to the scaling function which characterizes thecurrent MRA analysis and the latter to a function of namely

The wavelet can generate the wavelets just asgenerates the Indeed eq.(7.24) tells us that the dilated and translatedversions of namely spanthe space Shortly we will see that the form an orthonormal basis inas a consequence, since the sum of the yields (see eq.(7.23)), the family

actually represents an orthonormal basis in This familyis generated by so that the basis is structured and the wavelet rightlydeserves the name of mother wavelet.

To demonstrate the orthonormality of the we start fromThe may be written as

From eq.(7.33) we get

20 From (7.51) it appears that may be interpreted as the component of the basisvector of the space along the vector of the subspace i.e.

21 Proof: if then is a vector inAlternatively, if is already in then and


From eqs.(7.49) and (7.25), and taking into account the orthonormality of weget

The proof of the orthonormality of the is completely analogous to thatof the (see eq.(7.7)).In terms of the coefficients the may be written as

The orthonormality condition implies

Substitution of eq.(7.46) yields the corresponding condition on the coefficients inviz., (see eq.(7.60)). Moreover, since the vectors

are orthogonal to the vectors From the aboveconsideration it follows that the orthogonal projection operatormay be written as 21

7.6 The recursive relations for the coefficients of the WT

Let us consider a signal viewed as the orthogonal projectionof an unknown signal to The recursive relations for the coefficientsof the WT of are easily obtained in terms of the expressions resulting from theapplication of the operators H,G and their adjoints H*,G*. This is done by takinginto account that the application of the operators and to vectors in yieldssame results as in (eqs.(7.35) and (7.38)), viz.,

operator (eq.(7.37)) restricted to (indeed, from(7.32), For we get

From eq.(7.38) it follows that the terms for which is odd do not contributeto the sum; when is even, e.g. we get


operator (eq.(7.36)) restricted to

operator restricted to (indeed, from (7.31),From Figure 2 it is easily seen that for

Substituting eq.(7.50) for and proceeding as in the above case ofyields

operator (eq.(7.26)) restricted to From Figure 2 it iseasily seen that for

Proceeding as in the above case for we get

From eq.(7.29) we get the identitywhere and Eqs.(7.52) and (7.54)

yield

From the definitions and and (7.40) it turns out thatand, analogously, Then

Proceeding in this way, we successively obtain the sequences of vectors in andin viz.,

Since (again from eqs.(7.27) and (7.28))and, analogously, we get

Let us now consider the inverse operation of gettingfrom the sequence and From eq.(7.20) we get

In terms of the polynomials and this expression reads (see eqs.(7.56))

and then

The first term on is a vector in and the second term is a vector in applyingto the above equation once and once yields

Summing these two expressions (recall that and )

where the last equality follows from the fact that In terms of thecoefficients of the polynomials this expression reads (see eqs.(7.53)and (7.55))

so that we finally get

7.7 Computational procedures

The stage decomposition–reconstruction of a signal may be schema-tized as follows:

Decomposition given

22 The convolution of the two sequences and is defined asthen


i. Compute the convolutions 22 and whereand


7.8 Choice of the mother wavelet

In the preceding sections we have seen that the mother wavelet is defined interms of the scaling function (eq.(7.51)). The first and simplest scaling functionsuitable for generating an orthonormal wavelet basis goes back to Haar (1910), viz.,

where

ii. Apply to the above convolutions the downsampling dyadic operator i.e. retain

only the even terms of each convolution, thus obtaining and

At this point, the components and may be computed fromeqs.(7.56) and (7.57).Starting from computed in step ii) we may go further in the decomposition.

Reconstruction given

i.

ii.

Apply to the given sequences and the upsampling dyadic operatori.e. define two new sequences each obtained by inserting a zero

after every element of the corresponding original sequence. By inserting the zerosin the odd positions of the resulting sequences, we get

andCompute the convolutions and

and then sum them to get

The above computational scheme is sketched in Figure 3.

and

Note that obeys the orthonormalization condition (7.6). A disadvantageof this scaling function is that it is poorly localized in frequency: indeed

which slowly decays as From the definition we get


In particular

The above coincides with the dilatation equation (7.16) provided is defined as

Eq.(7.44), then, yields and, finally, the Haar motherwavelet is

Coming back to the general problem of finding the scaling function, recall thatdepends, up to a normalization factor, on the translation operator through thedilatation equation (7.17). The FT of this equation reads

where is the trigonometric polynomial

Writing (7.58) for and substituting in (7.58), yieldsDoing this over again yields

In the limit and we finally obtain in terms onlyof the function, viz.,

The convergence of the above infinite product is detailed in Kaiser (1994, pp.183–190).Since and, as we shall see immediately below,the factors in the above product tend to unity and the product can be suitablytruncated. Thus may be computed, at least numerically. Inverse FT, finally,yields the required

From the above discussion it appears that the determination of the scaling func-tion and then of the mother wavelet from eqs.(7.51) and (7.46), rests on the

23 The first constraint is obtained by substituting as obtained from the dilata-tion equation (7.17) in the normalization condition (7.5). The second constraintfollows from the generalization of the dilatation equation (7.16). From eq.(7.3) weget Then


determination of or The coefficients of this operator obey the followingconstraints 23 :

Moreover so that

and then, from the second of eqs.(7.60)

Let us move to the frequency domain. Since the FT of is whereand we get

so that eq.(7.61) becomes

From (7.60) we get substitution in theabove equation yields and, then,Thus

The above conditions, together with eqs.(7.51) and (7.46), allow us obtaining thezero–mean condition for the mother wavelet viz.,

The eqs.(7.63) represent the basis for the construction of a sequence of Finite ImpulseResponse (FIR) filters constituted by polynomials of degree M (M = 0, 1, . . .)with real coefficients. Further assuming that the first and the last coefficient of each


polynomial, namely and have non zero values it follows that M must be odd.Indeed in case of an even M the condition (7.60) written for would become

in contrast with the above assumption. Writing M = 2N – 1 we thenhave

To determine the 2N coefficients in we resort to eqs.(7.60) which give usN + 1 conditions: one from the first equation and the remaining N from the secondone written for

For N = 1, we have no degrees of freedom and the above mentioned conditionsyield then

which is the polynomial operator of the Haar wavelets.For N > 1, we have N – 1 degrees of freedom and, correspondingly, we need

N – 1 additional conditions. These may be chosen in such a way that the low–passcharacteristics of are improved. These characteristics follow from the followingtwo features of the FT of taken in correspondence of the multiples ofthe fundamental frequency

i. The FT of the dilatation equation, i.e. eq.(7.58), yields

For odd, then so that Foreven, we get

Finally, for then Summarizing these results

ii. In correspondence of any function we get, from Parseval’s theorem,

where is the periodic version of viz.,


The function may be expressed as a Fourier series

In case of then and the coefficients so thatand, finally,

The eqs.(7.64) and (7.65) tell us that attains its maximum at and then itrapidly decreases, being zero in correspondence of all Thus it is a low passfilter. This feature may be further enhanced by using the N – 1 degrees of freedom insuch a way that smoothly attains the zero values at the frequencies i.e. byimposing the vanishing of the derivatives of at the To do this, a possiblerecipe consists in assuming that is given by the product of a polynomial

of degree N – 1 (normalized so that ) times the product of NHaar–filters, viz.,

Thus as required by the first constraint (7.60), and alsoa zero of order N. The problem is now that of selecting a such that

also obeys the orthonormality condition represented by the second of eqs.(7.60) or byeq.(7.62). To this aim, define

where and From the identitywe get

Since and it follows that andThe above identity then reads

and then

where

where

is a polynomial of degree 2(2N – 1) in Since is a real quantity,we may write and then

is a polynomial of degree 2(N – 1) inThe above analysis shows that and share the same analyticalform and obey the same analytical constraints. Then andthe determination of amounts to find the square root of i.e.to find a polynomial of degree N – 1 such that

where and As in the preceding case,and are the minimum– and the maximum–phase filters, respectively.


where

In the engineering jargon, this square root extraction is also called spectral factoriza-tion.

We shall restrict ourselves to the analytical determination of for N = 2and N = 3. A detailed discussion of these cases may be found in Kaiser (1994, pp.180–183).

N = 2: The second–order polynomial reads

The–first order polynomial to be determined reads The unknown

out that there are two possible solutions, viz.,

and, correspondingly, there are two polynomials

Apart from the zeros at has a zero outside the unit circle andhas a zero within the unit circle: following the usual engineering jargon, they willthen be called minimum– and maximum–phase filters, respectively.

N = 3: The fourth–order polynomial reads

The second–order polynomial to be determined reads Theunknown coefficients are determined by the condition (7.66) and

It turns out that there are two possible solutions, for namely

coefficients are determined by the conditions (7.66) and It turns

The filter coefficients for cases from N = 4 to N = 10 are given in Daubechies (1992,p.196).


8. SUB–BAND FILTERING

The sub–band filtering of a signal is a decomposition–processing–reconstructiontechnique widely adopted by electrical engineers. In its simple, two–channel version,suitable filters are applied to the signal, which is decomposed in a low frequencyand a high frequency component, each relating to half of the entire frequency range.Since these components have different Nyquist frequencies, correctly sampled versionsare then obtained by discarding every other sampled value (sub–sampling). Thedecomposed signals are then suitably processed according to the specific problem athand, and finally the reconstruction stage is carried out by means of a procedurewhich is the inverse of that adopted in the decomposition stage.

Let us consider a band limited function i.e. a function such thatfor As mentioned in §2.2, may be represented by a Fourier

series in terms of the discrete sequence of values sampled at timeswhere is the time step resulting from the Nyquist criterion. From eq.(2.4) weget

Consider the signals resulting from low-pass (LP) and high-pass (HP) filtering

8.1 Ideal case

Low–pass filtering. Let be an ideal LP square filter centered athaving a bandwidth half of that of viz.,

Inverse transforming, we get

From eq.(2.4) it follows that can also be expressed by its Fourier series in termsof sampled with a time step less or equal to Therefore we canuse the same time step selected for so that

where or, more specifically

Application of the LP filter to yields In the frequency domain weget, from eqs.(8.1) and (8.3),

where


so that, from the first of eqs.(2.4), we get

The expressions so far obtained have been derived by sampling and withthe time step However by definition, has the same bandwidth

of therefore the Nyquist criterion is satisfied also if is sampled witha time step Then, from eq.(2.4) we get

where

Knowledge of the coefficients i.e. essentially of sampled with a time stepof allows the determination of the function (see eq.(2.5)), viz

From a computational point of view, let us assume that we have a fast code forcomputing convolutions. The coefficients appearing in the expression (8.6) for

may then be computed as follows:

i. compute the convolution between the coefficients and viz.,

ii. retain only the even terms of the sequence (downsampling of the convolu-tion)

Very simple and important expressions may now be obtained by resorting to the24. We define

Retaining only the even terms in (downsampling), yields

24 For i.e. for on the unit circle, is the Fouriertransform.


where

Sampling with a time step yields

so that

Moreover,

where or, more specifically,

Following a procedure quite analogous to that given in the LP case, we end up withthe following expressions

Inverse transforming, we get

High–pass filtering. Let be an ideal HP square filter having a bandwidthwhich is the complement of that of in the bandwidth of viz.

Finally

The function may be expressed in terms of and as follows


Knowledge of the coefficients allows the determination of (see eq.(2.5)),viz.

The recipe for computing the coefficients runs as before:

i. compute the convolution between the coefficients and viz.,

ii. retain only the even terms of the sequence ( downsampling of the convolu-tion)

Going to the we define

Retaining only the even terms in (downsampling), yields

Finally, the function may be expressed as the in eq.( 8.10), viz.

Reconstruction of f(u). From the definitions (8.2) and (8.11) of the idealsquare filters, it follows that so that, from eqs.(8.5) and (8.13)we get

Evaluating for

Inverse transforming (see eqs.(8.7) and (8.15)), we get


For even, i.e. we get

so that

Since the above expression may be formally written as

For odd, i.e. we get

so that

Substitution of eqs.(8.4) and (8.12) yields

Taken together, eqs.(8.14) and (8.15) yield

Thus, knowledge of the coefficients and appearing in the expressions (8.7)and (8.15) of the filtered signals, allows the determination of the coefficientsrequired for the reconstruction of the original signal

From a computational point of view, the reconstruction formula (8.20) may beviewed as obtained through the following steps:

i. Starting from the adimensional sequences introduce two new adi-mensional sequences each obtained by inserting a zero after everyelement of the corresponding original sequence. This operation is called ”upsam-pling” of the original sequences. By inserting the zeros in the odd positions ofthe new sequences, their elements are


ii. Compute the convolutions of the new,”upsampled”, sequences with the constantsof the two filters

iii. Add the results obtained in item ii., viz.,

where and

Let us now consider the reconstruction formula (8.20) or (8.23) in terms of theWe define

and the of the convolution (8.23) reads

8.2 Real case

The choice of the ideal square filters and for sub–band filtering agiven signal has the drawback that the filter constants, namely andslowly decay with Then a large number of them is required for performing thecalculations pertaining to the filtering procedures. A possible remedy could consistin performing the filtering action with a pair of new filters obtained bysuitably smoothing respectively. Correspondingly, with respect to theoriginal filters, the new ones will have larger bandwidths therefore theyare more concentrated in time, which means that the constants decay fasterwith and the burden of the convolution computations is lower. However, the largerbandwidths of the new filters also imply that the time step adopted in the idealcase for sampling the filtered signals should be correspondingly reduced, the maximumallowable value compatible with the Nyquist criterion being Theuse of the time step then gives rise to the aliasing phenomenon. Nevertheless wesubstitute the ideal filters with their smoothed version, viz.,

and introduce the following new notation

and


i.e. is delayed by In case of ideal filters, the filtered signals havesupports so that the addition of their versions shifted by does not change theirshape. Viceversa, in case of real filters, the supports are larger than and the abovementioned additions corrupt the filtered signals. Nevertheless the true signal can berecovered provided the filter constants are selected in such a way that the second termon of eq.(8.26) vanishes, i.e. provided

On we have written instead of because of the aliasing effects whichprevent the exact reconstruction of the true signal. The sub–band filtering proceduregiven in this section is schematically represented in Figure 4.

Let us come back to the aliasing effects present in the reconstruction formula(8.26). More specifically, the aliasing effects is present in the second term onindeed, for

Correspondingly, the decomposition expressions (8.10) and (8.17) become

and the reconstruction formula (8.24) becomes

Substitution of eqs. (8.25) finally yields

and

In both the above schemes, the relating filters are functions of the basic filterfor computational reasons, this is usually of the FIR–type, i.e. it has only a finitenumber of nonzero terms. Within the Esteban and Galand scheme, no FIR filter

has been found suitable for an exact reconstruction of (i.e. such that thefactor in brackets in eq.(8.28) be equal to 2), even though and may be keptvery close to each other. In the Mintzer’s scheme, instead, it is possible to find FIRfilters such that the above reconstruction is exact.

We end this section by comparing a decomposition/reconstruction stage of thesub–band filtering with that of the MRA in the wavelet transform. In the following weshall assume real valued sequences and work in the domain.The two procedures, namely sub–band filtering and MRA coincide provided that:

i. The operator coincides with Since


In pursuit of satisfying this condition, we report the two following schemes:

a) The Esteban and Galand scheme (Esteban and Galand, 1977). The variousinvolved function are expressed in terms of as follows

With this choice, the condition (8.27) is satisfied and eq.(8.26) becomes

The corresponding filters are called ”quadrature minor filters” (QMF) for thefollowing reasons:The condition implies that so that forwe get

Thus equals delayed byIn addition, if is symmetric, and then

which means that is the mirror of with respect tob) The Mintzer’s scheme (Mintzer, 1985). The various involved functions are ex-

pressed in terms of as follows

With this choice, the condition (8.27) is satisfied and eq.(8.26), in correspondenceof and real yields

9. CONCLUSIONS

A very simple indication of the variability of a stationary signal is representedby its variance, i.e. by a single number. After Fourier – 175 years ago! – we knowthat this single number may be resolved in a function of the frequency, namely thepower spectrum, which tells us how the variance is distributed among the harmonicsof the signal. By so doing, a lot of additional information is gained from the data:for example, a peak in the spectrum of a signal measured on a plant indicates theproneness of the plant to resonate at the peak frequency and it may suggest techniquesfor favouring or avoiding that resonance.

A further, important, improvement in signal analysis is constituted by the Win-dowed Fourier Transform, by means of which the above analysis may be extended tonon stationary signals. At each time, the Fourier transform is now restricted to aportion of the signal enveloped within a window function contained in a small timeinterval around the given time. By so doing, the signal is more deeply investigatedsince, instead of the above function of one variable, we now deal with a function oftwo variables, namely time and frequency, which allows us to examine the frequencycontent of successive pieces of the time signal. However the constancy of the width of


the condition becomes

ii. The operator coincides with or Since (see eq.(7.46))

the condition becomes

iii. The operators and coincide with and respectively. Since

the conditions become

and

In terms of the subband filtering functions are

which, apart from a change of sign in and coincide with the choice of theMintzer’s scheme. The change of sign is inessential since the above definitions alsosatisfy the condition (8.27).

the selected window makes the approach too rigid: indeed the window width may betoo large when the signal exhibits fast transients and too small when it slowly varies.

The Wavelet Transform overcomes this drawback by using windows of differentwidths: a mother wavelet, a little wave suitably localized in time and frequency, isselected and at the various scales the signal is viewed by wavelets which are dilatedand shifted versions of the mother wavelet. Thus, the procedure extracts the localdetails, e.g. the fast transients, at small scales and the signal trend at large scales.An additional feature of the wavelet transform is the possibility of selecting a motherwavelet which fits the problem at hand: in the windowed Fourier approach we alwaysmake use of complex exponentials and the freedom is limited to the choice of thewindow which envelopes the exponentials.

In spite of the fact that nowadays the wavelets have become a relatively popularmathematical technique, they are still scarcely utilized by nuclear engineers. A field ofprimary interest is certainly that of the early detection of system failures: indeed thewavelet transform seems the right methodology for detecting the warning spikes orfast transients of short duration randomly occurring in some plant signals before theoccurrence of important failures. Another field which may be profitably tackled bythe wavelet methodology is that of signal denoising, handled by suitably thresholdingthe wavelet coefficients. Among the standard applications of the wavelets, we shouldmention the attempts to improve the solutions to some mathematical models of phe-nomena of interest to the nuclear power plants: an example is the open question of thefully developed turbulent flow, generally performed by means of the Navier–Stokesequations with a dominant non linear advection term. Another standard applicationis the data compression, which is of importance in many different nuclear engineeringfields ranging from the reactor operation and maintenance to the safeguards.

We hope that the present review/introduction will stimulate our professionalcommunity towards the use of the wavelet methodology.

Acknowledgments: I would like to thank Dr. S. Tarantola for useful discussions andsuggestions.

MARZIO MARSEGUERRA112

REFERENCES

Chui, C., 1992, An introduction to Wavelets, Academic Press, San Diego.Cohen, A., and J., 1996, Wavelets, the Mathematical Background, Proc. IEEE, 84, 4.Daubechies, I., 1992, Ten Lectures on Wavelets, Soc. for Industrial and Applied Mathematics,

Philadelphia.

Esteban, D., and Galand, C., 1977, Application of Quadrature Mirror Filters to Split–BandVoice Coding Schemes, Proc IEEE Int. Conf. Acoust. Signal Speech Process., Hartford,Connecticut, pp. 191–195.

Gabor, D., 1996, Theory of Communication, J. IEE (London), 93, pp. 429–457.Haar, A., 1910, Zur Theorie der Orthogonalen Funktionen–Systeme, Math. Ann., 69, pp. 331–371.Kaiser, G., 1994, A Friendly Guide to Wavelets, BirkhäuserMallat, S., and Hwang, W.L., 1992, Singularity Detection and Processing with Wavelets, IEEE

Trans. on Information Theory, 38, 2.

Mallat, S. and Zhong, Z., 1992, Characterization of Signals from Multiscale Edges, IEEE Trans. onPattern Analysis and Machine Intelligence , 14, 7.

Mintzer, F., 1985, Filters for Distortion–Free Two–Band Multirate Filter Banks, IEEE Trans.Acoust. Speech Signal Process., 33, pp. 626–630.

Special Issue on wavelets, Trans. of the IEEE, April 1996, 84, No. 4.

GENETIC ALGORITHMS FOR INCORE FUEL MANAGEMENT ANDOTHER RECENT DEVELOPMENTS IN OPTIMISATION

1 INTRODUCTION

The number of different optimisation schemes available to the nuclear industry hascontinued to grow. In the last review, Parks1 identified Simulated Annealing (SA) as auseful tool and Genetic Algorithms (GA) as tools with potential but needing further de-velopment. Since that review, two simpler derivatives of SA have been developed – theGreat Deluge Algorithm (GDA) and Record to Record Travel (RRT) – and the GA havebeen successfully applied to fuel cycle management. This paper reviews these develop-ments in optimisation and also describes some other new optimisation tools available:the Tabu Algorithm (TA) developed by Glover2, the Population-Based IncrementalLearning Algorithm (PBIL) proposed by Baluja3 and the Multi-point ApproximationMethod (MpA) being developed by Frandsen4 for use in the petroleum industry. To-gether, this collection of techniques form a powerful toolkit for the nuclear engineerseeking to solve difficult optimisation problems.

2 OPTIMISATION PROBLEMS

Optimisation is the process of finding the optimum (i.e. best) solution possible to aproblem. This solution is often required to meet a number of additional constraints. Inmathematical terms this can be formulated as:

Advances in Nuclear Science and Technology, Volume 25Edited by Lewins and Becker, Plenum Press, New York, 1997

Jonathan N. Carter

Centre for Petroleum StudiesImperial College(ERE)Prince Consort RoadSouth KensingtonLondon SW7 2BPUnited Kingdom

113

are the decision variables, i.e. those things that can be changed. is theobjective function which is usually a numerical, rather than an analytical, functionof the decision variables. If one has to minimise a function this is equivalentto maximising So clearly maximisation and minimisation problems can betransformed one into the other.

In numerical optimisation, an optimum, either local or global, is assumed to havebeen found when all points in a small region around the current point, which meet all theconditions, have function values that are worse than the function value at the currentpoint. In practice we often cannot test all points, and so substitute ‘a sufficiently largenumber’ in place of ‘all’. As a consequence one can never be completely certain thatan optimum has been identified. Neither can one be sure whether the optimum is theglobal optimum or just a local optimum.

An example of such a problem might be to minimise the cost, of runninga power station, subject to the requirement that the power output will be 1 GW andemissions will not exceed the safe limit:

114 JONATHAN N. CARTER

Subject to

Minimise

Subject to emissions safe limit - actual emissions > 0power output = 1 GW

The decision variables might be the number of staff at different grades, or thetemperature of the core.

In practice we are often satisfied with finding a ‘better’ solution, rather than thetrue (global) optimum solution. How much ‘better’ a solution needs to be, dependson the economics of the problem. How much does it cost to find a better solution, toimplement it and what is the benefit? We may not even know if a better solution exists.

2.1 Type of Problems

Many optimisation problems can be classified as one of the following:

Continuum problems having decision variables which are continuous numbers.For example, the outlet pressure of a steam turbine.

Discrete problems having decision variables which can only take discrete values.For example, the number of cooling towers to be built.

Combinatoric problems are those where the decision variable is the order of afinite set of objects, which determines the value of the objective function. Forexample, the Travelling Salesman Problem (TSP) or the PWR reload core designproblem described below.

Optimisation methods have been developed to tackle problems in each of these cate-gories. Table 1 gives a list of some of these methods.

However, many practical optimisation problems encountered in industry do notfall neatly into one of the above categories as they involve a combination of continuous,discrete and/or combinatoric variables. These are much harder to solve with traditionalmethods. In these cases, the problem is often approximated into a form which can behandled by traditional methods. For example, a problem involving continuous and

GENETIC ALGORITHMS AND OTHER RECENT DEVELOPMENTS

discrete decision variables, might use continuous variables to approximate the discretevariables.

In this paper, several different optimisation methods are described. Some areparticularly suited to one category of problem, while others can be used on all three,singly or in combination.

2.2 The PWR Reload Core Design Problem

Most of the optimisation methods described in this review can be applied to combi-natoric problems such as the PWR incore fuel management problem described in thissection. As this problem is used to illustrate most of the optimisation methods reviewedin this paper, the problem and previous studies are described in some detail below.

The Problem. For a Pressurised Water Reactor (PWR) incore fuel managementinvolves specifying the arrangement of fresh and partially burnt fuel in the core and theburnable poison loadings in the fresh fuel assemblies. Most PWRs use a three batchloading scheme, i.e. at each refuelling, one third of the fuel is replaced. The remainingfuel assemblies, having resided in different regions of the core for one or two cyclesalready, are generally all different. Hence for a PWR with 193 fuel assemblies, the fuelplacement problem alone (assuming there are 64 identical new fuel assemblies and thecentre assembly, which is generally the assembly with the highest burn-up, has beenspecified) has (128!) possible solutions (since there are ways ofchoosing where to put the new fuel assemblies and 128! ways of arranging the others inthe remaining spaces). The search space is usually reduced by assuming rotational orreflective quarter-core symmetry or one-eighth symmetry.

However, even if quarter-core symmetry is assumed, fuel assemblies from equivalentlocations in different quadrants of the core are not identical. A fuel assembly from onequadrant is identical to the corresponding assembly in a neighbouring quadrant rotatedby 90°. Hence when working in (rotational) quarter-core symmetry, each fuel assemblycan be considered to have four possible orientations each of which represents exchangingthat assembly with the corresponding one from another quadrant. New fuel assembliesof course have uniform cross-section and therefore do not need to be re-orientated.Thus, if we consider the fuel placement problem in rotational quarter-core symmetrythe number of possible solutions is (since in rotational symmetrywe consider 48 fuel assemblies 16 of which are new).

Although all the new fuel assemblies are identical the fuel manager may choose toload Burnable Poison (BP) rods with some of the new fuel assemblies that are not in

115

Previous Studies. Early fuel management studies used few-zoned one-dimensionalreactor models with axial symmetry and tried to determine into which of these zonesfuel assemblies should be placed (although not where exactly in the zones different fuelassemblies should be located) for a series of cycles. In the first of these, Wall andFenech5 considered a PWR model with three equal volume zones, identified by increas-ing distance from the centre of the core, and three fuel batches which each occupied one


control rod positions. It is usual to load each quadrant of the fuel assembly identicallywith a maximum of 24 BP rods per fuel assembly. Thus there are seven possible BPloadings for each assembly (0-6 BP rods on each fuel assembly quadrant). If we includeburnable poison loading the search space contains about possible solutions.

To summarise, to specify a core reload pattern, the fuel manager must specify foreach location:

the fuel assembly to go at that location

the orientation of that fuel assembly (if it is a partially burnt assembly)

the burnable poison loading on that assembly (if it is a new assembly).

Objectives and Constraints The fuel manager’s objective is to minimise thecost of the energy generated by the power plant and early studies considered suchfactors as the interest on fuel, the re-processing costs and operating charges (e.g. Walland Fenech5, Lew and Fenech6). However, because these costs may vary and there is nosimple formulation of the cost function, it is general practice to work with cost-relatedobjective functions such as:

maximum End-of-Cycle (EOC) reactivity (equivalent to maximum cycle length)(e.g. Mélice7, Colleti et al. 8)

maximum burn-up of fuel (e.g. Stover and Sesonske9, Hoshino10)

minimum radial power peaking (e.g. Chen et al. 11, Naft and Sesonake 12, Federow-icz and Stover13, Chitkara and Weisman14).

The last of these is not a true economic objective (see Downar and Sesonske15) butas it is consistent with safety considerations, is frequently used as an objective in fuelmanagement studies. Where it is not used as the objective function, it must be incor-porated into the problem as a constraint. Some early studies sought to minimise thefresh fuel consumption (e.g. Suzuki and Kiyose16, Goldschmidt17) but as this must bedetermined well in advance, it is properly an out-of-core decision. The actual objectivefavoured in practice will depend generally on the utility but a practical optimisationtool should be flexible enough to work with any of the above.

The constraints include limits on:

maximum moderator temperature coefficient;

maximum burn-up of discharged fuel;

maximum average burn-up of each batch of discharged fuel;

maximum power peaking;

minimise pressure vessel damage.

zone. At each refuelling, one of 28 options (since each batch could be replaced or movedto another zone) had to be chosen. They used dynamic programming (see Bellman18)to minimise the cost of the power produced subject to power-peaking and burn-up con-straints. Stover and Sesonske9 used a similar method to identify Boiling Water Reactor(BWR) fuel management policies. Their model was similar to that of Wall and Fenechexcept the two interior zones were scatter-loaded and they optimised the fraction ofeach zone to be refuelled at the end of each cycle. Similarly, Fagan and Sesonske19

used a direct search to maximise the cycle length for a scatter-loaded PWR with fuelshuffling between zones permitted, and Suzuki and Kiyose16 used Linear Programmingto place fuel in five radial zones to minimise the fresh fuel consumption of a Light WaterReactor (LWR). These studies did not address the poison management since parallelwork was directed at solving this part of the problem independently for a given fuelloading (e.g. Terney and Fenech20, Motoda and Kawai21, Suzuki and Kiyose22). IndeedSuzuki and Kiyose16 used this decoupling to assign the problem of the power-peakingconstraint to the poison management problem.

Mélice7 took a different approach to the problem by surveying analytical tech-niques for determining reactivity profiles which maximised core reactivity or minimisedpower-peaking. The available fuel could then be used to synthesise the optimal reac-tivity profile as closely as possible. This approach also allowed the decoupling of thefuel placement and poison management scheme that would maintain the desired powerprofile. Mélice sought flat power profiles since these would enable the extraction ofmaximum power from the core. These led naturally to the out-ini fuel managementschemes which were widely used at the time. Mélice’s fuel placement problem thenwas reduced to seeking the arrangement of partially-burned fuel in the core interior(the periphery being loaded with fresh fuel) that would best match the optimal reac-tivity profile. Thus, Mélice was the first to consider the detailed placement of fuel in a2-dimensional lattice which he did by trial and error, guided by experience.

Several authors extended this work using variational techniques (e.g. Wade andTerney23, Terney and Williamson24) or even interactive graphics (Sauer25) to identifyideal power distributions and Linear Programming to automate their synthesis (Sauer25,Chen et al. 11, Terney and Williamson24). These methods however either distributedfuel amongst but not within the several zones or restricted fuel to an out-in scatter-loadedii core.

Meanwhile the problem of determining optimal detailed placement of fuel assem-blies was being tackled by considering the effects of successive binary exchanges offuel assemblies. Having used heuristic rules to restrict the possible exchanges, Naftand Sesonske12 proposed ranking binary exchanges by the improvement in the objec-tive function and then performing each in order of ranking, keeping the improvements.Similarly Stout and Robinson26 formulated shuffling rules for fuel within zones of aout-in scatter-loaded PWR. These direct search approaches reduce to a hill-climbingsearch since the best solution is preserved after each exchange. They are therefore proneto locating local rather than global optima.iii A simple direct search on all such ex-

GENETIC ALGORITHMS AND OTHER RECENT DEVELOPMENTS 117

iOut-in refuelling is when old fuel is removed from the centre of the core, partially burntfuel is moved from the outer edge to the centre of the core and new fuel is added to theedge. In-out refuelling is similar but the new fuel is placed in the centre. These different fuelloadings produce different performance profiles.

iiScatter loading is when new fuel is distributed throughout the core.iiiThe objective functions used are non-linear functions of the decisions variables, this means

that finding optima is difficult and that multiple local optima often exist.

changes is clearly too time-consuming so Chitkara and Weisman14 used a direct searchto fine-tune the result obtained by linearising the objective and constraints and usingLinear Programming. However, because they allowed fuel movement to neighbouringlocations only, this method was also prone to convergence onto a local optimum. Themethod was later extended to BWR fuel management by Lin et al. 27 and Yeh et al. 28.

Binary exchanges are one of the smallest changes that can be made to a loadingpattern and are therefore useful for fine-tuning a reload core design. Small changes froma reference loading pattern also lend themselves to evaluation by perturbation theoryat the cost of being limited to certain small perturbations only. This was exploited byFederowicz and Stover13 who used integer linear programming to find the best loadingpattern close to some reference pattern. This pattern could then be used as the newreference pattern and the optimisation continued. Similarly, Mingle29 and Ho andRohach30 used perturbation theory to evaluate the effects of binary exchanges duringa direct search, enabling more candidate solutions to be examined without calling atime-consuming evaluation code more frequently.

The size of the fuel placement problem prompted many researchers to divide itinto sub-problems which could then be solved using the most appropriate optimisationtechnique. Motoda et al. 31 developed an optimisation scheme for BWRs using themethods of Sauer25, Suzuki and Kiyose16 and Naft and Sesonske12. The basic ideawas to allocate fuel in three zones, expand this to a five zone model and finally use adirect search to shuffle the fuel within those zones. The poison management problemwas decoupled by assuming the constant power Haling depletioniv32. However, theinflexibility of PWR control methods at that time precluded the use of this method forPWR fuel management.

In the early 1980s, the development of burnable poison rods for PWRs prompted in-terest in increasing the burn-up of the fuel to further minimise costs. Ho and Sesonske33

considered changing from a 3-batch to a 4-batch scheme while maintaining the cyclelength. Clearly higher enrichment fresh fuel would be required and this could causepower-peaking problems. Reload core design would thus be dominated by the powerpeaking constraint. They rose to this challenge by assuming a modified out-in loadingand using a direct search similar to that of Naft and Sesonske12 to place fuel assem-blies in one-eighth symmetry. Burnable poisons were used where necessary to suppresspower-peaking but the orientation of fuel assemblies was not considered.

However, the combination of out-in loading with higher enrichment fresh fuel leadsto increased neutron leakage and thus an economic penalty as well as greater irradiationdamage to the pressure vessel. An alternative way to increase the burn-up of fuel isto use low-leakage fuel loadings where the fresh fuel is allowed in the core interior. Tokeep the power-peaking within safety constraints, burnable poisons can be loaded ontofresh fuel assemblies. The resulting centre-peaked power distributions had already beenshown to imply minimum critical mass (see Goldschmidt17 or Poon and Lewins34).

By 1986, interest in extending pressure vessel life by reducing neutron leakagemeant that low-leakage loadings were being used in over half of all PWRs (see Downarand Kim35). However, the optimisation methods described above assume the conven-tional out-in loading patterns. Many of them assign fresh fuel to the periphery andshuffle only partially burned fuel in the core interior. Low-leakage loadings requirefresh and partially burned fuel to be shuffled and burnable poison loadings to be de-termined for the fresh fuel assemblies increasing considerably the size of the problem.


ivHaling’s principle states that the minimum power-peaking is achieved by operating thereactor so that the power shape does not change appreciably with time.

In addition, because out-in loadings deplete to give flatter power profiles, the powerpoisons in low-leakage designs meant that even if the Beginning-of-Cycle (BOC) powerprofile was feasible, it did not imply that there would not be power-peaking problemslater in the cycle.

By this time, emphasis had shifted from optimising the arrangement of fuel overmany cycles to optimising the single-cycle problem. Suzuki and Kiyose16 had shownearlier that cycle-by-cycle optimisation gave results that were within 1% of the multi-cycle optimum.

Chang and Sesonske36 extended Ho and Sesonske’s direct method to optimise low-leakage cores by fixing the fresh fuel positions to enforce low-leakage. In a differentapproach, Downar and Kim35 separated the fuel placement from the burnable poisonloading problem by assuming a constant power Haling depletion. A fuel loading couldbe chosen in the absence of poison, using a simple direct search of binary exchanges andthen the core was depleted in reverse back to BOC to find the burnable poison loadingthat would satisfy power-peaking constraints. However, the inflexibility of PWR poisondesign meant that the optimal poison management schemes and the constant powerHaling depletion were often not realisable.

In practice the simplified models used by researchers made nuclear utilities reluc-tant to use their optimisation methods and actual reload cores were designed by expertengineers using heuristic rules and experience (see Downar and Sesonske15). As earlyas 1972, Hoshino10 investigated the incorporation of these heuristic rules into an au-tomated optimisation procedure by using a learning technique to deduce the relativeimportance of various rules in refuelling a very simple reactor model with four equalvolume zones and one fuel batch in each zone. More recently, there have been severalinvestigations of heuristic search and the use of expert systems to apply heuristic rules(e.g. Moon et al. 37, Galperin et al. , 38, Tahara39, Rothleder et al. 40). While thisapproach has the potential to reduce unpredictability and variability of human design-ers and to retain the expertise of these fuel managers, it does not necessarily produceoptimal loading patterns, being limited by the knowledge of those same people. It alsobecomes unreliable when faced with situations not previously encountered.

Thus the challenge to produce an optimal Loading Pattern (LP) using a realisticreactor model remained. Computer codes available today include LPOP (Loading Pat-tern Optimisation Program) and FORMOSA (Fuel Optimisation for Reloads: MultipleObjectives by Simulated Annealing).

Following the practice of Mélice and others whose work concentrated on findingand synthesising optimal power and reactivity profiles, LPOP was developed at West-inghouse by Morita et al. 41 and Chao et al42 , 43. Starting with a user-specified targetpower profile, a backward diffusion calculation is performed to find an equivalent reac-tivity distribution. This is then matched as closely as possible with the available fueland BPs using integer programming. However, in general it is not possible to matchexactly this power profile, which being typically based on low-leakage cores and designexperience is not necessarily optimal.

FORMOSA was developed by Kropaczek44, 45 at North California State University(NCSU) where the search for practical optimisation tools which make a minimum ofassumptions led to the use of Monte Carlo direct search techniques (e.g. Comes andTurinsky46, Hobson and Turinksy47). Any reactor model could thus be used and treatedas a ‘black box’ and any combination of objective function and constraints used. Con-tinuing along the lines of Federowicz and Stover13, perturbation theory was used todetermine the effects of fuel shuffling from a reference LP. This approach tended torestrict searches to around the reference solutions but was refined by White et al. ,



who introduced first-order48 and then higher-order49 Generalised Perturbation Theory(GPT). The latter is able to predict the effects of quite large perturbations includingthe movement of fresh fuel. FORMOSA combines GPT with the Monte Carlo searchalgorithm, Simulated Annealing, which is described later in this paper. Poon50, 51 hassince shown that Genetic Algorithms can be used to speed up the identification of near-optimal LPs. This work is described in more detail section 3.6. Since Poon’s work,the GA has been applied to incore fuel management by DeChaine and Feltus52, 53 andTanker and Tanker54. These studies are discussed in section 3.7.

2.3 A Simplified Reactor Model

To illustrate how all of the algorithms, except the Multi-point Approximation method,described in this paper work, we will use a simplified, idealised reactor core containinga total of 61 fuel assemblies in quarter-core rotational symmetry, as shown in figure 1.v

The optimisation problem is to arrange the fuel within the core to maximise someobjective function while satisfying the necessary constraints. For the purpose of thisstudy, we assume that the objective function is calculated by a ‘black box’ reactorsimulator and that there are no constraints. This last assumption is not restrictive sinceany constraints could be incorporated into the objective function through a penaltyfunction.

We will assume that the central fuel element has already been selected (it is usuallythe oldest assembly, or the assembly with the highest burn-up) and we will use quarter-core rotational symmetry. We will also make the following three simplifications to theproblem. Firstly, rotational orientations are neglected (i.e. we do not allow fuel elementsto be shuffled from one quadrant to another). Secondly, we give each new fuel element inthe same quadrant a unique identifier, as if they were not identical (in practice, aroundone third of the fuel elements will be new and therefore identical. By giving each newfuel element in the quadrant a unique identifier, we avoid the complications of havingto avoid exchanging two new, identical fuel elements). Thirdly, we neglect burnablepoisons. These simplifications reduce the problem to finding the optimal arrangementof the fifteen fuel assemblies in the quarter-core. There are then differentways of rearranging the fuel in the core. Figure 2 shows one possible rearrangement.

vThe use of bold characters in figure 1 is just a naming convention and does not implyanything about the properties of the fuel elements.


This simplified incore problem is useful for the didactic purposes of this paper.In practice, it is not difficult to incorporate rotational orientations, identical new fuelelements and burnable poisons when using the optimisation schemes described in thispaper. For example both Kropaczek44 and Poon50 show how this can be done, forSimulating Annealing and Genetic Algorithms respectively.

For the purposes of this paper all of the examples will assume that we are tryingto maximise some function. It should be remembered that any minimisation problemcan easily be converted into a maximisation problem.

3 OPTIMISATION ALGORITHMS FOR COMBINATORIALOPTIMISATION

3.1 Hill-climbing Algorithms

Many optimisation algorithms proceed by making a small change to the ‘current’ so-lution to obtain a ‘trial’ solution. If this change results in an improvement, then the‘current’ solution is replaced by the ‘trial’ solution. Otherwise the ‘current’ solution isretained. An algorithm of this type is known as a hill-climber. The problem with algo-rithms that follow this procedure is that they tend to get trapped by local, sub-optimumsolutions.

The two important decisions that have to be made when implementing a hill-climbing algorithm are firstly, how to define a ‘small change’ and secondly, how tochoose the ‘trial’ solutions. A ‘small change’ will have different meanings for differentproblems. For the idealised incore problem that we are considering there are severalpossible definitions of ‘small’ change that define the local neighbourhoodvi relative tothe ‘current’ solution. One choice might be to exchange two adjacent fuel elements, eg

An alternative might be to exchange any two fuel elements, eg

viThe local neighbourhood of a solution consists of all the solutions that can be obtainedby making a small change to the solution. The neighbourhood is therefore dependent on theallowable ‘small change’.


There are other possible choicesvii, but these are the simplest. The trade-off overthe size of the local neighbourhood is that for small neighbourhoods (adjacent fuelelement swap), the search converges rapidly to a local optimum. For larger neighbour-hoods, convergence is slower but the final solution reached is likely to be better.

The method employed to choose the fuel elements to be exchanged can either bedeterministic or random (stochastic). These two methods lead to either deterministichill-climbers or random hill-climbers.

3.2 Simulated Annealing

The major problem with hill-climbing algorithms, is that when they converge to alocal optimumviii they are then unable to escape and find the global optimum. Thisis because, starting at the local optimum, all moves in the local neighbourhood areworse and so are rejected. Figure 3 illustrates the concept of a local optimum for twocontinuous decision variables.

The Simulated Annealing (SA) algorithm differs from simple hill-climbers by al-lowing some ‘trial’ solutions worse than the ‘current’ solution to be accepted. In thisway the algorithm may be able to extricate itself from one local optimum and pass intoan adjacent region with a better local optimum. The algorithm is shown in figure 4.The term ‘T ’ in the exponential probability is known as the temperature. This refers tothe origin of the method in the simulation of annealing processes in solidsix. In general

vii For example, if we were using a more realistic incore fuel management problem, we mightchange a fuel assembly’s orientation or its burnable poison loading.viiiA local optimum is any solution which has a better function value than any other solutionin its local neighbourhood.

ix In 1953, Metropolis et al55 described a statistical mechanics algorithm to find the equi-librium configuration of a collection of atoms at different temperatures. Kirkpatrick et al. 56


T is not a constant and decreases with increasing number of trials. The way that Tdecreases is known as the cooling schedule. A common cooling schedule is to reducethe temperature by a constant factor after ‘N ’ trials.

The temperature controls the search by making it more or less difficult to escapefrom one local optimum to another. At the start of the search, a high temperature willbe chosen; this allows the algorithm to search widely for a good area. Later, as thetemperature decreases towards zero, fewer non-improving solutions are accepted, andthe algorithm behaves more like a stochastic, or random, hill-climber.

The strength, and weakness, of simulated annealing is the cooling schedule. The-oretically, it can be shown that given a sufficiently high initial temperature and aslow enough cooling schedule, then reaching the global optimum can be guaranteed57.However, in practice, there is insufficient time available to meet these conditions andtherefore SA users are forced to select initial temperatures and cooling schedules whichdo not guarantee reaching the global optimum. Nevertheless, the choice of these twoparameters has to made with care, since they have been shown to affect the qualityof the solution obtained and the rate of convergence. To obtain a good result requiressome experience of working on the problem under consideration.

There are many references to applications of simulated annealing. For example,

“invented” SA by recognising that the same algorithm could be used as a general optimisationtechnique.


Aarts and Korst57 and Reeves58 discuss SA applied to combinatorial optimisation ingeneral, and Parks59 discusses its application to fuel management in an advanced gas-cooled reactor.

3.3 Great Deluge Algorithm

The Great Deluge Algorithm has been proposed by Deuck60 as a simpler alternativeto Simulated Annealing. Instead of simulating the annealing process, this algorithmmimics the wanderings of a hydrophobic creature during a flood.

Consider a landscape consisting of hills and valleys, where the hills represent goodsolutions. Initially the landscape is dry and the ‘current’ solution can wander anywhere.It then starts to rain and the lowest areas start to fill with water. The current solutionis allowed to wander, but it does not like to get its feet wet, so it avoids areas that havebeen flooded. As the water level rises the current solution is forced on to ever higherareas of the landscape. Eventually the rising water level will cause the appearance ofislands, at which point the current solution will become trapped near a local optimum.

One might expect this algorithm to work less well than Simulated Annealing, sinceonce the solution has been trapped on an island then it cannot escape to another island.x

However, practical experience has shown this not to a problem, and further that theresults do not strongly depend on the ‘rain’ algorithm used. A good rain algorithm has

xIt is not possible to construct a proof that shows that the global optimum can be reachedunder certain conditions (unlike Simulated Annealing).


been suggested by Deuck60, and confirmed by Carter61 as:

The best known will always be above the current water level.The GDA has been applied to several combinatorial optimisation problems, in-

cluding the travelling salesman problem60, balancing hydraulic turbines61, cutting stockproblems61 and chip placement60.

3.4 Record-to-Record Travel

The basic algorithm for Record-to-Record Travel (RRT), also proposed by Deuck60, isthe same as for GDA – the difference is that the water level is a fixed level below thebest known solution, i.e. the record solution. The water level is only raised if a newbest solution is found.

When Deuck compared the performance of SA, GDA and RRT, he found that bothGDA and RRT performed better than SA. Although it is known that given sufficienttime, an appropriate cooling schedule and initial temperature SA is guaranteed to reachthe global optimum, no such proof can be produced for GDA and RRT. Comparativetests between GDA, RRT and other algorithms by Sinclair62 and Carter61 have shownthat GDA is a good choice as a ‘black-box’ combinatorial optimisation algorithm andCarter61 suggests parameters for the inexperienced user.

3.5 Tabu Search

Tabu Search, proposed by Glover2, is quite different in its philosophy. Rather thanmaking a random variation to the current solution to obtain the next point, all theneighbouring points are examined and the best, subject to some tabu, is chosen as thenext current point.

In figure 7 a flow diagram for Tabu Search is given. The basic ideas are explainedwith the example given below.

Before we embark on the example, it is necessary to define four items:

the neighbourhood of the current solution is defined by all possible two-elementswaps.

the ‘Tabu Rule’ states that a swap is tabu if either of the fuel elements have beenused in either of the last two iterations.

the ‘Aspiration Criteria’ is that a swap is non-tabu if it would make the functionvalue better than any previous solution.

the penalty function is given by

We start with an initial ‘current’ solution and evaluate its function value, Nextwe evaluate the function valuesxi for all possible ‘trial’ solutions in the neighbourhood

0.1 × (sum of frequencies for two swapped fuel elements)

xiAn arbitrary penalty has been included in the calculation of so as to penalise theswapping of two new fuel elements. Alternatively this could have been included as part of theTabu Rule.


of the current solution. These are then ranked according to the improvement they offercompared to the ‘current’ solution. We also have an empty tabu table.

The tabu table records two pieces of information for each fuel element; how manyiterations it is since a fuel element was last tabu (the recency) and the number oftimes a fuel element has been involved in a swap (the frequency). The recency andthe frequency are both a form of memory. The is calculated from minus thepenalty function.

At the start, see figure 8, all swaps are acceptable, i.e. there are no tabu options.We therefore take the top option in our ranked list to obtain a new current solution.The tabu table is updated as follows: the recency line is incremented by ‘1’ and fuelelements D and J are set to ‘-2’, indicating that they have just been used; fuel elementsD and J of the frequency line are also incremented by ‘1’. Finally, we evaluate and rankthe new neighbourhood ‘trial’ solutions.

At iteration 1, see figure 9, swaps involving either of the fuel elements D and J are


now tabu. So three of top five ‘trial’ solutions are marked as tabu. This leaves the swap(N ,K ) as the best option. However, the swap (J,H ) would make the current solutionbetter than any previous solution, and so it meets the ‘aspiration criteria’ and has itstabu status removed. Now the best non-tabu swap is (J,H ), and we can generate thenext ‘current’ solution, update the tabu table and generate the new neighbourhood.

At iteration 2, see figure 10, we find that the best of the non-tabu swaps has anegative i.e. the move would make the function value worse. None of the tabu

swaps meet the aspiration criteria, so we are left with selecting a non-improving move.We calculate the for each of the non-tabu ‘trial’ solutions, and choose the optionwith the least bad (best!) In this case none of the acceptable swaps have non-zeropenalties. However, this is not generally the case. The effect of choosing a swap withgood is that the search is pushed into regions of parameter space not previouslyexamined. It is clear that choice of the penalty function is critical in getting the correct


balance between search and convergence. So in this case we select swap (N,A) thefunction value drops to and we proceed to iteration 3 (figure 11).

For this example it has been decided that fuel elements should be tabu for twomoves. So any fuel element that has a recency value of ‘0’ or more can take part invalid swaps. So the fuel element D is no longer tabu. Swaps involving fuel elements A,H, J and N are tabu. So the best non-tabu option is (D ,F), and none of the tabu movesmeet the aspiration criteria. The (D,F) option is chosen, and the process proceeds tothe next iteration (figure 12).

When implementing Tabu Search there are a number of issues that have to beaddressed. The first is the choice of neighbourhood. In the example the neighbour-hood was any ‘trial’ solution that could be reached by swapping two fuel elements.Alternatives would be to swap adjacent fuel elements, or to allow four fuel elements tobe moved. As the size of the neighbourhood is increased, so is the effort required toevaluate at all the points. Against this is the likely improved search pattern, withfewer local optimum and faster convergence. It may be possible to reduce the effortrequired to search a large neighbourhood by using a fast, but less accurate, model toinitially rank the trial solutions and then to use a slower, but more accurate, model onjust the best trial solutions.

A second issue is that of the tabu table. The table represents a form of memory.Part of the memory is short term, in that it defines moves in the local neighbourhoodwhich are tabu. In the example the rule was that a swap involving a fuel element movedin either of the last two two iterations was tabu. We could have had a longer/shorterrestriction, and/or changed the rule to make tabu only the exact pairs swapped. Therule could easily have been one of many other possibilities.

Then there are the aspiration criteria, which allow the tabu status of moves to beover ridden. In the example the aspiration criteria was that if a tabu move would resultin a function value that was better that all previous ‘current’ solutions, then it was tobe accepted. Obviously this rule is open to modification.


Finally there are the rules that govern the penalty; how the penalty is defined andwhat forms of memory are used. There is no particular reason for using the frequency-based memory of the example.

The issues of what rules, memory and neighbourhoods should be used are widelydiscussed in the literature. For an expanded introduction the reader is referred tothe paper by Glover63, which includes a discussion on how to apply Tabu Search andreferences to over 30 different applications.

Tabu Search is a much more complex algorithm to implement than simulatedannealing or any its derivatives. It requires a degree of experience both in using TabuSearch and an understanding of the function that is being optimised. The benefit isthat when correctly used the performance is better than SA like algorithms.64, 65, 66, 67 Itshould be possible to construct the rules for Tabu Search such that the global optimumcan be reached. However the author is not aware of any formal proof to this statement.In short, Tabu Search should NOT be used as a ‘black-box’ routine. While the author


is not aware of any current applications of this technique in the nuclear industry, TabuSearch could be applied to incore fuel management and related problems.

3.6 Genetic Algorithms

Genetic Algorithms is the name given to a group of optimisation algorithms that drawtheir inspiration from the ideas of evolutionary genetics. There are a number of signif-icant differences between the group of algorithms and those that have been discussedso far in this review. Rather than having a single ‘current’ solution, genetic algorithmshave a population of many current solutions. The method proceeds by ‘breeding’ new(offspring) solutions from two parent solutions from the current population. Breedingis a way of exchanging information between the parents to produce the offspring. Allof the decisions in this process are made in an essentially random way. It might seemsurprising that such a scheme would work, but experience shows that it works extremelywell.

In Parks68 review he gave a description of a simple genetic algorithm using bi-nary strings. Below we discuss the basic algorithm and then expand on the details ofimplementation. Finally we discuss a specific application to incore fuel management.

The basic algorithm is given in figure 13. To implement it a number of decisions

Encoding the Problem. Encoding your problem, choosing how your problem isrepresented, is probably the single most critical issue in the implementation of the GA.Get it right and the GA will perform well; get it wrong and the GA will perform badly.It is for this reason that when used as a ‘black box’ optimisation tool the results areoften poor. There is a close relationship between the representation and the possiblecrossover operators, which can lead to the choice of representation being led by thewish to use a particular crossover operator. This is not the correct way to proceed.

The general rule that has to be followed is that the representation should be as‘natural’ and ‘simple’ as possible and preserve important relationships between ele-ments. For example, in a curve fitting problem we aim to find optimum values for a listof independent parameters, (a, b...). There are no relationships between the parame-ters and the natural form is a simple list. The only outstanding decision is to whetherto use a binary representation for the numbers or a decimal representation. A binaryrepresentation is generally preferred as it is simpler than a decimal representation.

If the problem is the travelling salesman problem (TSP), where we find the sequencein which a group of cities should be visited to minimise the total distance travelled,then the most natural representationxii is to list the cities in the order that they shouldbe visited. It is possible to find ways that use binary or decimal notation, but they arenot natural and do not preserve the relationship between elements.

For a problem like that of fuel management described in this paper, the naturalrepresentation is that of a 2-D grid. It is possible to express the grid as a linear sequence,but this does not preserve all the relationships, or a decimal notation, losing even moreinformation about relationships between elements.

Other problems have their own representations, which fall into none of the de-scribed categories.

Crossover Operators. The purpose of a crossover operator is to take informationfrom each of the parents and to combine it to produce a viable (or feasible) individualoffspring. An ideal crossover operator will take any two parents in the natural encodingand generally produce a viable offspring. If this is not possible, then you either have to


.

have to be made. In order of importance these are:

1.

2.

3.

4.

5.

6.

7.

8.

The problem of encoding, or representation;

Crossover operator;

Selection scheme for parents;

Construction of the new population;

The mutation operator;

Population size;

Initial population;

Parameter settings.

xii There is no formal definition of a natural representation. The guiding principle is that therepresentation should be as simple and as meaningful as possible, and that under crossoverand mutation, relationships between elements should be preserved

test each offspring solution to see if it is viable, or the representation has to be changedso that guaranteed crossover can be performed.

It is at this point that many of the people who use the GA as a black box algorithmgo wrong. They often select a crossover scheme based on binary strings (since mostintroductory texts use this form) and force their problem to be represented by binarystrings. They may be able to guaranty the viability of the offspring, but the encodingis not natural and does not preserve relationships. Hence, the algorithm fails to givegood results, and the GA maybe dismissed as ineffective.

Crossover on binary strings. We start with two parent strings, P1 and P2, andrandomly select a number of points along the strings. The number of randomly chosenpoints may be one, two, some other number, or randomly chosen. Then to produce anoffspring we copy the binary digits from P1 until we reach the first marker, we thencopy from P2 until the next mark, and then copy from P1 again, switching at eachsuccessive marker.


Clearly one can easily produce a second offspring by starting with P2 instead ofP1, and this is often done.

Crossover on ordered lists. For problems like the TSP, where the natural rep-resentation is an ordered list, simple crossover, as described above, does not lead toviable offspring, e.g.

The result is both an incomplete list and one with duplication.To overcome this problem many ingenious operators have been proposed: Par-

tially Mapped Crossover (PMX)69, Order Crossover (OX)70, Order Crossover #271,Position Based Crossover (PBX)71, Genetic Edge Recombination (GER)72, EnhancedEdge Recombination (EER)73, Tie-Breaking Crossover #1 (TBX1) and Tie-BreakingCrossover #2 (TBX2)74, Intersection Crossover (IX) and Union Crossover (UX)75, andpermutation Union Crossover (UX2)74. Figure 14 gives an example of one, known asthe Partially Mapped Crossover (PMX).

Crossover on grids. Many of the crossover operators can easily be extended towork on grids, see figure 15 for a version of PMX.

General considerations for crossover operators. A consequence of designinga crossover that generates viable offspring is that the offspring solution tends to receivemore information from one parent and some randomisation occurs. Both of these aregenerally considered bad. In the example of fuel management that is discussed belowwe describe a representation that includes information not only on the position of fuel


elements but also other properties (their reactivities). The crossover operator then usesboth positional and reactivity data to produce offspring. At first sight the offspring seemto have suffered a lot of randomisation. However, this is not the case and the crossoveroperator combines the information in each parent to obtain a balanced, viable offspring.

Selection scheme for parents. To produce an offspring solution, we need twoparents. These parents are selected from the ‘current’ population. There are variousways in which the parents could be selected. All of the methods are essentially randomin their nature, but each introduces ‘selection pressure’ (or bias).

The simplest method would be to assign each individual in the population an equalprobability of being selected and then to choose two randomly (this is known as the ran-dom walk selection). The problem with this method is that ‘bad’ individuals are as likelyto be selected as ‘good’ individuals. Hence there is no selection pressure, which shouldfavour better individuals and improve their chances of breeding. In nature, organismsthat are well suited to their environment are likely to produce many offspring for thenext generation, while poorly adapted individuals may produce few, or no, offspring.There are two distinct methods for introducing selection bias: that canbe based either linear ranking or proportional selection, and tournament selection.

In the earliest applications of the GA the probability thatan individual was selected was based on some fitness function76.

The difficulties with this procedure are three:

all the need to be positive with having the highest value if is the bestindividual. To obtain an acceptable the calculated model functionmay need to be scaled and translated,

If the variation among the is too small then the selection probabilities willbe become uniform, and selection pressure will be lost.

If the best has a function value then only one individualwill be selected as a parent.

It therefore requires a degree of care in deciding on how is calculated fromThis usually will required a good understanding of the function

An alternative scheme based on linear ranking was introduced by Baker77. In thisalgorithm the probability that an individual is selected depends on its rank wherethe best solution has rank

is a parameter in the range [1,2] which modifies the selection pressure. cor-responds to a uniform selection probability while gives a value of

proportional to the rank In most studies that use Baker’s ranking scheme,a value of is normally taken. This method has been extended, by Bäck78 andCarter79, so that only part of the population is used to breed from. It is known as the

scheme

(NB at this equivalent to Baker Ranking with )It has recently been claimed, on theoretical grounds, that tournament selection

is the best method81. A major advantage for tournament selection over rank-basedselection when is large and is small, is that it is much quicker to compute.

Construction of the New Population. Once function values for each of theoffspring solutions have been calculated, we need to construct a new population froma combination of the previous population and the children. We have a total ofindividuals to choose from to create a population of size It is quite possible thatsome of these individuals are identical, so we may wish to remove most of theduplicated individuals before proceeding to create the new population. The principalmethods, discussed below, are either generational replacement (with or without elitism)or steady state replacement.


µ is the size of the population from which parents are selected. Carter79 has shown forseveral problems that choosing gives good results.

Tournament Selection. Tournament selection for parents was introduced byGoldberg, Korbel and Deb80. To select a parent we first select members of thecurrent population, using a uniform probability. Then the best of these is selectedas the parent. This is repeated for each parent required. It can be shown that theprobability of selecting a particular individual as the parent is given by

Generational Replacement. In a pure generational replacement scheme, nomember of one population is allowed to pass into the next population. The populationis made from the best of the children produced. A variation on this is to allowan ‘elite group’ to be guaranteed survival from one generation to the next. Clearly

where is the size of the elitest group. A commonly used value iswhere just the best individual is kept. Another possibility is to set and N = 1,where the worst members of the population are discarded at each generation andreplaced. A third possibility is to set so that the previous breeding populationis carried forward but after ranking, some of these individuals may not be in the newbreeding population.

Steady-State Replacement. In this case N is normally set to N = 1, andall members of the previous population (except the parents) are carried forward. Theparents and the children then compete, either deterministically or randomly to completethe new population. This approach ensures that parents are always selected from thebest available population.

Mutation Operators The type of mutations that are possible will depend on therepresentation that you are using. For the incore fuel management problem, the readeris referred to section 3.1. It should be noted that in the GA, the mutation operator isonly rarely used. The author often uses one mutation per twenty offspring solutions.

Population Size and Initial Population. The size of the population and itsrelationship to the performance of the GA is one of the least well documented areas.Initial work82 suggested that populations should be quite large and that the initial

population should be randomly chosen. However more recent work83, 79 has suggestedthat it is quite possible to work with small breeding populations, but that it mayneed a non-random initial population to ensure good convergence. In some problems83

it is fairly clear how to select initial populations. In others, such as the incore fuelmanagement problem, it is not obvious how the initial population should be selected.If a random initial population is used, it is worthwhile including a best guess for thesolution. However, care should be taken so as to avoid biasing the results.

Parameter Settings. Having read the preceding discussion, one might thinkthat setting up a Genetic Algorithm would be a complex task. However most of theparameters are for fine tuning the performance and, a default set of parameters willgive acceptable results for many problems.

In the author’s experience the following represent an acceptable initial parameterset:


population size

sets of parents

number of children per set of parents

mutation rate, one mutation per 20 offspring

elitism with

selection probability either selection withor tournament selection with

initial population: 20% user specified, 80% random

problem representation: this can only be decided by the user. The aim should beto choose a ‘natural’ representation.

crossover: this needs to work effectively with the representation, but the authorsuggests the following as a starting method:

binary strings, either 2 point crossover or random-k point crossover

ordered strings (with no element information), the fast implementation ofUnion crossover (original Fox and McMahon75, fast version Poon and Carter(UX2)74).

ordered strings (with element information): HTBX as described below.

higher-dimensional representation, generalisations of the above.

Heuristic Tie-Breaking Crossover for Ordered Strings (HTBX). Manyproblems can be represented by an ordered string of a finite set of objects. The archety-pal problem of this type is the Travelling Salesman Problem, where the string representsthe order in which the cities are to be visited. Many crossover operators have been sug-gest to tackle strings of this sort and ensure that any offspring is viable. These includePMX, OX, OX2, PBX, CX, IX, UX and UX2. In a comparative study Poon & Carter74

showed that averaged over six problems UX2 was the best performer.There are ordering problems where additional information is available. Such a

problem is the balancing of Francis turbines. A “Francis” hydraulic turbine is con-structed from a central shaft and an outer annulus. Between these are fixed a number

(typically 20) of curved blades. As water flows from the outer edge to the centre theturbine turns, rotating the shaft which is connected to an electric alternator. Thesemachines can be very large with diameters of 10 m and blade masses of about 18 tonne.Blade masses can vary by ±5%. As the position of the blades is fixed, the system isbalanced by welding the blades in an order that minimises the distance of the centre ofgravity of all the blades from the centre of the shaft axis. Final balancing is achievedby adding lead weights to the outer annulus. The additional information available isthe mass of the individual blades.

In a standard implementation of an ordering problem a list of objects (a,b,c...) hasto be placed in some order, so as to optimise a function that depends on the order.In many problems the elements have some property, such as their weights, which allowthem to be ranked. So if the element labelled ‘e’ is the eighth heaviest, then instead ofreferring to element ‘e’ we refer to the eighth heaviest element. It is also easy to findelements similar to element ‘e’, these being the seventh and ninth heaviest elements.The HTBX operator, described in figure 16, uses these principles.

In the example included in figure 16, after the parents have been encoded usingthe ranks of the elements and crossover performed, the first offspring is (5,1,3,6,4,4).In this offspring the second ranked element (if ranked by weight then this would be thesecond heaviest) is not present and the fourth ranked element appears twice. After theoffspring have been re-mapped, this first offspring becomes (5,1,2,6,4,3). The elementswith ranks 1, 5 and 6 have retained their positions. The element with rank 3 has beenreplaced by the element with rank 2, leaving the two places occupied by the elementwith rank 4 to be filled by the elements with ranks 3 and 4. This final decision is madein a random way. Comparing the offspring to its parents one might be surprised tofind that what had been the element ranked 3 in both parents has been replaced by theelement with rank 2. This is just a consequence of trying to meet the preferences forthe element with rank 4. In a situation like this the element is always replaced with a‘similar’ element.

In larger problems the operator does not introduce variations as large as those seenhere, since if we have 100 elements then the sixth heaviest will be much like the secondheaviest, etc.

HTBX for incore fuel management. To apply HTBX to the incore fuel man-agement problem, we need to generalise the crossover operation to two dimensions andto identify the property by which to rank the fuel elements. The ranking propertythat we choose to use is that of the fuel element reactivity. Where two or more fuelelements have the same reactivity the ties are broken randomly. The substrings used incrossover for ordered strings are replaced by ‘chunks’ from the two dimensional space,as illustrated in figure 17.

We start with two randomly chosen parents, P1 and P2. The fuel element identifiersare replaced by the fuel element rank to obtain R1 and R2. A offspring, C1*, is createdby copying one chunk from the first parent and another from the second parent. Thechunk boundary was chosen randomly; there is no particular reason for the chunks tobe simply connected as in this example. A random map is generated and combinedwith C1* before the resulting grid is re-ranked, Finally the ranks are replaced by fuelelement identifiers to obtain the offspring C1.

Comparison of the original descriptions of the parents (P1 & P2) and the finaldescription of the offspring (C1) might make you think that the operation had beenvery destructive. However if one compares parents and offspring using rank labels (R1,R2 & C1**) we can see that most positions have changed their rank by at most one.



3.7 Comparison of Algorithms in PWR reload core design

In this section we compare the performance of SA and GA in a realistic PWR reloadcore design problem. This work is reported more fully by Poon50.

Test Problem. The problem is to find a loading pattern in rotational symmetrythat minimises the power-peaking, starting from a reference loading pattern. There are48 assembles to shuffle of which 20 are identical fresh fuel assemblies. These may beloaded with up to three ‘Burnable Poison’ rods per quadrant, so there are four possibleburnable poison loadings for each fresh fuel assembly which is not in a control rodposition. The 28 partially burnt fuel assemblies may be orientated in four ways.

Simulated Annealing. The Simulated Annealing (SA) results for the aboveproblem were generated using FORMOSA84. FORMOSA is a simulator based on Gen-eralised Perturbation Theory, coupled to some highly tuned SA algorithms.

Genetic Algorithm. The results presented here were generated using a tunedversion of the GA with the HTBX crossover operator. The simulator used by FOR-MOSA was used to produce the reactor simulations required for the GA. The samereference loading pattern was used throughout by both algorithms.

Discussion of the Results. Figure 18 shows the average performance over tenindependent runs, for the GA and two of FORMOSAs search algorithm. The GA findssolutions as good as those found by SA in approximately 1/3 of the function evaluations.The GA was stopped at 20000 function evaluations, when it seemed to have stoppedconverging. This is a well documented problem with the algorithm. Whilst the GA is

fine tuning that estimate. This has been demonstrated theoretically by several authors(e.g. Sulomon85). The advised response is to switch to a hill-climbing algorithm whenthe GA starts to lose performance.

Applications of the GA to Nuclear Engineering. Apart from the work byPoon et al. 50, 51 on the application of the GA to incore fuel management, the author isaware of only three other papers. DeChaine and Feltus describe their CIGARO systemin two papers52, 53, the third paper is by Tanker and Tanker54.

DeChaine and Feltus describe how they use a binary bit string (genome) to repre-sent the beginning-of-cycle (the ratio of neutron numbers in successive generationsin a source-free nuclear reactor of infinte extent) at all the loading positions. Theythen try to match the required distribution with the available fuel type, beforesubmitting the core loading to a reactor physics code. The GA is quite standard fora bit based genome (although the crossover rate of 0.45 might be considered slightlylow). They justify their choice of representation and GA by the statement: “The GAcannot work with the real optimisation variables, i.e. the assignment of fuel types toloading position.”52 The author hopes that the preceding discussion has indicated howthe GA can be directly applied to the assignment of fuel types to loading position.

Tanker and Tanker54 work with the fuel loading pattern, but only allow threefuel types. They use what is a modified version of the Partially Mapped Crossover,described in figure 14, to re-order fuel elements. They report that linear programmingis about 30 times quicker than their GA, although the result is not as good. Linearprogramming is a very efficient way of solving problems where the objective function isa linear combination of the decision variables and all the constraints are linear as well.The linear programming algorithm is able to exploit the high degree of redundancydue to having just three fuel types. The Genetic Algorithm is unable to exploit thisredundancy or all of the constraints, so it has to search a much larger solution spacethan the linear programming algorithm. When the objective function is linear thenthe GA would not be the optimisation algorithm of choice since it performs best onnon-linear functions.

General Applications of the GA. Genetic Algorithms have been an activearea for research for the last two decades. Since the algorithm was first introducedby Holland76 there have been six international conferences86, 87, 88, 89, 90, 91 on the topicand applications of the algorithm are also widely discussed in other journals and con-ferences. There are many introductory texts now available, although two of the betterones, Davis92 and Goldberg82, are now a little dated. Since these were published muchprogress has been made, particularly in the area of combinatorial optimisation.

3.8 Population-Based Incremental Learning Algorithm

The population-based incremental learning (PBIL) algorithm, proposed by Baluja3,combines a hill-climbing algorithm with evolutionary optimisation. It is different fromall the other algorithms considered in this paper, in that it does not proceed from


good at finding the approximate position of an optimum solution, it is not suited to


a current solution(s) to a trial solution by some re-combination or mutation method.Instead a prototype probability vector is used to generate random solutions, which inturn are used to update the prototype vector. The PBIL algorithm operates on binarystrings. A flow diagram is given in figure 19.

The prototype probability vector P is a vector of numbers in the range [0.0, 1.0].Initially every element of the vector is set to 0.5. To generate a random trial vectorT, a binary string, we first generate a vector Z with random elements in the range[0.0, 1.0]. If the value of an element in the random vector Z is greater than the elementin the prototype vector P, then the element in the trial vector T takes on the value ‘0’,otherwise a value of ‘1’ is taken. For example

If, after generating M trial vectors, the best trial vector is B and the worst is W,with another random vector then the prototype vector P is updated according to thefollowing rule

Prototype vector PRandom vector Z

Trial vector T

( 0.4 , 0.2 , 0.7 , 0.7 , 0.2 )(0.76, 0.21, 0.87, 0.56, 0.32)

( 0 , 0 , 0 , 1 , 0 )

where and are numbers in the range [0.0, 1.0]This moves the prototype vector slightly towards the best vector, slightly away from

the worst vector and adds a small random variation. The size of the three parameterscontrols the search and convergence properties.

For a problem that can be expressed as a binary string, this method can be applieddirectly. For combinatoric problems, the problem can be encoded using the followingscheme. Gray codingxiii is used to represent a number in the range where

is greater or equal to the number of elements. For example, in our test problem with15 elements, S = 4 so The length of the binary string is then S timesthe number of elements.

For a six-element problem, as in figure 16, so one would needa total of 6 x 3 = 18 bits. If at some stage the prototype vector P and random vectorZ are:

4 OPTIMISATION METHODS FOR CONTINUUM PROBLEMS

The previous section concentrated on methods for combinatorial optimisation, althoughmost of the methods can be adapted to work on problems involving continuous deci-sion variables. In this section a new method, developed in the petroleum industry byFrandsen4, for continuum problems is described. It should find applications in manyother industries.

4.1 A Trust Region Multi-point Approximation Method

In many fields of science and engineering we wish to optimise a function that dependson the results from a slow numerical simulation (e.g. simulation of a reactor core). Thisoften results in a trade-off between the quality of the solution (as many optimisation

xiiisee the appendix on Gray coding


Prototype Vector P(0.74, 0.52, 0.80, 0.30, 0.67, 0.68, 0.53, 0.51, 0.27,

0.61, 0.22, 0.67, 0.39, 0.34, 0.50, 0.84, 0.69, 0.90)

Random Vector Z(0.52, 0.73, 0.75, 0.66, 0.66, 0.51, 0.01, 0.02, 0.75,

0.35, 0.13, 0.12, 0.89, 0.02, 0.86, 0.75, 0.83, 0.91)

then this will generate the following trial vector T of Gray codes

(1,0,1,0,1,1,1,1,0,1,1,1,0,1,0,1,0,0,)

which decodes to give (6,2,4,5,3,7). Clearly this needs to be re-mapped and any conflictsresolved. This could be done as in the HTBX crossover algorithm. Finally we obtain(d,c,f,b,a,e) as a trial solution.

The PBIL algorithm is quite a new method and initial tests93 on some standardcombinatorial optimisation problems (TSP, Knapsack, bin packing, job-shop schedul-ing) look promising. In particular, although PBIL is much simpler than the GA, it cangenerate comparable results for many problems. It is therefore expected that there willbe a growing interest in applications of PBIL in industry.

where and are linear scaling parameters, and is a quickly evaluatedapproximation model with (L – 2)xiv adjustable parameters. These parameters wouldbe physical parameters of the model such as the neutron absorption cross-section orthe thermal utilisation factor. In a standard approximation model these parameterswould be fixed by the design of the reactor. We allow them to change, accepting that itimplies a slight modification of the reactor. It is entirely possible that by adjusting theseparameters one would obtain a reactor that could not physically be built. For instancethe parameters might imply moderator characteristics that can not be provided by anyknown moderator. However, if a moderator with those characteristics did exist, thenthe reactor could be constructed.

The parameters α are used to adjust the generic model so that it matchesas closely as possible at a set of points at which is known. This isachieved by minimising a weighted error squared function.

xivThe number of parameters (L – 2) in the approximation model is dependent on the user’schoice of model


schemes require numerous function evaluations) and the computational resources used.This method attempts to alleviate this problem.

It is often the case that the objective function cannot be directly calculatedfrom the decision variables In these cases the objective function depends on a vector

The elements of are themselves complex functions of the decision variables.Mathematically this can be written

The elements of will often be the output of some comprehensive and slow numericalsimulation, such as a reactor simulation.

In many optimisation methods for continuous decision variables, implicit use of ageneric function to approximate the objective function around a current pointis made. For example the Newton-Ralphson method assumes a linear model. Thesegeneric models have a number of parameters that are adjusted to achieve the approxi-mation

where are tunable parameters.As an alternative to using a generic model one might use an approximation model

to replace the comprehensive model, ie

The approximation will normally be quick to evaluate, at least compared withcapture the relevant physics and be qualitatively correct. These models generally donot have tunable parameters. The use of Generalised Perturbation Theory (GPT) forincore fuel management would be an example.

Multi-point Approximation The multi-point approximation (MpA) combinesthe generic model approach with the approximation model approach. A generic modelis defined which uses an approximation model with tunable parameters.


where the sum is over all the points at which is known. The are weightsassociated with each point.

If is the current estimate of the optimum of and a given radius, then

as shown in figure 20. is chosen such that there are at least L points (L beingthe number of tunable parameters ) within the hyper-sphere Theregion is known as the “trust region”, i.e. the region where we trust

to be a good approximation toHaving defined a trust region and fixed the generic model we now find

the point within or on the boundary of the trust region that minimises the genericmodel function. For this point we then run the slow numerical simulator and calculatethe following three quantities

and are used to assess the success of the approximation.The multi-point approximation can be considered as a function that takes and

as arguments and returns and the algorithm is defined in table 2.The radius of the trust region, is adjusted as the algorithm proceeds.

Having calculated and a step can be classified as either a ‘poor step’ or a ‘goodstep’ or an ‘indifferent step’. A step is classified as poor if either or itis classified as good if and otherwise it is classified as indifferent.

If the step is classified as indifferent then is replaced by is left unchanged,and a new multi-point approximation is made.

If the step is classified as poor then the number of points at whichhas been calculated inside the trust region are counted. If C < L then a new randomlychosen point is added to the trust region and evaluated at this point before anew multi-point approximation is made. If this is not the case then is reduced by afactor of 2, and if is positive then is replaced by

If the step is classified as good then a new multipoint approximation is made withthe radius of the trust region increased by a factor where is the number of successive‘good’ multipoint approximations. This sequence is stopped if either the MpA is ‘notgood’ or stops improving. New values for and are the selected.

All the details of this algorithm are given in figure 21. The success of this approachwill depend critically on the choice of the fast approximation model and the parametersthat are chosen to be adjustable.

The MpA method is very new and is still under development for the petroleumindustry. One of its advantages is that it can smooth-out a noisy objective functioncalculated from a comprehensive simulation due to computation effects withinthe simulations. It has been successfully tested on the standard analytic test functionsused by the optimisation research community. For this reason it has been includedin this review. The difficulty for the petroleum industry has been in the selection ofappropriate approximation models for the problems being tackled. This should notdeter researchers from applying the technique to other problems.

5 CONCLUSIONS

In this paper six algorithms for combinatorial optimisation problems, such as PWRreload core design, have been reviewed. Five of these algorithms have been tested byseveral researchers on a range problems. Of these five, GDA is the best choice for aninexperienced user wanting a “black-box” routine for occasional use.

If you need to perform optimisation regularly, then the author suggests that theGA would be most efficient. It would need to be coupled with a simple hill-climbingalgorithm for final optimisation. Time spent on choosing a representation and crossoveroperator will be repaid by better performance. PBIL is a new algorithm, which requiresfurther testing. It is much simpler to implement than the GA, and does not suffer fromthe local optimum trapping problems of the GDA. Tabu Search has produced some verygood results, but does seems complicated to implement. It has a definite down side inthe requirement to evaluate the function for the complete local neighbourhood. Theauthor is not aware of a good comparative test between Tabu Search and the GA. Allof the methods, except Tabu Search, can be adapted for problems involving continuousdecision variables. They are most useful for problems where many local optima exist.Interested readers should consult the relevant literature (see references) to find theappropriate implementation methods.

The Multi-point Approximation algorithm is a new method that may make op-timisation possible for problems for which optimisation has not been practical before.Most iterative search algorithms need hundreds or thousands of function calls to solvefor real problems. If each function call takes 2-3 hours to compute, then optimisation is


not practical. The method is allowing progress on such problems in the petroleum in-dustry. The critical element is the selection of the approximation model for the genericfunction. It is expected that this algorithm will have many applications in industry.

Acknowledgements

I would like to thank Drs J.D. Lewins and G.T. Parks for their valuable assistance inthe preparation of this paper, and in particular for the benefit of their knowledge ofthe relevant nuclear technology. Any mistakes are however mine. I would also like tothank Dr P.W. Poon for the review of previous work on the reload core design problemand for permitting the use of results from her PhD thesis.

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J.J. Grefenstette (ed), Proceedings of the Second International Conference on GeneticAlgorithms and their Application, Lawrence Erlbaum Associates, Hillsdale, (1987).

J.D. Schaffer (ed), Proceedings of the Third International Conference on GeneticAlgorithms and their Applications, Morgan Kaufmann, San Mateo, (1989).

R.K. Belew and L.B. Booker (eds), Proceedings of the Fourth International Conferenceon Genetic Algorithms and their Applications, Morgan Kaufmann, San Mateo,(1991).

S. Forrest (ed), Proceedings of the Fifth International Conference on Genetic Algorithmsand their Applications, Morgan Kaufmann, San Mateo, CA, (1993).

L. Eshelman (ed), Proceedings of the sixth International Conference on GeneticAlgorithms and their Applications, Morgan Kaufmann, San Mateo, (1995).

L. Davis, A Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York, (1991).S. Baluja, An empirical comparison of seven iterative and evolutionary function

optimization heuristics, CMU-CS-95-193, School of Computer Science, CarnegieMellon University, (1995).

D. Whitley, K. Mathias, S. Rana and J. Dzubera, Building better test functions,presented at ICGA 6 see reference 91, 239 (1995).

Appendix: GRAY CODING

A Gray code is a binary representation of an integer number such that adding orsubtracting one from the integer, changes the Gray code at only one bit in the repre-sentation (unlike a standard binary number). Table 3 lists the Binary and Gray codesfor the integers 0-32.

If one looks at the change in representation between integers 15 and 16, one findsthat the binary representation has changed at five locations, while the Gray code haschanged at just one location. This is repeated at many points in the sequence. Thebinary coding has proved difficult for some algorithms to exploit successfully. Mostalgorithms have found Gray coding easy to exploit. This has been discussed for the GAby Whitley et al. 94

To convert between binary and Gray coding we make use of binary matrix multi-plication


85.

86.

87.

88.

89.

90.

91.

92.

93.

94.

where B is a binary coded vector, G is a Gray coded vector and is a upper triangularmatrix of the form


To decode we use

for example 26

where is an upper triangular matrix of the form

Using the previous example

THE COMPUTERIZATION OF NUCLEAR POWER PLANTCONTROL ROOMS

Dr. Bill K.H. SunSunutech, Inc.P. O. Box 978, Los Altos, California 94023, USAandDr. Andrei N. KossilovExitech CorporationStavangergasse 4/9/4, A-1220 Vienna, Austria

1. INTRODUCTION

1.1 Operation of Nuclear Power Plants

The major goals of using computers in operation of nuclear power plants (NPPs) are(1) to improve safety, (2) to reduce challenges to nuclear power plant, (3) to reducethe cost of operations and maintenance, (4) to enhance power production, and (5) toincrease productivity of people.

In the past decade, there has been a growing need to address obsolescence, improvehuman performance, and to comply with increasingly stringent regulation require-ments. The needs to use computers arise in the on-line, real-time processes of controland protection, alarm detection and display in the control rooms, and in the on-lineassessment of processes needed for operation of a power reactor. In addition, theyarise equally in the semi on-line operation needed for daily or hourly assessment ofoperation; for example, monitoring of detailed reactor flux distributions, etc. As aresult, NPPs have implemented plans to replace ageing analogue systems with digitalsystems and have developed comprehensive and accessible information database andmanagement systems. These systems support operations for an overall improvementin quality assurance and productivity. Advances in information and communicationtechnology have been proven to help utilities operate power plants more efficiently byintegrating computer resources and increasing the availability of information to meetNPP staff needs and corporate business strategy [1-3].

A major difficulty with the application of computers in the control rooms is that therequirements of a NPP are always stringent and specific. Much on-line software must,therefore, be specifically written and the hardware configuration is generally difficultto obtain off the shelf. A major problem with procurement of on-line systems hasbeen and continues to be the need for internal redundancy to ensure high availabilityof the computer functions. These problems have often prevented the designintentions from being fully achieved, and the economic benefits expected have notalways been fulfilled in consequence.


1.2 Benefits of Computerization of Control Rooms

An increasing use of computers and information technology in general in manyexisting and all new nuclear stations have enabled a relatively high degree ofautomation and allowed dynamic plant state to be represented in digital computermemory and logic [4,5]. Exploiting this advantage and the rapid evolution of digitaltechnology, nuclear power plants have achieved substantial safety and operationalbenefits. Some of the most potential significant features and benefits forcomputerization of control room systems are the following:

2.1 Importance of Human-Machine Interface in Computerization

Many functions in NPPs are achieved by a combination of human actions andautomation. Computerization is basically an automation process that allocates certainhuman functions to computer machines. To understand the roles of human, machines,and their interactions is of critical importance in computerization of nuclear powerplant control rooms.

The importance of human-machine interface for ensuring safe and reliable operationof nuclear power plants had been recognised by the nuclear energy community longbefore the Three Mile Island Unit 2 (TMI) and the Chernobyl accidents

The concept of operator support and human factors have been increasingly used tobetter define the role of control rooms. In the late 1970s, the impact of analysis

156 BILL K. H. SUN AND ANDREI N. KOSSILOV

Substantial reduction in panel complexitySubstantial reduction in instrumentation complexityElimination of error-prone tasksIntegrated emergency response information systemProcedure driven displays to guide operator actionsCritical alarms with diagnostics messages dealing with plant disturbanceIncreased intelligent information for operators to plan and execute correctly

The above features have been applied to control room systems to provide operatingnuclear utilities with tools that substantially reduce power plant operating,maintenance and administration costs. This has been achieved in the following ways:

Reduction of plant forced outagesFaster recovery from forced outagesAvoidance of plant equipment damage and extension of service life due to earlydiagnoses of equipment malfunctionFaster start upsAutomation of labour intensive operation, maintenance and administrationprocesses

1.3 Scope and Purpose of the Paper

The paper describes the status and philosophical background of computerization ofnuclear power plant control rooms. The paper also provides information describingthe history of operational experience, lessons learned, and a summary of majorcomputer systems. The state-of-the-art and future trends in the use of computers incontrol rooms, including man-machine interface, life-cycle management, verificationand validation, etc., are also considered in view of the fast evolution of thetechnology.

The paper provides a resource for those who are involved in researching, managing,conceptualizing, specifying, designing, manufacturing or backfitting power plantcontrol rooms. It will also be useful to those responsible for performing reviews orevaluations of the design and facilities associated with existing power plant controlrooms.

2. HUMAN-MACHINE INTERFACE

results from the TMI accident considerably accelerated the development ofrecommendations and regulatory requirements governing the resources and dataavailable to operators in NPP control rooms. One important outcome was theimplementation of computer driven safety parameter display systems in control roomswith the objective of providing human operators with on-line and real-time display ofcritical safety information to aid operators in emergency management.

Among the human-machine interface parameters, the ergonomics of control boardsand panels, resources and facilities to deal with abnormal situations and data displayfrom instrumentation, are all important for design improvements made to computerizecontrol rooms of nuclear power plants.

COMPUTERIZATION OF CONTROL ROOMS 157

Existing practicesOperational and design experienceRegulatory factorsFeasibility

CostTechnical climatePolicy matters

Cultural and social aspects

1.2.3.4.

5.6.7.8.

The various factors may differ between applications and may be affected by whethera new design or a modification to an existing process through retrofit is beingconsidered. In the retrofit case, the implementation of computers has less flexibility,owing to existing plant designs, operating practices, the need for replication, etc.

2.3 Application of Human Factors in Control Room Design

Human factors efforts in design of computerized control rooms has been based on afirm analytical foundation. Human factors efforts complement those of other controlroom design team participants, resulting in an integrated design that supports tasksperformed by control room personnel. Human factors principles and criteria, alongwith information resulting from analyses, are applied in selecting panels and consoles,configuring them in relation to other furnishings, and establishing ambientenvironmental conditions (light, sound climate) to promote effective personalperformance. In addition, ability features (personal conveniences, aestheticconsiderations, safeguards against common hazards) are specified to promotepersonnel comfort, morale, and safety [5,7].

The primary human factors objective in control room design is to increase operationaleffectiveness by ensuring that capabilities and needs of personnel are reflected in co-ordinated development of interactive design features. Human factors recommend-ations are intended to ensure that the design and placement of consoles and othermajor items support effective task performance during all operating modes.Recommended layout alternatives facilitate visual and physical access to display andcontrol instruments.

2.2 Allocation of Functions between Automata and Humans

Increasingly, computer-based systems are used to support operations and maintenancepersonnel in the performance of their tasks in the control rooms. There are manybenefits which can accrue from the use of computers but it is important to ensure thatthe design and implementation of the support system and the human task places thehuman in an intellectually superior position, with the computer serving the human. Inaddition, consideration must be given to computer system integrity, softwarevalidation and verification, consequences of error, etc. To achieve a balance betweencomputer and human actions, the design process must consider each operationalfunction in regard to either computer, human operation, or more commonly in nuclearplants, a combination of human and computer [6].

The following factors will govern the relative weighting used in allocating functionsbetween humans and computers:

3.1 Application of Computer Technology in Control Rooms

The use of computers in nuclear power plants date back almost to the beginning ofcommercially applied nuclear power. At the time, a central computer served as a datalogger. With the progress of technology, smaller dedicated computers have beenintroduced which now serve for data acquisition, exchange of data throughout theplant, information generation by means of simple logic or more complicatedanalytical functions, and by providing the desired information to the operator in thecontrol room, usually by means of video display units.

In parallel, computers began to be used for open and closed loop control and are alsoapplied for the protection of the plant. Early protection applications often werelimited to calculating safety-relevant parameters such as departure from nucleateboiling; later applications included all signal handling, trip detection and redundantmajority voting functions.

The application of computers in control rooms can be distinguished between twofunctions:


3. COMPUTERIZATION IN NUCLEAR POWER PLANT CONTROLROOMS

for storage of operational data in order to have available historical data for latercheck-up or comparison andfor real-time data management in order to serve all needs of on-line monitoringand automation.

The basic functions usually provided for a nuclear power plant by on-line computersystems are plant monitoring and recording, and the display of information andalarms.

On a typical reactor plant, between 3000 and 7000 analogue signals from instrumentsmeasuring temperature, pressure, flow, neutron power levels, voltage, current, andother special parameters exist. In addition, between 10 000 and 20 000 state signalswill be used. These state signals provide information on switch gear states, valvestates, and alarm states and may include the states of control room switches.

The main use of the computer system is to read in these signals at intervals, typicallyone second, and to form an internal data base representing the plant condition. Thatinternal database is then available for software to perform checks of analogue signalsfor high, low and other alarms, and checks of state signals for alarms. In the alarmcheck process, the computer system provides recorded time of detection on printedlogs and on magnetic media for off-line analysis. Video Display Units (VDUs) allowthe alarms to be displayed to the operators, with clear language titles, the instrumentor plant contact identity, and other information.

Typical computer functions are listed in Table I. This table shows a great variety ofcomputerized functions exists in various nuclear power plants. These functionsinclude information and operator aid systems, computerized control and automation,and computerized protection systems.

It is worth mentioning, however, that while some sort of information and operator aidcomputer system exist at almost every nuclear power plants, the application ofcomputers to closed-loop control is more recent and the application to the protectionsystem has been done only to very recent plants and current designs of light water andCANDU reactor plants. Particularly for the protection systems, safety implicationsand regulatory concerns have been major challenges to implementation.

3.2 Evolution of Control Room Designs

In the past two decades, rapidly evolving computer and communications technologyhas revolutionized control room system designs. The technology include computers,


data highways, communication devices, many different information displaymechanisms, human input/output facilities, software, voice annunciation and voiceactuation systems.

Control Room designs consider various factors including cost, operational reliabilityand safety. This rapid technological development in computers and electronics iscoupled with significant progress in the behavioural sciences that greatly increasesour knowledge of the cognitive strengths and weaknesses of human beings.

In nuclear power stations, as in most complex industrial plants, control room systemsdesign has progressed through three generations.

First generation systems consist entirely of fixed, discrete components (handswitches, indicator lights, strip chart, recorder, annunciator windows, etc.). Humanfactors input was based on intuitive common sense factors which varied considerablyfrom one designer to another.

Second generation systems incorporate video display units and keyboards in thecontrol panels. Computer information processing and display are utilized. There issystematic application of human factors standards and guidelines. The human factorsare applied mainly to the physical layout of the control panels and the physicalmanipulation performed by the operators.

Third generation systems exploit the dramatic performance/cost improvements incomputer, electronic display and communication technologies of the 1990s. Furtherapplications of human factors address the cognitive aspects of operator performance.

All new nuclear plants and most operating plants now utilize process computers toimplement part of the control room information system. As computer costs decrease

and reliability increases, computers are being used more extensively. The increasedfunctionality provided by computer systems yields significant benefits. The use ofcomputers also creates problems. For example, additional costs must be justified,information overload must be avoided and provision must be made to deal with thepossibility of rapid obsolescence because the technology is changing so rapidly.

3.3 Computer Architecture

There are three classic forms of computer hardware architecture as they are applied tocontrol room system design:


1. Centralized Redundant Computers

Typically a dual redundant centralized computer system is specified where onecomputer is in control and the other one is running in "hot standby". When a fault isdetected by self checking in the controlling computer, the "hot standby" takes over.

2. Distributed Computing (functional distribution)

Functional distribution provides for multiple computers to dynamically share the totalcomputing load. The computing tasks are allocated among a number of separatecontrol processing units which are interconnected in a communication network.Although the computing tasks are distributed, the processors are not geographicallydistributed to achieve cost saving and simplification in wiring, cabling andtermination.

3. Distributed Control (geographic distribution)

A distributed control architecture allows the processing to be geographicallydistributed. The processors are located close to the inputs and outputs to the plant.This architecture can provide substantial cost savings and reliability benefits becausethe conventional wired analogue and relay logic is replaced by more highlystandardized self checking digital system modules. Because such configurations arerelatively new and because they require greater performance and faster response,there is more technical risk in such an architecture.

Because of the improvements in computer and VDU functionality, applicationsoftware and their development tools have been developed that provide the basicsoftware building blocks for the design, implementation and validation of thedetailed control room system software. The existence of these tools makes it possibleand desirable for plant staff to undertake the detailed design, implementation andvalidation of the control room designs. With the aid of the full scope trainingsimulators that are now required before start-up, the plant personnel can carry out thedesign validation and training of operators for the new designs.

Software quality is essential for successful implementation and licensing of ControlRoom Systems for new plant and retrofit applications. Software quality is achieved bycareful attention to the following requirements:

Properly trained, and experienced software development staff;Comprehensive, clearly documented software requirements and specifications;A well organized and clearly documented and understood software developmentprocess based on a pre-established software life cycle;Use of proven, up to date software development tools such as compilers, editors,graphics interpreters, de-bugging facilities, and file revision control tools;Documented validation and verification to the level required in the softwaredevelopment process in accordance with the degree of nuclear public safetyfunctionality in the software;Thorough, well organized testing;Comprehensive software configuration control.

3.4 Control Room Operator Support Systems

Operator support systems are discrete computer systems or functions of the plantprocess computers that are based on intelligent data processing and they draw inputsfrom plant instrumentation and control systems. Applications are mostly real-timeand on-line [8-10].

In addition to control room operators, users of the support systems include operationsstaff management, technical specialists (e.g. engineering reactor physicists),maintenance staff, emergency management and sometimes safety authorities.

In practice, the operator support systems have been implemented as functions of plantprocess monitoring systems, or stand-alone applications for monitoring and diagnosis,such as materials stress monitoring, vibration monitoring and loose part monitoring.In the following, the function and purpose of major operator support systems aredescribed with examples of practical applications and their operational status.

1. Task oriented displays

The function is primarily to present relevant plant information to support operators inspecific tasks such as start-up, shut-down and other transients by optimizinginformation type, form and presentation. Typical examples are operating pointdiagrams and curves for optimum operation in transients indicating operating area andpossible limits and their violation.

2. Intelligent alarm handling

The function is to support operators to understand the information given by the alarmsespecially in plant transients, where the alarm overflow often is a problem. This isdone by logical reduction and masking of irrelevant alarms, synthesizing them,dynamic prioritization based on the process state, first alarm indication, displaying thealarm state of subsystems or functional groups of the plant, etc.

3. Fault detection and diagnosis

The function is to alert operators to problems and to aid them to diagnose those beforethe normal alarm limits are reached, where simple alarm monitoring is impractical orwhere complex situations cannot be revealed by alarms or alarm logic. Examples are:


Fault monitoring of protection logic and associated electrical supplies, fuel pinfailure detection and prediction.Detection and identification of leakage, e.g. mass balancein the primary circuit.Model-based fault detection for components (e.g. preheaters) and measurementloops.

4. Safety function monitoring

Examples include critical safety function monitoring, safety parameter displaysystem, etc. Their function is to alert the operators to the safety status of the plant.This is based on the monitoring of derived critical safety functions or parameters, sothat operators can concentrate on maintaining those safety functions. The severity ofthe threat which challenges functions as well as guidance in the recovery are alsogiven in some applications. In these cases, relevant emergency procedures arereferred and implementation of corrective actions are supervised.

5. Computerized operational procedures presentation

The function is to complement written operating and emergency procedures bycomputerized operator support. For instance:

Guiding the operator to the relevant procedure.Presentation of procedures dynamically and interactively on displays.Follow-up monitoring of actions required in the procedures.

•••

•

••

6. Performance monitoring

The function is to calculate and monitor the efficiency and optimum operation ofmain pumps, turbine, generator, condenser, steam generators, preheaters, etc. in orderto detect developing anomalies. The reactor thermal energy can be calculated as wellas heat, electricity and mass balances. The computation is based on physicalequations and plant measurements which must be accurate enough to guaranteereliable results.

7. Core monitoring

The function is to calculate and monitor the operation of reactor and fuel, for instancein order to maximize the energy output of the fuel but still keeping adequate operatingmargins. Examples are:


Load following and simulation/prediction.Reactor power distribution and burn-up.Prediction of Xenon, critical Boron.

Computation is based on reactor physics and in-core measurements, such as neutronflux, and temperature.

8. Vibration monitoring and analysis

The function is to reveal, in an early phase, failures of rotating machines such asturbines and main pumps by monitoring the shaft vibration using Fourier analysismethods. Systems are operational in most countries. This is to aid the technicalspecialists to analyze the often voluminous data from the monitoring instrumentation.They are typically stand-alone or common; with loose parts monitoring they might beconnected to the plant process computers to submit information also to the controlroom operators.

9. Loose part monitoring

The function is to detect loose parts in the reactor circuit based on noise analysismethods. Systems are operational in most plants.

10. Materials stress monitoring

The function is to monitor and predict cracks in pipes, tanks, vessels, etc. This isbased on counting the thermal transients of the critical points, on the results/specialarrangements, and calculation of stresses and cracks using physical or empiricalalgorithms. They are mostly dedicated stand-alone systems.

11. Radiation release monitoring

The function is to monitor in plant emergencies the radiation release to the plantenvironment for the plant emergency staff, authorities, etc. The evaluation is based ondeviation models using radiation measurements of the plant and meteorologicalmeasurements as the source data.

3.5 Emergency Response Facilities

The TMI action plan called for improvements in emergency preparedness through theprovision of three separate facilities to be utilised in support of emergency operations,namely:

1. Technical support center, a room near to but separate from the control roomthat will be the focus for technical and strategic support to the control roomoperations staff. The room must provide a plant status information system andcommunication facilities.

2. On-site operational support center, a marshalling area for operational supportpersonnel (maintenance, security, auxiliary operators, etc.). This facility also mustcontain a plant information status system and communications facilities.

3. Near site emergency operations facility, the central focal point for planningand co-ordinating all on-site and off-site emergency activities including evacuation,communications with news media organizations, co-ordinating with government andcommunity organizations. A plant information status system and adequatecommunication systems are required.

An essential requirement of the control room system design is that the plantinformation system that is used by the main control room staff should be the same onethat provides plant information in the emergency response facilities. The intent is toprovide facilities for use in normal routine operations which will also be useful inemergency situations. If station staff are not accustomed to using a particular facilityin routine operation, they will be unfamiliar or uncomfortable with it for emergencyuse.

4. SAFETY AND LICENSING

4.1 Safety Considerations

Safety considerations are critical in the design and operation of control room systems.The human-machine interface provides the media for communicating the plant stateto the operators and, the mechanisms for the operator to alter the state of the plant. Ifinformation is misrepresented because there is a fault in the display systems, theoperator may respond incorrectly during a plant upset. Consequently, there may besituations where the correct operation of these systems is critical to ensure publicsafety [3, 11-13].

It is important to identify a small subset of the control room systems that are requiredto respond correctly to the "design basis accident" and Probabilistic Risk Analysis(PRA) scenarios that are analyzed as part of the licensing process for the plant. Thereis a portion of the control room system that are dedicated as "safety systems" that arephysically, functionally and electrically isolated from the other systems and subjectedto more stringent design requirements. The challenge for the control room systemdesign is to provide an interface to the safety and non-safety systems that alleviateany human factors problems resulting from the differences in design.

4.2 Control Room Function with Increased Complexity

From the production point of view, the economic operation of NPPs is emphasized.For maintaining the high availability of the plant, the design of control room systemsshould support the operators in the following:


Normal operation including pre-analyzed transientsAbnormal transients, especially in early fault detection and diagnosis in order toprevent the situation leading to reactor scrams on the initiation of safety systemsOutage operation

The increased size and complexity of nuclear power plants has greatly influenced theoperational requirement for the design of the control rooms and their systems. Plantoperation is centralized in the main control room. More extensive monitoring of theplant is needed to achieve high availability. As a consequence, the number ofindicators, alarms and manual controls, etc. in the control room has grownsubstantially. Load following of the electrical grid is a factor in the operationalrequirements for utilities in geographical areas with a high percentage of nuclearpower supply to the grid.

4.3 Operational Experience and Lessons Learned

The operational experience of plants shows, that for safety and productivity of nuclearpower, operator action is very important. Investigations indicate that human error isthe main contributing factor of the incidents which occurred.

The scenarios of the TMI accident in 1979 and the Chernobyl accident in 1986 arewell known, with lessons learned. The following are worth particular attention:


The initiatives to solve the problems of growing complexity and information overflowin control rooms are:

Higher automation levels, i.e. automation of some operator actions.Utilization of computer technology, e.g. by:

reducing irrelevant information by means of hierachization, prioritization,condensing, suppression etc.

supporting operators by further data processing.

At TMI, because the operators had to base their decisions on a situation whichwas not clear, many of the actions they took to influence the process during theaccident significantly exacerbated the consequences of the initiating events. Oneof the factors, which led to actions being taken which were both inadequate andtoo late, was poor use of the data made available to the operators in the controlroom. They were unable to satisfactorily process the large amounts of dataavailable to them and had difficulty distinguishing between significant andinsignificant information.

At Chernobyl, the main cause of the accident was a combination of the physicalcharacteristics and safety systems of the reactor and the actions and decisionstaken by the operators to test at an unacceptably low power level with thedisabling of automatic trips. Their actions introduced unacceptable distortions inthe control rod configuration, and eventually led to the destruction of the reactor.The root cause of the human error relates to the lack of a safety culture in thestation which in turn led to, among other things, inadequate knowledge of thebasic physics governing the operational behaviour of the reactor.

During the last three decades of reactor operations, the role of control room operatorshas been shifting from the traditional equipment operator to a modern dayinformation manager. As such, the cognitive requirements on control roomoperations personnel to improve availability and reliability and improve safetychallenges to the plant have increased. These personnel are working with morecomplex systems, and responding to increasing operational and regulatory demands.

As the demand and requirement on the operators intensified, diagnostic andmonitoring errors have all occurred in power plants, causing reductions in availabilityand substantial cost consequences. Plant safety has been challenged due tomisinterpretations of data and incorrect assumptions of plant state. Since the ThreeMile Island event, a number of diagnostic aids have been implemented such as criticalparameter displays, saturation and sub cooling margins and symptom basedemergency operating procedures. These have all been useful in assisting humans inmaking their decisions. A number of human factors studies on human-machineinterfaces have also been performed. Therefore, reliable, integrated information foroperation use is a critical element for protecting the nuclear plant capital investmentand increasing availability and reliability.

With appropriately implemented digital techniques, human capabilities have beenaugmented substantially in their capacity to monitor, process, interpret and applyinformation, thus reducing errors in all stages of information processing. Takingadvantage of technological and human engineering advances will continue to helpoperations personnel to reduce errors, improve productivity, and reduce risk to plantand personnel [14-16].

In recognition of the problems and needs from the operating experience, there aremajor industry efforts underway to take advantage of experience. One is the designand construction of new plants with modern control room systems, such as the French1450 MW N4 plant and the Japanese 1300 MW advanced BWR plant. The other isthe upgrading and backfitting of existing control room systems including digitalcontrol and instrumentation as well as human-machine interface systems.

4.4 Challenges to Control Room Retrofitting

Design of a Control Room System should also involve careful consideration oftraining issues such as training programs and instructions. There is always risk thatinstructions and training programs neglects human factor issues. For instance, ifinstructions are written without participation of the users they may only reflecttechnical issues. Thus user participation is required.

The way a control room system is organized from a ergonomical and human factorspoint of view effects the way the operators learn to handle the system. For instance,clear labelling, coding and demarcation facilitates the learning process and theoperator can spend more time in learning "mental models" about the system ratherthan be occupied with unnecessary cognitive activities related to bad ergonomics.

Maintenance outages may deserve special attention with respect to training as ageneral remark. It is noted that outage operations have been found to create specialproblems from the control room point of view due to high activity in the station.

When retrofitting upgrades are made it is important that the operators are giveninstructions and procedures related to these changes before they are made.

The economic lifetime of instrumentation and control systems is much shorter thanfor the major process equipment and structures such as turbine and pressure vessels.

The main factors which affect the useful life of instrumentation and control systemsare technical obsolescence and functional obsolescence. Increased functionality isachieved mainly through software upgrades. Consequently, there is an increasingneed to be able to modify existing software and build in new software modules.

The retrofit of control rooms in many plants in the world will be a challenge in thenear future. The cause of this is not only the ageing but also the safety modificationsand operational improvements available from new technology.

In the replacement of equipment and systems, developments, technical trends andsupplier policies should be considered particularly with digital instrumentation andcontrol standardization, compatibility and open system architecture making gradualupgrading possible.


5. IMPLEMENTATION AND MAINTENANCE ISSUES

5.1 Quality Assurance

Computerization of control room systems should be developed according to arecognized Quality Assurance (QA) plan and properly defined project plan,describing the purpose of the system, the responsibility of each member of the projectteam, the project segmentation, reviews, hold-points, end-user approval, etc. [17-20].

The development should be divided in defined phases (e.g. definition,implementation, configuration), including for each phase, the required output (i.e.documentation and test results). In addition, in-service maintenance, developmentand upgrading should be considered.

Standardization in development helps in obtaining compatibility with suppliers, easiermaintenance and longer life. Proven methods and tools should be used especially insoftware development, and new methods should first be tested with prototypes.Modular design eases the management of program units.

5.2 Verification and Validation (V&V)

In the functional design phase, the correct assignment of control room functionsbetween operator and automation should be verified. This functional assignmentshould be validated to demonstrate that the whole system would achieve all thefunctional goals. The V&V of functional assignment is related to the design of newcontrol rooms and major retrofitting projects, where the role of the operator willchange. The procedure of V&V should, however, be applied to the design offunctional requirements of all new systems or functions installed in the control rooms.The output of this phase is an input to the specification of control room systems.

In the specification phase the functional specifications are verified and validated inorder to make sure that they fulfil the design principles and technical requirementsand the control room systems really support safe, reliable and economical operation.

The use of flexible computerized human/machine interface techniques and simulatorsmakes it possible to perform the final validation in the implementation phase. Evenin the commissioning phase of the implementation in the real plant, it is possible tomake modifications to the human/machine interface such as display pictures oroperator support systems.

The process of V&V of control room systems is described in more detail in IEC 964[14]. The main considerations are:


V&V should be planned and systematic;Evaluation should be based on predefined criteria and scenarios;The evaluation team should consist of specialists with various expertise, who areindependent from the designers.

A basic requirement for computer systems is that system documentation is verified ateach stage of production. Each document which defines the design should beproduced by a top-down process. For the highest requirement of safety, theverification should be independent, and formally conducted using check lists anddocuments with recorded resolutions of any discrepancies.

After completion of design, in the form of detailed specifications of hardware andcomputer codes, system validation is needed to confirm that the integrated hardwareand software perform correctly. The definition of functions given in the originalrequirement for the system must be taken, and interpreted into the form of detailedfunctional tests, test instruction, and expected test results. The computer system mustthen be systematically tested to those requirements, and the results recorded andanalyzed. Any discrepancies of performance should be formally recorded andcorrected through change notices, in accordance with the QA procedures.

5.3 Configuration Management and Change Control

An important concern for control room systems is the accuracy and correctness of thedata which they use as input. Techniques are implemented which assure that thecorrect data is being accessed and used. Information sources and documents, such asplant drawings, plant models, computer-aided design data bases, equipmentdescriptions and procedures, etc., must be kept up-to-date. On-line real-time datashould be time stamped and should be checked to ensure that the correct parameterand time step are used. Similarly, plant archival and trend data should be checked toensure that the correct data is being used. Software configuration control is alsoimportant to assure that the proper version is being utilized.

The importance of supplying the correct information to control room systems cannotbe overstated. These systems will perform control and safety functions which affectthe plant directly. They will also perform monitoring, display, diagnostic anddecision aid functions. The output of these functions will be used by the plant staff tomake their decisions for operating the plant. If the input to the control room systemsis not correct and accurate then the output to these systems will be faulty and thewrong actions will be taken.

6. FUTURE TRENDS

There is a continuing trend in the nuclear power industry to apply computertechnology to various engineering, maintenance and operations functions.

The next generation of nuclear power programs has placed significant emphasis onthe appropriate use of modern technology for the Human-Machine Interface System.Considerable experience has been accumulated regarding computerization of controlrooms. This experience provides opportunities to improve fundamentally the safetyand operability characteristics of nuclear plants. Breakthroughs in information andcommunication technology provide a real opportunity and challenge to exploit thesecapabilities in a manner that will provide benefits to operation of nuclear powerplants.

Computerization in existing plants will bring powerful computing platformsand will make possible the integration of new sophisticated applications with thebasic process monitoring systems in the control room. Specifically, the integratedcontrol room will include the following aspects:


diagnostic and monitoring functions to operations stall;operator aids and advisory systems;dynamic plant monitoring information systems and information management suchas electronic document management, automated procedures, and plant equipmentdatabases;human-machine interface environment which is common for all systems andallows the integration of all capabilities rather than using several different human-machine interfaces;incorporation of human factors engineering and human reliability in the designand development of systems for plant operation;The communication of real time plant information to off-site personnel fordispatch or monitoring functions.

Some specific trends are discussed in the following sections with more details.

6.1 Diagnosis and Prognosis Systems

The control room operators will be equipped with diagnosis and prognosis systemswhich integrate a broad knowledge of the dynamic behaviour of processes andsystems, disturbance sequences and failure mechanisms with recovery methods andoperating procedures, etc. Those systems will provide easy access to the storedknowledge and the ability to explain to the user how solutions and answers have beenachieved. One of the most important enhancements to diagnosis and prognosissystems is to ensure reliable and accurate signal input into the systems.

This calls for development of on-line simulation models with the ability forseparation of process anomalies and sensor faults, filter techniques to connect themodels to the process and coupling of simulation with cause-consequencerelationships.

6.2 Distributed Systems

The use of distributed on-line systems, with input and output equipment local to plantand connected by local area network methods will increase and will integrate datacollection, control room controls and automatic closed-loop control functions. They

will operate autonomously, and provide their information to other systems for on-linedisplay and operator support in the control room as well as for off-line engineeringanalysis. The advantage of such systems is the reduction in cables and the provisionof comprehensive information while preserving total electrical and physical isolationwith optic fibre.

6.3 Communication Systems

Communication systems will play a key role in a power plant to inter-connectcontrollers, plant monitoring and data acquisition systems, and engineering,maintenance, and operator workstations and diagnostic systems for plant operators.These networks can best be integrated by the most widely recognized InternationalStandards Organization Open Systems Interconnections.

6.4 Fault Tolerance Design

Fault Tolerance design for hardware and software will become a common practice incontrol room systems to satisfy the requirement of high reliability and availability.Fault tolerance design will protect against single failures, able to identify faults asthey occur, isolate the faults, and continue to operate.

Since the failure modes for software are different from hardware, fault-tolerant designfor software will consider the failures caused by common modes effects which maybe introduced during designing or programming. While the assurance of fault-freesoftware is difficult to achieve, regardless of formal verification and validation,software fault-tolerance may require the consideration of functional diversity. Thismeans that duplicate software would be designed, programmed, verified, andvalidated independently and separately. Alternately, fault-tolerant software mayrequire a backup system or a fail-safe option so that if the software fails, it will fail toa safe state as the overall system requires.

7. CONCLUSIONS

In summary, computerization of nuclear power plant control rooms will increase andwill bring significant advantages of safety and economy. Computerization has greatimportance both for existing plant upgrades and for future plants. The study leads tothe following conclusions:

168

1. The integration of human factors knowledge and practices with newinformation system technology is leading to significant improvement in thenuclear power plant human-machine interface.2. The control room computerization effort has realized the need toaccommodate station staff operating philosophy, procedure implementationprinciples, operation work control, operations work organization and personalcommunication among the plant operations staff.3. The use of computer systems for on-line display of plant state to the operatorsis now common, and retrofit systems with improved performance are beingimplemented. The flexibility of color video display units and the use of structuredhierarchies of displays have overcome most of the problems of early systems.The use of such systems has major advantages of providing information on thecomplete plant, with the added ability to present summary and calculatedinformation to operators.4. Fault tolerant digital control systems have significant advantages, and havebeen successfully applied to many nuclear plants. These controllers useredundant microprocessors and signal validation methods, and provide wide rangealgorithms with more optimized performance and higher reliability than theprevious analogue controllers. They have been shown to reduce plant outages andtrips, and reduce safety challenges to the plant.5. Computers are being used increasingly to provide integrated and multiplexedoperation of control room controls, with connection to the plant equipment using

BILL K. H. SUN AND ANDREI N. KOSSILOV

local area network systems. This provides significant advantages of cablereduction, space reduction and improvement of safety.6. Computer-based systems for protection have been implemented successfully

in several countries. The lessons learned from extensive verification andvalidation required for assurance of software accuracy and integrity havestimulated effort in the production of high quality software to address concerns byregulation and licensing.7. The flexibility of computers for information processing has resulted in their

application for diagnosis and prognosis purposes, such as disturbance analysis,success path monitoring, computerized operation instructions and vibrationmonitoring, etc.

REFERENCES

1. Taylor, J.J., Sun, B.K.H., Application of Computers to Nuclear Power PlantOperations, Nuclear News, October (1990), pp.38-402. Computerization of Operations and Maintenance for Nuclear Power Plants, IAEA-TECDOC-808, IAEA, Vienna, July (1995).3. Safety Implications of Computerized Process Control in Nuclear Power Plants,IAEA-TECDOC-581, IAEA, Vienna, (1991).4. Control Rooms and Man-Machine Interface in Nuclear Power Plants, IAEA-TECDOC-565, IAEA, Vienna (1990).5. Control Room Systems Design in Nuclear Power Plants, IAEA-TECDOC-812,IAEA, Vienna (1995).6. The Role of Automation and Humans in Nuclear Power Plants, IAEA-TECDOC-668, IAEA, Vienna (1992).7. Human Factors Guide for Nuclear Power Plant Control Room Development, EPRINP-3659, EPRI, Palo Alto (1984).8. Computer Based Aids for Operator Support in Nuclear Power Plants, IAEA-TECDOC-549, IAEA, Vienna (1990).9. Functional Design Criteria for a Safety Parameter Display System for NuclearPower Stations, Standard IEC-960, Geneva (1988).10. Computerized Support Systems in Nuclear Power Plants, IAEA-TECDOC-912,Vienna, October (1996).11. Programmed Digital Computers Important to Safety for Nuclear Power Stations,Standard IEC-987, Geneva (1989).12. Nuclear Power Plants-Instrumentation and Control Systems Important of Safety-Classification, Standard IEC-1226, Geneva (1993).13. Safety Related Instrumentation and Control Systems for Nuclear Power Plants: ASafety Guide, Safety Series No. 50-SG-D8, IAEA, Vienna (1984).14. Design for Control Rooms of Nuclear Power Plants, Standard IEC-964, Geneva(1989).15. Guidelines for Control Room Design Review, NUREG-700 (1981).16. Nuclear Power Plants - Control Rooms - Operator Control, Standard IEC-1227,Geneva (1993).17. Quality Assurance Organization for Nuclear Power Plants, A Safety Guide, No.50-SG-QA7, IAEA, Vienna (1983).18. Establishing the Quality Assurance Programme for Nuclear Power Plant Project,and Safety Guide, No. 50-SG-QA1, IAEA, Vienna (1987).19. Code on the Safety of Nuclear Power Plants: Quality Assurance, No. 50-C-QA,IAEA, Vienna (1988).20. Manual on Quality Assurance for Installation and Commissioning ofInstrumentation, Control and Electrical Equipment in Nuclear Power Plants, Tec nicalReports Series No.301, IAEA, Vienna (1989).21. Control Points for Reactor Shutdown with Access to Main Control Rooms,Supplementary Standard IEC-965, Geneva (1989).22. Software for Computers in the Safety Systems of Nuclear Power Station, StandardIEC-880, Geneva (1987).23. Standard for Software Configuration Management Plans, IEEE-828 (1983).


CONSEQUENCES OF CHERNOBYLA VIEW TEN YEARS ON

1. INTRODUCTION

On the night of April 26th, 1986, at 1:23, the mistakes of personnel operating Unit 4 of theChernobyl Nuclear Power Plant (ChNPP), multiplied by the mistakes of the reactordesigners, resulted in the biggest accident in the history of atomic energy. The explosioncompletely demolished the active core and the upper part of the reactor building. Otherstructures were significantly damaged. Barriers and safety systems protecting theenvironment against radionuclides in the nuclear fuel were destroyed. Radioactive releaseat the level of around 1016 Bq per day continued for ten days (26.04.1986 to 06.05.1986).After that, the release rate became a thousand times less and continued to decreasegradually.

When the active phase of the accident finished, it was clear that a relatively small amountof nuclear fuel (~ 3.5%) and a significantly larger (by an order of magnitude) amount ofvolatiles had been released. If fuel contamination defined the radiation situation near theChNNP (within a 30 km zone), then was the source of the long term contaminationfor a surrounding area of many thousands of square kilometres.

The Chernobyl accident has influenced, to a certain extent, the lives of millions of people.Hundreds of thousands were evacuated from the contaminated areas, a further hundredthousand participated immediately in the creation of the "Sarcophagus" (Russian Ukritiye)over the damaged Unit 4. Many were involved in the decontamination work at the siteadjacent to the ChNNP. Others were engaged in activities to prevent the contamination ofthe Pripyat and Dniepr Rivers.

Ten years have elapsed since the accident. Work on the mitigation of its consequences hasnot stopped for a single day. Nevertheless as the twentieth century passes into the twenty-first century, many problems of Chernobyl remain unsolved.

In our opinion there remain three main problems:

Advances in Nuclear Science and Technology, Volume 25Edited by Lewins and Becker, Plenum Press, New York, 1997

A. Borovoi and S. BogatovRussian Research Centre "Kurchatov Institute"123182, Kurchatov Squar, Moscow, Russia.

Sarcophagus safety;Revival of the area for residence and operation;

171

About 180 te of irradiated uranium is still in the Sarcophagus, having at present more than750 GBq of radioactivity. This radioactivity can be hazardous to the environment.

The extent of the hazard is not yet comprehended due to inadequate information about theproperties of the Sarcophagus, but it is clear that the hazard increases with time. It isnecessary to transform the temporary storage of the nuclear and radioactive materials intoan ecologically safe system. That is the problem of Sarcophagus safety.

Huge resources have already been spent in cleaning up and remediating contaminated areas,but the success attained so far is no more than meagre. Until recently, people were stillbeing evacuated from the contaminated areas. This area is now being widened and peopleare still being moved from their homes. What can be done to allow people to return andresume their normal lives is one more unsolved problem.

In the framework of the International Chernobyl Project (ICP), experts from dozens ofscientific institutions all over the world carried out extensive work in an attempt to clarifythe medical consequences of the accident. Their modest results were contradicted by somespecialists (and even more by some non-specialists) in Belarus, Ukraine and Russia. One ofthe arguments was that screening was only carried out on the population of thecontaminated areas and not on the liquidators working at the site of the ChNPP. Thisquestion was not considered by the ICP and, consequently, this group was not studied. Thisis only one example of the medical problems still to be solved.

The problems mentioned above are far from being all within the competence of the authorsof this review. Sarcophagus safety is our main field of interest. Remediation problemswere looked at from time to time and medical problems were studied in as much as wewere involved in the preparation of the input data for professionals working with us. Inaccordance with this remark, we invite the interested reader to consider the significance ofour conclusions in this review.

2. THE ACCIDENT

Before the accident

In the 1960s, the programme of rapid development of atomic energy in the USSRencountered a significant obstacle. The pressurised water reactor (the so-called VVER-typereactor) needed the production of a very large, hardened, containment or pressure vessel.But the USSR industry available at the time was not able to produce such vessels, as well asa number of other necessary elements. This was the reason that another type of NuclearPower Plant (NPP) was started without the necessity for such containment. This type ofreactor was called RBMK- the Russian abbreviation of "channel type reactor of highpower"

The principle of this type of reactor, where graphite is used as a moderator and boilingwater is used as a coolant, was well known since the Russian NPP had been put intooperation in Obnisk. There was experience of two dozen years of operation, for uranium-

Long term medical consequences for the still irradiated population and the "liquidators",participants in post-accidental mitigation activities.

172 A. BOROVOI AND S. BOGATOV

graphite military reactors had been used for plutonium production. Therefore thedevelopment of a new type of power reactor was completed rather quickly.

The active core of RBMK-1000 (1000 corresponds to the electric power in MW) looks likea cylinder 7 m high and 11.8 m in diameter, composed of graphite blocks. 1661 verticalchannels of internal diameter 80 mm, made of zirconium-niobium alloy, pass through theblocks. Heat-generating fuel assemblies, composed of fuel rods, are situated within thechannels. The total amount of nuclear fuel in the reactor was of 190.2 te of uranium as

enriched in the isotope. Water comes in from below the channel, washes aroundthe assemblies and is heated up to boiling point. Having been separated from the water, thesteam generated goes into the turbine. As the energy is transferred, the steam is condensedand pumped back into the reactor.

To control and stop the reactor, there are 211 control rods. To stop the reactor, 187 controlrods are introduced from above into the active core through special channels. Another 24shortened control rods, intended to smooth the axial energy field, are introduced into theactive core from below.

Without considering the reactor design in more detail, let us note that a kind of a "delayed-action mine" was put into the design and that "mine" finally caused the accident. Thescientific name of this mine is "Positive Reactivity Feedback". That this happens may beexplained as follows.

Water plays a double role for the RBMK design - it is simultaneously a neutron absorberand moderator. But the main physical manifestation for water is neutron absorption, ratherthan moderation; that is why the graphite is used as a basic neutron moderator. When boiledto steam, water reduces its density, so that fewer neutrons are absorbed by the water. Thusthe rate of nuclear fission increases together with heat generation. The temperatureincreases, accelerating steam generation, and so on. Positive Feedback or "reactivityrunaway" occurs, resulting in a rapid increase in the power of the reactor. It was asignificant feature of the Chernobyl reactor, that the Positive Feedback effect increases withtime of operation, as the fuel burnt up. Did the designers of the RBMK know about it? Yesthey did and they tried to provide the means not to lose control of the reactor. These meanswere described in Technical Regulation Rules, a document to be executed by personnelunder all circumstances.

In order not to allow reactivity runaway to evolve rapidly, and to allow the operator toundertake the necessary actions, at least 26-30 control rods had always to be situated withinthe active core. In special cases, in accordance with the special order of the NPPadministration, it was permitted to operate at a lesser number of control rods. If the numberof control rods was reduced to 15, the reactor had to be stopped immediately. In TechnicalRegulation Rules, a mode of operation was considered when the power level might forsome time be less than 50% of the designed one. In this case the reactor was in its so-called"xenon poisoning" position and subsequent power increase is allowed only when at least 30of the control rods are within the core. Otherwise the reactor must be stopped until xenonpoisoning completion; the time necessary for the decay of the xenon-135 being some 48hours.

In an emergency, the reactor might be stopped by the same control rods. Control rods weretripped into the core by the emergency protection system at the rate of 0.4 m/s within a total18-20 s. The second "mine" occurred here.

CONSEQUENCES OF CHERNOBYL 173


Being inserted from above, the control rods had an absorbing part about 6 m long and agraphite displacer 4.5 m long situated at the end of the rod. When the control rod wascompletely withdrawn, the graphite displacer was located at the middle of the core. Whenthe rods are first inserted into the core, the graphite displacer (a neutron moderator) firstsuperseded the water column (a neutron absorber) located beneath the displacer. At thistime, for several seconds the effect opposite to the one desired took place; the absorbingwater was removed and the moderator was inserted. As a result the power in the lower partof the reactor began to increase until the absorbing part of the rod had gone down. Theeffect of the positive reactivity overshoot was amplified when a lot of control rods werelowered simultaneously. However, the effect of the temporary power increase was notconsidered dangerous when sufficient control rods were situated in the core. Thisunpleasant feature of the reactor had been familiar to the designers since the first Unit ofthe Ignalia NPP (Lithuania) had been commissioned 13 years before the accident. Howeverthe effect was underestimated and no protective measures were undertaken.

Causes and development of the accident

On April 26th 1968, it was planned to stop Unit 4 of the ChNPP for maintenance service.During the stoppage some tests were planned to clarify some questions about reactor safety.We will not spend time describing the test programme and discussing its necessity as thedisputes about it continue to this day. Just note that the fatal mistakes made by thepersonnel during the test, resulted in the two incipient defects blowing up the reactor.

First, xenon poisoning happened due to the long reactor operation at 50 % power level (Fig.1). Then, at 00:28 (less than an hour before the accident) power dropped almost to zero asa result of operator mistake or technical failure. At this time the number of control rods inthe core was less than 30 and in accordance with regulatory rules, the reactor should havebeen stopped and the test put off. Instead the power was increased to perform the test. Thiswas done by the withdrawal of a number of operating control rods; subsequent calculationsshowed that only 7-8 control rods remained in the core. Therefore, protection againstPositive Feedback was reduced to impermissible levels. At the same time the automaticemergency reactor stoppage system had been disconnected so that the tests could go ahead.

At the beginning of the test at 01:23:04, the power level of the reactor, brought to anextremely unstable position, increased suddenly up to 115 % (of nominal level) and hadbeen increasing continuously. At 01:23:40 the rod drop button (EPS-5) of the emergencyprotection system was pressed to shut down the reactor. It was a moment when the second"mine" was fired. Graphite displacers superseded water in the control channels, the neutronflux along with the heat generation in the lower part of the reactor increased. The amountof steam increased along with the power level and that in turn increased the steam content.Control rod lowering occurred slowly and Positive Feedback had time to manifest itself to alarge extent. In accordance with different assessments, the power level was increased by3.5-80 times. Two explosions followed each other in a few seconds; Unit 4 of the ChNPPceased to exist. Disputes about the development of the accident continue, but for the mostpart scientists follow the version presented in broad outline here.1-3

Safety improvements for RBMK reactors undertaken after the accident

What was to be done after the accident? The two major questions were: shut down theRBMK immediately and whether to try to get rid of any shortcomings in the RBMKreactors elsewhere? These were questions having not only a technical but economic basis.So therefore let us quote some numbers.3

There were 14 operating Units of the RBMK-1000 in the USSR (10 in Russia and 4 in theUkraine) at the moment of the accident and another Unit of the RBMK-1500 in Lithuania.Their power was 50 % of the total power of all the other NPPs. Several years after theaccident another Unit of the RBMK-1500 in Lithuania and one Unit of the RBMK-1000 inRussia were put into operation.

Ten years after the accident, in 1996, the total energy produced by nuclear energy inRussia, was more than 1800 million MW*hour. In 1995 NPP's generated ~ 12% of the totalelectric power in the country. The share of the RBMK was 55% of nuclear energyproduction. These reactors generated and still generate the cheapest electrical energy. Theforecast of energy production for the RBMK reactors to the year 2020 was estimated to bean additional 1700 million MW*h at a total cost of $68 billion ($40 billion by Russianreactors) at the rate of electricity 4 ¢ per kW*h. Safety improvements for these reactorswere assessed to cost an order of magnitude smaller. That is why shutting down theRBMKs has been regarded as unacceptable.

Safety improvements have been done as follows: first, administrative ones, prohibitingoperations in conditions similar to those at Chernobyl; second, technical modifications ofthe reactor, reducing Positive Feedback in particular during the control rod trip, andcreating a rapid emergency system for reactor shutdown.


Positive Feedback reduction was achieved by inserting into the active core eighty to ninetyadditional control rods instead of fuel assemblies and by increasing the number of controlrods in the core (43-48) required during the operation. Further Positive Feedback reductionwas provided by an increase in the fuel enrichment (2.4 % of instead of 2 %previously). Control rods were modified to avoid the graphite column entering the lowerpart of the channel. Therefore the effect of positive reactivity due to lowering the graphitedisplacers was eliminated. Automation was modified to reduce the time of total control rodinsertion into the core from 18 s to 12 s and a rapid emergency system with an operationtime of 2.5 s was created. Other safety improvements have been made and are beingplanned.

In the reports submitted to the international forum of the IAEA One decade afterChernobyl: nuclear safety aspects2,3, it was declared that "... objective indices for NPPswith the operating RBMK reactors witness a safety level comparable with Western NPPsbuilt at the same time."

First countermeasures undertaken and their efficacy

Let us come back to the first days after the accident. Hundreds of urgent questions had to beanswered, the main problems had to be identified and formulated appropriately.

As far as the nuclear fuel was concerned, there were three types of associated hazard:nuclear, thermal and radiational. The nuclear hazard is usually considered as a self-sustaining chain reaction (SCR). It can happen only if several conditions are met. The maincondition is a large enough fuel accumulation in some space and availability within the fuelcontaining material (FCM) of the neutron moderator, graphite and water, etc. SCR couldtake place, for example, in the remaining reactor structure, if the structurewas conserved after the explosion(s). As a matter of fact, SCR is possible in approximately1/10 of the active core, if the control rods are removed. How dangerous would be theconsequences of SCR within the damaged Unit-4? For a long time this damage wasoverestimated and continues to be overestimated to the present day. First, it happened dueto the mistrust towards specialist assertions (the Chernobyl accident itself did not fostertrust); later it was due to personal interests and mass-media misinformation. The term"nuclear hazard" is associated by ordinary people with a nuclear explosion and,consequently, with a huge flash of light and a shock wave. Nothing similar was expectedinside Unit 4. In the case of SCR the fuel would be heated, the dangerous compositiondispersed and the reaction stopped. The main hazard in this case would be due to the releaseof radioactivity generated for the time of the operation of this "self-made" reactor. Allestimates showed the release could not be comparable (at least 1000 times smaller) than theone that had taken place during the initial accident. But, having been influenced by thecatastrophe, the members of the Governmental Commission did not trust such predictions.

That is why on the first day after the accident, some attempts were made to measureneutron fluxes near the ruins of the Unit. It was supposed that big neutron fluxes couldindicate continuing reactor operation. The measurements have failed, but the attemptsresulted in noxious consequences for the men participating.

As well as the nuclear hazard, a thermal one (the so-called China Syndrome) caused fear.This term taken from a movie of the same name, implied that molten nuclear fuel, havingbeen heated by nuclear decay energy, would flow down, burning through the floors of thebuilding and would reach and contaminate subsoil water.

A. BOROVOI AND S. BOGATOV176

Finally the radiation hazard became greater with each hour as every puff of smoke spreadthe radioactivity to new areas.

It was necessary to create barriers against all the mentioned menaces. Resolute measureswere undertaken involving thousands of people. For the past decade accounts of thesemeasures appeared in many publications. But the efficacy of these measures was notestimated anywhere, taking into account the material resources spent and the collectiveexposure of the participants. There are first estimates of "benefit" of the countermeasures4,5 as well as more detailed analysis6.

We would like to discuss here only two actions taken involving the effort. Dumping ofdifferent materials into the reactor compartment had been started since April 27, 1986. Partwere boron compounds, mainly which acted as the neutron absorber to providenuclear safety. The other part (clay, sand, dolomite) was intended to create a filtrating layerand to reduce the release of radioactivity. In addition, dolomite in the hightemperature regions, was able to generate carbon dioxide during its decomposition and thusprovide gas cover that prevented oxygen access to the burning graphite. Finally the last partwas to absorb the heat being generated. From the 27th of April to the 10th of May, 5000 teof different materials were dumped, including 40 te of 800 te of dolomite, 800 te ofclay and sand and 2400 te of lead. Dumping of materials continued after the active phase ofthe accident had finished. According to the registers of helicopter crews, about 14 000 te ofsolid materials, 140 te of liquid polymers and 2500 te of three-sodium-phosphate had beendropped in the period before June 1986. In accordance with the initial plan, the active corehad to be gradually covered by a loose mass. That reduced radioactivity but made heatremoval harder. According to expert calculation, the simultaneous influence of these twomechanisms would stop the release initially; then it would increase for a short time (hotgases breaking through) and stop the release eventually.

For many reasons it was very difficult to measure adequately the activity release; the errorof measurement was too high. Nevertheless, the measurements showed first a reduction inrelease and then an increase. Finally on 6th of May the release had become a hundred timessmaller. It seemed practice confirmed the theory. It was considered true for three years andin some works7 it continues to be asserted today. But it became clear to the scientistsworking at Unit 4 between 1989 and 1990, that the bulk of the material did not get into thereactor at all4,5.

Let us consider the facts. First is a picture of the former Central Hall of the reactor. It isfilled with the dropped material which formed mounds many metres high. It could be seenfrom the helicopters before the completion of the Sarcophagus erection and this wasconfirmed by groups of carefully prepared scouts who reached the Hall. But for fairnesssake, it can not still exclude that a significant part of the materials got to the reactoraperture. Second in the middle of 1988 it was possible with the help of optical devices andTV cameras to see inside the reactor shaft. None of the dropped materials were detectedthere. One can argue that the materials on reaching areas of high temperatures, melted andflowed throughout the lower rooms of the Unit. Indeed, masses of solidified, lava-likematerial ("lava") containing the nuclear fuel, were detected on the lower floors. Thirdly,any presence of lead in the lava could indicate it to consist not only of the reactor materials(concrete, uranium, steel, etc.) but the dropped material as well. No lead was found withinthe reactor or beneath, whereas 2400 te were dropped from the helicopters. Having studieddozens of samples, it became clear that negligibly small amounts of lead (less than 0.1 wt%) is contained in lava, i.e. there is less than 1 te of lead in lava weighing about 13 000 kg.



This means that virtually no lead went into the reactor. There were suppositions that thelead had evaporated, but soil samples showed no significant lead content.

Therefore, dropped materials even if they got into the reactor; have not been able toinfluence the release materially.

Those are the known facts. What prevented the pilots fulfilling their orders? It is possiblethat a bright spot (burning graphite) glowed near the reactor aperture. It could easily betaken by non-specialists for the reactor mouth. This is discussed in the work of Dr. Sich6

and we just present a figure from this work (Fig .2).

Maybe the upper biological shield of the reactor (its upper lid), thrown by the explosion andlanding in an almost vertical position, as well as the hundreds of steel tubes located on thelid, played the role of a "shield" which "reflected" all the dropping material. It is difficult tojudge now.

However, dumping the materials in the Reactor Hall cannot be regarded as useless. Boron-containing materials have been found in the Central Hall where many active core fragmentsas well as fuel dust were thrown. Landing on the fuel, the material ensured its nuclearsafety. Sand, clay and dolomite created a thick layer over the radioactive debris and helpedto make more safe the subsequent work of the builders and scientists. Some small part of

the materials could still get to the reactor and facilitate lava formation. It took three years,however, to accumulate and realise these facts. No one was able to foresee this in May1986.

A second example of the post-accident misunderstanding concerns the perceived need toprovide a forced cooling system. To prevent the China Syndrome, a heat exchanger wascreated under the basement of the Unit for a very short time (towards the end of June)

under hard working conditions. Forced cooling was foreseen for the concrete slab. It wassupposed at the time that if the hot nuclear fuel got onto the concrete, it would interact with

it like a hot iron thrown onto ice. Really the fuel, melting the ambient materials, began todissolve in the melt, reducing in such a way the specific heat generation of the mixture - asif iron would be soluble in water, like salt. As a result, only one floor was burnt through, inthe room immediately under the reactor, but three other floors remained in front of the meltcorium on the way to the earth's surface. However, the real hazard of the China Syndromecould not be estimated with the knowledge available in May 1986.

Nuclide inventory of the reactor

The immediate questions that had to be answered to the Governmental Commission were asfollows: what type and amount of radionuclides were stockpiled in the reactor at the time ofthe accident; how much nuclear fuel and radionuclides were thrown out of the damagedUnit and how much of it remained inside? It was clear, that without having information onthe nuclide inventory and the radiation from nuclear fuel, it was impossible to assess eitherthe radiation hazard for the release or the nuclear and thermal hazard of the fuel inside theUnit.

Usually, appropriate calculations are carried out during the reactor design process, but forUnit 4 they were either not carried out or were not accessible even to the GovernmentalCommission. During the first days, the general calculations of the Moscow PhysicalEngineering Institute for the uranium-graphite reactors were used. Then the crude results ofour group in the Kurchatov Institute were used as they had been made just for theChernobyl reactor. At the next step, calculations were carried out where the nuclideaccumulation history was considered for every one of the 1659 fuel assemblies that hadbeen in the reactor before the accident8. Finally, inventory calculations have been carriedout taking into account neutron field inhomogeneity inside the core during the reactoroperation time. It is significant, for example, for transuranics to be generated due toconsecutive neutron capture9. Other scientific groups carried out similar calculations. It isworth noting three of them10-12. First is the work of E. Warman10, who made hisassessments on the basis of Soviet reports to the IAEA. Second is an article by G. Kirchnerand C. Noak11. The authors had no detailed information on fuel burnup and neutron fieldsin the core; nevertheless they were able to extract the necessary information from the ratiosof nuclide activities in the hot particles detectednear Munich. The work of A. Sich12 is based on data the same as the Russian work9, and ituses the same calculation method. That is why the results are close to each other andpractically similar for radiologically significant nuclides. Comparison of the results,mentioned above, is given in Table 1. Which of the radionuclides enumerated are the mosthazardous for the longest period of time? Among gamma emitters, it is

Its half life is 30 years, which means its activity becomes an order of magnitude smalleronly 100 years later. Activities of the main gamma emitting nuclides in the fuel are shownin Fig 3. Seven years later, virtually only has notable significance

Number one among the pure beta-emitters is Alpha-emitters have changed their leaderfor these years and will have changed again in the next ten years. At first it was

Then plutonium isotopes became the most intensive alpha emitters.However, one of the plutonium isotopes is transformed by beta-decay

into being an alpha emitter. Having been accumulated, this isotope surpasses theplutonium isotopes alpha-activity (Fig. 4).



Active phase of the accident (26.04.86 - 05.05.86). "Lava" formation

After an explosion and loss of all means of surveillance, a damaged reactor for a finite timehas become invisible and we can only observe the final result of many multi-stage physicaland chemical processes. Besides the natural complexity of the processes, theirunderstanding is complicated to a large extent by a number of post-accident activities. Thefollowing hypothesis seems to us to be the most comprehensive and encompasses all theobserved facts.

1. The first explosion, which had a clear enough mechanism, resulted in the dispersal ofthe nuclear fuel and a rupture in the lower part of the fuel channels. The secondexplosion is supposed to be due to the loss of water in the active core. After these twoexplosions, the active core looked like a homogenous mixture of fuel fragments,zirconium and graphite. About 90 te of the nuclear fuel within the reactor shaft tookpart in this lava formation.

2. During the following days, three main processes took place in the core: filtration ofatmospheric air from the lower rooms through the gap after the lowering of the reactorbasement plate (BP); graphite and fuel fragments oxidising in the air, resulting in the


fuel being dispersed into micro-particles. And finally, as a result of the graphite burningup, the fuel descended into the reactor basement plate and was compacted. Thisincreased the specific heat generation of the residual mixture. These processes couldincrease the release rate both due to the increase in the temperature of the fuel-containing mixture and its compression, resulting in the thinning of the filtration layer.

3. Finally the specific heat generation of the mixture was high enough to melt throughthe damaged south-eastern quadrant of the BP. Serpentine, metal of the BP and lowerwater communication tubes had been involved in the melting process, as well as morethan 900 te of sand and concrete. The sand involved was likely to have come from thefilling of the lateral biological tank shield. The concrete had come partly from the floorof the sub-reactor room and partly from the wall slabs of the steam separators rooms.Presumably these plates were thrown into the reactor vault by the explosive wave.

3. AREA POLLUTION

Estimates of total radioactive release

In principle, there were three approaches to assess the release. The first way was to measurethe radioactivity as it was going out of the reactor. Secondly, the radioactive fallout afterthe accident might be defined and measured. The third way was to define the residualactivity within the Unit.

These three processes led to the following situation. The transformation of a dry form offuel containing materials into a liquid one after the melting was, presumably, the maincause of the sharp decrease in the release rate on May 6th (Fig. 5).

The lava generated was a melt of silicon-containing materials from the active coreenvironment that comprised the active core fragments. The average fuel content of the lavais about 7 % (wt), 2 % of which is dissolved in a silicon matrix and another 5 % associatedwith micro-inclusions. The total amount of lava is estimated to be about 1300 te.


Consider the first way. The day after the accident, the first attempts were made to takesamples of the aerosols above the damaged Unit and to study their radionuclidecomposition. Subsequently, air samples were taken regularly with helicopters or speciallyequipped aeroplanes above the reactor as well as at the site adjacent to the Unit. For manyreasons (unstable nature of the release, change in weather, high radiation fields, workaround the damaged Unit, etc.), accuracy of the measurements was not very good - as seenin Fig. 5. At present, attempts are being carried out to establish a data bank on the releasesand to assess their validity.

Despite all the intrinsic uncertainties, these first measurements of the release made itpossible to obtain a significant result - besides the volatiles (noble gases, iodine, caesium,tellurium and some others), other radionuclides in the release were bound inside small fuelparticles. Ratios of observed and expected (in fuel) radionuclide activities for one of thefirst air filters, taken above the damaged reactor (27.04.86), are shown in Table 2.

The ratios are similar (within methodical uncertainties) for all nuclides other than caesium.The result was considered as evidence that the caesium release occurred independently ofother nuclides, having been significantly more intensive. Subsequent analyses of other airfilters and soil samples confirmed this conclusion.

Let us discuss the second possible approach. Large scale studies of area pollution hadstarted the first day after the accident. But if the dose rate could be easily measured bydifferent type dose meters, the composition of gamma-emitting nuclides was not so easilymeasured by different gamma-spectroscopy methods. The gamma measure-ments could beconsidered adequate but the identification and quantitative measure-ment of pure alpha-(Pu) and beta-emitters (Sr) in fallout needed complicated radio-chemical analyses and wereinappropriately delayed. The method proposed was based on the fact that the main part ofthe nuclides, having been little measured, is bound in the fuel matrix where the radionuclidecomposition is relatively constant. That is why, instead of a long and difficult chemicalanalysis, it was possible to define the quantity of only one chosen gamma-emitting nuclideand subsequently to use known correlations between the activities in the fuel for the chosen


Really, there was a three-stage system of fallout studies. Firstly, aero-gammareconnaissance had been carried out as a first crude approximation. Secondly, soil sampleswere taken and measured by semiconductor gamma-spectrometers to define the content ofthe Finally, radio-chemical analyses were carried out intermittently to be confidentin the stability of the correlations.

It was soon clear that the main part of the fuel fallout occurred in the nearest zone (about 10km) around the ChNPP. "Independent" (of fuel matrix) caesium fallout gave aninsignificant contribution to the total (nearest) pollution. This is illustrated in Table 3,where the averaged ratios of activities and corrected to 26.04.86, are presentedfor different distances from the damaged reactor. The activities ratio deviated from the"fuel" ratio (0.066) only at distances of more than three km.

The data on fuel fallout made it possible to give the first estimate of the fuel release out ofUnit 44. In accordance with this estimate, about 0.3 % of the total fuel amount were thrownout on the site around the ChNPP, ~1.5 % - within the exclusion zone, ~ 1.5 % on otherareas of the USSR and less than 0.1 % occurred outside the USSR. Therefore the total fuelrelease according to this estimate was about 3.5 %. This number was reported by the headof the Soviet delegation V. Legasov at the IAEA meeting in August 19861.

Work on the third way to estimate the release (the assessment of fuel remaining in Unit 4)has been carried out. Huge radiation fields and ruins hindered the work. The firstassessment had been obtained in August-September 1986, when a programme of thermalmeasurements on the surface and around the reactor ruins had been completed. Comparingthe experimental heat generation with the calculated one, it was possible to assess that notless than 90 % of the initial fuel amount remained inside (only an upper assessment wasavailable)14.

hard connection with the fuel matrix due to high evaporation temperature;relatively long half-lifeconvenient energies and yields for gamma-rays.

nuclide and the desired nuclide. For example, the following correlation was used to defineplutonium isotope content in samples:

where A(Pu) is the total alpha activity of the plutonium isotopes kis a correlation coefficient at the moment of the accident, and is theactivity of the cerium-144 isotope. This radionuclide was taken for the following reasons:

where is a ratio of activities of isotope (i) and of in the sample and isthe same calculated ratio for the fuel. It is clear that the fractionation coefficient equals 1for pure fuel fallout and is more than 1 in case of sample enrichment by the isotope (i).

Significant enrichment by caesium isotopes was observed for the northern direction of theradioactive trace (Gomelskaya, Gitomirskaya, Chercasskaya regions), as well as for thesouthern periphery of the western trace (northern regions of Gitomir and Rovno). Table 4shows the fractionation of the radionuclides for typical areas of caesium enrichment(caesium spots)35.

The feature of radionuclide composition at caesium spots is a significant enrichment(regarding ) by virtually all nuclides except cerium and plutonium. Caesiumcontamination was extremely inhomogeneous. That was due to both the variations in the

At the present time on the basis of these analyses for hundreds of thousands of samples, itcan be regarded as established (0.68 confidence level) that more than 96 % of the initialfuel amount is located within the Sarcophagus (Ukritiye)14.

Features of contamination: Caesium spots

The first release of radioactivity immediately following the reactor explosion was directedtowards the Southwest. Subsequent releases lasted for many days (chiefly for the first tendays) and looked like a radioactive plume, whose elevation was due to the burning graphiteand the processes of heating the active core materials. The most intensive plume wasobserved 2-3 days after the explosion in a northerly direction, where radiation levels wereup to 10 mGy/h on April 27th and 5 mGy/h on April 28th, at an altitude of 200 m, 5-10 kmaway from the damaged Unit. The plume elevation on April 27th exceeded 1200 m abovesea level at a distance of 30 km to the Northwest. On the following days15 the plumeelevation did not exceed 200-400 m.

The basic method of radioactive fallout surveillance was the aero-gamma survey. Figure 6shows schematically the dose rates (corrected to May 10th) around the ChNPP obtained byaero-gamma survey 35. This map was used as the basic one for decision making. Inaccordance with the data, borders for different zones were defined: evacuation zone (doserate (DR) more than 50 µSv/h); exclusion zone (DR> 200 µSv/h); and a zone of rigidcontrol (30 µSv/h<DR < 50 µSv/h), where partial evacuation (children and pregnantwomen) has been carried out.

It was decided to organise the radiation survey of the contaminated area in accordance withthe radionuclide composition in the fallout. The following zones were chosen: the nearestzone (from 10 to 100-150 km from the source), where sharp gradients in fallout wereobserved; peripheral regions towards the northern and southern directions, where highenrichments by volatiles was observed; and "background" regions outside. The area withinthe radius of 10 m in the immediate vicinity of the ChNPP was called the "special" zone.

To characterise quantitatively the enrichment or depletion of the sample by someradionuclide (i) with respect to fuel composition, the so-called fractionationcoefficient, was used:



release rate and weather conditions during the transport of the radioactivity. One of themain reasons for caesium spot generation was "wet" fallout (accompanied by rain). It isemphasised16 that content in wet fallout was 15-20 times more than dry fallout.Figure 7 shows dose rates at different distances from the ChNPP for wet and dry fallout. Itis seen how fallout is more extensive in the areas where rain occurred.

Problems of caesium fallout estimation. As was mentioned above, fuel fallout is locatedimmediately near the ChNPP; outside the 30 km border, it presents no danger. It wascaesium fallout that caused the pollution of vast areas far from the ChNPP. The urgentquestion was (and remains), what total amount of caesium, especially its radiologicallysignificant isotope was released during the accident? Let us

recollect that its amount in the core at the time of the accident was Inthe Legasov report1 to the IAEA, a release of (13±7) % was presented. It was a firstassessment, when a detailed survey of the contaminated areas had not yet been completed.Subsequently this number was considerably modified. Many studies have been carried out,where caesium and other volatile releases were considered. First we would like to discussthe results of two of them (see Tables 5&6), where the authors prove that the volatilerelease was significantly larger than was previously1 reported. The first work (E.Warman10) was done in 1987, the second one (Ilyin et al.17) in 1990. It will be recognisedthat the initial release had been significantly underestimated.

A specially devised method of estimation was used by us18 in 1990. It was based on thestudies carried out inside the Sarcophagus. As we mentioned above, one of the fuelcontaining material (FCM) modifications is solidified lava5. It was discovered that onlyone-third of the caesium amount remained in the lava compared to the amount that wouldbe expected from the uranium content. Thus having been heated, the deficient caesium wasreleased from the FCM. Other FCM modifications indicated a caesium-fuel correlation. Thetotal amount of fuel in the lava is estimated to be about 100 te An estimation ofcaesium release can be done as follows:


Level of radiation is measured in units of natural background supposed to be 0.1 µSv/h. 1 -dryfallout, 2-wet fallout.

Finally in 1995 at the Conference "Radiological, medical and social consequences of theaccident at the ChNPP: Remediation of the areas and population", it was said in the reportpresented by Yu. Israel19: "The total amount of detected at the nearest trace, was

in the European part of the USSR, outsidethe nearest trace and all over Europe11." The data were obtained afterthe integration of caesium fallout over all the available maps of pollution.

The Iodine Problem. The iodine problem first of all relates to the thyroid gland radiationinjury, especially for children. The main source of this radiation was the isotope with ahalf-life of 8 days. It was reported by V. Legasov1, that the release of iodine was of(20±10) % of the total a5). On the whole, all the assessments done later are morepessimistic, but neither the early assessments nor the subsequent ones have any validity.One of the ways to get a valid quantitative assessment would be to define the amount of thelong-lived isotope in the FCM inside the Sarcophagus,especially in the lava. If we know the residual activity of inside the Sarcophagus, wewould be able to assess the release knowing the initial activity of in the fuel

The method was developed, but proved to be very expensive and has not yetbeen realised. It is reasonable now to suggest the iodine release to be 50-60 % of the initialamount, or 1.5 to

There were of in the active core (total fuel amount 215 te ), thus,

aboutIt is known that about 66 % of caesium left the lava. That corresponds to

or say 65/250 ~ 30 %.

As a result of the clean up as well as radioactive decay, dose rates near Unit-4 did notexceed 15 mSv/h at the time of completion of the "Ukritye" encasement (November 1986).

"Ukritye" encasement building

The necessity for the erection of a casement for Unit 4 had been clear from the first daysafter the accident. This construction had to prevent dispersion of radioactivity from thereactor ruins and to protect the adjacent area against gamma-radiation. Among 18 projectsconsidered, one solution proposed the erection around the Unit of an independent air-tightbuilding. Another solution proposed utilising to the maximum extent the remainingstructures of the damaged Unit. The second approach was finally chosen. It had theadvantage in cost and terms of building construction. The design and building wereompleted in 6 months (an unprecedented example in world practice), but this approach hadits negative features. There was a lack of information about the rigidity of the old structure


4. CREATION OF THE SARCOPHAGUS:ITS ADVANTAGES AND SHORTCOMINGS

Work on the site

After the explosion, the area in the immediate vicinity of the damaged Unit wascontaminated by dispersed active core fragments, pieces of graphite and fuel rods andradioactive structural elements. They occurred on the roof and inside the turbine hall,deaeration stack, metal supports and roof of Unit 3, etc. Near Unit-4, gamma radiation waslimited by the reactor ruins.

During the active phase of the accident, radioactive dust (fuel particles) was deposited onthe site, roofs and walls of the buildings. First measurements of dose rates around thedamaged Unit gave big values - hundreds and thousands of mSv/h (see Figure 8, where theresults of dose rate measurements at the site are presented, taken on the afternoon of26.04.8620). This circumstance substantiated a long-lived myth, that almost all the fuel wasthrown out of the Unit. Really the large dose rates were due to the extremely high specificactivity of the fuel. Calculation showed that if 0.3 % of the total fuel was disperseduniformly over the site, the dose rate on May 6th at a level of 1 m above the ground wouldbe of 0.5 Sv/h. Due to the uneven location of the sources, dose rates observed varied fromparts to tens of Sv/h (near the active core fragments).

Before commencement of mitigation activities, it was necessary to provide passage-waysand to clean up the site. For this purpose, military machines IMR-2, equipped withadditional shielding were used. From the very beginning the work was hard. Standardmilitary machines do not have the necessary protection; there were neither remote devicesto search for local radioactive sources, nor vehicles for the transportation of radioactivewaste. During the work, radiation protection for the drivers was increased from between ahundred to a thousand times and remote controlled vehicles were developed.

Clean-up work was carried out in the following manner:cleaning the site of contaminated garbage and equipment;decontamination of the outdoor surfaces of the buildings;excavation and removal of the upper soil layer (50-100 mm);laying concrete slabs and "clean materials" (sand, gravel) on the ground;covering the surfaces by film forming compositions.


used as supports for the new one, a necessity to utilise remote concreting, the unavailabilityin some cases of welding, etc. All these difficulties were due to huge radiation levels nearthe damaged Unit. All these obstacles have resulted in two significant shortcomings for theconstruction - uncertainty as to the strength of the supporting structures and lack of leak-tightness. The total area of openings over the new encasements after building completionwas about During construction, significant amounts of concrete (called "fresh")flowed into the destroyed Unit building. This made it difficult or impossible to pass intoand observe many of the rooms. On the other hand, partial covering of the FCM with aconcrete layer improved the radiation situation and facilitated passage into other rooms.

5. RESEARCH ACTIVITIES ASSOCIATED WITH THE SARCOPHAGUS

On completion of the Sarcophagus encasement, information about fuel location wasrestricted to the data available from the periphery of the Unit. Penetration into the room,located close by the reactor, was complicated by high radiation levels, damaged structuralelements and fresh concrete. To continue the studies and to get information on nuclear andradiation hazard from the Sarcophagus, a programme was developed by the KurchatovInstitute5. In accordance with the programme, the rooms to the west and south of the Unitwere cleaned up, drilling equipment was installed inside and bore holes were drilledthrough the concrete and steel to the places of potential fuel location. By means of visualmethods (TV-cameras, periscopes) and newly developed thermal and radiation detectors,many measurements have been taken. Samples were studied simultaneously. It was thenpossible to assess the distortions inside and to strengthen the emergency structures which, ifthey fell, would result in additional destruction and radioactive dust release.

It had been clear in 1988 that the reactor shaft was empty. Subsequent studies showed thatthe fuel inside the Sarcophagus had the following modifications14:

Active core fragments, which were supposed to be located mainly on the upper floors ofthe Unit, in particular in the Central Reactor Hall, where they were found under thelayer of materials dumped in 1986. Up to this day, there is very little information aboutthis fuel;

Finely dispersed fuel (dust), called "hot fuel particles". Their dimensions vary fromparts to hundreds of microns. This fuel modification forms all the fuel surfacecontamination. The total amount of fuel dust inside the Sarcophagus is assessed to be 10te including 1 te located immediately under the roof of the Sarcophagus (both values areknown no better than to the order of magnitude);Solidified lava-like fuel containing materials (LFCM). We described above the origin ofthis form of the FCM during the active phase of the accident. There is relativelysatisfactory information on lava at the lower floors, but the high radiation field andfresh concrete hinder as before the precise estimation of the amount of fuel locatedinside. The range in the assessments of fuel amounts in lava is now within 70-150 te.This estimate is influenced by the environment, especially by water; lava destructs fast;The last modification is presented by water solutions of uranium, plutonium, etc. Forfairness sake, let us note insignificant uranium concentrations - about 1 mg/1.

During the work many FCM locations have been identified. Physical and chemicalproperties of the FCM have been studied and constant surveillance of the FCMaccumulations is established. The surveillance is organised for radioactive aerosol releaseout of the Sarcophagus and for water inside. The data on the FCM location in theSarcophagus are presented in Table 7.

6. WHAT IS THE THREAT FROM THE SARCOPHAGUS?

What are the hazards represented by the Sarcophagus? They are the radioactive dust releaseat the fall of the buildings; the penetration of radioactive water out of the Sarcophagus intothe environment; the beginning of a self-sustaining chain reaction (SCR) inside anaccumulation of fuel-containing masses; and the release of radioactivity to the outsidethrough openings. What is the likelihood of these processes and their aftermath?

The experience of recent years. The release of radioactive aerosols out of the Sarcophagusis being controlled with the help of plane-tables mounted on its roofing. The most probablepath for aerosol release was determined, to choose suitable control points. Special attentionwas paid to air streams passing through the places of main accumulations of fuel-containingmasses, reactor space and ruins in the Central Hall.

The upper estimations of radioactive release through the openings of the Sarcophagus wereof: in in in 1992, in 1993,

in 1994, and in 1995. The plutonium fraction in the total activitywas limited to 0.4-1.2 %. Figure 9 shows the average concentration of alpha-active nuclidesin the air at the site near the Sarcophagus.

A considerable decrease in radioactive aerosol concentration was caused by the work of adust-suppressing installation, mounted in the Central Hall. It is noticeable that the air nearthe Unit is being decontaminated with time. Work in 1987-1989 on strengthening theaccessible inner construction, which was damaged to a large extent in the accident, hasavoided further destruction.

Oservation of the building for subsidence did not display any anomalies. Seismic waves ofthe magnitude 3.4 - 4, which were felt in the region of the ChNPP, from the Romanianearthquake on the 30-31st of May 1990, did not cause noticeable external damage andmovement. Inside the Sarcophagus some increase in cracks in the walls was observed.


193

The general conclusion is that up to this time the Sarcophagus has exerted no negativeinfluence upon the neighbouring territory. And up to this time it has been possible to avoidemergency situations.

The most hazardous structures of the Sarcophagus. All main bearing structures of theSarcophagus, such as beams B1 and B2, roofing over the Central Hall, steel shields of thecovering, the beam "Mammoth" and others are designed and constructed in accordance withbuilding requirements. That is why the durability of these structures themselves does not

CONSEQUENCES OF CHERNOBYL


cause any doubt. The supports for the main structures are another matter. The question oftheir durability has already been discussed in many works. There is no common opinion onthis question at present and we have to consider the most pessimistic forecasts. Accordingto these forecasts, lasting stability of the old constructions which support beams Bl and B2(see Fig. 10), is estimated as 10 years at usual climatic conditions (snow, wind,temperature). For comparison, the lasting stability of the other Sarcophagus constructions isestimated to be 80 years14.

Possible consequences of the building collapsing. Efforts to estimate the initial events,the sequence of their passing, the most direct influence and further consequences of theaccident were made in many works21-24. The most hazardous radiation accident, associatedwith the fall of Bl and B2, along with the upper structures of the building was consideredin these works.

As a result of the roofing fall, large amounts of dust from the surfaces beneath would beinvolved in the turbulent air trace behind the fallen structures. The calculations carriedout23 indicated that under these accident conditions, about 5 te of dust, containing about 50kg of fine dispersed fuel, could be involved in the turbulent trace. The altitude of the cloudelevation over the earth surface would be 100 m (the height of the building being about 60m), its diameter would be 20 m. According to these estimates, about 20 % of the releasewould be involved in a so-called "aerodynamic shadow" behind the building. The extent ofthis shadow is about 200 m.

Even taking into consideration average optimistic estimations, the surface contamination inthe region of the aerodynamic shadow at the site could reach many tens of

for Cs-137 and Sr-90 and about for Pu+Am isotopes. At slow windspeeds and at short distances from the Sarcophagus (hundreds of metres), the inhalationdoses due to transuranics would be very high up to the lethal effects causing lung cancerinduction. With increasing distance, doses drop quickly and at a distance of 10 km they areless than the permissible ones.


The model chosen was based on a supposition that there was no strong wind. In theopposite case, at very strong wind speeds, tornado or hurricane, for example, which couldcause the collapse of the Sarcophagus, dust clouds are formed mainly by other mechanismsand a greater amount of dust along with the fuel particles could be involved.

Recently, Dr. Pretch and others24 published calculations of doses to which the peopleworking at the site could be exposed in case of Sarcophagus collapse. The results of theseestimations are presented in Fig. 1124.

The results23,24 cause clear anxiety. However it should be mentioned that the necessaryinformation about the source term is not good enough to make accurate quantitativeforecasts. And in the work mentioned above, "white spots" are interpreted as maximumpessimistic suppositions.

Water in the Sarcophagus. Water is the main enemy for the safety of the Sarcophagus. Itcan

destroy the FCM, increasing the quantity of "loose" radioactivity inside;contribute to the destruction of the building elements;cause an increase in the criticality of the FCM and in the course of time, as their coolingand destruction proceeds, causes the generation of nuclear hazardous compositions;contaminate ground waters with radionuclides.


Apart from these direct influences upon the Sarcophagus, water has an indirect influence.It destroys the normal operation of diagnostic systems, makes obstacles to theinvestigations, transforms the premises into especially hazardous ones (from the standpointof electrical safety), and so on.

That is why the most important task for Sarcophagus safety is to take measures for thereduction in the quantity of water penetrating the premises, and, in the event of waterpenetration, to organise the constant control of its location, of its radionuclide composition,of the presence of fissile materials in it and to take active counter-measures if necessary.

Investigations display several possible ways of water penetration into the building. Theyinclude: natural precipitation penetrating through the openings and water in dust-suppression composition being regularly spread over the Central Hall. There is one moresource of water, the influence of which is increasing as the Sarcophagus cools. This iscondensation water originating as a dew inside the cold areas of the building afterpenetration of the moist and warm outdoor air into the inside.

As a result of work on hermitisation, many cracks in the coverings of the Turbine Hall anddeaeration stack, as well as a part of long cracks in the inclined parts of the covering of thereactor compartment, were closed.

The work on the roofing hermitisation has decreased water penetration, but not to a greatextent. Radioactive water accumulates gradually in the lower storeys of the Sarcophagus(total amount estimated to be ) and then leaves them by ways still not well known.

Systematic studies of water masses in the Sarcophagus were started in 1991. In 1995 about40 different areas of the Sarcophagus were monitored. The total beta-activity for the watersamples ranged within in 1995. The main contribution to totalactivity was due to caesium isotopes being present mainly in dissolved forms (74 - 100%).

Activity of the samples, caused by Sr-90, ranged within Uranium isotopecontent varied within 5-20 000 µg/l, mainly in soluble forms. Plutonium activity does notexceed 3000 Bq/l. Among the areas being observed, Room 009/4 (level 0.0) stands out .The concentration of caesium and uranium in this room has increased by two orders of

that at the composition of lava studied, (the neutron multiplication coefficient in aninfinite medium) is always less than 1 (again, according to the results for surface samples).This is illustrated in Fig. 12, where the nuclear-hazardous area is depicted withinco-ordinates (content of uranium in lava) - (fuel burnup) for the most hazardoushomogenous mixture of the LFCM with water. The area of observed parameters of LFCMstands far from the hazardous border.

In addition, two more barriers prevent water penetration into lava accumulations:


thermal barrier (large lava accumulations had temperatures at the surface of 60-70 °Cand, according to the estimations, temperatures inside had to exceed 100 °C);waterproof glass-looking surface of the lava.

An additional safety barrier was the content of the neutron absorbers - boron andgadolinium salts - in the samples of the water studied in the Sarcophagus. The salts were

magnitude since 1991 (caesium from to uranium from 10 to 4300µg/L).

What is the level of danger for water penetration out of the sarcophagus borders? Toanswer this question, it is necessary to remember that several hundred kg of fuel (500-700)are buried in the soil immediately under the Sarcophagus and about 3 te of fuel is situatedout of the site, inside the exclusion zone. This fuel is washed out by rain and other naturalwaters and, in our opinion, must play a considerably more important role in thecontamination of subsoil waters.

The possibility and aftermath of a further nuclear accident. The hazard of the SCR inthe active core of the Unit 4 Reactor bothered scientists up to the middle of 1988. It wasdue to the fact that for the RBMK-1000 reactor, a comparatively small part of its core(more than 154 fuel channels with graphite moderator in the absence of control rods) wasable to go critical. Such a part could have been left after the accident. But, as it was alreadymentioned, in May 1988 investigators managed to look inside the reactor shaft and theydetermined that the reactor stack (active core as it was ) did not exist any more.

They had to answer the question - could the SCR start with the accumulation of fuel-containing materials mentioned previously. The answer was given in "Technical basis ofnuclear safety for the Sarcophagus encasement (TBNS)", which was published by theKurchatov Institute towards the end of 199025. In this work, experimental and calculateddata, obtained up to the middle of 1990, were generalised and conclusions on nuclear safetywere made. The basic conclusion of the TBNS was as follows:" Since the moment of theend of the active phase of the accident, the aggregate of diagnostical measurementsindicates subcriticality of all the FCM located inside the Sarcophagus." At the same time itshould be mentioned that all the measurements were made at the surface of the fuel "lava",because of an absence of a so-called "hot drilling" technique which is the extraction ofhighly radioactive core samples. Calculations confirmed also that all modifications of theFCM are deeply subcritical in every geometric combination in the absence of water.Estimates of the criticality of mixtures, composed of lava-looking FCM and water, showed


dissolved in water when it passed through the materials located in the Central Hall or weredissolved intentionally in water used in technological operations.

Over the last five years, a lot of new results have been obtained, including ones on the longterm stability of the LFCM. These results revealed that many of the safety barriers havebecome lower (as predicted in the TBNS).

Two barriers prevented earlier water penetration into lava-looking FCM: its hightemperature and the water-proofness of the substance itself. Calculations (Fig. 13) andexperiments displayed a considerable cooling of the lava.

Cracking also transforms lava into a water permeable structure. The amount3 of condensedwater formed on the cold surfaces has increased noticeably during recent years. This waterdoes not pass through much material and does not contain neutron absorbers. Moreover, letus remember that the so-called "nuclear safe" parameters for the lava are suitable for itssurface. Further studies of samples, taken from subreactor room 305/2 in 1992-1993,revealed the presence of active core fragments in a non-melted state.

Thus it has become necessary to take into consideration a new composition "lava + activecore fragments + water", being in some cases more hazardous than a composition "lava +water".

But what is the degree of real hazard of the consequences of SCR in the FCM inside theSarcophagus? First, we should mention that an initial SCR is not equal to the explosion ofsome specially constructed mechanism. Calculations and estimates show that at the existinggeometry of the LFCM, neither an explosion nor a blast wave should be expected. More


probably, the heating up and decay of conglomerates, accompanied by radioactive release,would take place. Thus it is convenient to estimate the nuclear accident only in terms ofradiological consequences.

Nowadays, the most dangerous scenario for the development of a nuclear accident isconnected with any rapid pouring of water over the fuel-containing materials. Evenneglecting the presence of protective barriers, the consequence of such an accident might bepersonal irradiation in the immediate vicinity of the Sarcophagus up to doses of tens ofmillisieverts. That is many hundreds of times smaller than similar values for theSarcophagus collapse.

7. NECESSITY AND STRATEGY FOR THE TRANSFORMATION OF THESARCOPHAGUS

Our knowledge of the Sarcophagus is still very limited and our fears can be exaggerated toa great extent, but until our information is complete, the general approach in a science ofsafety is to consider maximum conservative forecasts. If we cannot describe the real hazardfor the Sarcophagus quantitatively, a tendency to increase is nevertheless definitely seen.

What shall we do? Should we continue our investigations intensively until it is possible toestimate the hazard associated with the Sarcophagus? But our investigations become moredifficult with time and safety requirements become stricter. We must refuse without anydoubt to allow people to operate within the hazardous premises, as was allowed during thepost accidental years 1986-1987. The corresponding robotics and remote methods are onlynow being developed and they are very expensive. Objective factors, financial andtechnical, as well as a great variety of subjective ones, led to the situation during recentyears for the study to be shortened rather than intensified.

Should we take preventive methods and rebuild the Sarcophagus? Having providedecological safety for hundreds of years, we might under the shield of Sarcophagus-2,without hurrying and very accurately, take radioactive materials apart, arrange themaccording to their activity and eventually dispose of them.

However this is a very expensive measure. Still in 1989, Academician S. Belyaev and oneof the authors proposed a concept for the Sarcophagus transformation and a Sarcophagus-2creation. The concept was discussed and modified, but gradually everybody came to the

conclusion that the problem cannot be solved by our own forces. It is necessary to ask forhelp from the international community - ideological, technical, but mainly financial help.

In June-July of 1993, an international tender to transform the Sarcophagus into anecologically safe system was held in Kiev. About 400 projects, proposals, and ideas weresubmitted. Six of them have become the winners. They were well-considered projects fromFrance, Germany, England, Ukraine and Russia. None of the projects was able to satisfy thejury totally. But from the work that had been done, an assessment of the cost of theSarcophagus-2 was about 2 billion dollars.

In the spring of 1995 the Kurchatov Institute proposed its "Concept for work at theSarcophagus for the years 1995-2000". The main task was considered in the Concept to beto undertake immediately measures to stabilise the state of the Sarcophagus. It was said inthe Concept: "It seems to be unjustified optimism for such a short period as 3, 5 or even 7years, to rely on the Sarcophagus transformation to isolate it totally from the environmentwith the help of Sarcophagus-2. This was not only from the technical standpoint, but from afinancial one. The solution of the Sarcophagus problems has to include for the years 1995-2000 the following tasks:

to ensure the current safety of the Sarcophagus;to ensure its longterm safety (stabilisation);to prepare for the transformation (to Sarcophagus-2);

At the stabilisation phase, measures need to be undertaken to minimise the influence of theSarcophagus on the environment for a sufficiently long time, more than 15 years. Thesemeasures will allow us to carry out the transformation safely and thoroughly." The conceptwas approved in general by the Ukranian Institutions, enlarged and transformed into thedocument "Basic directions of assurance for Sarcophagus safety for 1995-2000".

At present the international organisations give much help in work on the Sarcophagusstabilisation and transformation. There are several projects funded by the Commission ofEuropean Communities. But time slips by and the Sarcophagus remains one of the mosthazardous structures of atomic energy as well as the symbol of the Chernobyl tragedy.

8. REMEDIATION OF CONTAMINATED AREAS

Measures undertaken in agriculture

A considerable part of the contaminated agricultural areas are located in a region of marshyscrub, where sandy soils with low humus and acid pH are widespread. Under theseconditions and have an increased migrational ability in the soil-plant chain thatlead to increased content of the radionuclides in agricultural and stock-breeding production.

Intensive reclamation measures were undertaken:deep ploughing (5 cm deeper than usual) over all contaminated arable lands in theautumn of 1986 and spring 1987 along with inverting the upper soil layer. This resultedin a significant lowering of the nuclide concentration throughout the plant root region,the cessation of dust resuspension and a lowering by 3-4 times the dose rates at thefields;


Practical experience in the reclamation activity over arable and fodder lands contaminatedby Chernobyl fallout make it possible to get stable harvests and to decrease radionuclideaccumulation in plant-growing production by 1.5-10 times. The introduction of reclamativeagents into the sandy soil reduces nuclide mobility and restricts their transport by 2-4 times.

Decontamination of territories and buildings

To lower radioactive contamination of populated areas, where evacuation has not beencarried out, measures were undertaken to remove and to bury radioactive substances aswell as to cease or to make lower their migration in the environment. A brief description ofsome of the activities in the republics of the former USSR is given below.

Ukraine. The first clean up was carried out over places of attendance, such as schools,hospitals and stores as well as the most contaminated dwelling houses. From May toOctober 1986, 1229 localities were cleaned up in the Kievskaya and Gitomirskaya regionsand 479 for the years 1987-1990. Some characteristics and scope of the clean up activitiesare given in Table 826.

The removal of contaminated soil was a difficult operation. This was due to lack ofpackaging, dustproof loaders and means of compaction, as well as special vehicles for thetransport of the radioactive waste. During decontamination a large amount of liquid andsolid municipal waste, with different levels of radioactivity, were generated, as well asbiomass. During the first months, the radioactive waste was buried in temporary storagesites without isolated linings.

Although some populated areas were decontaminated two, three or more times, the resultsachieved were not satisfactory. One of the reasons was that secondary radionuclides weretransported by wind from forests and dirty roads as well as poor quality decontamination.Experience has shown that decontamination by means of:

clean up of solid surfaces with special mechanisms and chemicals;use of cheap absorbers to decrease the surface contamination;replacement of roofs of one storey buildings;

are of low efficacy from the standpoint of a cost-benefit analysis. Relatively more effectiveare:

removal of contaminated soil from homesteads;ploughing the soil in private gardens along with the introduction of fertilisers.

Figure 14 shows the results on dose rate reduction achieved during remediation activities atfour settlements in the Kievskaya and Gitomirskaya regions27. The remediation included:

evacuation of contaminated soil in gardens and drains;concreting around houses;removal of contaminated ground to places of burial.

The efficiency of the remediation was not high; the dose rate was reduced by 26 % onaverage.


liming all arable land of acid conditions. This measure, being a necessary part of crop-growing, decreases considerably the transport of radionuclides from soil to plants;annual fertilisation of contaminated land with increased amounts of mineral fertilisers,mainly potash and phosphoric fertilisers.


Byelorussia. To reduce dose rate, more than of contaminated soils wereremoved from populated areas, more than 4500 ramshackle buildings were demolished andmore than a million square metres of roofing were replaced. Much work has been done toasphalt streets and pavements. To supply the population with clean water, more than 3000wells were cleaned up, additional artesian wells have been used in the municipal watersupply, and new pumping stations mounted.

Productivity of the central water supply was increased by per day; more than900 km of piping were laid. As a result the central water supply for the most contaminatedareas was increased by 90-100 %. The big anxiety was due to the Gomel water supply fromthe river Soge flowing through the contaminated areas. In this connection new artesianwells were built and the Gomel water supply was switched to underground sources.


Russian Federation. The main work has been done by military units, civil defence,volunteer detachments and inhabitants. For the years 1986-1989, 915 km of roads werecleaned up, of contaminated soil were removed, of clean soil weredelivered, 74 km of water piping were laid, 5762 wells (including artesian ones) were built,2334 flats were supplied with natural gas. In western regions of the Bryanskaya region, 302localities were decontaminated; 50 of them were decontaminated two, three or six times.These works have not given satisfactory results; the coefficient of decontaminationachieved was of 1.2-1.6. This was mainly due to protraction in remediation (time elapsedafter contamination was long enough to allow the radionuclides to penetrate into the soiland materials) as well as the low efficiency of decontamination techniques available.

9. MEDICAL CONSEQUENCES: RESIDENCE IN THE CONTAMINATEDAREAS

Early consequences (acute injury)

Large groups of personnel and population were subject to screening from the first days afterthe accident. Patients had clothes and body surfaces contaminated; increased levels ofradioactivity of the thyroid gland were registered. Because any information on the externalirradiation of people was absent, it was difficult for medical staff to arrive at an adequatediagnosis. All patients, without exception, who appealed to the hospitals in Ukraine,Byelorussia and Russia about their health after the Chernobyl accident, were examined orsubjected to hospitalisation.

The main group who received acute radiational disease (ARD), were treated in Moscow(clinical hospital of the Institute of Biophysics) as well as in hospitals in Kiev. Generaliseddata on the patients with ARD are presented in Table 928

.

Therefore 134 persons who were at the site of the ChNNP at the time of the accident,incurred ARD. Almost one-third of the patients had a heavy (III) or extremely heavy (IV)degree of ARD. Twenty eight people who died, were patients who incurred external andcombined irradiation (large radiational burns of the skin, of underlying tissues and wholebody irradiation) in doses incompatible with life. It was due to the efforts of the doctors that

it was possible to save several patients with a heavy degree of ARD and in particular, onepatient irradiated by an absolutely mortal dose.

"Liquidators"

The main work to clean up the industrial site adjacent to the Chernobyl Nucler Power Plant(ChNPP) has been carried out by soldiers. Within two days of the accident, the mobilisationof soldiers and officers from the reservists had been declared through the militaryregistration and enlistment offices. These people contributed more than 80 % of the totalnumber of the military men. We have no official information on the number of militarymen involved in the mitigation activities and we present here only a crude assessment28.From the end of April to November-December of 1986, more than 90 000 military menwere irradiated within the 30 km zone, including the site of the ChNPP. Taking intoaccount the work during the years 1987-1989, this number has to be increased by 3-4 times.The number of persons sent to the ChNPP from other NPPs, either from another facility ofthe former USSR, personnel of the ChNPP, or the basic organisation of the Ministry ofMiddle Machinery (US-605), was 52 778 in 1986.

Generalised data on the Republics of the former USSR, based on State StatisticalCommittee information, were obtained in 1991 for the first time. Starting in 1990, 316 553people taking part in the mitigation, were surveyed in the USSR. Among these, 112 952were from the Russian Federation, 148 598 from the Ukraine, and a further 37 346 persons(11.8 %) were surveyed in the other republics.

According to the data of the All-Union Distributive Registry (Obninsk), to the end of 1991the number of "liquidators" (the clean-up troops) involved were:

138 390 in 1986;85 556 in 1987;26 134 in 1989;43 020 in 1989.

Thus an approximate number of liquidators is estimated to be 300 000 persons.

One of the urgent problems related to the liquidators is the level and kind of irradiation.Deficiency, or sometimes absence of, emergency dose-meters did not allow the assessmentof doses of external irradiation - to say nothing of the intake of radionuclides or beta-irradiation of the skin. Fortunately, doses due to radionuclide intake usually did notcontribute too large (~5%) a part to the total dose absorbed. The main body of soldiers andsergeants did not have individual dosemeters and a so-called "collective dosimetry" methodhad been put into practice. Scout-dosimetrists measured dose rates at the workplace inadvance and then the person responsible for the work calculated the time to be spent in thiszone taking into account the permissible levels of irradiation. The accuracy of such acalculation cannot be regarded as high, because radiation fields were extremelyinhomogeneous due to the presence of local sources (active core fragments) with extremelyhigh dose rates in the vicinity.

In general, the officially registered doses to liquidators are in doubt. Figure 15 shows thedistribution of documented doses received by liquidators according to the data of theRussian National Medical Dosimetric Registry (RNMDR).

Sharp peaks (or drops) attract attention at doses 25, 10 and 5 cGy, corresponding toemergency levels in 1986, 1987 and later. The authors are not able to explain this effectfrom a physical standpoint and state it without making any comment.



In 1986, immediately after the Chernobyl accident, the Ministry of Health of the USSRinitiated a programme to create an All-Union Distributive Registry for persons incurringirradiation. At the moment of the USSR break-up, the data base for liquidators comprisedmedical and dosimetric information on about 284 919 people. At present only residents ofthe Russian Federation are included in the RNMDR. On 01.12.94, the RNMDRencompassed 370 120 persons, including liquidators (43 %), evacuated men (2.2 %),residents or (former residents) of the contaminated areas (50.4 %), descendants ofliquidators of the years 1986-1987, and migrants from the evacuation zone (0.2 %). At thepresent the RNMDR seems to be the most informative source to forecast the long termconsequences of irradiation to the health of the liquidators.

The forecast30 of additional mortality for liquidators 20 years later due to malignanciescaused by radiation is shown in Table 10, with an account of age distribution (average age33 years).

The accuracy of mean dose assessments is about 50 %. In particular, it is seen thatexcessive radiation-induced mortality (attributive risk) due to all malignancies reached 2.8%. A similar index for leukaemia is 23.6 %. Mortality for liquidators for the years 1990-1993 is shown in Fig. 16. It is seen not to be detectable at the back-ground of controlindices.

The situation seems to be much more difficult concerning the interpretation of all diseasesand disabilities among the liquidators. It is known29 that indices of diseases of theliquidators in many cases exceed similar ones throughout the Russian population. Forexample, the diseases of the endocrine system are 18.4 times more frequent, mentalderangements are 9.6 times more frequent and the mean index for disease is 1.5 times morethan corresponding indices throughout Russia. Certainly the quality and completeness ofsurveillance for the liquidators is much better than the usual practice in Russia; the mostexperienced specialists are involved in the survey. Registered diseases are, on average,several times more often revealed in special institutions than in ordinary clinics. For thisreason it is extremely difficult to get an adequate control group for comparison.


It has been revealed that social and psychological factors connected with the accident havea considerable influence on the diseases and mental state of the liquidators. All this togetherwith radiational injury could be defined as the "Chernobyl syndrome" and attempts toextract from this complex the radiation influence by itself, are very difficult. One of theseattempts has been done30, where disease and disability indices were assessed through thegroups corresponding to ranges of doses received of 0-5 cGy, 5-20 cGy and > 20 cGyaccording to the data from the RNMDR. Liquidators receiving doses of 0-5 cGy wereconsidered as the internal control group. Within the framework of standard multi-factor

analysis, two factors influencing morbidity were analysed: the dose (with three gradationsof 0-5 cGy, 5-20 cGy and more than 20 cGy) and date of entering the zone of radiation(also with three gradations: 1986, 1987, 1988-1990). On the basis of the analysis of threekinds of diseases (endocrine system, blood circulation system and mental derangement), itwas revealed that entering the zone is the prevailing factor from the standpoint of influenceon the morbidity.

A similar situation is observed concerning disability indices. The disability index forliquidators is 2.8-3.2 times more than a similar index for all Russia. It is noticeable thataccording to the frequency of occasion and the gravity of the disease, the persons engagedin mitigation activities at the age of 18-30 years and now about 40 years old, correspond tothe age group of 50-55 years old in the Russian population in general.

Therefore two conclusions can be derived from the data on liquidators:virtual data for the time after the accident as well as the forecast of total mortality,obtained from the radiational risk coefficients of the ICRP, are in good agreement withthe observable indices; they do not exceed corresponding control indices throughout theRussian Federation;the liquidators of the years 1986-1987 represent a group at especially high risk.

Population of the contaminated areas

As far as the population of the contaminated areas is concerned, the question to be asked isabout the influence of low doses of radiation. Within the range of doses 20-50 mGy, andaccording to some estimations, up to 100 mGy, radiation does not cause immediatedeleterious effects to people's health. At the same time, according to the concept of athreshold-free influence of radiation, any irradiation may in principle, be a cause ofinduction of late consequences experienced as an increase of malignant damage amongirradiated persons, as well as detrimental hereditary effects affecting their descendants.

According to the recommendations of the International Commission on RadiologicalProtection (ICRP)31, the likelihood of occurrence for all kinds of fatal malignancies amonga population for all life-time at low dose irradiation is estimated to be some(i.e., 5 % per sievert). It is worthwhile comparing this value with a similar index for"usual", spontaneous mortality due to all cancers, about 20 % for developed countries.

Let us consider expedient radiological consequences to the population that resides in the"zones of rigid control" (ZRC) over the territory of 9 regions of Russia, Ukraine andByelorussia contaminated by radioactive fallout as a result of the Chernobyl accident. Thecollective dose equivalent for all life-time, to the population (273 000 persons) isestimated32 to be 31 000 person*Sv, unless any restrictions on the manner of life areintroduced. Calculation gives, if spontaneous mortality due to malignancies is supposed tobe 20 %, an increase due to expected radiation induced cancers will be 0.56 % to give atotal of 20.56%.

15 617 000 persons resided in the territory of these nine regions (including the ZRC in fiveof the regions). The committed dose equivalent for this population due to Chernobyl falloutwas assessed to be 192 000 person*Sv. Calculation shows that in similar circumstances,radiation induced mortality due to all malignancies may be increased by 0.06%, from 20 %to 20.06 %.


Considering this excess mortality, it should be taken into account that the annual incrementin malignancies for developed countries can be up to 1.5 %, exceeding the assessmentsmentioned above. Besides that, it does not seem possible to observe radiation inducedcancers (except for malignant tumours of the thyroid gland) at this expected level of allfatal cancers.

Firstly, consider any statistical restrictions. For example, to get statistically valid dataon deleterious effects due to an irradiation by doses of about 10 mSv per person, boththe considered and reference groups have to consist of 5 million people.Secondly, the natural fluctuation in oncological diseases is about ± 10 %, exceeding theexpected effect.Thirdly, radiation induced tumours do not differ from those of other origins.

Within a framework of all cancers known to be radiologically induced, leukaemia(malignant tumour of blood-creating tissue) is of major interest. The peculiarity ofleukaemia, especially the acute forms, is their short latent period. It is known that theminimum latent period of leukaemia is 2 years, and the maximum within 5-15 years afterirradiation. The life time risk of leukaemia induction at low intensive irradiation is

according to the ICRP data, that is, one-tenth of the total risk coefficient for all fatalcancers. Studies carried out in Russia, Ukraine and Byelorussia show the lack ofstatistically valid differences in leukaemia induction during the pre- and post- accidentalperiods. Even though leukaemia is the earliest evidence of stochastic effects of irradiation,the time elapsed is not sufficient to produce a final conclusion. However no leukaemiaepidemic is expected.

Malignant tumours of the thyroid gland are rarer than leukaemias in humans and fatalitydue to these tumours lies within 5-10 %. Taking account of the low frequency of thyroidmalignancies of natural induction, it could be expected that radiation induced cancers wouldexceed those occurring spontaneously. According to world statistics, the minimum latentperiod for thyroid tumour manifestation is about ten years after irradiation. However, threeyears after the accident, a sharp increase in thyroid cancers was reported from Byelorussia.There is still no consistent opinion on this subject among scientists.

During the first weeks after the accident, measurements of content in thyroid glandswere carried out on 31 000 people in the most contaminated regions of Russia. Thesemeasurements revealed a strong dependence on age. Children younger than 3 years oldreceived doses 5-8 times higher than adults under the same conditions. Mean doses todifferent age groups varied within 1-20 cGy, but individual doses could be 10 Gy or higher.The forecast30 of excessive numbers of thyroid cancers for the population of contaminatedKalugeskya and Bryanskaya regions (population 105 300 and 466 900 respect-ively) isgiven in Table 11.

As seen from the Table, the attributive risk for children in these regions is 45 % and 26 %respectively; that is, every second or third cancer respectively will be radiation induced.

The next deleterious effects proved to be due to radiation are hereditary effects. It is knownthat radiation induced damage occurs in the gonads. Such damage can manifest as someimpairment for the descendants of irradiated individuals. The likelihood of such hereditaryeffects was clarified by the ICRP in 1990 (Publication 60). It adopted a coefficient for



serious hereditary effects of for all populations. According to this number, it canbe estimated that the expected risk of serious hereditary ill-health occurring for subsequentgenerations of people residing in the 9 contaminated regions of Russia, Byelorussia andUkraine, will be something more than 100 occurrences per 1 million. Taking account of thehigh spontaneous level of clinically significant hereditary diseases for humans (60 000innate anomalies and 15 000 genetic diseases per 1 million live-born children), it becomesclear that there are insuperable practical difficulties in detecting these theoretically possibleextra hereditary effects. There is a hazard of radiation injury to embryo and foetus. Mostsusceptibility to radiation occurs between the 8th and the 15th week of pregnancy andirradiation by doses above 1 Gy can result in mental backwardness; the ICRP believes thatthese effects can have a threshold. But such doses have never been reached in women as aresult of the Chernobyl accident.

Finally, some words about the radiological significance of the so called "hot particles" ofthe Chernobyl accident. As a result of complicated physical and chemical processes of fueldestruction and volatiles condensation, a large number of particles of high specific activitywere generated. If a hot particle occurs in the human body, local doses in the immediatevicinity of it can be extremely high; this was regarded as an ability for increased cancerinduction. It is worth noting that if we suppose a linear dose-response dependence for anyorgan as a whole, there is no problem concerning hot-particles; the energy deposited andthe mass of the organ are similar both in the case of hot-particle and the same uniformlydistributed activity. An increased response is possible only for effects that are non-linearwith dose. The comparative hazard of the Chernobyl fuel hot-particles and similaruniformly distributed activity was assessed34 according to a model of non-linear dose-response dependence. Withiout considering the calculation in detail, it can be noted that thehot-particles proved to be several times less hazardous than could be expected from thesame amount of gaseous radio nuclides intake.

According to the assessments, the dose rate of on the 10th of May 1986,corresponded to an annual dose equivalent of 50 mSv (5 rem). That was half thepermissible dose limit set by the Ministry of Health for the first year after the accident.

To facilitate planning the protective measures, criteria were developed on the basis ofsurface contamination by long-lived radio nuclides. Thus the urgent task was in borderingregions, where contamination exceeded accepted limits. The idea was to set theinstrumentally measured limits of contamination, providing permissible levels of annualintake and external irradiation by The following limits were chosen as the criteria:

for for andfor Pu (alpha-emitting isotopes).

The basic criterion, having been 100 mSv for a population for the first year after theaccident, was mainly met (average doses being 3-4 times lower). It was the reason for theNational Commission on Radiological Protection (NCRP) to set annual dose limits asfollows: 30 mSv/y in 1987, 25 mSv for 1988-1989 and 173 mSv as a total dose up to 1st ofJanuary 1990. Theoretical assessments by the Ministry of Health indicated, that the groupof people for which irradiation received was more than the emergency limit (set at 173mSv), was 0.4 % of the population.

The task was given to the NCRP of the former USSR to develop recommendations on thesubstantiation of permissible limits of irradiation to the population for a long time,including the restoration period. It should be noted that international practice had not yetdeveloped a distinct position on this question. The working group of the NCRP proposed toutilise, as a criterion for radiological protection, an individual dose to some critical group ofthe population. In this case, the critical group consisted of children born within the timeperiod of 1986 ± 2 years. It is worth noting that the usual distribution of dose is log-normal,and in the case of the Chernobyl accident, individual doses (within a single group) couldvary by 5 times.

As a result the NCRP proposed in October 1988:

Criteria of residence in contaminated areas

As mentioned above, by early May 1986 a generalised map of dose rates around the ChNPPwas obtained on the basis of aero-survey and terrestrial measurements, Fig. 6. This mapwas regarded as the guide for the evacuation of the population. According to proposalsmade by the Ministry of Health and the State Committee for Meteorology, adjacent to theChNNP area, the zones were set as follows:

exclusion zone (dose rate > );evacuation zone (dose rate > );zone of rigid control (dose rate > ), where temporary evacuation of childrenand pregnant women was carried out.

to set a quantity of 350 mSv as the sum of external and internal irradiation for 70 yearsof life, supposing the critical group as children;this limit must include doses for the previous years since the accident;to consider this foreseeable limit as an intervention level for planned and controllableevacuation of people from areas where this limit would be exceeded;to put the recommendations into practice from 01.01.1990.

A. BOROVOI AND S. BOGATOV210

According to the assessments, the dose limit of 350 mSv could be exceeded within the ZRCfor a population of ~ 56 000 in 242 populated areas if no measures to improve the radiationsituation were carried out. Mainly these were villages where the surface contamination wasmore than

For many reasons, the implementation of the above mentioned concept was very difficult.As a result the Academy of Sciences of the USSR has created a Commission to develop amore human concept. The objective of this new concept was a mainly social compromisedue to radiophobia (fear of radiation) in the population.

It was proposed to take as a criterion for any necessary resettling of the population, anannual dose of 5 mSv, and as a "criterion of intervention", a dose of 1 mSv/year wasproposed. For settlements where doses of irradiation ranged within 1-5 mSv/y, it wasrecommended to take protective measures to reach a dose of 1 mSv/y on the condition thatvoluntary resettlement was available, along with a payment of compensation at Stateexpense.

The articles of this conception were realised in the law "On the social protection of citizens,incurring radiation due to the accident at the ChNNP", passed by the Supreme Council ofthe Russian Federation on 15th May 1991.

On the breakdown of the USSR, the newly created Russian National Commission onRadiological Protection (RNCRP) approved the modified project for the "Conception ofprotection of population and economic activities at the areas affected by radioactivecontamination" and "Proposals on its practical realisation". The main difference of the newConception from the previous one, was a refusal to impose compulsory resettlement of thepopulation away from the contaminated regions. In the new variant, areas where doses of 1-5 mSv/y are likely, are referred to as zones of radiational control of the environment,agricultural production and doses to the population. Areas where the annual dose of 5-50mSv are likely are referred to as zones of voluntary resettling.

10. CONCLUSION

In September 1996, while this article was being written, the mass media of Ukraine andother states, reported on a supposed "self-sustaining chain reaction" (SCR) that occurred inthe lower part of the Ukritiye. This publicity sowed seeds of anxiety which fell on thefertile soil of fear and misunderstanding. The facts were that an increase of neutron counts,by a factor of 3-5 were observed over a period of several hours. It will require aconsiderable effort by the scientists and engineers involved to provide the population with areal explanation of the event.

Very properly, neutron monitoring had been maintained to detect a self-sustained criticalreaction. Neutrons would be observed even in a sub-critical system, however, due to theeffect of long lived delayed neutrons and natural fissioning in uranium-238, multiplied bythe remaining, dispersed fuel. The observed increase factor would not be consistent with adangerous SCR but could be due to more minor changes in conditions with theSarcophagus.

The event could in reality be due to two causes. Firstly, it could be due to trivial instrumentfailure. Though all possible tests were performed, this cause cannot be ruled out entirely,making the observation spurious. Secondly, it could be due to physical effects. It had rained


heavily on those days and large amounts of water penetrated the Sarcophagus. Additionalwater in the melted fuel (LFCM) would result in a change of neutron spectrum bymoderation and hence of the multiplication properties of the LFCM. Both of these effectswould then lead to an increase in count rate in the detectors. But the rise was perhaps abillion times lower than consistent with a self-sustaining reaction. And we might remindourselves that even if such an SCR occurred it would be dangerous only for those in theimmediate vicinity.

The hazard should not be overestimated and the publicity smacks of taking advantage of thetrue Chernobyl tragedy. But the event strengthens, we believe, our argument that it isnecessary to start work immediately on the stabilisation of the Sarcophagus and to improveits safety before the next century is reached.

When we speak of remediation, what should our target be? In our opinion, the Chernobylarea itself should be considered as a reserve and not the focus of remediation. Money andhuman health should not be wasted on the removal of fuel particles. The naturalenvironment has taken care to catch and restrain them successfully. Dissemination ofradioactivity in this area is very slow and does not require the evacuation of people. Thiswould leave it feasible, under proper care, to continue the operation of the remaining powerplants with their essential supply of electricity.

We cannot say the same of the caesium contamination that covers thousands of squarekilometres of valuable land. The loss of useful territory is too large and here activeremediation measures are necessary. It is essential to develop economically acceptableprocedures to return people to the land and to begin to re-use it.

The question of the long term consequences of the Chernobyl accident to the health of thepeoples of Russia, Ukraine and Byelorussia remains difficult to answer. Even though theexpected number of additional mortalities from cancers is unlikely to be seen against thestatistical background of natural death, the deleterious effect includes a wide range offactors which probably interact to strengthen each other. First amongst these we would putfactors of a social and psycho-emotional nature. While less tangible, we do not deny theirreality. It is difficult, however, to separate the consequences of Chernobyl from the generalchanges in the health and reported health of the former USSR.

Finally we have to say that all the power of the former Soviet Union and its ability toconcentrate such a huge force in this single endeavour have not served to overcome theproblems of the accident. Most of these problems remain to be solved by future generations.

REFERENCES

1. The accident at the Chernobyl NPP and its consequences. USSR State Committee on the Utilization ofAtomic Energy. IAEA Post Accident Review Meeting, 25-29 August, Vienna (1986).2. A.A. Abagyan, E.O. Adamov, E.V. Burlakov et al. Chernobyl accident causes; overview of studies overthe decade. IAEA International Forum "One decade After Chernobyl: Nuclear Safety Aspects". April 1-3,IAEA-J4-TC972, Working Material, 46:65, Vienna (1996).3. V.A. Sidorenko. Nuclear safety of the RBMK reactors: main results and prospects. IAEA InternationalForum "One decade After Chernobyl: Nuclear Safety Aspects". April 1-3, IAEA-J4-TC972, WorkingMaterial, 435:447. Vienna (1996).4. A.A .Borovoi. Fission product and transuranic release during the Chernobyl accident. Materials ofInternational Conference "The Fission of Nuclei - 50 Years". Leningrad (1989).


5. A.A. Borovoi. Inside and Outside the Sarcophagus. Preprint of the Complex Expedition of the KurchatovInstitute. Chernobyl (1990).6. A.R. Sich. Chernobyl accident management actions, Nuclear Safety, v.35(1), 1:23 (1994).7. Nuclear accident and RBMK reactor safety, report GRS-130, Berlin (1996).8. S.N. Begichev, A.A.Borovoi, E.V. Burlakov et al. Fuel of the Unit 4 of the ChNPP (short reference book).Preprint of Kurchatov Institute 5268/3. Moscow (1990).9. A.A. Borovoi, A.A. Dovbenko, M.V. Smolyankina and A.A. Stroganov. Definition of nuclear physicalproperties for the Unit 4 of the ChNPP. Report of the NSI AS 52/11-20, Moscow (1991).10. E.A. Warman. Soviet and far-field radiation measurements and an inferred source term fromChernobyl. Presented at the New York Chapter Health Physics Symposium, April 3, Brookhaven NationalLaboratory, New York (1987).11. G. Kirchner and C. Noack. Core history and nuclide inventory of Chernobyl core at the time of theaccident. Nuclear Safety, v.29(1), 1:15 (1988).12. A.R. Sich. Chernobyl Accident Revisited, Ph.D. dissertation, MIT, Massachusetts (1994).13. Sandia National Laboratories. Technical report on item 6 - development of a model of fuel-surroundingmaterial interactions. Contract AL-9407. Albuquerque (1995).14. A.A. Borovoi. Analytical report (Post-Accident Management of Destroyed Fuel from Chernobyl) IAEAWorking Material, Vienna (1990).15. Yu. A. Israel, V.N. Petrov, S.I. Avdiyushin et al. Radioactive pollution of the environment at theaccident zone of the ChNPP. Meteorology and Hydrology, No 2,5:18 (1987).16. Ch. Hohenemser and Renn Or. Chernobyl's other Legacy. Environment. 30:4 (1988).17. L.A. Ilyin et al. Radiocontamination patterns and possible health consequences of the accident at theChNPP. Journal of Rad. Protection, v.10(1) (1990).18. S. Beliayev, A. Borovoi et al. Radioactivity Releases from the Chernobyl NPP Accident InternationalConference. Comparison of Consequences of Three Accidents: Kyshtim, Chernobyl, Windscale, October 1-5,Luxembourg (1990).19. Yu.A. Israel, E.D. Stutkin, I.M. Nazarov and A.D. Fridman. Radioactive pollution of the environmentafter the Chernobyl accident. Radiological, Medical and Social Consequences of the Accident al theChNPP. Remediation of the Areas and Population. 21-25 May. Theses of the report, Moscow (1995).20. S.V. Ilyichev, O.A. Khochetkov, V.P. Kriyuchkov et al. Retrospective Dosimetry for the Participants ofMitigation Activities after the Accident at the ChNPP. "Seda-Style", Kiev (1996).21. M.V. Sidorenko et al. Forecast of lasting stability of the "Ukritiye" encasement of the Unit 4 of theChNPP. Final Report on Contract No. 877. Kiev (1994).22. V.P. Beskorovaynyi, V. Kotovich, V.G. Molodykh, V.V. Skurat, L.A. Stankevich and G.A. Sharovarov.Radiation consequences of collapse of structural elements of the Sarcophagus. "Sarcophagus Safety '94".The State of the Chernobyl Nuclear Power Plant Unit 4. Proceedings of International Symposium, 14-18March, Zeleny Mys (1994).23. Preparing and Expert Assessment of Input Materials for a new issue of the "Base for Radiational Safetyof the "Ukritiye". Report 09/39 of State Enterprise "Expertise" Moscow (1992).24. G. Pretzsch. Analysis of the accident "Roof Collapse" for the "Ukritiye" Encasement. Report GRS-A-2241, Berlin (1995).25. S.T. Beliayev, A.A. Borovoi, V.G. Volkhov et al. Technical Basis of Nuclear Safety for the "Ukritiye"Encasement (TBNS), Complex Expedition of Kurchatov Institute, Chernobyl (1990).26. Chernobyl. Five Difficult Years: Collection of the Materials. IzdAT, Moscow (1992).27. A.V. Kretinin, A.F. Landin. Efficiency analysis for countermeasures on the reduction of irradiation topopulations residing in the radioactively contaminated areas. Problems of Chernobyl Exclusion Zone,Minchernobyl, Naukova Dumka, Kiev (1995).28. L.A. Ilyin. Realities and Myths of Chernobyl. "ALARA Limited", Moscow (1994).29. V.K. Ivanov, A.P. Konogorov, A.F. Tsyb, Eu. M. Rastopchin, M. A. Maksyutov, A.I. Gorsky, A.P.Biryukov, S.Yu. Chekin. Planning of long-term radiation and epidemiological research on the basis of theRussian National Medical Dosimetric Registry. Nagasaki Symposium on Chernobyl Update and Future,Amsterdam (1994).30. A.F. Tsyb, L.A. Ilyin and V.K. Ivanov. Radiational risks of Chernobyl: an assessment of mortality anddisability on the basis of data of the National Radio-epidemiological Registry (1995). Radioecological,Medical and Social Consequences of the Accident at the Chernobyl NPP. Remediation of the Areas andPopulation. All-Russian Conference, 21-25 May, Moscow (1995).31. Recommendations of the International Commission on Radiological Protection. Publication 60.


International Commission on Radiological Protection. 1990. Pergammon Press, Oxford and New York(1991).32. L.A. Ilyin. Time -limits for radiational influence, irradiation of the population and medical consequencesof the Chernobyl accident. Med. Radiology. No 12. 9:17 (1991).33. V.F. Stepanenko, A.F. Tsyb, Yu.I. Gavrilin et al. Doses of internal irradiation to the thyroid for theRussian population as a result of the Chernobyl Accident. Radioecological, Medical and SocialConsequences of the Accident at Chernobyl NPP. Remediation of Areas and Population All-RussianConference, 21-25 May, Moscow (1995).34. S.A. Bogatov. Method of comparative assessment of the radiological significance of inhalation of "hotparticles" of the Chernobyl accident. Kurchatov Institute Preprint 5601/3, Moscow (1992).35. Yu.A. Israel, S.M.Vaculovsky, V.A.Vetrov, V.N. Petrov, F.Ya. Rovinsky and E.D. Stukhin. Chernobyl:Radioactive Pollution of the Environment. Hydrometeoizdat, Leningrad (1990).


DYNAMIC RELIABILITY

Jacques Devooght

Service de Métrologie NucléaireUniversité Libre de Bruxelles50, av. F.D. RooseveltB - 1050 Brussels

1. INTRODUCTION

Dynamic reliability is a term coined for a chapter of reliability theory linked to dynamicsystems. If reliability is the ability of an industrial system to perform a given task on amission time without failure, dynamic reliability emphasizes the fact that such systemsevolve dynamically and that failures (and repairs) can influence the dynamics andreciprocally, that dynamics (and its associated state variables) can affect failure or repairrates. Since the evolution of the system can in general branch at any time from onedynamics to another, the resulting event tree is infinitely ramified : hence the alternate termcontinuous event tree used for dynamic reliability. The term "probabilistic dynamics" is alsoused to make it clear that a system switches randomly from one dynamics to another.

The industrial and scientific context under which such studies appear is clearly the fieldof PRA (Probabilistic risk assessment) and PSA (Probabilistic safety assessment) used forinstance in nuclear reactor safety studies, whose backbone is the event tree/fault treemethodology1 . The current use of this methodology has been reviewed recently2.

On the other hand, dynamic reliability as a specialized field, has been reviewed recentlyby N. Siu3. The interested reader is encouraged to read first this thorough review which isa bottom-up approach to dynamic reliability in the sense that it shows clearly how theclassical approach can lead sometimes to erroneous results and how new methods wereprogressively developed to remove the defects. On the other hand we will adopt here acomplementary, top-down approach where we try to formulate the problem in a generalmathematical setting, introduce the approximations needed to cope with the extensivenumerical problems met, and try to unify existing approaches.


2. PHYSICAL SETTING

Ideally an event tree should start from a given top event (characterized by the currentstate of the reactor and eventually its past history) and yield probabilistic statements on thefuture state of the reactor, e.g. on the physical state variables, (like power, temperature,isotopic concentrations, etc.), on the component (or hardware) state of the system taking intoaccount the protection devices, controls, operator’s interventions including all possiblefailures, malfunctioning, operator’s errors, etc. These probabilistic statements are used eitherto determine damage to installations and public (as in level 2 and level 3 PSA studies4) orto pinpoint defects of design, procedures, etc. Obviously no method is able today, nor is itlikely in the near future, to yield such sweeping statements, essentially on two counts : (1)the considerable complexity of the problem due to a combinatorial explosion of possiblesituations combined with uncertainties on most data, not to mention the complexity ofhuman modelling, (2) the sheer volume of numerical calculations needed for any realisticappraisal. The classical event tree is essentially static in the sense that it takes into accountthe chronology of events, not their actual timing. More precisely, to quote N. Siu3, thereare "a number of structural characteristics of this static, logic-based approach for modelingaccident scenarios which are of interest. First, if variations in the ordering of the successand failure events are possible, these variations do not affect the final outcome of a scenarioor its likelihood. (If ordering does make a difference, the event tree would have to beexpanded in order to handle possible permutations of events). Second, variations in eventtiming do not affect scenario outcomes or frequencies, as long as these variations are notlarge enough to change "failures" to "successes", or vice versa. Third, the effect of processvariables and operator behavior on scenario development are incorporated through thesuccess criteria defined for the event tree/fault tree top events. Fourth, the boundaryconditions for the analysis of a given top event (or basic event, when dealing with cut setrepresentations of accident scenarios) are provided in terms of top event (basic event)successes and failures; variations in parameters not explicitly modeled are not treated".

216 JACQUES DEVOOGHT

The plan of the paper is the following :

I. METHODS

Introduction.Physical setting.Chapman-Kolmogorov equations.Reduced forms.Exit problems.Semi-Markovian generalization.Subdynamics.Application to event trees.

1.2.3.4.5.6.7.8.

II. SOLUTION TECHNIQUES

Semi numerical methods.The Monte Carlo method.Examples.Prospects and conclusions.

9.10.11.12.

Models of the system under study.

The subject of nuclear dynamics is the evolution with time of state variables such aspower, temperature and pressure, etc. (as well as neutron or precursor densities) and thestudy of "forces" such as reactivity.

The state of the reactor in a dynamics problem is described by a vector whosecomponents are fields, such as power density and temperature, which are functions of the

position and time t. A huge amount of work has been devoted to the definition ofcondensed descriptions of the state of the reactor. From continuous to nodal and fromnodal to point descriptions, we have a whole set of approximations to which aprobabilistic description can be appended. They are all characterized by the fact that allparameters are assumed to be known, and in that sense, reactor dynamics is adeterministic problem. The knowledge of the parameters amounts to the knowledge ofthe structure of the reactor : components function properly, or else in accident analysis,their mode of failure is given. In general, whole subsystems are compactly described bylaws that do not disclose their internal structure. For instance, the safety control rodsmay be introduced through ramp reactivity laws, etc.

Time scale of the accident.

Needless to say the knowledge of the initial state of the reactor is essential for thedevelopment of the accident. However this development is dependent on failureshappening after the initiating event but also on failure or errors developed before theinitiating event as in the case of passive safety devices which lie dormant. Usually theinterval of time over which accidents develop precludes significant additional failures -e.g. besides those already present before the start - except those induced by the operators,or else some hardware failures which increase rapidly when state variables grow out oftheir nominal range or when sensors are destroyed.

Change of state of hardware components.

These changes of state are either provoked by protection devices, or by operatorintervention or by the physical influence of state variables inducing failure (e.g., ruptureof a vessel by overpressure). They are either deterministic or probabilistic.

Human error modelling.

Recovery actions by operators are essential parts of dynamic event trees and arecharacterized by time distributions (see 6).

Uncertainty of data and models.

The treatment of model uncertainty is sometimes reduced to the treatment of theuncertainty of parameters involved in the models - which is not the whole story5. Failurerates are uncertain and characterized by a distribution. Latin hypercube sampling is oftenused to treat uncertainties in parameters for models which are otherwise deterministic.

1.

2.

3.

4.

5.

DYNAMIC RELIABILITY 217

However if we plan a top-down approach we should at least be able to list thecharacteristics of a realistic model.

The evolution of a transient under the influence of a mix of deterministic andprobabilistic laws (and where transition points are not predetermined) is a bona fideproblem of nuclear safety; therefore, our objective is to give a proper rigorousmathematical setting for this problem, a starting point from which methods ofsolutions and adequate data collections can be examined.

Transition laws already exist that depend on state variables, such as the probabilityof ignition of a hydrogen explosion as a function of its concentration or theprobability of containment break as a function of overpressure8. Manyexperts’opinions are expounded in this form9. Moreover, sensitivity studies can beused to bracket the uncertainties10.

3. THE CHAPMAN-KOLMOGOROV EQUATIONS

3.1. Forward and backward equations

Since our objective is the description of reactor transients where change of states of thereactor can occur randomly we have naturally to look for what the theory of stochasticprocess has in store for us.

The most developed theory is based on a Markovian hypothesis, which loosely speakingamounts to saying that the failure evolution of a system is dependent only on its presentstate. If a stochastic variable x has taken values at time the conditional probability


However there is no difficulty to introduce parameter uncertainty - at little cost - insimulation methods of calculation6 which are essential and practically unique tools fordynamic reliability studies. Most of the reluctance met for the treatment of uncertaintyis unjustified from a Bayesian point of view : if experts disagree, probabilistic statementsincorporating their opinion are after all belief statements that should incorporate whateverknowledge we have.

Objections have been made that failure probabilities are subject to uncertainties andtherefore that the dependence of on physical variables is subject to evengreater uncertainties, if not to outright ignorance. Therefore, any theory using thisconcept would be useless or premature. We do not believe the objection to be valid ontwo distinct grounds7 :

a.

b.

6. Time delays.

Contrary to the standard event tree methodology where time plays only an ordering role(before, after), time plays an essential role in dynamic reliability where competingprocesses are at play and where the next branching is determined for instance by thefastest process. This is of course true in deterministic dynamic processes but in our casewe must add the fact that certain processes, like human intervention, have uncertainduration. On a shorter scale, uncertainty on time delays in breakers, relays, etc. mayinfluence the outcome of a fast transient.

Finally, time distribution is at the root of the distinction between Markovian and semi-Markovian modelling where in the former it is necessarily exponential.


If is a vector in space the state of the reactor can be given as a point in a spacewhich is the Cartesian product i.e. M replicas of A Markov processcan be either continuous or discontinuous. Let us assume that is continuous but thatthe hardware state index jumps from one integer to another. Symbolically we can describea sample trajectory in phase space as in Fig. 1.

Let us remark that for a general stochastic process, need not be continuous, althoughit is in our case. It could also be either deterministic or stochastic. In the first case we havea so-called partially deterministic process12 and in the second case a Wiener process, betterknown as a Brownian motion. In the first case, for each state i, the reactor obeysdeterministic dynamics

Standard dynamic models use fields like temperature or power field whereis a point in the reactor, and yield partial differential equations. We assume in writing

(3.1) that these equations have been discretized and reduced to a set of ordinary (non) linear

with can be written i.e. the past can beignored. This is a strong assumption and strictly speaking most physical systems do notobey it, only because aging of a system obviously influences its actual behaviour. Howeverit is very often adopted because departures from a Markovian hypothesis are usually smalland if not, there is a well known trick11 which amounts to enlarge the state space, e.g. tointroduce auxiliary variables to reduce the problem again to a Markovian formalism. Thisis a last resort attempt because it is extremely costly from a numerical point of view.However as we shall see below, the case of "semi-Markov" systems can be handled withlittle more difficulty.

To describe the state of a reactor, we need at each time t

A state vector whose components are power, fluxes, temperature, etc., i.e. whatevervariables that are needed to describe adequately the neutronic and thermohydraulic stateof the reactor and its auxiliary equipment. The choice and definition of is consideredto be adequate for the purpose of PRA analysis.

The component (or hardware) state of the reactor labelled by an integer i. If for instancethe reactor has n lumped components, each of which being in m possible different states,we have a total of states, i.e. index i runs from 1 to

1/

2/

differential equations. Our objective is to find the conditional probability densitywhere the initial state is at time and the Chapman-Kolmogorov

equation yields13 the set of partial differential equations

and is the conditional probability of transition by unit time from component statei to component state j. All the knowledge necessary to develop a PRA analysis is in fact

summarized in and Let us note however that concerns not onlyspontaneous transitions but also transitions induced by controls and operators, although inthe latter case we must enlarge our phase space (see § 6). From a mathematical point ofview, (3.2) is a system of M linear partial differential equations in and is, fora given t, a multivariate distribution in a space of dimensions. Thereforeno standard numerical scheme (except Monte Carlo) is ever likely to solve (3.2) and themain purpose of (3.2) is to lead to new concepts and introduce simplified models. Let uspoint out however that the number of distinct dynamics (3.1) is usually considerably smallerthat M. Finally if we forego the deterministic dynamics assumption, we will obtainadditional second-order (e.g. diffusion-like) terms in (3.2), a situation we will meet latter(§ 7) when uncertain parameters are involved in the dynamics.

To the "forward" Kolmogorov equation (because "future" variables are involved) wecan append an equivalent equation, the "backward" Kolmogorov equation

where


which gives identical results, the initial condition for both being eq. (3.5)

Equation (3.4) will play an essential role in the definition of generalized reliability and exittimes.

3.2. Particular cases

Let us remark first that the forward Kolmogorov equation contains two importantparticular cases :

a. e.g. the reactor does not change in any way its component state. Equation(3.2) reduces to a Liouville equation with solution


where is the solution of the O.D.E. system (3.1) with initial condition

This is the usual deterministic reactor dynamics case.

b. i.e. the dynamics is uninfluenced by the component state of the reactor and it

is initially (and stays) in a steady state. Then

which is the standard Markovian model used in reliability, except that which is herea parameter does not appear explicitly.

Equation (3.2) has the form of a conservation equation for a probability fluid and takinginto account case (a) we can rewrite it in integral form7.

To simplify notation we shall no more write explicitly the initial condition unless it isnecessary to do so, and assume we have :

The physical meaning of (3.9) is clear : the contribution to is given either by

trajectories in phase space from to with a probability of

unchanged component state, or by trajectories that start from at some time havea transition from j to i and do not change anymore this state in the remaining intervalThe analogy with neutron transport is evident: if we write the monoenergetic Boltzmann

equation with discretized angular fluxes in its standard notation

we have the immediate correspondance

4. REDUCED FORMS

4.1. Discretization of physical variables

We can define two auxiliary probability densities :


It is even more apparent in a context of statistical mechanics, using a Hamiltonian

formalism with where corresponds to (3.1) and

where the Liouville theorem gives

the component state i being for instance a label for a type of particle (for instance itscharge), each type of particle having its own dynamics.

The system of integral equations (3.9) will be generalized in § 6 allowing for a semi-Markovian formulation, but as much it is a point of departure for Monte Carlo methods ofsolution.

Summing (3.2) over all states

with

Not surprisingly we recover the two special cases (3.6), (3.8) since we have "projectedout" variables or i, without making however any special assumptions. But we observe

that parameters are time dependent contrary to (4.5) even if is time independent. Assuch these equations are useless because their very definition rests on the detailed knowledgeof through (4.4) and (4.6).

However we can partition space in cells and integrate (3.2) over with


Integrating (3.2) over

with

and

Then

where is the exterior normal to the surface of and

with

Let be the subset of such that for

Then

where is the common surface of and its neighbors withTo exploit (4.12) numerically we must make the additional assumption


with

Therefore

What we obtain is a pure jump process13. Equation (4.15) is the basis for the CCCMTmethod14 which is an outgrowth of the CCMT method developed by Aldemir15,16.

4.2. Moments and marginal distributions

We define, using eq. (3.2)

To obtain differential equations for moments, we need some assumptions on the

dependence of on the variable as well as for If we assume17 a quadratic

dependence on of these functions, the system obtained for willinevitably depend on higher moments. This problem is usually solved with the introductionof an approximate closure relation. The choice made in refs. 17,18 is the closure relationof a Gaussian distribution.

The structure of the system obtained is :


where and µ are vectors and a matrix. The explicit form of is very complexand will not be given here.

Methods to solve (4.18) and later to synthesize from the moments will beexamined in § 9.

Obviously the large number of variables is an important numerical obstacle and we maybe tempted to work only with marginal distributions and later try to synthesize the

distribution from its marginals,

Let us point out however that marginal distributions do not uniquely determine the fulldistribution19. The literature on this subject yields many suggestions each with itsadvantages and defects. For instance, let be a marginal distribution of itscumulative distribution function.Then

is an interpolant to a distribution with the same univariate marginals. Coefficientsetc. may be fitted to obtain the conservation of covariance and higher moments.

An alternate solution, based on the use of Gordon’s transfinite interpolation yields20 :

where the and are auxiliary distributions linked by

The covariance matrix and some (but not all) third moments may be conserved using freeparameters Unfortunately positivity of (4.20) cannot be guaranteed under allcircumstances although numerical experience so far (§ 9) did not show too much difficulty.Substituting (4.20) in (3.2) with the same quadratic dependence assumed above, adding anindex i to each function in (4.21) to label Markovian state i, we obtain the following system

for the marginal distribution


We have now N independent systems of hyperbolic partial differential equations for

The explicit expressions for all coefficients appearing in (4.22) are given in18.

4.3. Benchmarks

Finally to close this analytical chapter we ask ourselves if there is no closed form solutionof (3.2) available to serve as benchmark to test approximate methods of solution. Forinstance the trivial example of two Markovian states with state graph

and dynamics initial state in state i=1, yields21

Other benchmarks have been solved by P.E. Labeau22.

5. EXIT PROBLEMS13,23

5.1. The event tree as an "exit" problem

One of the fundamental problems of the theory of stochastic differential equations is thecalculation of the probability that the representative point in phase-space will exit (andwhen) from a given domain D. For instance, if we superpose a noise to a deterministicprocess, even if all trajectories point away from the boundary of D, a probability existsthat, due to the random perturbations, or noise, the point will cross the boundary. The exitproblem is concerned with the distribution of the exit point and with the exit time. Theproblem is of relevance for transmission over potential barriers in the field of chemicalkinetics, and the reader is referred to ref. 24 for other examples.

The exit problem is fully relevant to the mathematical theory of continuous event treesand its safety consequences. For instance, consider the event tree associated with a loss-of-coolant accident. Some branches are associated with a return to a safe state and others areassociated with catastrophic outcomes, such as the failure of containment followed byradioactive releases.

The safety analyst wants to know the probability that will cross the boundary of asafety domain D, usually a polyhedron determined, for instance, by maximum allowablepowers, maximum allowable temperatures or pressures, etc., or by the fact that somedangerous states i are entered.


Let Then is the distribution of the time of passage in state whichis not, in general, the distribution of the time of first passage of interest in exit problems.To study exit problems, we must convert eq. (3.2) into a boundary value problem.

Indeed, the initial value problem that describes reactor dynamics involves no boundaries.

The values of are constrained only by laws of physics (for instance, positivity of

temperature), and these constraints are embodied in the certain domains of the phase-

space being inaccessible.

The definition of (Fig. 2) is the analyst’s business and, therefore, a function of his

objectives. Let us examine the sign of where is the exterior normal to

If and if then When varies

continously as well as is a set of lines on i.e. to pass from toon we cross a line. However, if D is, for instance, a box with plane faces, we can

have everywhere on a face. The exit problem involves only

The partition of may be different from each component state i. Forinstance, let i=1 be the state where all components function properly. In principle, on we

should have and No trajectory should cross the safety boundary

An exit of D necessarily involves a transition of We transform the problem byforbidding trajectories leaving to re-enter through by imposing

where we introduce explicitly the initial condition We define the mean escape timethrough by


The time of crossing of is the time of first crossing because there are no secondcrossing due to eq. (5.1).

Condition (5.1) is equivalent to the vacuum boundary condition for

used in transport theory. The domain outside D is made fully absorbing, and no return isallowed. We remark that condition (5.1) is more involved because it applies only to the part

which is not necessarily the same for all i.

We introduce conditional probabilities by Let

The probability that the escape time T > t when the component state is i is

Let be the escape rate of D in state i :

The mean escape time in state i is

If we introduce an average escape time irrespective of

then7 obeys

5.2. Generalized reliability

For the case where is independent of t we have time translation invariance, e.g.


which generalizes a well known expression for the mean-time to failure in

Markovian reliability analysis, if we write (5.8) as

where the last term in (5.9) expresses the fact that during the stay of the system in state j,its dynamics will move from to a neighboring point where the MTTF is modified if

The rate of escape through the surface summed over all states is

Integrating (5.11)

If we evaluate (3.4) for and apply operator we obtain

with the initial condition if and

We can generalize this problem further by introducing a damage function when, insteadof scoring a failure (independent of whenever we cross we score instead a damage

function of 25,26. The use of the backward (or adjoint) equation always appear when westudy the outcome of a stochastic process as function of its initial state.

We can transform (5.14) into a set of integral equations

6. SEMI-MARKOVIAN GENERALIZATION

6.1. Definition and formulation

Let us start from eq. (3.9) and define


is a generalized reliability function, giving the probability that at time t, asystem which starts at in state will not have "failed" by crossing the safety

boundary. If for all i and all the system can never fail and we have thesolution However we can also introduce failed Markovian states.Following the usual definition, this amounts to partitioning the component states inwhere Y is the set of failed states with for Then the system will faileventually at some time either by crossing the surface or by entering Y, i.e.and the generalized reliability function obeys

A formal proof is given in ref. 27 although we can justify (5.15) heuristically in thefollowing way. The reliability of a system starting at time t=0 in is the sum of twoterms. The first is the probability that the system does not leave state i in interval [0,t]

if the system starts in and does not leave D

is the characteristic function of D). The second term is explained likewise as the sum overof the probability of staying in (i,D) for time and have a transition to a working state

and survive for the rest of the time

(a) is the conditional probability that the stochastic process will remain in

state i during time t if the state vector is initially equal to in other words it is theholding time distribution in state i;

(b) is the conditional probability that if a transition occurs when in stateit will be to state with the normalization


Therefore

where

We can understand as the probability to have in j a transition to state i and

to stay in that state up to a time in interval The rate of transition out of state

is and

with

which is the rate of transition out of state without any previous transition out of theinitial state.

The physical significance of (6.6) is therefore clear since the rate of transition is now the

sum of two rates according to the fact that a transition to a intermediate state occursor not. As such, nothing prevents us to use (6.6) in the general case where is the

cumulated distribution function (c.d.f.) of the transition time t when the initial state isrelaxing the exponential distribution (6.1) assumption. Admittedly this extension is heuristicand must be validated through the theory of piecewise deterministic stochastic process12.

In the same way we can generalize (5.15) to

This is a generalization to the semi-Markovian case of the standard system of equations28for reliability functions. We obtain the classical system if we omit all references to

6.2. Human error modelling29

The generalization to non-exponential transition time distributions is particularly usefulin the case of human error modelling where the exponential distribution for the time ofaction (for instance) of an operator is notoriously insufficient. Although an Erlang

distribution can be obtained by the addition of r fictitious states in Markovmodelling, it is necessary to have a large value of r to model dead time distributions like

From the point of view of Monte Carlo (see § 10) it is not much more difficult to treatarbitrary c.d.f. However to model correctly the action of the operator we must

enlarge our state space. On one hand we have a reactor R in a state on the other handwe have an operator O in a state defined for instance as the direct product of "diagnosticstates" and "stress states" (or levels) i.e.

We define the combined state of the system Reactor plus Operator : as sincethe evolution of is deterministic when is fixed and we define also


to be substituted to (eq. 6.5), etc.

We can use the same formalism as defined in § 6.1. We decompose

since we have three possibilities : transition of the reactor (first term) without change ofstate of the operator, transition of the operator without change of the reactor (second term)or simultaneous transition which has a probability of second order (third term).

We can further decompose where the first term relates to

spontaneous transitions independent of the action of the operator (for instance hardwarefailures) and the second term relates to transitions of the reactor induced by actions of theoperator. Similarly

where change of state of the operator is due in part in reaction to sensors activated by thereactor and, for instance, changes in stress level due to its past errors, etc.

where

The human error modelling involved is quite heavy and it is not our purpose to dwellhere on this subject. What we wanted to show here is that the difficulty of the problem liesnot so much with the computational difficulty of solving this kind of problem (for instanceby Monte Carlo29) as with the definition of an adequate human error model30.

7. SUBDYNAMICS

An event tree may be defined as a collection of paths. Each path is characterized by

the set which means that the dynamics is until the

transition time when it becomes until etc., the probability to complete the path being

and we write for short for the compound trajectory with the vectorof components

Therefore for a given path and a given transition time vector the trajectory isfully deterministic and the corresponding conditional probability density of is


the notation being self-evident. The trajectory in phase space is

If we know the probability density of we obtain the unconditional probability

density

A classical event tree analysis is a pruning of an (hypothetical) complete tree withand, when it is done at all, the choice of an average vector will give the probabilityof a damage : as

where is the characteristic function of the unsafe domain

equation (7.7) reads


Equation (7.5) is formally a solution of the Chapman-Kolmogorov equation, with theunderstanding that we have a continuous set of paths and is probably impossible to

find if we take into account all possible transitions. However this is not any more true ifwe recall (§ 2) that most transitions are not failures but transitions provoked by protection -devices, or operators and that is a distribution peaked around an expected value

If the fluctuations of are large enough, the chronological order of transitioncould change, but in that case it would define another path

We can consider as a solution of the Liouville equation

which is technically a parametric stochastic differential equation where

Can we find an equation which obeys

To answer this question we must first define a restriction operator

and a prolongation operator

with the identity operator and a projection operator. What we want to findout is the equation obeyed by the projection If we define

and

with

and

where and where we do not show the dependence of explicitly. Equation(7.18) is the point of departure of an analysis, which after due "Markovianization"(elimination of the convolution product) leads33,34 to a Fokker-Planck equation

and the diffusion tensor has components

Another way to obtain is to use the first-order Taylor’s development of

in terms of which can be obtained by Peano’s theorem, and which gives

identical results for the case of a Gaussian distribution for Since the Fokker-Planckequation (7.20) is an advection-diffusion equation we can easily surmise the qualitativebehaviour of its solution : the probability cloud centered on the trajectory defined


The problem is technically the problem met in statistical mechanics when we look for"subdynamics", i.e. when unwanted variables are "projected out" (i.e. the Zwanzig projectormethod)31,32.

One obtains33

where is solution of

where the flow vector is

with

and

by will progressively spread out. What is less easy to predict is that the

diffusion tensor will suddenly grow in the vicinity of each transition time and decrease after.An important conclusion of the analysis is that despite this behaviour, the probability density

remains highly singular. Indeed, looking at equations (7.2 and 7.3) the locus ofis a one-parameter (t) family for a two parameters family for and ingeneral a n parameter family after n-1 transitions. This means that the support of thedistribution is an hypersurface of dimension n after n transitions, where n < N inpractice. This is a clue for approximation techniques that need only to operate in a spaceof (much) lower dimension than N.

Extension of this procedure to other parameters is possible. For instance, we considersystem (3.2) as describing transport of isotopes in a geological medium, where is

the probability of finding isotope i at position If the dynamics depends on anumber of uncertain parameters describing the variability of the medium, the ensembleaverage will yield diffusion-like equations. The same conclusion appears if we take intoaccount uncertainties on failure rates or on the dynamics of the reactor transients. Theapplication of any restriction operator blurs the image, i.e. introduces a smearing or adiffusion around the convective solution. The same result obtains also if one aggregatesMarkovian states into macrostates.

8. APPLICATION TO EVENT TREES

8.1. Dynamic event trees

The technique of event tree plays a cardinal role in probabilistic risk assessment studies.The reader can find up to date accounts of their use in PRAs in ref. 4 as well as in thereports NUREG-1150 : "Severe accident risks : an assessment for five US nuclear powerlants". A full analysis of the shortcomings of the current use of event trees as far as thetime variable is concerned has been given by N. Siu3 as well as attempts to remedy theseshortcomings.

One of the acknowledged deficiencies is the existence of many uncertainties.

"The importance of considering uncertainty in the inputs ... goes beyond just accountingfor the inherent imprecision in inputs. Some calculations in PRA involve thresholds, which,depending on whatever or not they are met, can have a large influence on subsequentresults... An example is the treatment of the probability of early containment failure in anAccident Progression Event Tree

J. Helton and R. Breeding35 remark that the treatment of uncertainty involves two types:"(1) uncertainty due to physical variability, which results in many different accidents beingpossible, (2) uncertainty due to lack of knowledge which results in an inability to determinethe properties of individual accidents".

These two categories could be qualified as parametric uncertainties. Indeed the firstcategory involves the uncertainty on rate of failures and the second is often related to thepresence in physical models of parameters which are uncertain and determined by expertopinion. Monte Carlo is often used (or its stratified version : latin hypercube) to treat theseuncertainties. However we should stress that a third category is no less important (and



usually underestimated if not ignored completely). It relates to branching time uncertaintyand generally speaking needs to explicit knowledge of the dynamics. This is particularlytrue of sequences of event trees involving recovery operations with human intervention36,or loss of offsite power (and its subsequent eventual recovery) with the ensuing transientstriggered by the protection devices.

Let us examine for instance a simple example described by the following elementaryAPET : water vapour mixed with fission product gases pours out of reactor vessel (sequencea); HR (Heat Removal) starts before Spray (b) which branches to (b’) : Spray startseventually or to (b") Spray never starts; Spray starts before HR (c) which branches to (c’): HR starts eventually or to (c") : HR never starts. Let and be the cumulateddistribution function of the starting time of HR and Spray respectively with and

the probabilities that HR and S (respectively) ever starts.

Elementary calculations give the time dependent probabilities for each state a,b’,b",c’,c"as displayed on Fig. 3. The asymptotic values for b",a,c" are those obtained withoutconsideration of time distributions. However for b’ the probability that this sequence will

be obtained asymptotically, i.e. is dependent upon the explicit

knowledge of and

Similarly we can obtain the distribution of the physical state vector in accordance withthe method exposed in § 7. A complete exposition of the method of calculation of theunfolding of an event tree, essentially by iterative solution of the system of integralequations given in § 6 is given in ref. 37. If we limit ourselves to sequences a,b,c forsimplicity, i.e. either HR or S can happen, which ever is faster :

with

and i=a,b,c are the respective dynamics which in this case correspond to aninvolving, pressure and temperature of water vapor, isotopic concentrations, etc. In aclassical event tree, even if one takes into account the two options (H before S or S beforeH) the consideration of a single value of could fail to capture the probability of crossinga threshold level, a failure more likely and more sensitive if it involves tails of distributions.

If the knowledge of the can be obtained from the deterministic dynamics, thedistribution and will depend on various factors :

Eq. (8.4) is true in general because the probability that both groups change their state ininterval is of order and not Therefore only one group at a time can changeits state. Although rates of failure may be dependent on most transients do not involvethe extreme values of state variables that justify the dependence. If we associate with thesecond group and if we assume moreover that the passive events due to the failure of thesecond group are uninfluenced by the states of components of the first group, we have


operator actions;state dependent failurestransitions on demand resulting from threshold crossings for control or protectionsystems.

(1)(2)

(3)

The latter case will be examined in § 8.4.

Distributions and in this example, embody the uncertainties on transitiontimes. They could be treated in principle as parametric uncertainties although very seriouscomputational problems do arise as we shall see in part II. If we bar the case (3) oftransitions on demand where uncertainty is limited, cases (1) and (2) can lead to a largeoverlapping of transients. Even if we examine only the order of appearances of transitions,the fact that, for a given order, we could have very different probabilities of damageaccording to the specific value of the transition times. The current use of adjectives inAPET studies (level II event trees) like "early", "late" or "very late" is to say the least, avery crude way of discretizing the time variable.

8.2. Decoupling passive and active events37

Let us assume that we partition the Markovian states in two groups : where labelsthe first group and the second. In the protection domain context, components may changetheir state either because they fail or malfunction, or because the change is triggered by asignal when they act as a protection device. We assume that the first group can have bothtypes of transition and the second only failures. We write therefore

We can write in general

where is a conditional probability density.

Substituting (8.6) into the differential formulation of probabilistic dynamics7

with

and

integrating (8.7) over and summing over gives

preaccident period, the reactor operates in a steady state with and we havea graceful degradation of the system through random failures obtained by solving (8.11).

Once the initiating event of the accident happens at time T, the evolution during theaccident transient can be described by


Since

and therefore the conditional probability density of given is the solutionof

which is now the evolution equation for the first group of (active) components only.Moreover the consideration of time scales introduces further simplifications. During the

8.3. Reduction of the number of Markovian states

One of the objections raised against a classical Markovian analysis of the type (8.11) isthe large number of states growing exponentially with the number of components. The firstpractical approach to the reduction of the number of states was introduced by Papazoglouand Gyftopoulos6 where they show that the consideration of symmetries betweencomponents or groups of components allows exact aggregation. The factor of reduction canbe substantial, but not important enough to allow the consideration of practical problemswith a few dozens of components. Approximate methods can be applied after the firstreduction due to symmetry. When components are independent, e.g. when their failure orrepair rates are independent of the state of all other components, the solution of the generalproblem can be obtained from the state probability of each component. In practice we haveinfluences on these rates due to groups of other components, for instance for failure ratesdue to common cause effects, or for repair or maintenance rates due to repair policiesinvolving priorities between components, etc. The influence graph of the system synthesizesby directed arrows between component nodes the fact that some components influenceothers. Although it allows the solution of system (4.5) in block triangular form it does notby itself reduces significantly the size of O.D.E. systems to be solved. However we canusually partition the state space in "contexts" or groups of states, such that the conditionalprobability of transition of a component is approximately the same for influencingcomponents belonging to the same context. For technical details see refs. 38 and 39 wherethe technique used is similar to the one given in § 8.2.


since on a time scale much shorter than T, the only transitions to be examined are those ofthe first group influenced by In practice the last term of (8.12) may very well benegligible because since protection devices, operator actions and

failures strongly influenced by are much more important.

Let us remark that the influence may be strictly limited to a few states. If we assume atransition is dependent only on the availability of an auxiliary electric source thestates of other components are irrelevant. The number of distinct contexts under which weneed to compute is usually drastically reduced. We may also relate theconditional probability dependence on to the mixed structure of an event tree where weappend to a branch of the event tree an associated fault-tree involving states.

8.4. Transitions upon demand

We will assume in the following, transitions upon deterministic demand. Thereforeintermediate failures, other than those eventually met at setpoint surfaces, are excluded fromthe analysis developed below. The demand is triggered if any one of the setpoints isreached. In general the setpoint corresponds to a surfacedepending usually on a single variable but possibly on more than one. For instanceexcessive power or temperatures, or insufficient coolant flow, define setpoint surfaces.Often, however, they are hyperplanes involving several variables, as in

the case of the overtemperature-overpower reactor trip for Westinghouse Nuclear PowerPlants, defined by a linear combination of Reactor Coolant System pressure, core inlet andoutlet temperatures. We assume that the demand is instantaneous, e.g. that the transitiontime is negligible compared to the characteristic time constants of the dynamic transient


Since we need to calculate expressions like we may represent setpoint

transition by a rate given by a sum of Dirac functions

Therefore

The crossing of the surface will happen at time s, solution of

and we assume here a single crossing. Therefore the Dirac function depends on the singlevariable s and can be expressed by the well-known relation

where with n=l by our assumption.

This allows us 37 to express

where is the time necessary to go in state j from the initial point to the point

of crossing of

Eq. (8.18) means that the probability for the system of not having a transition out of

component state j is a decreasing step function. The number is defined by the fact

that

is the probability of failure of the demand, in the sense that the change of state has notoccurred. These data can in principle be obtained directly from the FMEA of the protectionsystem.

9. SEMI NUMERICAL METHODS

9.1. Discrete dynamic event trees methods (DET)

One of the oldest and most important tools to deal with dynamic reliability problems isthe discrete dynamic event tree method, as developed in various codes like DYLAM40,DETAM41, ADS42 and their subsequent offsprings.

Basically it amounts to discretizing the time variable, e.g. approximating in eq.(6.4) by a staircase function25,43.

We define the rate of transition out of i, if m transitions have alreadyoccurred. Then from (6.6)

A similar analysis is made in ref. 37 where the transition times are defined by the

crossing of set point surfaces (see § 8.4).

The general structure of can be easily obtained : transitions occur at times

the trajectory is etc.


We approximate by

with Therefore

The probability of a given branch is given by where i(n) is the

state reached before the transition.

The most important problem is the choice of the We emphasize the fact that the

are potential transition times but not necessarily actual transition times. Therefore we are

facing the following dilemma: either the grid is dense to have a good approximation

to and we have a combinatorial explosion of the event tree; or the grid is

sparse to avoid this memory problem, but we expect a loss of accuracy due to the poor

agreement of with

It can be shown that for a semi-Markovian reliability problem with no physical variables

the error incurred by the choice of is bounded by

On the other hand in DET methods the transition times are fixed at the start. The choice

made in DYLAM is not identical to (9.6) because the time are regularly spaced (with

decreasing values of as t grows).


where X is the set of working states and which means that P

is substochastic (and exists). Obviously a good criterion for is

In the case of a Monte Carlo method, the generation of a sample of n transition times

will correspond to The stochastic variable

appears in the Kolmogorov-Smirnov non-parametric test44 and for instance

The probability of a sequence up to a branching point is the product of all probabilitiesof the ordered set of transitions in the sequence before the branching point. When do weneed to generate another sequence at the branching point? Two parameters regulate theapparition of new descendant sequences :

(1) the time step (which is also used for the integration of the physical variables).

(2) the probability threshold which defines if a new branching point is generated.

If is the current probability of a sequence, branching point k included, we examine∆ t later if

As such, system (9.10) does not allow for transitions on demand, i.e. transitions inducedby the control or protection devices. A simple modification14 allows such transitions if onesubstitutes


where is the probability later that no transition has occurred. If (9.9) is verifieda subsequence is generated, or more precisely, the physical variables and the initialprobability of the subsequence is stored in a stack to be retrieved when needed. The currentsequence is explored up to and the software goes back to the last branching point andstarts again, the whole search being done by a depth-first algorithm74. The dynamic tree islimited by two characteristics : (1) if the probability of the current sequence is below athreshold it is deleted or grouped with others (2) or a Russian roulette saves 1 sequence outof a given number. The growth of sequences is examined in ref. 45.

The combination of DYLAM with simulation packages extends the possibilities of themethod46.

9.2. Cell-to-cell methods

Cell-to-cell methods were developped by T. Aldemir et al.15,16,47 at Ohio University forapplication to reliability and safety problems, as an outgrowth of a general methoddevelopped by Hsu48. They can be derived from the Chapman-Kolmogorov equations alongthe lines set in § 4.1.

We can rewrite the final system (4.15) in the form

in the matrix of eq. (9.10), where is the probability that the protection device putsthe system in state i if it was in state j, while the physical variables move from cell to cell

Large linear systems of type (9.10) are commonplace in classical Markovian reliability.Objections raised concerning the practical difficulty of writing are not warranted;these matrices are very sparse and can be built as needed from a knowledge of the structure

of the system38 . The stiffness of the matrix due to the large ratio of physical time constantsto components mean time to failure needs a particular treatment49. Some results arediscussed in § 11.2.c. A similar scheme was proposed by Aneziris and Papazoglou50.

9.3. Other methods

We may, as an alternative approach, use the techniques sketched in § 4.2. This approachis a bottom-up method using low-dimensional marginal distributions. A few problems havebeen studied by P.E. Labeau51 who has observed that the fact that many multivariatedistributions may be associated to a given set of marginal d.f., which may lead todifficulties, the main one being the lack of a priori information on the support of thedistribution On the other hand, synthesis methods like (4.19),(4.20) or others canbe valuable to handle Monte Carlo results in a concise manner.

10.1. The transport analogy

Monte Carlo techniques have been much developed in neutron transport theory wheretreatment of shielding problems has shown the necessity to improve the accuracy ofestimates of the transmitted fraction when this fraction is very small. A well documentedand up to date analysis of the Monte Carlo technique in linear transport theory has beengiven recently by I. Lux and L. Koblinger52 and we shall adopt the same notation as wellas refer to its proofs in the sequel below. Among other valuable references we find theolder text of Spanier and Gelbard53.

Let


a point in phase space;

T(P’,P), the transport kernel such that T(P’,P)dP is the probability that a particle havingits last collision with final characteristic, P’ will have its next collision in P;

the collision kernel such that is the mean number of particlesresulting from a collision in and produced with final characteristics in dP’ about P’.

If is the collision density in P, it obeys the integral Boltzmann equation :

where Q(P) is the external source density.

For later use we define the full kernel

A formal analogy exists with reliability if we establish a correspondence between eq.(10.1) and eq. (6.6).

10. THE MONTE CARLO METHOD

is the conditional probability that a "collision", e.g. transition in will lead to a stateThe Dirac functions express the fact that neither nor changes during the

transition.

Finally

which expresses the probability of "transport" along the deterministic trajectory withoutleaving state i.

Since

we have

and

The Neumann development of the solution of the transport equation

with


Let

e.g. the rate of disappearance from state i.

Then

where

is interpreted in neutron transport as the contribution of the successive collisions until theneutron either is absorbed or escapes of the reactor. We can give the same interpretationin reliability37 as the contribution to of the successive transitions until the systemfails or crosses a safety boundary (cfr. (9.1)).

10.2. Sampling methods

The generation of non uniform random variates is the key problem of Monte Carlo. Ahuge literature exists related to this subject The reader is referred to the excellent book ofL. Devroye54 for a thorough "state-of-the-art" report. The most useful methods for dynamicreliability are given below :

(a) The inverse method

Let F be the c.d.f. of a random variable x with inverse defined by

If u is a uniform [0,1] random variable, the has distribution function F.If x has a c.d.f. F then F(x) is uniformly distributed on [0,1].

For example if

We see already that unless the solution of (10.14) may not be readily available.

(b) The rejection method

If the density f(x) is bounded by

for all x, and if it is easier to sample from g(x) than f(x), we can use the followingalgorithm: "generate two independent random variates x (from g) and u (from a uniformdensity on [0,1]) and accept X if

If not, repeat".

Numerous variants exist and the quality of a rejection method depends, among others things,on the expected number of trials to obtain x.

(c) The discrete inversion method

To sample from a discrete distribution where state i (i=1,2,...) has probability wegenerate a uniform [0,1] variate u and set


The search for x can be sequential, e.g. by computing successive values of and could

therefore be time consuming. Improved methods (binary search, guide-tables, etc.) may beused54.

We should at this stage stress the important conceptual difference between the solutionof Markovian problems by the solution of O.D.E. and the solution of the same problems byMonte Carlo. In the first case we have for instance N components, each of which may be

A Monte Carlo method will not select a next state after a transition outof state i, by a sampling of the state from the discrete distribution. Instead if componenthas a total transition rate (failure or repair) the time of transition is selected from

and the component which has failed is identified by algorithm (c) with with the

advantage that now we have generally

However we cannot shed the state concept altogether because the may depend on the

current state of the system which means that the set must be updated after

each transition. In all fairness, the objection raised against conventional Markovian methodsas to the need to store is unwarranted, since it can be computed when needed fromthe logic of the system and its operational rules38. The situation is more complicated innon-Markovian problems.

Let where is the c.d.f. of the sojourn time in its present state

of component when the state of the system is The hazard rate is

Semi-Markovian models can be studied with this type of c.d.f. as well as non-Markovianmodels, provided we enlarge suitably the phase space by addition of new variables. Thishappens for instance when hazard rates are function of the time spent before t in each stateof the system. If we have N independent components, each of which will have its firsttransition at time then


in two states (working or failed), and therefore we have states with a transition matrix


is the c.d.f. of the transition time of the system. It cannot be handled as (10.18) but by thefollowing method :

(d) The composition method54

The algorithm amounts to solve sequentially for

and retaining the minimum value of

Various improvements like the thinning method of Lewis and Shedler54 can be used ifwe know a simple function for all t.

If we examine the C(P”,P’)T(P’,P) obtained in (10.6, 10.8)

all three transitions rates can be treated by one of the algorithms given above, the last onebeing deterministic. We should stress the fact that in dynamic reliability, the computationalburden rests essentially on the evaluation of e.g. the solution of the dynamics ofthe system, a sobering fact if one is tempted to look for refined sampling methods for i andt. This problem will be examined in § 10.5.

10.3. Scoring a Monte Carlo game

Once we select an initial state :

(a) we follow the trajectory in space according to probability law T(P’,P) until we reachthe transition time obtained from the c.d.f.

e.g. we generate a uniformly distributed random variable in [0,1] and solve

(b) if we are in state the next Markovian state will be j if

where again is uniform in [0,1] , and we return to (a).

If we keep track of the number of "particles" in state i, cell k of centered

in then we obtain an estimate of if N is the size of the sample.

This "analogue" Monte Carlo, so called because we follow strictly the analogy of thetransport of a particle, is usually inefficient because variance decreases slowly (as ).

The purpose of a Monte Carlo game is to compute a functional of say

Although we may optimize the estimation of one functional (for instance minimize itsvariance) we cannot do so for more than one, a fortiori for an uncountable infinity of them.We are essentially interested in the unreliability of (or in the damage incurred by) a system.Let us point out however that to know the unreliability of a system with a relative accuracyof say 1 % is considerably more complicated than to know the reliability with the sameaccuracy. Since most industrial systems are (very) reliable most of the sampling time willbe wasted unless one biases the sampling towards interesting events, e.g. those leading toa failure.

The unreliability of a system, where the probability density of state is isgiven by

This expression is obtained from the definition of

from the relation where A is the event "to have left X ininterval (0,t)" and B the event "to be outside D", barring the possibility to leaving X in D

If we choose to evaluate the unreliability at time we obtain from (10.25)

with

and


Let be the expected value of the score when the process starts from point P.Therefore

since the distribution of the starting points is Q(P).

It can be shown52 that obeys an integral equation adjoint to (10.1). Indeed

or

with

and the kernel e.g. eq. (10.32) is the adjoint of eq. (10.1). Theinterpretation is familiar to reactor physicists conversant with the concept of importance55,since here the "importance" of a starting distribution is the expected score. If we modifyf(P) we modify but not kernel L(P,P").

We may now explicit the collision kernel and write

with

where

is the probability that the particle be absorbed in P’ (with any point outside ofand where is a conditional probability.

The score will be f(P,P’) when we have a free flight between P and P’, if there isan absorbtion in P’ and if we have a scattering with transfer from P’ to P".Then I(P) is transformed into

The solution of this integral equation is uniquely determined by Therefore we maychoose our non-negative score functions at will provided we preserve e.g. as long as

Four possible choices, among others, are given in Table 10.1. with



The first has the inconvenience of a possible singularity where The fourth isless complicated than the third and they both avoid the singularity. With the fourth we scoreat each free flight, contrary to the first and second where we score only at the last event, aninconvenience for very reliable systems. The first two are called last-event estimators andthe last two free-flight estimators. For short time compared to MTTF, free-flight estimatorsare more efficient since they score in any state; on the other hand for long times comparedto MTTF, last-event estimators are to be preferred since most histories will stop beforereaching the mission time56.

10.4. Non-analogue games and zero-variance schemes

To assess the quality of a Monte Carlo estimate, it is customary to use a figure of meritor an efficiency factor S given by

Indeed for a given game the variance of an estimate varies asymptotically as the inverse ofthe sample size. Therefore a non-analogue game will be more efficient than another if, fora given sample size the variance is smaller, or for a given variance the needed size issmaller.

A non-analogue game is a Monte Carlo game with T(P,P’) and C(P’,P") altered

respectively to and with the obvious requirement that the expected valueof the score remains unbiased, but with a reduced variance for the same sample size.Statistical weights are attributed to "particles" e.g. in our case to systems transients, in orderto compensate for the altered outcome of the game. For instance if we want to estimate theunreliability at some time we can favor transitions towards failed states - the largernumber of failures reducing the variance - but we have to multiply by a weight W < 1 toconserve the correct expected value. This problem is examined in § 10.5.

A zero-variance scheme is ideally possible and will be discussed below. It means thatall successive histories yield the same (correct) score, which is only possible if we know thisscore in advance... which means that the Monte Carlo game is unnecessary! The importantpractical result, however, is the fact that any approximate knowledge of the functional to beestimated can be used to obtain a scheme with at least a strong reduction of variance.

Let us specialize our results to conventional reliability. The Markovian differentialsystem for is

which gives the integral form

with

The unreliability of a system starting at t=0 in state i is and can be obtainedimmediately as a particular cas of (5.15):

which by differentiation gives

We remark that if we take into account the fact that for system (10.44)is adjoint to system (10.40). It is curious that conventional reliability theory makes little useof (besides and even less of the adjoint character.

Let the unconditional reliability function R(t) of the system be

Therefore

with

is obtained.

Importance sampling52 is an analogue game without the introduction of statistical weights.However the non-analogue kernels are given by


If one chooses the estimator then the transformed game (10.49,50) is

unbiased, e.g. gives the same final expected score (10.24) as the analogue game52.

We have however an arbitrary function U(P), or V(P). If we choose V(P) such that weincrease the probability to meet interesting events (and contributions to the score) we mayexpect to reduce the variance. In fact if the variance will be zero. Sincethe game is unbiased even if it is sufficient to choose V(P) as a bestapproximation of The case for reliability is given in Table 10.2.

We may however proceed in a different way and ask for a zero-variance whatever be P.In other words if is the second moment of the score then a zero-variance method is

defined by asking for all P.

However the source must be modified to

The modifications of the kernels are given in Table 10.3. with the corresponding resultfor reliability20

’27

’52

.

Although the functions are not known, any approximation can be substitutedin the right columns of Tables 10.2. and 10.3. and yield Monte Carlo games with varianceswhich are not zero but generally strongly reduced even for crude approximations (see forinstance § 11.1.). We show now (§ 10.5) that standard biasing techniques are related to thegeneral schemes given hereabove.


10.5. Biasing and integration techniques

10.5.1. State selection

Let be a random variable equal to 1 if the transition from state i reaches state k, andzero otherwise. We want to estimate the expected value of the random variable

where

yields

The weight is constant and it suffices to fix the weight W of the starter as

where are given weights. We partition the state space E into mutually exclusive setsand we use conditioning to write

where is the conditional probability that when a transition occurs in state the nextstate will be in the subset and where is the conditional probability that if thenext state is in J, it will be j.

Therefore

and

If we define a modified Monte Carlo game with probabilities and

with a score then

The expected value is unchanged if We can therefore

choose freely provided

Since we minimize the variance

under constraint (10.62) if we take

with



For the particular case

Nothing prevents us to make the partition of E dependent on the initial state. The biasingof Monte Carlo games for reliability evolution was used by Lewis et al.57 and their choiceamounts to write where R is the set reached after a repair of a component andF the set reached after a failure. Since the objective is to evaluate the rare event (for areliable system) of a system failure they bias the transition towards F, e.g. and

In their case is defined implicitly. A substantial improvement was proposedby Marseguerra and Zio58 where class F was again split in two classes : say C is the set ofstates which is a basic event for one or more cut-sets and NC for the remaining states.Probabilities are chosen in an ad-hoc manner and set C is further splitinto subsets with associated probabilities which are inversely proportional to the "distance"

Although the values chosen in this example for can be considered as an optimalgame for values of chosen as in refs. 57 and 59 we could turn the problem around andset an objective defined by (10.59). A natural example would be to define as theunreliability. If we put for the failed state, then (10.43) can be written

Therefore for a discrete one-step

and we observe that the Monte Carlo game defined above by (10.55)(10.59) relates to(10.66).

Note : N : number of trials; t: computer time (s); U : unreliability; : average numberof transitions per trial; S : figure of merit.

from the current state i to the closest cut-set. This amounts to choosing inverselyproportional to that distance. These heuristics are excellent, as examplified in Table 10.4.by the figure of merit S (see eq. 10.39). The example chosen59 is a ten-component system.

A look at Table 10.2. shows that the modified transition rate is given by

with since

The analogy of (10.67) with (10.65) will be complete if the sets contain only one statei, in which case,

10.5.2. Transition time selection

The choice usually made is

to force a transition before the mission time

The analogy with the zero-variance result is striking if we compare to the same result ofTable 10.2

with the understanding that if but the result is however different, even if oneadopts an approximation

Special problems are met when we have transitions on demand. The discussion so farconcerns standard reliability evaluation, but it is not essentially different for dynamicreliability. The choice made by Labeau20 is

with

and the greater the state has a chance to contribute to failure.


close to 1 and with

10.5.3. Integration techniques

The greatest challenge for dynamic reliability, by far, is the cost of integration of thedynamics (3.1). Let us remember however that accuracy is not essential. Indeed roughscoping in dynamic PRA is certainly a substantial progress compared to methods whichignore dynamic variables altogether. Considering the data uncertainty, a relative accuracyof ten percent, and certainly of a few percents is acceptable. Oddly, this is completely atvariance with the culture of ODE-solvers designers and a degraded standard ODE-solverwith large time steps may not be the optimal solution considering that anyhow the numericalstability criterion should be satisfied. Limited experience shows that a RK 2 solver, withadaptative time step, is satisfactory, but the problem remains open. However substantialimprovements are needed and one possibility is the memorization technique developed byLabeau56. Indeed for reliable systems - which is in general the case for industrial systems -most trajectories in phase space coincide with the most probable trajectory. Memorizationof that trajectory saves effort time if we have the possibility of branching out at any time.

Let be the probability that a system starting in state i has no transition between

the and kth control surface; is the probability of transition at the kth control

surface towards the most probable trajectory. Then

and

are respectively the probability to reach the kth control in the most probable trajectory, andthe probability to have, moreover, a correct functioning of the kth control. Sampling

with the algorithm 10.2 (c) (eq. 10.17) gives the interval where a stochastic transitionmay occur, and forced by a law of type (10.68) on which control device will be out of work.

If we have including a fictitious one for the end of mission control surfaces met

along the most probable trajectory, a proportion

of stories lead to a known score. We can therefore force (see § 10.5.1.) transitions towardsless probable trajectories. This corresponds to one of the rules of Monte Carlo lore, e.g. tosubstitute to a stochastic variable, whenever possible, its expected value. The expected score,

if is the score of the mth modified game (e.g. the unreliability at time and

the score obtained following a most probable trajectory completed up to mission

time, is given by

This amounts to an effective number of histories which is a substantial

improvement for safe systems where


Another possibility is the neural network technique of Marseguerra and Zio59,60. Theyuse a three-layered, feedforward neural network which acts as a nonlinear mapping. Thevalue of connection weights are obtained through a training, obtaining a set of input andoutputs by running the full problem. The choice of the variables is important: for instance,the authors recommend that the output variables should be only a part of the input variables,the rest being obtained by simple algebraic relations. The reported increase of integrationspeed compared to a standard numerical method is 20. However there is a certain numberof questions like the number of networks that should be in principle equal to the number ofdifferent dynamics and the time necessary to identify the right variables compared to thetime of integration for the Monte Carlo trials. The pros and cons of each method arecurrently under investigation. Use of sensitivity coefficients as an alternate way to reducevariance has been examined in ref. 62.

Let us point out as a final remark that a distinction between Monte Carlo and discreteevent simulation (DES) seems to us completely unnecessary (see ref. 3). DES deals withobjects, here components. Whenever the term Monte Carlo is used here it is meant that thesampling is done as in DES. The use of the symbol does not signify that in anactual simulation its explicit knowledge is required.

11.1. Classical Markovian and semi-Markovian problems

None of the references quoted here in this section deal specifically with dynamicreliability. They deal however with some aspects of reliability calculation through MonteCarlo and they have a potential of generalization to dynamic problems.

The pioneer work of Papazoglou and Gyftopoulos6 showed the importance of dealingwith transition rates uncertainties, Monte Carlo has a decisive advantage over other methodsin this respect. The technique used is "double randomization"63 which means that for everynth history a new sampling of the transition rate is made. Optimal choice of n is an openproblem and interesting results have been obtained by Lewis and Bohm57,64. NonMarkovian problems have been treated by Dubi et al.65,66 and Lewins et al.67,68,69. Thetreatment of non-Markovian problems, like those taking into account the age of thecomponents or the time a component has been working, etc. involves generally theintroduction of supplementary variables11,70. The mathematical difficulties associated withnon-Markovian problems show again the unmatched ability of Monte Carlo to deal withthese problems.

"Zero-variance" methods in reliability have been little explored so far. The generalalgorithm of § 10.4 has been applied recently by Delcoux et al.27 to a system of 7components. Results for the variance and the quality factor are given in Fig. 4 and Fig. 5.

Case I corresponds to analogue simulation. Cases II and III correspond to an algorithmwhich minimizes the unreliability variance at time the simulation giving unbiased

estimates for Cases II and III are associated respectively with and asdefined in Table 10.3. In order to average the unreliability, cases IV and V correspond to

a minimization of the variance of with and where


11. EXAMPLES OF NUMERICAL TREATMENT OF PROBLEMS OF DYNAMICRELIABILITY

in Table 10.3 we substitute to It is interesting to note that

the variance has a minimum for h and that improved sampling of the time iscostly and unnecessary.

Attempts to generalize the method to the tank problem are still unconclusive20.

11.2. Dynamic reliability

11.2.a. Fast reactor transient

The first full-scale analysis of a dynamic reliability problem by Monte Carlo was doneby C. Smidts and J. Devooght71 for a reactivity transient in a fast reactor. The EUROPAmodel was the same as the one treated by Amendola and Reina by an earlier version ofDYLAM72. The components were : two channels, one pump, four sensors, three controllers,



a scram logic and a scram actuator leading to states. The dynamicinvolves 12 differential equations. The Monte Carlo simulation was run on a CRAY X-MP/14. The vector structure allows to run histories in parallel. Up to stories could berun in parallel, but the initial group of stories must be divided into a limited number ofsubgroups of reasonably large size that undergo the same treatment. A dozen such groupswas identified, and their size was maintained constant by delaying the calculation until thesize was sufficient to run a batch in parallel.

Biasing techniques were of type discussed in 10.5.1 and 10.5.2. Two categories oftransitions were considered for biasing : failed safe transitions (leading to precocious scramand therefore to success) and pure failures. Ninety percent of the transitions were forcedin the second category.

The time distributions of the success and failure time obtained by Monte Carlo hadrespectively two and one peaks that could be identified as the times necessary to completesequences identified in the classical event tree analysis. One of the most interestingconclusions is the proof that a dependence of a transition rate on the physical variables

can affect the outcome of the calculation, here the failure probability. Fig. 6 shows thevariation of the (hypothetical) failure rate of the scram actuator as function of thetemperature of the sodium channel temperature, the results corresponding to the 4 laws beinggiven in Table 11.1.

The same problem was examined recently by De Luca and Smidts73 from the point ofview of a "zero-variance" sampling. The method used was importance sampling (see Table10.2).


11.2.b. Comparison between Monte Carlo and DYLAM

A comparison was made between DYLAM and Monte Carlo by C. Smidts43. Thecomparison is meaningful if we apply it to a problem for which we have a ready-madesolution available. Analytical benchmarks are not very numerous but multidimensionalproblems can be generated by using the direct product of one dimensional problems. Thesimple example given in § 4.3 was used and the object of the calculation was theprobability that for any Two values were used the systembeing in two states (1,2) with a failure rate and a repair rate

Various errors were examined like for instance the maximum absolute errorwhere are respectively the analytical and

numerical (DYLAM) probabilities.

The comparison between the performances of DYLAM and Monte Carlo is difficult tomake because many parameters are involved in the fine tuning of each method; moreoverthe results are problem-dependent. The author43 reaches the following conclusion for asingle component: DYLAM is superior to Monte Carlo for low failure rates and the reverseis true for high failure rates. However for many components (and certainly for a largenumber) Monte Carlo proves superior if it used with biasing, whatever are the failure rates.The use of failure on demand is different: the number of sequences is limited and in generala DET method is suitable. However Monte Carlo can be easily adapted to such situations75

(see § 8.4) and is not sensitive to loss of events of very low probability. Its accuracy is alsoself- improving with the number of trials, which is not the case of DET methods whereis common to all sequences and cannot be adapted to each state.

11.2.c. The heated holdup tank

One of the most popular problems in dynamic reliability is the heated storage tankproblem treated by Aldemir76, Deoss and Siu77, Marseguerra and Zio78, Tombuyses and


Aldemir14, Cojazzi40, Labeau20, Matsuoka and Kobayashi79. Unfortunately the problem wasnot defined as a benchmark : many variants were used to test algorithms which are notstrictly comparable.

The holdup tank system is described in Fig. 7. Level should be maintained betweenHLA=6 m and HLB=8 m. Failure occurs either by dry-out when the level is below HLV=4m or by overflow when the level exceeds 10 m. Control logic is exercised on two pumps(one normally in standby) and a valve, each component having 4 states. Heat is added tothe liquid at rate W.

The dynamic of the system is

with h the level, the temperature, being the Boolean variable of the ithBcomponent (1 for ON or stuck ON, 0 for OFF or stuck OFF). We have also

and

The value Q is the flow common for the three components, is the inlet temperatureand W a constant proportional to the heat flux. All failure rates have a common temperaturedependence given at Fig. 8. Analytical solution being possible, the accuracy of testalgorithms can be checked. The Monte Carlo analysis of Marseguerra and Zio shows clearlythe impact of the failure (and repair) rates on the probability of crossing the safety boundary.Fig. 9 show the p.d.f. and c.d.f. for dry-out and overflow. The solid line corresponds to areference case with no failure on demand and equal transition rates and the dashed line tothe case in which the failure rates of the components in states opposite to the initial oneshave been increased by a factor of 10 for transitions towards the state "stuck-on" and 100for transitions towards the state "stuck-off". The first case can be treated by classical fault


tree analysis and the effect appears clearly when failure rates are different, as well as whenfailures on demand, are introduced. A solution of eq. (10.14) could be found in this casewithout much difficulty due to the choice made for but this is not necessarily true inall cases.


The same problem was treated by Tombuyses and Aldemir14 by the continuous cell-to-cell mapping method (CCCMT) (§ 4.1). A comparison of the results was made with MonteCarlo converged to 0.15 %. The space is divided in cells when and arerespectively the number of intervals for and Results were compared to the CCMTmethod which is a discrete time Markov process and on the whole the results were favorableto the continuous version. Accuracy may be insufficient if the number of cells is too low(see Fig. 10 and Fig. 11) but the error lies below one percent for

P.E. Labeau has examined the same problem20 (without temperature) to test variousbiasing strategies for Monte Carlo. Since the objective was to run many Monte Carlosimulations, the variable was omitted. Some results are given in Tables 11.2 and 11.3.Two sets of stochastic failure rates (S1) and failures on demands (S2) were used, S2 beingone order of magnitude lower than S1. Free-flight estimators lead to better statisticalaccuracy but have a longer calculation time. For S2 the efficiency factor is in favor of free-flight estimators. Table 11.4 shows the results for four methods; although the computationtime for the fourth (corresponding to the scheme of eq. (10.73)) is higher, the efficiencyfactor defined by is better for this scheme for reliable systems.


,20

The same problem (without temperature) was treated by Cojazzi by DYLAM40 with themain purpose of showing the difference between a dynamic approach and a classical faulttree method, as well as by Matsuoka and Kobayashi by the GO-FLOW method79.

11.2.d. The pressurizer

One of the most complex system treated so far by Monte Carlo (with 11.2.a) is thepressurizer of a PWR treated by Labeau80. The same problem has been modelled by theneural network technique by Marseguerra et al.81 but no Monte Carlo results have beenpublished so far. The model used in ref. 80 is a system of 12 nonlinear differentialequations. Safety boundaries are a lower and upper water level and a lower and upper


pressure. Failure occurs when these boundaries are crossed or when heaters are failed orwhen we have massive water discharge with valve stuck-open. The total number of statesis 900. Mission time was equal to 115 s necessary to reach a steady state after a fewopenings and closing of the relief valve.

Superiority of the biased method with memorization (eq. 10.73) appears clearly at Fig.12 where low-probability events are not even detected by the two other methods. Moreoverit is much more stable (as shown by the constancy of the standard deviation (Fig. 13)). Afairly good estimate of the unreliability is estimated with only histories which is not truefor the other two. Discontinuities in Fig. 14 are due to the occurence of rare events. Table11.4 shows that the single memorization technique and the biasing with memorization havecomparable efficiency factors.

We reach the general conclusion that this sizeable and realistic problem is treated byMonte Carlo with a small computation time (on a RS-6000 workstation) and good relativeaccuracy, the problem being well out of reach of any other methods discussed in this article.

11.2.e. Optimization of the Steam Generator Tube Rupture emergency procedure

An application83,84 of eq. 8.14 has been successfully implemented as a variant of theDET methods described in 9.1, that is free of the discretization assumption of as astaircase function, using eq. 8.18 instead to calculate Although restricted to setpoint


transitions, it has as an important advantage that the treatment of the dynamics has beenmade through an engineering simulator of the same capability as those traditionally used inthe classical safety cases by the main US vendors, as well as those used off-line to setup thesuccess criteria of classical static PSAs. The simulator has been coupled to the schedulerof the DYLAM code in order to be able to follow the dynamics of all the branches of adynamic event tree in an optimum way. The branching is generated every time a setpointfor the demand of a safety system is reached as calculated by the simulator83. This "treesimulation" package also incorporate a software able to execute procedures as if they wereautomatic pilots, potentially inducing transitions based on alarm setpoints for entry intoprocedure and/or procedure actions. Branching at these procedure setpoints allows then tostudy the impact of human errors or system malfunctions while executing procedures.

In Figures 15, 16 and 17 we can see results of this type of tree simulation in the case ofa steam generator tube rupture of a real PWR (see ref. 83, 84 for details). All the maincontrol and protection systems have been included. A set of important safety systems havebeen selected as potential headers of a dynamic event tree, the headers becoming active asa result of the simulated set point crossing branching conditions. In Figures 16 and 17 wecan see the effect of procedure execution including cycling instructions.


With the general implementation of parallel processors this method seems most promisingparticularly because efficient algorithms to handle larger and larger size trees are beingdeveloped.

Another important feature is that the calculation of the probability of each branch can bemade, as the sequence is developed, with exactly the same techniques of Boolean functionsthat are now implemented in classical PSA packages allowing the use of the same faulttrees. Finally, the implementation of Monte Carlo roulettes at the branching setpointconditions both, for the condition itself or for the time distribution of operator actions afteran alarm setpoint is reached, seems very possible indeed, thus removing in practice the main

implications of the restrictions to transitions upon demand and rising new expectations forconvergence with the general Monte Carlo techniques, providing at the same time naturalextensions of classical PSAs.

12.1. Synthesis of available methods of solution

We have shown in this review paper that dynamic reliability can be studied from anunified point of view. Not all problems have been examined, like for instance the optimalcontrol of piecewise deterministic processes12, or diagnostics82.

Fig. 18 is a summary of the methods used in dynamic reliability (except GO-FLOW).The cross at the center of the figure shows a partition of the plane in four regions. Belowthe horizontal line we have differential formulations and above integral formulations. Rightof the vertical line we have direct or forward formulations and left we have adjoint orbackward formulation. Our starting point is the differential forward Chapman-Kolmogorov(DFCK) equation (1) (eq. 3.2). However we have added a random vector parameter

which means that is a conditional distribution (see last paragraph of § 7). Thetransformation of DFCK into its integral form IFCK (2) is made like in eq. (3.9). Howeverwe could also extends IFCK to the semi-Markovian case (see eq. 6.4). The reverse is nottrue, since we cannot in general give a differential formulation from a semi-MarkovianIFCK. We have also the IBCK (2*) counterpart. The four equations in the dotted squareare equivalent formulations in the Markovian case.


12. CONCLUSIONS

If we project out along the lines used in § 7 (subdynamics) we obtain an equation for

which in general may be approximated by a Fokker-Planck

equation (3) (eq. 5.70). If is non existent, then (1) and (3) are identical as well as (2) and(4). The most general technique available for solving (2), hence (1) is Monte Carlo (12).No approximation whatsoever is needed since we sample from the dynamics, (i,t) fromthe semi-Markovian c.d.f. and random vector (for instance the failure rates) from a givendistribution IFCK (4) is the starting point of discrete dynamic event trees methods (6)

like DYLAM, DETAM, ADS, etc. replacing

Integrating (3) (or (1) without ) over cells we obtain (see eq. 4.15) of theCCCMT method (7). Projecting out Markovian states from (1) or (3) we obtain a Liouvilleor a Fokker-Planck equation for (8) (see eq. 4.3).

A particular case of (7), which is exact if is independent of is the MarkovianO.D.E. system (9) which is the backbone of standard reliability theory.

Another possibility is to obtain the marginal distribution (13) (eq. 4.22) and themoments (14) (see eq. 4.16,17) and synthesize the multivariate distribution (15) (eq. 4.20).

The integral backwards equation IBCK (4*) leads to a Monte Carlo evaluation of thereliability (or any damage function). The DBCK (3*) leads to the differential formulationof the generalized reliability function (eq. 5.14) leading also to generalized MTTF functions.

12.2. Prospects

What are the prospects for computational methods in dynamic reliability? Realisticproblems involve m=10 to 100 physical variables with a number n of components involvedin active events of a few dozens (see § 8.2).

These problems are computationaly intensive since they involve systems of a largenumber of partial differential or integral equations. No conventional numerical

method is ever likely to solve these equations since if we admit p intervals for each physicalvariable, we have unknowns. The only method which has the potential to solve suchproblems is the Monte Carlo method. It is rather insensitive to the size of the system, atleast compared to other methods. Indeed the computational burden varies linearly with thenumber of components n (and not with ) and with a low power of N, corresponding tothe task of solving O.D.E. equations (3.1). Some of the examples treated by Monte Carlo(see § 11.2.a, 11.2.d) show that certain problems are already reachable with workstations.The resources of biased Monte Carlo have been barely tapped and idle workstationcomputing power is available almost everywhere. The obstacles to Monte Carlo algorithmsis the fact that they are less scrutable than others on one hand and that they are often foreignto engineering culture on the other hand.

Other methods like discrete dynamic event tree methods or cell-to-cell methods are byno means out of consideration. First of all the first one provides a hands-on approachessential for a physical understanding of a dynamic reliability problem, and with an obviouslink to conventional event tree analysis. Monte Carlo is a last resort method that should beused on a restricted set of problems where dynamics plays an essential role. DET methodscan be used to identify these problems. A second reason for their future use is the fact thata biased Monte Carlo is blind without a rough chart of the domain to be explored and DETas well as cell-to-cell methods may be able to provide these charts. It is likely that in a not-too-distant future dynamic reliability software will incorporate these complementaryapproaches.


ACKNOWLEDGEMENTS

I would like to thank first Prof. C. Smidts, Drs. B. Tombuyses and P.-E. Labeau with whomin order I have enjoyed the fascinating and challenging problems of dynamic reliability andwho have contributed to many interesting solutions.I owe much to Prof. M. Marseguerra’s critical acumen and thanks are extended to Prof. T.Aldemir and A. Mosleh, Drs. J.M. Izquierdo and N. Siu as well as others who knowinglyor not have shaped my ideas on this subject. I am also indebted to Prof. J. Lewins for histhoughtful suggestions on a first draft.Mrs. J. Immers is to be thanked for her tireless efforts to bring the typescript into goodshape.

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INDEX

A-bomb: see survivorsAccident Progression Event Tree (APET), 233Accute doses, 13Adjoint, 56, 248Admissibility conditions, 71ADS code, 239Advanced neutron source (ANS) reactor, 32Aldermaston, 2, 3Analog(ue) Monte Carlo, 40, 247Autocorrelation, 33Auto-power spectral density (APSD), 38

Basisdual, 55reciprocal, 57standard, 54

Bayesian theory, 213Beginning-of-cycle (BOC), 98Biasing (Monte Carlo), 253, 265Biological tank shield, 180Black, Sir Douglas, 1

Advisory Group, 2Boiling water reactor (BWR) fuel management,

95Boltzmann transport equation, 39, 216, 241Branching time uncertainty, 233British Nuclear Fuels (BNFL), 1Brownian motion, 214Burghfield, 2, 3Burkitt’s lymphoma, 17Burnable poison (BP), 94

Caesium contamination, 183Caithness study, 10Causality criteria, 7Cause-and—effect, 7Cell-to-cell method, 219, 241, 264Cf-252 source, 40Change control, 167Chapman–Kolmogorov: see KolmogorovChernobyl control room, 156, 164

Nuclear Power Plant (ChNPP), 170China Syndrome, 175Chronic doses, 13Closure relation, Gaussian, 220Combinatoric problems, 93Committee on Medical Aspects of Radiation

(COMARE), 2, 21Computer architecture, 160Continuous cell-to-cell CC(C)MT: see Cell-to-

cell

Cross-correlation, 33Crossover operators, 113

partially mapped, 124Cross-power spectral density (CPSD), 34, 38,46CWT (continuous wavelet transform), 70

Data management, 158storage, 158

Decontamination, 198Delay times, 46Design base accident, 163DETAM code, 239Detectors, 42Deterministic dynamic event tree (DDET),

268DFCK, 270; see also Chapman–KolmogorovDilation equation, 74, 98Discrete event simulation (DES), 259Dose rate (DR) zones, 184Doses, accute v chronic, 13

doubling, 14Dounreay nuclear establishment, 2, 10Downsampling, 104, 105DYLAM code, 239, 260, 265, 268Dynamic event trees (DET), 239

Earthquake, Romainian (1990), 190Egrement North, 13Emergency response, 162Encoding, 113End-of-cycle (EOC), 95Ensemble, 32Epstein–Barr virus (EBV), 17Ergodic process, 33Erlang distribution, 218Errors, systematic, 7Escape times, 224Esteban and Galland scheme, 109EUROPA model, 260Event tree, continuous, 210

exit problem, 222Expert system, in core design, 118

Factor X, 12–15Fault tolerance, 168Film badge, 4, 11Filtering, 81

FIR-type, 110high-pass, 104low-pass, 103sub-band, 102

279

280 INDEX

Foker–Planck equation, 231FORMOSA (code), 99, 123Fragile gene sites, 20Frame, continuous, 65

operator, 66Francis turbine, 119Free-flight estimator, 249FT (Fourier transform), 58 et seq.Fuel assemblies, 94

containing material (FCM), 174

Gabor, 58, 60Gardner report, 4Gaussian, 53

closure relation, 220Generalised perturbation theory (GPT), 123Generation time, 35, 45Genetic algorithms (GA), 93, 111GO-FLOW method, 265, 270Gordon interpolation, 220Gray coding, 126, 137Great deluge algorithm (GDA), 91, 104

Haar wavelets97, 100Haling’s principle, 97Hamiltonian, 217Harwell, 3Health and Safety Executive (HSE), 11Heavy water, reactor pool, 44, 45Helicopter drops, 176Hereditary component, 16Hermetian conjugate, 56Heuristic rules, 98

HTBX crossover, 119tie-breaking, 120

Hilbert space, 59, 66Hill, Sir Austin Bradford, 7Hölder index, 74

regular, 73, Hope v BNFL, 5Hot particles, 199HP filter: see FilteringHuman error, 212

factors, 157modelling, 218

IBCK 270; see also KolmogorovIFCK 270; see also KolmogorovIFT (inverse Fourier transform), 59Imperial Cancer Research Fund, 3Importance (adjoint), 248

sampling, 251Infection, leukaemia cause of, 21In-out loading, 96International Chernobyl Project (ICP), 171

International Commission on Radio-logical Protection (ICRP), 5, 203

Iodine release 187IWFT (inverse windowed Fourier transform),

63

Jump process, 219

KENO code, 32Kolmogorov equations, 213 et seq.

-Smirnov test, 240

Last-event estimator, 249Latin hypercubes, 212, 233Laurent polynomials, 84, 91Lava-material, 176 et seq.

-like fuel containing materials (LFCM),189

Legasov report, 185Leukeamia incidence, 1 et seq.

projections, 202Linear energy transfer (LET), 18Liouville equation, 216, 230Liquidators, 200LP filter: see Filtering

Maintenance outages, 165Mapping, inverse, 68Mammoth beam, 191Markovian modelling, 213Mean time to failure (MTTF), 225Medical Research Council, UK (MRC), 6Memorization technique, 258, 267Mintzer scheme, 110Monte Carlo calculations, 215, 217Mother wavelet, 53, 69, 93, 97Mouse data, 13–14, 18MRA (multiresolution analysis), 81Multaplicative factors, 13Multi-point approximation (MpA), 128Mutations, minisatellite, 19

National Commission on RadiologicalProtection(Russian) (NCRP), 200

National Radiological Protection Board(UK)(NRPB), 1

Neumann series, 39, 243Neural networks, 258Noise analysis, 37Non-analogue Monte Carlo, 249Non-Hodgkin’s lymphoma, 2NPP (nuclear power plant), 158, 171NUREG-1150 report, 232Nyquist frequency, 60, 70

INDEX 281

Operator support, 161Out-in loading, 96

Parseval’s theorem, 77, 79, 80Peano’s theorem, 232Phase filters, 102Plutonium correlation, 182Point kinetics, 31 et seq.Poisson distribution 38, 41Population-based incremental learning algo-

rithm (PBIL), 125Positive feedback, 174Pressurised water reactor (PWR), core reload,

93Probabalistic risk assessment (PRA), 163

safety assessment (PSA), 210, 268, 269

QMF (quadrature minimum filter), 110

Radiation Effects Research Foundation (Japan),9

Radioactive release assessment, 186Rain algorithm, 105Random variates, 244RBMK-1000 (Russian) reactor, 171Reactivity, positive feed-back, 172Reactor dynamics, 212Reay v BNFL, 5Reclamation, 197–8Record-to-record algorithm (T), 105Reproducing kernel, 64Resolution of unity (identity), 57

continuous, 63Rose review, 9Russian National Medical Dosimetric Registry

(RNMDR), 201

Safety domain, 224Sampling, up and down, 88Sarcophagus, 170 et seq.

predicted collapse, 192water entry, 193–2, 196

Scintillation detectors, 42Seascale, 1Selection schemes, 116Self-sustaining chain reaction (SCR), 174, 194,

207Semi-Markov: see MarkovSimulated annealing (SA), 91, 102Smoking, 8Source transfer function, 36

Spectral densities, 34factorisation, 102

Spencer’s distribution (Cf-252), 41Statistical mechanics, 217Stochastic processes, 214Survivors, atomic bombs, 9, 13, 17Symmetry reduction, 237

Tabu search algorithm, 105Technical basis of nuclear safety (TBNS) report,

194Thinning method, 246Three Mile Island (TMI) control room, 156, 164Threshold crossings, 235Thurso, 2Time delays, 46, 51Top event, 211Training, 165Transfer functions, 31Transition times, 240, 257Travelling salesman problem, 119Trust region approximation, 127Twins, monozygotic, 15

Ukritiye see sarcophagusUnited Kingdom Atomic Energy Authority

(UKAEA), 6United Nations Scientific Committee on the Ef-

fects of Radiation (UNSCEAR), 14, 21Unsafe domain, 230

Variance reduction, 40Vector computing, 261Verification and validation (V&V), 166Volterra equation, 39VVER (Russian) reactor, 171

Water supply, 199Weapons testing, 4WFT (windowed Fourier transform), 58 et seq.

metric operator, 63Wiener process, 214Window, band limited, 76, 78Windscale, 1WT (wavelet transform), 89, 94

Xenon poisoning, 172

Zero-variance schemes, 250 et seq.Zone of rigid control (ZRC), 204Zwanzig projector, 231

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