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KATHOLIEKE UNIVERSITEIT LEUVEN

FACULTEIT INGENIEURSWETENSCHAPPENDEPARTEMENT ELEKTROTECHNIEKAFDELING ELEKTRISCHE ENERGIE ENCOMPUTERARCHITECTURENKasteelpark Arenberg 10, B-3001 Leuven (Heverlee), Belgie

EFFECT OF DISTRIBUTED GENERATION ON

FAULT DETECTION AND RIPPLE CONTROL

Jury: Proefschrift voorgedragen totProf. dr. ir. P. Van Houtte, voorzitter het behalen van het doctoraatProf. dr. ir . R. Belmans, promotor in de ingenieurswetenschappenProf. dr. ir. D. Van Dommelen, promotorProf. dr. ir. J. H. Blom door

(Technische Universiteit Eindhoven, Nederland)Prof. dr. ir. G. Deconinck Pieter VERMEYENProf. dr. ir. W. DhaeseleerProf. dr. ir. J. DriesenProf. dr. N. Hadjsaid

(Institut National Polytechnique de Grenoble,Frankrijk)

Prof. ir. W. L. Kling(Technische Universiteit Delft,Technische Universiteit Eindhoven, Nederland)

Prof. ir. L. van der Sluis(Technische Universiteit Delft, Nederland)

UDC 621.316.31

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c Katholieke Universiteit Leuven - Faculteit IngenieurswetenschappenArenbergkasteel, B-3001 Heverlee (Belgium)

Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/ofopenbaar gemaakt worden door middel van druk, fotokopie, microfilm, elektronischof op welke andere wijze ook, zonder voorafgaande schriftelijke toestemming van deuitgever.

All rights reserved. No part of the publication may be reproduced in any form byprint, photoprint, microfilm or any other means, without written permission from thepublisher.

D/2008/7515/86

ISBN 978-90-5682-976-6

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Summary

The introduction of large amounts of distributed generation (DG) affects the distri-

bution grids normal operation. The effects of DG on the various aspects of gridoperation have to be studied in order to make accurate predictions and appropriategrid designs. In this study, the effects of DG on two aspects of grid operation areanalysed: short-circuit detection and ripple control.

Effect of DG on short-circuit detection

The effect of a local generator on short-circuit detection at the beginning of thegenerators feeder is investigated by means of simulations of a simplified system. Ifthe detected current is lower than the tripping current of the circuit breaker, theprotection schemes of feeders have to be adapted when embedded generators are

installed. By varying the values of the system parameters in a structured way, alarge number of system configurations are covered. Two generator types are used:induction generator and synchronous generator.

Four types of short circuit are considered: three-phase fault, line-to-line fault, doubleline-to-neutral fault and single line-to-neutral fault. The current values produced bythe simulations are analysed by comparing them with two values: the original faultcurrent IOR, i.e. the current detected without a generator, and the required minimumshort-circuit current IMSC. A comparison of the effects of both generator types showsthat the effect of the synchronous generator is the most pronounced.

It is also found that the effect of both generator types on the detected current isstrongest for a single line-to-neutral fault. In case of a single line-to-neutral fault,

a large number of configurations result in a detected current lower than 90 % ofthe minimum short-circuit current. The lowest detected current is 66 % of IMSC.For these configurations, a single line-to-neutral fault may be detected too slowly.The fault current may be interrupted too late, resulting in damage to parts of thepower system. The reduction of the detected current due to a local generator maybe compensated by reducing the relays tripping current, either permanently or bycontinuously adapting it to the state of the generator. With the second possibility,nuisance tripping occurs less frequently than with a fixed tripping current. Therequired reduction of the tripping current can be derived from the simulation results.

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Summary

Effect of DG on ripple control

In order to calculate the propagation of ripple-control signals in a distribution network,the impedances of the network components at the signal frequency are required. Theimpedance at signal frequency of three generator types is calculated: synchronousand induction generator, and power-electronic inverter-interfaced power source.

The calculation of the synchronous machines impedance is derived from the theorydescribing the machines normal operation at mains frequency. As the machinesinductances depend on the magnetization of the machine, the inductances of theequivalent circuits are adapted. Therefore, the magnetic characteristics of the machinevia the effects of saturation and hysteresis are discussed. The adaptation of theparameters is based on the fluxes and the reluctances. Fluxes and reluctances arecalculated by subdividing the machines iron in multiple geometric shapes and byapproximating these shapes by cuboids.

The calculation of the synchronous machines impedance Zrcs shows that, for smallmachines, Xrcs decreases with increasing active power and with decreasing reactivepower. For the larger machines, a reversal of these relations is found when the machineis operating at a fraction of its rated power (< 0.2 p.u.). It is also found that thechoice of the machines iron and the choice of the rated air-gap flux density have acertain effect on the machines impedance.

An adapted version of the equivalent circuit is used to calculate the induction ma-chines impedance. A signal-frequency slip srcs is introduced, frequency dependenceis added to the resistance representing the iron losses, and the skin effect is taken intoaccount to calculate the rotors resistance and leakage inductance. The effect of themachines operating state on Zrcs is small.

The impedance for ripple-control frequencies of inverter-interfaced power sources isinvestigated. The inverter used in the analysis is a voltage-source inverter with PWMcontrol. The results indicate that the impedance of the inverter and the dc circuitas a function of frcs is similar in shape to the ac impedance of the dc circuit. Theimpedance characteristic is shifted towards higher frequencies by 50 Hz. Its magnitudedepends on the inverters modulation ratio and switching frequency. For high valuesof ftri, e.g. higher than 10 kHz, the resonance frequency is higher than 3 kHz, andfor frcs < 3 kHz, Zrcs 0. Therefore, the inverter-interfaced power source discussedin this study can be represented by its filter impedance j Xf.All elements are combined to investigate the effect of DG on the transmission of ripple-control signals in a 15 kV distribution grid. For series coupling, it is found that thesignal voltages in the LV grids connected to the 15 kV grid decrease with increasing

amounts of DG, possibly resulting in detection problems. For parallel coupling, thesignal voltages increase. This increase should be limited, in order to avoid flicker.

The calculations with series coupling indicate that the largest decrease in signal volt-age occurs when synchronous generators are used; up to 42 % for SDG = 50 %. Withinduction generators, the maximum decrease is 40 % and with inverter-interfaced gen-erators 30 %, for SDG = 50 %. The calculations with parallel coupling indicate that,for SDG = 50 %, the largest increase in the maximum voltage occurs when inductiongenerators are used, i.e. 23 %. The increase is 21 % with synchronous generators and14 % with inverter-interfaced generators.

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The calculations can be extended by incorporating load and generation profiles. The

calculation tools developed to conduct the ripple-control calculations, can be used forother signal frequencies as well. If calculation results indicate excessive attenuationor amplification of the received signal voltage due to DG, the use of blocking filters,tuned to the signal frequency, can be required from the owners of the DG units.

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Voorwoord

Mijn jaren bij ELECTA laten een diepe indruk op mij na. Ik heb samengewerkt metaangename collegas, ik heb me in verschillende, boeiende onderwerpen verdiept en

ik heb een stukje van de wereld gezien. Ik heb een aantal jaren in Heverlee kunnenwonen, in de nabijheid van enkele goede vrienden.

Ik heb ook een doctoraat mogen maken. Hiervoor dank ik mijn promotoren prof.R. Belmans en prof. D. Van Dommelen. De gesprekken die ik met hen had en hunvragen en opmerkingen waren zeer nuttig en verhelderend. Ik dank mijn promotorenook voor de grondige en kritische lezing van mijn doctoraatstekst. Ik dank prof.P. Van Houtte, voorzitter van de examencommissie. Ik dank prof. J. Driesen voor degoede samenwerking en de constructieve gesprekken over mijn doctoraat. Ook dankik prof. W. Dhaeseleer, prof G. Deconinck, prof. W. L. Kling, prof. L. van derSluis en prof. J. H. Blom voor hun vragen en opmerkingen en voor het zetelen in deexamencommissie.

I would like to thank prof. N. Hadjsaid for his questions and remarks and for being

a member of the jury.

Ik dank de heer P. Lauwers (Eandis) voor de stimulerende gesprekken over mijnonderzoeksonderwerpen.

Ik dank mijn collegas Reinhilde Dhulst en Vu Van Thong voor de aangename sfeerin bureau 00.56.

Ik dank mijn ouders en mijn zussen Els en An voor alles. Op 21 juni is mijn zus Anmet Filip getrouwd. Ik wens hen het allerbeste.

Pieter Vermeyen

Zwijndrecht, september 2008

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Contents

Summary I

Voorwoord V

Contents VII

List of symbols and abbreviations XIII

Samenvatting XVII

0.1 Effect van DG op detectie van kortsluitingen . . . . . . . . . . . . . . XVII0.1.1 Inleiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII0.1.2 Werkwijze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII0.1.3 Resultaten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX

0.2 Effect van DG op centrale afstandsbediening . . . . . . . . . . . . . . XX0.2.1 Inleiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XX

0.2.2 Synchrone machine . . . . . . . . . . . . . . . . . . . . . . . . . XX0.2.3 Inductiemachine . . . . . . . . . . . . . . . . . . . . . . . . . . XXI0.2.4 Energiebron met invertorkoppeling . . . . . . . . . . . . . . . . XXIII0.2.5 Berekening van het effect van DG op het CAB-signaal . . . . . XXIII

0.3 Suggesties voor verder onderzoek . . . . . . . . . . . . . . . . . . . . . XXIV

1 Introduction 1

1.1 Distributed generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Protection of distribution grids . . . . . . . . . . . . . . . . . . 21.2.2 Ripple control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Protection problems 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Overview of protection problems . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Increased short-circuit currents . . . . . . . . . . . . . . . . . . 72.2.2 Undetected faults . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 Selectivity problems . . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Disconnection of generators . . . . . . . . . . . . . . . . . . . . 92.2.5 Islanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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2.3 Intelligent protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Effect of a local generator on short-circuit detection 11

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 General model of a distribution system . . . . . . . . . . . . . . . . . . 12

3.2.1 Analytical calculation . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Fault types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Short-circuit protection . . . . . . . . . . . . . . . . . . . . . . 163.2.4 Grid configurations . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.5 Rated power of the system components . . . . . . . . . . . . . 183.2.6 Cable impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.7 Feeder length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.8 Transformer parameters . . . . . . . . . . . . . . . . . . . . . . 283.2.9 Generator parameters . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Analysis of simulation results . . . . . . . . . . . . . . . . . . . . . . . 303.4 Feeder with an induction generator . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Three-phase fault . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.2 Line-to-line fault . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4.3 Double line-to-neutral fault . . . . . . . . . . . . . . . . . . . . 373.4.4 Single line-to-neutral fault . . . . . . . . . . . . . . . . . . . . . 383.4.5 Conclusion for the induction generator . . . . . . . . . . . . . . 41

3.5 Feeder with a synchronous generator . . . . . . . . . . . . . . . . . . . 433.5.1 Three-phase fault . . . . . . . . . . . . . . . . . . . . . . . . . . 433.5.2 Line-to-line fault . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.3 Double line-to-neutral fault . . . . . . . . . . . . . . . . . . . . 473.5.4 Single line-to-neutral fault . . . . . . . . . . . . . . . . . . . . . 49

3.5.5 Conclusion for the synchronous generator . . . . . . . . . . . . 513.6 Comparison of induction and synchronous generator . . . . . . . . . . 523.7 Implications for the protection design . . . . . . . . . . . . . . . . . . 543.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Ripple control 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Description of the technology . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.1 Signal injection . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.2 Signal transmission . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Impedance of network components . . . . . . . . . . . . . . . . . . . . 624.3.1 Cable Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.2 Impedance of transformers . . . . . . . . . . . . . . . . . . . . 65

4.3.3 Impedance of loads . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Impedance of synchronous machines 69

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Description of the synchronous machine . . . . . . . . . . . . . . . . . 695.3 Equivalent circuit and parameters . . . . . . . . . . . . . . . . . . . . 73

5.3.1 Equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3.2 Machine parameters . . . . . . . . . . . . . . . . . . . . . . . . 75

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5.4 Calculation of the synchronous machines state . . . . . . . . . . . . . 76

5.5 Calculation of impedance at signal frequency . . . . . . . . . . . . . . 775.6 Effect of saturation and hysteresis . . . . . . . . . . . . . . . . . . . . 825.6.1 Magnetic field strength and induction . . . . . . . . . . . . . . 825.6.2 Initial magnetization curve . . . . . . . . . . . . . . . . . . . . 835.6.3 Steady-state hysteresis loop . . . . . . . . . . . . . . . . . . . . 845.6.4 Open-circuit characteristic . . . . . . . . . . . . . . . . . . . . . 845.6.5 Hysteresis in the rotor iron . . . . . . . . . . . . . . . . . . . . 875.6.6 Minor loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.7 Saturation and hysteresis data . . . . . . . . . . . . . . . . . . . . . . 905.7.1 Rotor iron during normal operation . . . . . . . . . . . . . . . 915.7.2 Stator iron during normal operation . . . . . . . . . . . . . . . 915.7.3 Rotor iron during signal transmission . . . . . . . . . . . . . . 925.7.4 Stator iron during signal transmission . . . . . . . . . . . . . . 93

5.8 Calculation of reluctances and inductions . . . . . . . . . . . . . . . . 955.8.1 Flux paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.8.2 Dimensions of the synchronous generator . . . . . . . . . . . . 985.8.3 Calculation of flux from flux linkage . . . . . . . . . . . . . . . 1005.8.4 Reluctance fd . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.8.5 Reluctance ad . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.8.6 Reluctance aq . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.8.7 Reluctances d and q . . . . . . . . . . . . . . . . . . . . . . 1085.8.8 Reluctances 1d and 1q . . . . . . . . . . . . . . . . . . . . . 109

5.9 Adaptation of impedances of the equivalent circuits . . . . . . . . . . . 1115.9.1 Calculation ofLad,sat by means of the OCC . . . . . . . . . . . 1115.9.2 Inductances of the equivalent circuits . . . . . . . . . . . . . . . 112

5.9.3 Resistances of the equivalent circuits . . . . . . . . . . . . . . . 1135.10 Impedance of the synchronous machine at signal frequency . . . . . . 114

5.10.1 Xrcs as a function of active power . . . . . . . . . . . . . . . . 1145.10.2 Xrcs as a function of reactive power . . . . . . . . . . . . . . . 1155.10.3 Rrcs and Xrcs as a function of frcs . . . . . . . . . . . . . . . . 1175.10.4 Effect of the type of iron and the relation between ad and Bair 117

5.11 C onclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6 Impedance of induction machines 121

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2 Description of the induction machine . . . . . . . . . . . . . . . . . . . 1216.3 Equivalent circuit and parameters . . . . . . . . . . . . . . . . . . . . 123

6.3.1 Transformation factor . . . . . . . . . . . . . . . . . . . . . . . 1236.3.2 Impedances of the equivalent circuit . . . . . . . . . . . . . . . 1246.4 Calculation of the state of the induction machine . . . . . . . . . . . . 125

6.4.1 Calculation with constant impedances . . . . . . . . . . . . . . 1256.4.2 Effect of saturation . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.5 Adaptations to the equivalent circuit . . . . . . . . . . . . . . . . . . . 1286.6 Calculation of reluctances and inductions . . . . . . . . . . . . . . . . 129

6.6.1 Flux paths and dimensions of the induction machine . . . . . . 1306.6.2 Calculation of magnetization . . . . . . . . . . . . . . . . . . . 132

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6.6.3 Reluctance m . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.6.4 Reluctances s and r . . . . . . . . . . . . . . . . . . . . . . 1356.7 Adaptation of impedances of the equivalent circuit . . . . . . . . . . . 1376.7.1 Effect of magnetization . . . . . . . . . . . . . . . . . . . . . . 1376.7.2 Deep-bar effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.8 Impedance of the induction machine at signal frequency . . . . . . . . 1386.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7 Impedance of inverter-interfaced power sources 141

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.2 Description of the system . . . . . . . . . . . . . . . . . . . . . . . . . 1417.3 Discrete-time model of the system . . . . . . . . . . . . . . . . . . . . 1437.4 System parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.4.1 Parameters of the inverter . . . . . . . . . . . . . . . . . . . . . 148

7.4.2 Parameters of the dc circuit . . . . . . . . . . . . . . . . . . . . 1487.4.3 Parameters of the ac circuit . . . . . . . . . . . . . . . . . . . . 149

7.5 Time step and simulation period . . . . . . . . . . . . . . . . . . . . . 1517.5.1 Continuous Fourier transform . . . . . . . . . . . . . . . . . . . 1517.5.2 Fast Fourier transform . . . . . . . . . . . . . . . . . . . . . . . 152

7.6 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8 Effect of distributed generation on ripple control 159

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.2 Model of the power system . . . . . . . . . . . . . . . . . . . . . . . . 1598.3 Variables of the power system . . . . . . . . . . . . . . . . . . . . . . . 162

8.4 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1628.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.5.1 Series coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1648.5.2 Parallel coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 1668.5.3 Possible adaptations . . . . . . . . . . . . . . . . . . . . . . . . 167

8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9 Conclusions 169

9.1 Effect of DG on short-circuit detection . . . . . . . . . . . . . . . . . . 1699.2 Effect of DG on ripple control . . . . . . . . . . . . . . . . . . . . . . . 170

9.2.1 Synchronous machine . . . . . . . . . . . . . . . . . . . . . . . 1709.2.2 Induction machine . . . . . . . . . . . . . . . . . . . . . . . . . 1709.2.3 Inverter-interfaced power source . . . . . . . . . . . . . . . . . 171

9.2.4 Calculation of the effect of DG on ripple control . . . . . . . . 1719.3 Suggestions for future research . . . . . . . . . . . . . . . . . . . . . . 172

A Approximations for Kelvin functions 173

B Synchronous generator 177

B.1 Calculation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . 177B.2 Steady-state operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 178B.3 Magnetic circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

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Bibliography 183

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List of symbols and

abbreviations

Symbols

AC cross-sectional area of a conductor [m2]B magnetic induction or flux density [T]Bp peak induction [T]Br remanent induction [T]C capacitance [F]D distance between conductor axes [m]D geometric mean distance between conductors [m]Dij distance between conductors i and j [m]

Fc correction factor for increased resistancefm mains frequency [Hz]frcs frequency of ripple-control signal [Hz]fr rotor frequency [Hz]fs stator frequency [Hz]ftri frequency of the triangular waveform [Hz]gA air-gap length [m]h harmonic orderH inertia constant of rotor [s]

magnetic field strength [A/m]Hc coercive field strength [A/m]I current [A]

IMSC minimum short-circuit current [A]IN current in neutral conductor [A]IOR originally detected current [A]IR rated current [A]kim transformation ratio of the induction machineKwr winding factor of the rotorKws winding factor of the statorL inductance [H]lax axial length [m]

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List of symbols and abbreviations

F feeder length [m]

LF filter inductance [H]Lii indeterminate self-inductance of conductor i [H]Lij indeterminate mutual inductance of conductors i and j [H]ma amplitude modulation ratiomf frequency modulation ratiomr number of rotor phasesms number of stator phasesnrotor rotor speed [rpm]nsyn synchronous speed of an induction machine [rpm]Na equivalent number of turns of the armature windingNd equivalent number of turns of the field windingNLC,IG number of low-current configurations for the induction generatorNLC,SG number of low-current configurations for the synchronous generator

Pload active power of a load [W]p number of pole pairsQload reactive power of a load [var]R resistance [] reluctance [A turns / Wb]r radius [m]r geometric mean radius of a conductor [m]RF feeder resistance []RP resistance of a passive load []S apparent power [VA]s slip of an induction machinesa factor for stranding of conductors

sb factor for stranding of cablesSagg aggregate apparent power [VA]Sph apparent power per phase [VA]SR rated apparent power [VA]srcs signal-frequency slipSSC short-circuit power [VA]Sstart combined apparent power of a group of motors

with locked rotors [VA]t time [s]Tr number rotor turns per phaseTs number stator turns per phaseUR rated voltage [V]X reactance []

XF feeder reactance []Xii indeterminate self-reactance of conductor i []Xij indeterminate mutual reactance of conductors i and j []XP reactance of a passive load []Xp parallel reactance of an aggregate load []XS self-reactance []Xs series reactance of an aggregate load []yp correction factor for proximity effect

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ys correction factor for skin effect

ZF complex feeder impedance []ZP complex impedance of a passive load []

Greek symbols

20 temperature coefficient for resistance [K1]

i internal rotor angle or load angle [rad]0 permittivity of free space [F/m]r real part of the relative permittivity of a dielectric0 permeability of vacuum (= 4

107) [H/m]

avi average incremental permeability [H/m]dif differential permeability [H/m]inc incremental permeability [H/m]r relative permeabilityrev reversible permeability [H/m] resistivity [m] temperature [C]

angle between d axis and magnetic axis of phase U [rad] magnetomotive force [A turns]load argument of the complex impedance of a load [rad]start resulting argument of the complex impedance of

a group of motors with locked rotors [rad]

flux [Wb] flux linkage [Wb turns]tag total air-gap flux linkage [Wb turns] pulsation [rad/s]r electrical angular rotor velocity [rad/s]rcs pulsation of ripple-control signals [rad/s]rel relative signal pulsation [rad/s]

Indices

base base quantityU phase UV phase VW phase WN neutral conductorG grounda armature of the synchronous machineag air gap

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Samenvatting

Het gebruik van grote hoeveelheden gedistribueerde opwekking (distributed genera-tion of DG) benvloedt de werking van het distributienet. De effecten van DG op

de verschillende aspecten van het distributienet moeten onderzocht worden om hetgedrag van het net te kunnen voorspellen en om goede ontwerpen te kunnen maken.In deze studie wordt het effect van DG op twee aspecten van de werking van het netbestudeerd: detectie van kortsluitingen en centrale afstandsbediening.

0.1 Effect van DG op detectie van kortsluitingen

0.1.1 Inleiding

Het gebruik van generatoren in radiale distributienetten kan beveiligingsproblemenveroorzaken. De noodzaak om de netbeveiliging aan te passen zou een hinderpaal

kunnen vormen voor het gebruik van een grote capaciteit aan gedistribueerde op-wekking in het net. Een intelligent en flexibel beveiligingssysteem, bruikbaar in eenwillekeurig distributienet en met de mogelijkheid zich aan te passen aan veranderendeomstandigheden, zou een belangrijk hulpmiddel kunnen zijn om een verdere toenamevan het gebruik van DG mogelijk te maken.

Wanneer enkel gevalsstudies worden uitgevoerd, blijft het ontwerp van beveiligingssys-temen voor netten met gedistribueerde opwekking een ad-hocproces. Besluiten overbeveiligingsproblemen in een specifiek net dragen weinig bij tot de ontwikkeling vaneen algemene aanpak van problemen die ontstaan door DG. Een systematische werk-wijze is de aangewezen manier om een algemeen toepasbaar beveiligingssysteem teontwikkelen. In deze studie wordt een voorstel voor een meer algemene werkwijzevoorgesteld.

0.1.2 Werkwijze

Door middel van simulaties met een eenvoudig systeem (Figuur 1) wordt het effect vaneen lokale generator op de kortsluitdetectie aan het begin van een tak of feeder van hetdistributienet onderzocht. Als de gedetecteerde stroom tijdens een kortsluiting lageris dan de aanspreekstroom van het beveiligingsrelais, moet het beveiligingssysteemaangepast worden. Door op systematische wijze de parameters van het gesimuleerde

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Samenvatting

1

net

2 3 54

A generator fout

T

12kV/400 V

Figure 1: Eenvoudig netwerk waarmee gesimuleerd wordt: een MS-net, een transformator,een stroommeting A, een LS-tak, een generator en een kortsluiting op het einde van de tak.

stroom

tijd

IMSC

IOR

Figure 2: Karakteristiek van een beveiligingsrelais (type: 50/51). Stippellijn: invers verloop,

volle lijn: vaste uitschakeltijd.

net te wijzigen, wordt een groot aantal netconfiguraties beschouwd. De gebruikteparameters zijn het kortsluitvermogen SSC van het 12kV-net, de sectie AC van degeleiders (in mm2), de plaats van de generator in de tak, het relatieve vermogen van degenerator, en de minimale kortsluitstroom IMSC. Om bij een kortsluiting een snelleonderbreking van de stroom te hebben, moet IMSC in de uitschakelkarakteristiek vanhet beveiligingsrelais in het gebied met vaste uitschakeltijd gelegen zijn (Figuur 2).Op basis van IMSC wordt voor elke combinatie van parameterwaarden de lengte vande tak bepaald.

Twee generatortypes worden gebruikt: de inductiegenerator (of asynchrone genera-tor) en de synchrone generator. Vier kortsluittypes worden beschouwd: de driefasigekortsluiting, de tweefasige kortsluiting, de kortsluiting tussen twee fasen en de nul-geleider, en de kortsluiting tussen een fase en de nulgeleider. De stroomwaardendie voortkomen uit de simulaties worden geanalyseerd door deze te vergelijken mettwee waarden: de oorspronkelijke foutstroom IOR, namelijk de stroom die tijdenseen kortsluiting gedetecteerd wordt bij afwezigheid van een lokale generator, en deopgelegde minimale kortsluitstroom IMSC. De vergelijking van de effecten van beidegeneratortypes wijst uit dat de invloed van de synchrone generator het sterkst is.

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0.1 Effect van DG op detectie van kortsluitingen

1000

20

500

10

200

5

100

2

50

16IR

100%knoop1

300

240

185

150

120

95

70

50

35

25

16

10AC

SSC

0.1s

0.05s

0.2s

Figure 3: Voorstelling van de situaties met lage stroom (I < 90 % van IMS C), voor dekortsluiting tussen twee fasen en de nulgeleider, met IMS C = 6IR en een synchrone generatormet een relatief vermogen van 100 % in knoop 1.

0.1.3 Resultaten

De benvloeding van de gedetecteerde stroom door de lokale generator wordt in kaart

gebracht met behulp van roosters waarvan de kolommen overeenstemmen met de mo-gelijke waarden van AC en de rijen met de mogelijke waarden van SSC (bijvoorbeeldFiguur 3). De resultaten geven aan dat een inductie- of een synchrone generator geendetectieprobleem veroorzaakt in geval van een driefasige kortsluiting, een tweefasigekortsluiting of een kortsluiting tussen twee fasen en de nulgeleider. Het effect vanbeide generatortypes is het sterkst bij een kortsluiting tussen een fase en de nulgelei-der. Bij een kortsluiting tussen een fase en de nulgeleider resulteert een groot aantalconfiguraties in een gedetecteerde stroom die lager is dan 90 % van de minimale kort-sluitstroom IMSC. De laagste gedetecteerde stroom is 66 % van IMSC. Voor dezeconfiguraties is het mogelijk dat een kortsluiting tussen een fase en de nulgeleider tetraag gedetecteerd wordt. De foutstroom zal te laat onderbroken worden, waardoorbeschadiging van bepaalde onderdelen van het net kan optreden.

De afname van de gedetecteerde stroom door de aanwezigheid van een generator kangecompenseerd worden door de aanspreekstroom van het relais te verlagen. Men zouvoor een permanente aanpassing kunnen kiezen, of voor een aanpassing die voort-durend wordt bijgesteld, afhankelijk van de werkingstoestand van de generator. Metdeze tweede mogelijkheid zal ongewenste of onnodige uitschakeling minder voorvallendan met een vaste aanspreekstroom. De nodige vermindering van de aanspreekstroomkan afgeleid worden van de simulatieresultaten.

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0.2 Effect van DG op centrale afstandsbediening

daxis

q axis1

4

2

5

3

6 8

7

Figure 4: Doorsnede van een vierpolige synchrone machine met uitspringende polen. 1) sta-tor, 2) statorijzer, 3) statorwikkeling, 4) rotor, 5) rotorijzer, 6) veldwikkeling, 7) luchtspleet,8) fluxlijn.

magnetische eigenschappen van het ijzer en het effect van verzadiging en hysteresisbesproken. De aanpassing van de parameters is gebaseerd op de magnetische fluxendoor het ijzer en de magnetische reluctantie van de paden van de flux. De fluxen

en reluctanties worden berekend door het ijzer van de machine onder te verdelenin meerdere geometrische vormen en deze vormen te benaderen door balkvormigelichamen. Deze reluctanties vormen magnetische kringen waarmee de reluctanties bijsignaalfrequentie berekend worden (bijvoorbeeld Figuur 5).

De berekeningen van de signaalimpedantie Zrcs geven aan dat voor kleine machinesXrcs afneemt met toenemende injectie van actief vermogen en met afnemende injectievan reactief vermogen. Bij de grotere machines wordt voor een laag genjecteerd ver-mogen (< 0.2 p.u.) een omgekeerd verband vastgesteld. Ook blijkt dat de keuzes vanhet ijzer van de machine en van de nominale fluxdichtheid (inductie) in de luchtspleeteen zeker effect hebben op de signaalimpedantie van de machine.

0.2.3 Inductiemachine

Figuur 6 is een dwarsdoorsnede van een inductiemachine. De berekening van designaalimpedantie van de inductiemachine is gelijkaardig aan de berekening voor desynchrone machine. De regimetoestand van de inductiemachine wordt op iteratievewijze berekend. Een poging om verzadiging van Lm in deze berekening te verwerken,resulteerde in een divergerende berekening. Daarom wordt een veranderlijke waardevoor Lm bij normale werking niet in rekening gebracht.

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Samenvatting

RFF

RIJ

RIJ

RT q

RT q

RCD

RCC

RCD

FlqFaq

Fld

Fld

Fad

Fad

Qaq,rcs

Faq,rcs

Figure 5: Magnetische kring die overeenstemt met aq van de synchrone machine.

1

4

2

5

3

8

7

6

Figure 6: Doorsnede van een vierpolige inductiemachine. 1) stator, 2) statorijzer, 3) sta-torwikkeling, 4) rotor, 5) rotorijzer, 6) rotorstaaf, 7) luchtspleet, 8) fluxlijn.

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0.2 Effect van DG op centrale afstandsbediening

LfLdc

Cdc

Rdc

Udc

Urcs Ugridu+ v+ w+

u- v- w-

Figure 7: De invertor (bestaande uit schakelaars u+, v+, w+, u-, v- en w-) verbindt eendc-bron met een driefasig net. Het net bevat een CAB-signaalbron Urcs.

Een aangepaste versie van het equivalent schema van de inductiemachine wordt ge-bruikt om de signaalimpedantie te berekenen. Hiervoor wordt een signaalfrequen-tieslip srcs gebruikt. De weerstand waarmee de ijzerverliezen in rekening wordengebracht, wordt frequentieafhankelijk gemaakt. Het skineffect wordt in rekening ge-bracht bij de berekening van de weerstand en lekinductantie van de rotor. De aan-passing van de inductanties van het equivalent schema is gebaseerd op de fluxen enreluctanties van de magnetische kringen. Uit de resultaten blijkt dat het effect vande werkingstoestand van de machine op Zrcs klein is.

0.2.4 Energiebron met invertorkoppeling

De signaalimpedantie van energiebronnen met invertorkoppeling wordt bestudeerd.De invertor die hier bestudeerd wordt, is een spanningsbroninvertor met een regelingop basis van pulsbreedtemodulatie. Volgende elementen worden in rekening gebracht(Figuur 7): een inductief ac-filter, de invertor en de dc-kring, bestaande uit een LC-filter en een gelijkspanningsbron Udc voorzien van een inwendige weerstand Rdc.

De resultaten geven aan dat de impedantie van de combinatie van de invertor ende dc-kring in functie van frcs een gelijkaardig verloop heeft als de ac-impedantievan de dc-kring. De impedantiekarakteristiek is verschoven naar hoger frequentiesover een frequentieafstand van 50 Hz. De grootte van de impedantie hangt af van demodulatieverhouding en de schakelfrequentie ftri van de invertor. Voor hoge waardenvan ftri, bv. > 10 kHz, is de resonantiefrequentie hoger dan 3000 Hz. Voor frcs 1000

167 206 300 400 500 725 1350

168 210 316 410 582 735

175 217 317 420 750

180 228 383 425

183 232 390 430

190 270 396 435

194 283 483

198 287 485

291 492

297 494

frequency generator, the signals are converted into waveforms and injected into thegrid. The waveforms are detected by receivers and decoded. In order to have goodreception of the signal, its magnitude has to be at least 0.5 % of the supply voltage,according to information from a Belgian distribution system operator.

The duration of a transmission lies somewhere between 30 s and 3 min. It consistsof 10 to 60 pulses. The duration of a pulse lies between 0.1 and 8 s [55] (p. 63). Thevoltage of the signal injected into the grid by the transmitter is about 1 to 5 % ofthe distribution grids rated voltage [54] (p. 85). The rated power of a transmitter isabout 0.1 to 0.5 % of the load of the local distribution grid. This corresponds to apower of about 10 to 100 kW [55] (p. 60).

The signals are three-phase, symmetrical voltages. Because of this, they are notblocked by transformers. The signal voltage is transferred to the other side, in thesame way as the mains voltage. As the signal frequency is higher than the mains fre-quency, the transformers series impedance due to leakage inductance is proportionallyhigher. This results in a higher voltage drop at signal frequency.

4.2.1 Signal injection

Two main techniques for injecting ripple-control signals exist: series and parallelcoupling. Both techniques can be used to inject ripple-control signals at every voltagelevel.

Series (or inductive) coupling is achieved by means of an injection transformer ofwhich the secondary winding is connected in series with the secondary winding ofthe distribution transformer (Figure 4.2.a). The signal to be transmitted is appliedat the primary winding of the injection transformer. The current by which power isexchanged between the main grid and the local distribution grid passes the secondarywinding of the injection transformer.

Parallel (or capacitive) coupling is achieved by means of connecting the signal sourcein parallel with the three phases via injection capacitors (Figure 4.2.b). Usually thiscoupling system is connected at the bus bars of the substation. As the injection circuit

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Ripple control

the distribution network. If three-phase signal injection is used, the impedance of

the return path is zero. The entire circuit acts as a voltage divider. The voltageat the terminals of the distribution network is the useful signal. The voltage acrossthe distribution transformer and the HV grid is considered a voltage loss. Signalvoltages appearing in the HV grid are transferred to nearby distribution grids. Thesignal voltage appearing in neighboring distribution grids is considered a disturbance(spill-over).

At a higher signal frequency, reactances increase. As load impedances have a largeresistive component, the voltage drop across the HV grid and the transformer in-creases. The useful signal voltage in the distribution system decreases. Thereforeseries coupling is better suited for lower signal frequencies, e.g. frcs < 228 Hz [54](p. 95).

In the equivalent circuit for parallel coupling (Figure 4.2.b), the parallel signal source

injects the signal into the distribution network and the series connection of theimpedances of the HV grid and the distribution transformer. The current gener-ated by the signal source is divided between both circuits. The part flowing throughthe transformer and the HV grid is considered a loss and causes spill-over to neigh-boring networks. At a higher signal frequency, the reactances of the HV grid andthe transformer increase, resulting in a lower current loss. Consequently parallel cou-pling is better suited for higher signal frequencies. At higher frequencies, the effectof capacitance in the network increases (e.g. capacitance of cables, capacitors forcompensation).

4.3 Impedance of network components

In order to calculate the signal voltages at the nodes of the distribution network,the following elements of the network are represented by their equivalent circuit:cables, transformers, loads and generators. These circuits are combined as shown inFigure 4.3. The impedances of load L and generator G are connected to node N viatransformer T. RL, L

L, R

G and L

G are the equivalent resistances and inductances

of the load and the generator, referred to MV. ULV is the secondary voltage of thetransformer, referred to MV. The calculation of impedances of cables, transformersand loads is discussed in the following sections. The calculation of the impedance ofgenerators is discussed in the following chapters.

4.3.1 Cable Parameters

The impedances of the network have to be calculated for the signal frequency. In [66]detailed equations for the calculation of cable impedances at higher frequencies aregiven. The resistance increases with the frequency while the inductance decreases. In[67] a technique is described for estimating the impedance of the power supply as afunction of frequency. A voltage transient with a duration of 160 ms is injected inthe grid by means of an inverter. The resulting transient current is correlated withthe injected voltage to determine the frequency-dependent impedance. The compleximpedance is determined for frequencies between 0 and 1 kHz, with a resolution of

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R1

R2

L1

L2

ULV

C1

C1

C2

C2

2 2 2 2

RT

LT

RG

RL

LG

LL

N BA

Figure 4.3: Node N of the distribution network, connected to nodes A and B via circuitsrepresenting cables 1 and 2. The load L and generator G are represented by their equivalentimpedance. They are connected to node N via transformer T. U

LV

is the secondary voltageof the transformer, referred to MV.

6.25 Hz. Interpolation is applied between the calculated points. As the distributionnetwork and the connected systems have multiple possible states, the impedance asseen from a certain point in the network is not a fixed value, but a locus [64]. Foreach frequency a different locus for the impedance exists.

In this study, the effect of embedded generators on the transmission of ripple-controlsignals is investigated for a medium-voltage (15 kV) distribution network. The con-nections in this network are cables. The parameters of the cables are calculated usingdata1 for 20 kV cables with cross-linked polyethylene (XLPE). The network containsaluminium cables, with AC = 50, 95, 150, 240 or 400 mm2, and copper cables, with

AC = 25, 35, 50 or 95 mm2. For AC 95 mm2, a three-core cable is used. For largervalues of AC, three single-core cables are used, in a triangular arrangement.

-equivalent circuit

In [64] it is recommended to use -equivalent circuits to represent cables and lines forcalculations involving harmonic components. This can be applied to interharmonicfrequencies as well. The classic -equivalent circuit consists of a series circuit of theresistance R and an inductance L, combined with two shunt capacitors, one at eachend of the series circuit (Figure 4.4). Both capacitors have the same capacitance, i.e.half of the lines capacitance C.

Capacitance

The capacitance of a shielded power cable (F/m) is [44] (p. 179):

C =2 0 r

ln(b/a)(4.1)

1XLPE Cable Systems, Users guide, ABB

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J0 xj = ber x +j bei x (4.7)

J0

xj

1j

= ber x +j bei x (4.8)

Using these expressions, with x = m r, Zin can be rewritten as:

Zin = m

2 r j ber(m r) +j bei(m r)

ber(m r) +j bei(m r) (4.9)

=m 2

r ber(m r) +j bei(m r)

bei(m

r)

j

ber(m

r)

(4.10)

Zin can be split in a real and an imaginary part: resistance R and internal reactanceXin (4.11 to 4.13), both functions of .

Zin = R +jXint (4.11)

R =m 2 r

ber(m r) bei(m r) bei(m r) ber(m r)(ber(m r))2 + (bei(m r))2 (4.12)

Xin =m 2 r

ber(m r) ber(m r) + bei(m r) bei(m r)(ber(m r))2 + (bei(m r))2 (4.13)

In order to evaluate the functions for R and Xin, approximative equations are used.In [69] (p. 384-385) polynomial approximations are given for the following functions:ber x, bei x, ber x and bei x. The equations are listed in Appendix A.

4.3.2 Impedance of transformers

Transformers in the distribution grid are represented by their short-circuit impedance,i.e. the series circuit of a resistance Rk and a positive reactance Xk. The transformersparameters at mains frequency are calculated by interpolation with data provided in[51] (p. 234), consisting of average transformer impedances as a function of ratedpower and voltage. For calculations involving harmonic components, a harmonic

model of the transformer is used [64], [60]. In this study, the model presented in [64]is used:

ZT(h) = Rk

a0 + a1 hb + a2 h2

+jXk h (4.14)h is the harmonic order. It is assumed that 4.14 is valid for interharmonic frequenciesas well. h is replaced by the ratio frcs/fm. fm is the mains frequency. Typicalvalues for the parameters a0, a1, a2 and b are given in Table 4.2. It is required thata0 + a1 + a2 = 1.

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Ripple control

Table 4.2: Parameters for (4.14), for the calculation of the harmonic impedance of trans-

formers [64].

Small system transformers Large system transformers

a0 = 0.85 0.90 a0 = 0.75 0.80a1 = 0.05 0.08 a1 = 0.10 0.13a2 = 0.05 0.08 a2 = 0.10 0.13

b = 0.9 1.4 b = 0.9 1.4

4.3.3 Impedance of loads

Passive loads in LV distribution grids are typically domestic loads. These loads arean important element of the network impedance. They represent the main dampingcomponent and may influence the resonance conditions, particularly at higher fre-quencies [60]. They can be represented by the series circuit of a resistance and areactance [64]:

ZP(h) = RP

h +jXP h (4.15)

RP is the load resistance and XP the load reactance at the fundamental frequency.RP and XP are derived from the loads active and reactive power, respectively. Theharmonic order is represented by h. As for the transformers, h is replaced by frcs/fm.In [60] three additional models for aggregate loads are presented. The first model is

shown in Figure 4.5.a. It has been derived by the CIGRE2

, based on measurementsin different MV distribution systems in France. It consists of the series circuit of areactance Xs and a resistor R, connected in parallel with a reactance Xp:

R =U2R

Pload(4.16)

Xs = 0.073 h R (4.17)Xp =

h R6.7 tan 0.74 (4.18)

tan =QloadPload

(4.19)

UR is the rated voltage, Pload is the minimum active power of the load (at the fun-damental frequency) and Qload is the reactive power of the load. This equivalentcircuit can be used to represent loads in the frequency range from the 5th to the 20th

harmonic, approximately [60]. The circuits in Figures 4.5.b and 4.5.c are alternativeload models. The impedances R and X in the parallel circuit in Figure 4.5.b arecalculated as:

2CIGRE: International Council on Large Electric Systems - Conseil International des GrandsReseaux Electriques

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4.4 Conclusions

R

Xs

Xp R X

R

X

a) b) c)

Figure 4.5: Equivalent circuits for aggregate loads: a) empirical CIGRE model, b) parallelcircuit for loads in general and c) series circuit for motors [60].

X = h U2R

QloadR = U

2R

Pload(4.20)

The series circuit in Figure 4.5.c can be used to represent motor loads. The impedancesR and X are calculated as:

X =U2R

SstartR =

h

X

3(4.21)

Sstart is the motors combined apparent power, corresponding to the situation withlocked rotors. For the calculation of R the factor

h is used to include the skin

effect. The factor X/3 is used to have a cos start = 0.32. As the loads in the 15 kVdistribution system are aggregated loads, the model for the motor load is not used in

this study.

4.4 Conclusions

A general description of ripple-control is provided, including the main injection tech-niques: series and parallel coupling. In order to calculate the propagation of ripple-control signals in a distribution network, the impedances of the network componentsat the signal frequency are required. Equations for the impedance of cables, trans-formers and loads at the signal frequency are provided. In chapters 5 to 7, impedanceof generators is determined. Three generator types are discussed: synchronous and in-duction generator, and an inverter-interfaced power source. In chapter 8, all elements

are combined to investigate the effect of DG on the transmission of ripple-controlsignals.

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Impedance of synchronous machines

Steady-statemagnetizationdata

1 steadystatest

Machinesdimensions

Fundamentalparameters

Open-circuitcharacteristic

AdaptationofLad

Fluxlinkages Yi

Fluxes Fi

Inductions Bi

Equivalentcircuit

P Qand

2 steadystatend

Numberofturnsinwindings

Standardparameters

Figure 5.1: Flowchart for the calculation of the synchronous machines steady state. Solid

boxes represent calculation steps, dashed boxes represent data. After calculation of the 1

st

steady state, Lad is adapted and the steady state is calculated again, more accurately.

n0 = 60 fmp

[rpm] (5.1)

In Figure 5.3 the cross section of a four-pole synchronous machine with salient polesis shown. The main parts of the machine are the stator, consisting of laminatediron and a distributed winding, the rotor, consisting of laminated or solid iron and anexcitation winding or permanent magnets and possibly a damper cage, and the air gapbetween stator and rotor. The inner surface of the stator consists of teeth and slots.Inside the slots are conductors which make up the stator windings. These windingsare also referred to as armature [70] (p. 46). The windings form a three-phase circuit,mostly Y-connected to the mains voltage at the stator terminals.

The rotor has salient poles or it is cylindrical. The principal winding of the rotor isthe field winding [70] (p. 47). The rotor in Figure 5.3 consists of four salient poles.Each pole is equipped with a part of the field winding. In machines with a cylindricalrotor, the conductors of the field winding are situated in slots along part of the surfaceof the rotor. The discussion is limited to synchronous generators with a field winding;generators with permanent magnets are left outside of consideration.

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5.2 Description of the synchronous machine

Magnetizationdata:minorloops

Machinesdimensions

Fundamentalparameters

Adaptationofinductances Li

Reactances Xi

Machinesimpedance

atsignalfrequency

Inductions Bi

Adaptedequivalentcircuit

AdaptationofresistancesRi

Signalfrequency

Figure 5.2: Flowchart for the calculation of the synchronous machines impedance at signalfrequency. Solid boxes represent calculation steps, dashed boxes data and results from thecalculation of the steady state.

daxis

q axis1

4

2

5

3

6 8

7

Figure 5.3: Cross section of a four-pole synchronous generator with salient poles. 1) statoror armature, 2) stator iron, 3) armature winding, 4) rotor, 5) rotor iron, 6) field winding, 7)air gap, 8) flux line (based on [70] p. 46 and [71] p. 1).

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Rotors of synchronous machines usually have damper windings [70] (p. 47). Their

function is to damp out speed oscillations. In salient-pole rotors, damper windingsconsist of conducting bars embedded in the pole face, short-circuited at both rotorends, similar to the squirrel-cage rotor of induction machines. In cylindrical rotors,used for high speeds, the rotor iron is mostly solid steel. The damping effect isprovided by eddy current in the rotor steel and by the conducting wedges fixingthe conductors in the slots. Additional damping can be obtained by adding otherconductive parts.

The space between pole surface (or pole face) and stator surface is the air gap. Theproperties of a synchronous machine are determined to a large extent by the length ofthe air gap [72]. Due to slots at the surface of the stator (and the rotor, in case of acylindrical rotor) the effective air gap is larger than the geometric gap. In synchronousmachines the air gap is much larger than in equivalent induction machines [73] (p. 57).

Synchronous machines with multiple pole pairs can be described electrically by meansof an equivalent machine with one pole pair (Figure 5.4). The windings are representedas coils. In the actual machine, the spatial angles between the circuits are referredto as mechanical angles. In the four-pole machine in Figure 5.3 the angle between tosubsequent poles is /2 radians. In the electrical representation, all north poles andall south poles are represented by one north pole and one south pole, respectively. Asa result, the angle between the poles becomes radians. In general, all mechanicalangles are multiplied by p in the electrical representation.

In Figure 5.4 the three outer coils are the armature windings. They are Y-connected.Four rotor circuits are shown: the field winding connected to the voltage source efdand three short-circuited coils, i.e. the damper windings. Two axes are associatedwith the rotor: the d-axis, in the direction of the field windings magnetomotive force2

or MMF, and the q-axis, leading the d-axis by /2 electrical radians. These axes areused in the mathematical description of the synchronous machine. r is the electricalangular velocity of the rotor and is the angle between the d-axis and the magneticaxis of phase U of the armature windings:

= r t + 0 (5.2)

MMF is produced by each of the three phases U, V and W of the armature winding.The MMF of each phase is stationary in space and alternating in time. The electricalangles between the phases of the armature winding are 2/3 radians. The phasedifferences between the applied voltages eU, eV and eW and between the resultingcurrents iU, iV and iW are 2/3 radians as well. As a result, the combination of the

three stationary magnetomotive forces results in a MMF that is constant in value androtating inside the machine, if space harmonics are neglected. A dc voltage is appliedto the rotors field winding. This results in a dc current in the field winding and,associated with the current, a constant MMF. The interaction between the MMFsof stator and rotor results in the electromagnetic torque. In case of a generator,mechanical power is transferred from the prime mover (i.e. the energy source) to therotor via the shaft. The mechanical power is then transformed into electromagnetic

2Magnetomotive force is the general term for a cause or source of magnetic flux, such as a currentdistribution or a permanent magnet. The unit of MMF is ampere turns.

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5.3 Equivalent circuit and parameters

U

W

ViV

iU

iW

eU

eV

eW

wr

q

daxis

q axis

i1q

i1d

i2q

ifdefd

Figure 5.4: Circuits of a synchronous machine, with one damper circuit along the d-axisand two damper circuits along the q-axis (based on [70] p. 55).

power and transferred across the air gap, via the stator, and injected into the grid aselectric power.


Equations describing the operation of the synchronous machine in the three-phase

representation of Figure 5.4 are discussed in [70] (pp. 59-67). Each coil in the three-phase representation is characterized by a resistance and a leakage inductance. Eachpair of coils is characterized by its mutual inductance. As the air gap of a synchronousmachine is not constant, the rotation results in time-varying inductances. This com-plicates calculations with the system in Figure 5.4. To simplify calculations, thedq0-transformation is applied [70] (pp. 67-74), also known as Parks transformation.With this transformation, all machine circuits are referred to a coordinate systemrotating synchronously with the rotor. In the machine equations in dq0-coordinates,inductances are constant.

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Raid

ed Lad

iad

i1d

ifd

L1d

LldA

B

R1d

L -Lf1d adwYr q

pYd

efd

Rfd

Lfd

Figure 5.5: Equivalent circuit of the synchronous generator: d-axis, in p.u. [70] (p. 89).

5.3.1 Equivalent circuit

The d-axis equations derived in [70] (pp. 54-74) contain four mutual inductances:Lafd , Lfda, Lakd and Lkda. The q-axis equations contain two mutual inductances:Lakq and Lkqa . With the mutual inductance of two conductors, the flux linking oneconductor due to the current in the other conductor is calculated. By means of awell-chosen p.u. system, i.e. base values for the different quantities such as powerand current, arbitrary constants (from the dq0 transformation) can be removed andthe equations can be simplified. In [70] (pp. 75-88) the Lad-base reciprocal per-unitsystem is used. With this p.u. system all four d-axis mutual inductances are equalto Lad and both q-axis mutual inductances are equal to Laq. It becomes possible to

model the synchronous generator by means of two equivalent circuits: one for thed-axis and one for the q-axis (Figures 5.5 and 5.6). In order to be able to use thesecircuits taking into account the effects of saturation and hysteresis, it is necessary toassume that the four inductances represented by Lad and the two represented by Laqare altered in the same way and thereby remain equal in p.u.

At steady state, the equivalent circuits for both axes are dc circuits. Consequently,the voltage across the inductances is zero. The effect of the inductances is in the fluxlinkages d and q:

d = Lad iad Ld id (5.3)q = Laq

iaq

Lq

iq (5.4)

The circuits are not independent. d interferes with the q-axis circuit and q withthe d-axis circuit by means of the induced voltages r d and r q . In general,a voltage is induced in a circuit by a changing flux linkage. In this case the changeis caused by the rotation of the magnetic field. Therefore, the voltages contain thefactor r, i.e. the angular velocity of the rotor.

In Figure 5.5 the series circuit of L1d and R1d represents the d-axis damper circuit.The series circuit of L1q and R1q and the series circuit of L2q and R2q in Figure 5.6

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Raiq

eq Laq

iaq

L1q

i1q i2q

LlqC

D

R1q

wYr d

pYq

R2q

L2q

Figure 5.6: Equivalent circuit of the synchronous generator: q-axis, in p.u. [70] (p. 89).The second damper circuit (with L2q and R2q) is not applicable for salient-pole machineswith a laminated rotor.

represent two q-axis damper circuits. For a salient-pole machine with a laminatedrotor, which is the type used in this study, the equivalent circuit for the q-axis con-tains only one rotor circuit [70] (pp. 151-152). This is because the damping effect oflaminated iron is negligible, as opposed to the solid iron of cylindrical rotors. There-fore, the branch with L2q and R2q is connected to the rest of the circuit by means ofdashed lines in Figure 5.6.

5.3.2 Machine parameters

The impedance of synchronous machines at signal frequency is calculated for a seriesof industrial, four-pole synchronous machines3, in the power range of 7.5 to 2250 kVA.The parameters of the equivalent circuits are calculated from data provided by themanufacturer.

If the pulsation of the stator quantities (i.e. voltage, current and flux linkages) equalsthe base pulsation base, there is no difference between reactance and inductance inthe per-unit system [70] (p. 87). When voltages at signal frequency are introduced,the equivalent circuits are no longer operating as dc circuits. As a result, the voltagesacross the inductances are no longer zero. The impedance j Xi of the inductances isthen calculated as j i Li, with i the pulsation in p.u.The parameters of the equivalent circuits of the d- and the q-axis are the fundamentalparameters. They are calculated with the standard parameters using the equations in

Appendix B.1 [70] (pp. 144-159). The standard parameters describe the response of asynchronous machine to a disturbance. Rapidly decaying components of the reactionto the disturbance are described by means of the subtransient reactances Xd and X

q

(or inductances Ld and Lq ) and the subtransient time constants T

d0, T

q0, T

d and

Tq . Slowly decaying components are described by means of the transient reactancesXd and X

q and the transient time constants T

d0, T

q0, T

d and T

q. The reactances

describing the steady state are the synchronous reactances Xd and Xq . Additional

3Leroy-Somer: Industrial Range 4-pole alternators

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standard parameters are the armature leakage reactance X (or inductance L) and

the armature resistance Ra.In [70] (p. 82) it is stated that the armature leakage inductances of both axes arenearly equal, as the discussion of synchronous machines in [70] is focused on machineswith a cylindrical rotor. In machines with salient poles, the d-axis armature leakageflux passes through a longer section of air gap than the q-axis leakage flux does. Thisis explained in a following section. Therefore, different leakage inductances Ld andLq are used in this study. The armature leakage inductance is not specified in themanufacturers data. Therefore, the following assumptions are made: Ld = Xd /2and Lq = X

q /2. This choice is based on typical ranges of parameters listed in [70]

(p. 153).

The following is a brief description of the other parameters of the equivalent circuits.Lad is the mutual inductance of the d-axis of the armature circuit and the d-axis rotor

circuits. If the machines iron becomes saturated, the value of this mutual inductancedecreases. Laq is the mutual inductance of the q-axis of the armature circuit and theq-axis damper circuits of the rotor.

Ld = L + Lad is the d-axis self inductance of the armature circuit, Lq = L + Laqthe q-axis self inductance. In synchronous machines Ld > Lq as the reluctance ishigher along the q-axis4. In machines with salient poles this is due to the larger airgap along the q-axis. Normally Lq is situated between 50% and 80% of Ld [74] (p.332). In machines with cylindrical rotors this is due to the rotor slots containing theconductors of the field winding [71] (vol. III, p. 29). The difference between Ld andLq is smaller for a cylindrical rotor.

Lpl = Lf1dLad is the inductance corresponding to the peripheral leakage flux linkingthe field winding and the d-axis damper circuit but not the armature [70] (p. 149).

Lf1d is the mutual inductance of the field winding and the d-axis damper circuit.In [75] it is explained that, for synchronous machines with a cylindrical rotor, Lplis positive, indicating that the magnetic coupling between the field and the dampercircuit of the d-axis (expressed by Lf1d) is stronger than the coupling between statorand rotor circuits (expressed by Lad). For machines with salient poles, Lpl is oftennegative. A negative Lpl indicates that the magnetic coupling between the field andthe damper circuit of the d-axis is weaker than the coupling between stator androtor circuits. As information concerning Lpl of the machines under consideration isunavailable and as it is difficult to predict the value of Lpl at signal frequency, a valueof Lpl = 0 is used in this study.

Lfd is the leakage inductance and Rfd the resistance of the field winding. L1d, L1qand L2q are the leakage inductances of the d- and q-axis damper circuits. R1d, R1q

and R2q are the resistances of the damper circuit.

5.4 Calculation of the synchronous machines state

The goal of this study of synchronous machines is to calculate the impedance ofsynchronous machines for signal voltages superimposed on the mains voltage. The

4Ld < Lq in synchronous machines with permanent magnets.

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5.5 Calculation of impedance at signal frequency

calculation of the impedance is executed in two steps. The first step is the calculation

of the normal state of the machine. This is discussed in this paragraph. The secondstep concerns the behavior of the machine at signal frequency, discussed in the nextparagraph.

The state of the synchronous machine during normal operation (or steady state) iscalculated for a given electrical power Pt and a given power factor cos t (leadingor lagging). For the calculation of the steady state, the equations in Appendix B.2are used [70] (pp. 100-102). During normal operation, the rated, sinusoidal voltageis applied to the terminals, i.e. without superimposed signal voltage. Steady state iscalculated as it determines the magnetization of the machine, thus the level of satura-tion, affecting hysteresis. Saturation results in decrease of mutual inductance Lad andthe magnetization is an important element in calculating the machines impedance atthe frequency of ripple-control signals. These elements are explained further on.

At steady state, the d- and q-axis equivalent circuits are dc circuits. Consequently,the inductances are short circuits. The armature currents id, iq and the excitationcurrent ifd required to achieve the required output power Pt and power factor cos t,are:

id = It sin(i + t) (5.5)iq = It cos(i + t) (5.6)

ifd =eq + Ra iq + Ld id

Lad(5.7)

Expressions for It and i are given in Appendix B.2. i is the internal rotor angle orload angle [70] (p. 99). It is the angle by which the q-axis leads the stators voltagephasor Et. Under no-load conditions i = 0. In order to determine the magnetizationstate, flux linkages are calculated. The flux linkage of a certain inductance is theproduct of the inductance and the corresponding current. ad is calculated as:

ad = Lad iad (5.8)iad = ifd id (5.9)

In paragraph 5.6 it is explained how flux linkages are used to determine the syn-chronous machines magnetization state and how to adapt inductances accordingly.


The calculation of the synchronous machines impedance at signal frequency is dis-cussed. The current flowing through the machine as a consequence of the signalvoltage is calculated. The ratio of signal voltage and current phasors is the machinesimpedance Zrcs. The signal voltages eus(t), evs(t) and ews(t) are defined as:

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eus(t) = Ets cos(rcs t + ) (5.10)evs(t) = Ets cos

rcs t 2

3+

(5.11)

ews(t) = Ets cos

rcs t + 23

+

(5.12)

is the phase angle of eus(t) at t = 0. The first calculation step is applying thedq0 transformation to eus(t), evs(t) and ews(t). The voltages resulting from the dq0transformation are:

eds(t)eqs(t)e0s(t)

= 23 cos cos 23 cos + 23 sin sin 23 sin + 23

12

12

12

eus(t)evs(t)ews(t)

(5.13)e0s(t) is the zero-sequence voltage. As the signal voltages are balanced, e0s(t) = 0.Consequently, zero-sequence components do not appear. Here the transformationwithout the zero-sequence component is referred to as the dq transformation. eds(t)and eqs(t) are the resulting voltages:

eds(t) = Ets cos

(rcs r) t + 0

(5.14)

eqs(t) = Ets sin

(rcs r) t + 0= Ets cos (rcs r) t + 0 2 (5.15)

eds(t) and eqs(t) are applied to the equivalent circuit of the synchronous machine(Figures 5.5 and 5.6) excluding the excitation voltage source efd. It is omitted asits frequency (0 Hz) is not equal to the frequency of the transformed signal voltageand therefore does not influence the quantities at that frequency. This is done also in[63] in order to calculate the rotor current induced by harmonic currents in the statorwindings of synchronous machines.

In contrast to the dq-transformed mains voltage, eds(t) and eqs(t) are not dc values.They are applied to the d-axis and the q-axis circuits. Because of the transformation,the relative signal pulsation rel, relative to the rotating d-and q-axis, is lower thanthe pulsation of the three-phase signal voltages:

rel = rcs r (5.16)In the dq-plane, the d and q components of one quantity can be combined into onecomplex number, i.e. a phasor with r as the pulsation base. Edqs(t) is the signalvoltage in dq-coordinates:

Edqs(t) = eds(t) +j eqs(t) (5.17)

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ius(t)ivs(t)iws(t)

= cos sin cos 23 sin 23 cos

+ 2

3

sin + 23

ids(t)iqs(t) (5.26)The impedance of the synchronous machine at signal frequency is calculated by di-viding one of the phase voltages by the corresponding current. Which of the phasesis chosen, does not matter as the system is assumed to be symmetrical. Here eus(t)(5.10) and ius(t) are used. The exact expression for ius(t) is:

ius(t) = e

cos Ids ejrelt sin Iqs ejrelt

= e

cos I1 ej(relt+1) sin I2 ej(relt+2)

= I1 cos cos(rel t + 1) I2 sin cos(rel t + 2) (5.27)=

I12

cos((2 r rcs)t + 0 1) + cos(rcs t + 0 + 1)

I22

sin((2 r rcs)t + 0 2) + sin(rcs t + 0 + 2)

(5.28)

(5.28) follows from (5.27) by taking (5.2) and (5.16) into account. (5.28) showsthat the stator current ius(t) consists of components at two pulsations: rcs 2 r and rcs. This is also found in [63], where the effect of harmonic frequencieson synchronous machines is analysed in a different way. As the distribution gridis assumed to behave linearly, electrical quantities at different frequencies do notmutually interfere. Therefore, only the current component ircs(t) with pulsation rcsis required to study the propagation of ripple-control signals:

ircs(t) =I12

cos(rcs t + 0 + 1) I22

sin(rcs t + 0 + 2) (5.29)

=I12

cos(rcs t + 0 + 1) I22

cos

rcs t + 0 + 2 2

(5.30)

In order to calculate the impedance of the synchronous machine at signal frequency,eus(t) and ircs(t) are represented as phasors with rcs as the pulsation base:

Ercs = Et ej (5.31)

Ircs =

I1

2 ej(0+1)

I2

2 ej(0+2

2)

(5.32)

The impedance Zrcs of the synchronous machine is the ratio of Ercs and Ircs:

Zrcs = Rrcs +jXrcs =ErcsIrcs

(5.33)

=2Et ej(0)

I1 ej1 I2 ej(22 )(5.34)

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Zrcs can be split into its real and imaginary part, i.e. the resistance Rrcs and the

reactance Xrcs:

Rrcs = 2EtI1 cos( 0 1) I2 cos( 0 2 + 2 )

I21 + I22 2I1I2 cos

1 2 + 2

(5.35)Xrcs = 2Et

I1 sin( 0 1) I2 sin( 0 2 + 2 )I21 + I

22 2I1I2 cos

1 2 + 2

(5.36)As mentioned previously, 0 is the angle between Edqs(t) and the d-axis at t = 0and it is the argument of Eds(t) (5.20). In the expressions for Rrcs and Xrcs, 1 and2 are subtracted from 0. The phase difference between 0 and 1 or 2 isdetermined by the respective equivalent circuits. As a result, the value of Zrcs does

not depend on the value of 0.

5.6 Effect of saturation and hysteresis

In the case of normal operation of a synchronous machine, i.e. with only the mainsvoltage present, the mutual inductance Lad depends on the saturation level of the iron[70] (p. 113). The mutual inductance Laq is assumed to be constant in salient-polemachines due to the large air gap along the q-axis. By means of the machines open-circuit characteristic (OCC), the saturation factor can be calculated. The product ofthe mutual inductance and the saturation factor yields the saturated inductance. Inthis study, a similar approach is used to calculate new parameter values for determin-

ing the machines impedance at signal frequency. This is necessary, as the parametersat signal frequency depend on the magnetization state. This dependence is differentfrom the dependence at mains frequency.

According to [70] (p. 110), a rigorous treatment of the performance of the synchronousmachine including saturation is a futile exercise. Accounting for saturation effectsmust be based on approximations. The choices made, depend on the level of com-plexity and accuracy deemed adequate and on the availability of data. As the mag-netization dependence at signal frequency is a similar matter, it is reasonable to statethat these considerations apply to the calculation of the machines impedance as well.

The first section of this paragraph is an overview of the aspects of magnetic saturationand hysteresis relevant to the calculation of the saturation factor and the impedanceof the synchronous machine. Next the use of the OCC is discussed. In this study,

the idea of how to calculate the synchronous machines impedance at signal frequencyarose from studying the use of the OCC.

5.6.1 Magnetic field strength and induction

Electromagnetism describes the effect of current distributions on moving charges.Two important quantities in electromagnetism are magnetic field strength H andmagnetic induction or flux density B. The magnetic field strength H is a measure

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H

B

O

P

Q

R

S

Figure 5.9: Saturation curve and steady-state hysteresis loop of a ferromagnetic material.

for current distribution. To calculate the force experienced by a moving charge in amagnetic field, the field strength H at the location of the charge is converted intomagnetic induction or flux density B. With the induction, force can be calculated.Depending on the medium, the relation between H and B may be linear, e.g. in air,or non-linear, e.g. in iron. The relation between H and B is expressed by means ofthe permeability = 0 r:

B =

H = 0

r

H (5.37)

0 (= 4 107 H/m) is the permeability of vacuum and r is the relative permeabilityof the medium. In air r 1. In laminated iron, depends on the direction. Inferromagnetic material r 1 and it is a function of B or H. The relation betweenH and B inside a material is described by the materials magnetization curve. Itshows the magnetic induction B as a function of the applied magnetic field strengthH. Figure 5.9 is an example of the magnetization curve of a ferromagnetic material.Two phenomena are present in this magnetization curve: saturation and hysteresis.

5.6.2 Initial magnetization curve

In the initial state of a ferromagnetic material, H = B = 0. When an increasingfield strength H is applied to the material, the state of the material is moved alongcurve OP (Figure 5.9). This is the initial magnetization curve (IMC) [76] (p. 20).For low values of H, B increases approximately linearly with increasing H. Abovea certain field strength, the increase in induction becomes smaller, until B remainsalmost constant at high field strengths. This phenomenon is known as saturation.Beyond the region of linear change, the iron is said to be saturated. In the linearregion, r is approximately constant. When the iron is increasingly saturated, rdecreases.

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5.6.3 Steady-state hysteresis loop

If ferromagnetic material is in a state of initial magnetization, e.g. point P in Fig-ure 5.9, and H decreases, the new state of the material is no longer on curve OP,but on curve P Q if H > 0. For H < 0 the state is on curve QR. This is hysteresis:the relationship between H and B depends on the sense in which H changes and onthe recent evolution of the magnetization state. If H decreases, B decreases along acurve different from the curve along which it has previously risen with increasing H[77] (p. 31). If an alternating field strength H is applied to a magnetic material, theresulting curve is called a hysteresis loop. If H increases in point R (Figure 5.9), thestate moves along curve RP. Saturation occurs in the hysteresis loop as well. Thevariation ofB with H is large around the origin of the HB plane and becomes smallerfor increasing |H|.If the amplitude of the alternating field strength is constant, the state of the materialconstantly moves along the same loop, i.e. the steady-state hysteresis loop. Differentamplitudes of H result in smaller or larger loops, with slightly different shapes. Thepoints of maximum |H| and |B|, i.e. P and R in Figure 5.9, are always situated onthe initial magnetization curve. The shape of the hysteresis loop also depends on thematerial and the frequency of the applied field strength [77] (pp. 32, 228). A certainamount of energy loss is associated with each magnetization cycle, i.e. the hysteresisloss. The energy loss is proportional to the area inside the hysteresis loop.

Two characteristic values of a hysteresis loop are the coercive field strength Hc andthe remanent induction Br. H = Hc or Hc at both points in the loop where B = 0.B = Br or Br at both points in the loop where H = 0. There are two types ofmagnetic material: soft and hard. Soft-magnetic materials have a narrow hysteresisloop, i.e. Hc < 1000 A/m. Hard-magnetic materials (or permanent magnets) have awide hysteresis loop, i.e. Hc > 10000 A/m [77] (p. 34). In electrical machines a softmagnetic material is used to carry the magnetic flux. The most common material formotors and generators is non-oriented silicon-iron, with a silicon content of about 3to 4 % [72], [78] (pp. 269-272).

5.6.4 Open-circuit characteristic

Saturation during normal operation of a synchronous generator is taken into accountby calculating the saturated mutual inductance Lad,sat, from the unsaturated valueLad, using the saturation factor Ksd [70] (p. 113):

Lad,sat = Ksd Lad (5.38)As the air gap along the q-axis in a salient-pole synchronous machine is long, satu-ration along the q-axis is not taken into account. In general, the magnetizing curvesfor the d- and the q-axis are different [79]. Ksd = Ksq as the air gap is longer alongthe q- than along the d-axis. For a synchronous machine with a cylindrical rotorthe difference is small. The difference is caused by the presence of rotor slots alongthe q-axis of the rotor. For salient-pole machines, the difference is large due to thelarge variation of the air gap. With a longer air gap, the reluctance of the air gap is

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higher and saturation of the iron has a smaller effect on the overall reluctance of the

magnetic path. With a certain q-axis MMF aq , a longer air gap results in a highreluctance aq and consequently a lower flux aq:

aq =aqaq (5.39)

Ifaq is sufficiently high, aq is too low to result in saturation in the q-axis. In salient-pole machines, the q-axis air gap is long. According to [70] (p. 116) Laq does not varysignificantly with saturation of the iron in the flux path in salient-pole machines.Therefore it is assumed that saturation does not occur, i.e. Ksq = 1.

Saturation of the iron of a synchronous machine depends on the state of the machine,i.e. output power and current. As mentioned earlier, accurately including saturationeffects in calculations concerning synchronous machines is very complex. Thereforeaccounting for saturation effects must be based on approximations. In this study, thecalculation of the saturation level is performed according to the method described in[70] (pp. 112-116). This method consists of using the OCC or no-load characteristicto determine the magnetization state for operation under loaded conditions. Thefollowing simplifying assumptions are made [70] (pp. 112-113).

The leakage inductances are independent of saturation. A considerable part ofthe path of leakage fluxes passes trough air. The mutual inductance Lad is theonly element that can be saturated.

Saturation is determined by the total air-gap flux linkage tag =

2ad + 2aq.

The leakage flux does not contribute to the saturation.

The relation between the air-gap flux linkage tag and the MMF under loadis the same as under no-load. This allows the saturation characteristics to berepresented by the OCC, usually the only saturation data available.

Saturation is no cause of magnetic coupling between the d- and the q-axis5.

During open-circuit operation, the machine is not connected to the three-phase voltagesupply. The rotor is driven at synchronous speed by a prime mover and excitationcurrent ifd flows in the field winding. As a result of the rotating magnetic field, avoltage is induced in the armature winding; the open-circuit terminal voltage Et:

Et

= eq

= r

d

= r

ad

(5.40)

r is the angular velocity of the rotor. The OCC of a synchronous machine can beobtained by measuring Et as a function of the excitation current ifd . The OCC inFigure 5.10 shows ad as a function of the MMF ad. The air-gap line is the open-circuit characteristic of an imaginary synchronous machine without saturation. Forthis imaginary machine Lad is determined by the reluctance of the air gap for allvalues of ad. ad is equivalent to Et and ad to ifd . ad can be written as:

5The theory of cross magnetization is discussed in [80].

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Impedance of synchronous machin

effect of distributed generation on

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