analysis for power system state estimation

7/27/2019 Analysis for Power System State Estimation

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IEEE T r a n s a c t i o n s on Parer A p p a r a t u sa n d Systems vol. PAS-94, no 2 , M a r c h i A p r i l1 9 7 5BAD DATA ANALYSIS FOR POWER SYSTEM STATE ESTIMATION

E. Handschin* F. C.chweppe** J . Kohlas* Aiechter**Brown Boveri Research Cen ter, **Electric ower ystem EngineeringBaden, Switzerland Labora tory,Eept.,Mass. Institute of Technology,Cambridge, Mass.

SUMMARYThe state estimation problem in electric power systems consists

of four basic operations: ypothesize structure; stimate; etect;identify. This paper addresses the last two problems with respect to thebad data and structural error problem. T he paper nterrelates variousdetectionnd identification metho ds sum of squared residuals,weighted and normalized residuals, nonqu adra tic criteria) nd pre-sents new results on bad da ta analysis (probability of detec tion , effectof bad data).The theoretical esults are illustrated by means of a25 bus network.

1. INTRODUCTIONState estimation algorithms fit measurements made on the system

to a mathematical model in or der to provide a reliable data base forothe r monitoring, ecurity assessment and ontr ol functions. Thesealgorithms should be able to hand le four types of errors: 1. measure-ment (randommetering nd omm unication rrors), 2. parameter(un certa inty in model parameters such as line admittance), 3 . bad data(large unexpected meter and communication errors) and 4. structural(errors in the structural form of the model). This paper addresses solu-tions to the bad data and structure error problems based on residualanalysis andnonqu adratic estimation riteria. The tradeoffs be-tween these various approaches are compared to helpa system de-signer decide which combination of techniques will best meet his par-ticular needs. New bad data analysis techniques are presented whichenable: 1. evaluation of the bad data spreading effect and hence leadto the concepts of interacting noninteracting bad data and localredundancy, and 2. calculationof the probability of detecting baddata and of false alarm.

2. PROBLEM DEFINITIONThe purpose of a static state estimator is summarized in Fig. 1.To be mor e explicit defme:

System state : complex bus voltages, n-dimensional.Measurements 3 : telemetered line flows, bus njections,bus voltagesor pseudomeasurements, md imension al.Measurement error yz: zero mean random vector.Covariance matrix E: diagonal matrix whose i:th com pon ent ui2 givesthe variance of the i t h m easu rem ent e rr or E [ Y ~ y z T 1 = ~ ;i2 maydepend on the size of zi.Model parameter p: value of parameters used to de fii e: 1) individualline, transformer, etc. models (usually impedances of n-equivalentnetw ork ) and ) covariance matrix E. Obtained from systemdesign data.Model structure 2: telemetered or telephonedwitch and circuit

breaker positions which determine network structure and meterlocation.

Engineering Committee of the IEEE Power Engineering Society for presentationPaper T 74 309-1 recommended and approved by the IEEE Power System

at the IEEE PES Summer Meeting & Energy Resources Conf., Anaheim, Cal.,July 1419,19 74 . Manuscript submitted August 31, 1973; made available forprinting April 3,1974.

measurements 2 Rel iab lemeter) est imate of:s t r u c t u r a l n f o rmation g breakep o s i t i o n e t c . ) . s t a t e b u sv o l t a g e s )

Computer 2 ) models t ru ctu reandpararne-te r valu es )________E d es ignd ata)

Fig. 1. Purpose of static state estimationHypothesized model h 5): If the values of E nd 5 are known per-fectly, the nz, and are related by

- = b @ ) + 2 (2.1)where 5) is specified by p and 5 so that o be more precise

State estimate 2: estimate of the value of state x omputed from, p nd 5.Parameter errors Q: zero mean random vector describing m al l devia-tions of assumed values of parameter p from true model values

z =h x,2,9+h

&E=P+ p.Bad data b: unusually large measurem ent erro r added to caused bymeter-communication system failures)- = ~ ( x ) + ~ w i t h ~ = ~ + ~ .

If the i:th meter is bad then b =y with giT = [O,O,. . 1,. . O]and a = size of bad data.Structure error E: errors in system s truct ure5 .Detection: est to determinewhether bad dataor structuralerrorsare present.Identification: logic to determine which measurement(s) is (are) bad orwhich part of the structure is wrong.Redundancy I): ratio of number of measurements to number of un-knowns (states) = m/n .

Local redundancy ek: redundancy for each bus counting only measure-ments and unknowns at bus k plus at all buses up to two switch-yards away.Some of th e above definitions are no t necessarily standard ermi-nology b ut it s essential to have a set of selfconsistent terms.Using the above definitions, an expanded description of the stateestimator of Fig. 1 is shown in Fig. 2 in terms of four basic opera-tions: 1) hypothesize structure, 2) estimate, 3 ) detect, and 4 identify.Methods to hypothesize structure i.e. to convert E nd 1. n to &(&)are not discussed in this paper. They are conceptually straight-forwardnetwork theory but in practice they are nontrivial. Th e weighted leastsquares WLS) approach to estimate is summarized in Appendix A.This paper is primarily concerned with detect and identify.Thispaperconsiders the processing of only he observation zmade a t one time instant. In practice it is desirable to use time seriescurve fitting on the individual measurements and plausibility checkson the structural dat a2 and on the magnitude of the individual measure-

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SURMENTS INPUT Parameter inforpation Estructure information aIHYPOTHESIS MODEL

model ICE) and covarianceDetermine equations formatrix 8.I

next measure- , t Iment

Yes

criterion

becomesiniual.

r z-h g). mall-enough?If not, then there exist1) bad data g and/or2) structure errors 2

IDENTIFY Iiogic to find the locationof bad data in 5 and/orstructurerrors in E. I

modify inputs

Fig. 2. T h e four basic operations for state estimation in electric powersystems

ments. Such preprocessing is imp orta nt but is not discussed in thispaper to save space.Parameter errors vp are not discussed in this paper. If ~rps smalland modelled as a ero mean randomvector, heparameter rrorproblem can be viewed as being equivalent to a problem with-noparameter rrors but with ameasurement rrorwith covariancematrix larger than E. Of course it is difficult fo determine this new-; but fortunately WLS estimation has the property that the estima-tion,detectionand dentification logics tend to be insensitive to thevalue of E. Thus, provided the parametererrors are not t large,most of the pproaches to be discussed still apply. However, theanalysis of theperformance see sect. 5 can be influenced heavilyby the value of E.

m

Fig. 3. 25 busnetwork. T h e numbersofmetered witchyards areencircled. T h e measurements are given in kV, MW, MVar.munication inks are the mo st expensive items. All possible measure-men ts at a selected switchyard are taken with the constraint hat noline is measured a t bo th ends. All generator switchyards were selectedplus the minimal num ber of oth er switchyards equired for a com-plete state estimation (the minimization was done by eyeball,not rigorously). T h e redundancy r = 115/49 = 2.3. Bus 3 was chosenfor most bad data studies. Two m e te r c o n f i r a t i o n s will be discussed.Case 1 is the configuration of Fig. 3 and leading to a local redundancy3 = 2. In case 2, the voltage and injection measurements at bus 3 areomitted, giving a l o c a l redundancy 3 = 1.7. The standard deviationsui of the error re set equal to 0.02 p.u. (on a 1 00 MVA base) for realand reactive power measurements and 0. 002 p.u. fo r voltage magnitudemeasurements.

Many different types of numerical studies were done on thissystem. Because of space limitationsonlya few of these esults areexplicitly mentioned here. Ref. 22 contains more details.

4. BAD DATA ANALYSIST h e key to the analysis of bad data is the residual sensitivitymatrix of (A9) which was obtained by linearization. With it nd

relatedequations such as A6), (A81 and A1 l), a wide variety ofresults can be obtained.Nonlinear simulation howed the validity of the linearization.For example, the component of &, corresponding to P3-13 is shownin Table I for different sizes a of bad data.Numerical values of the residual sensitivity mat rix how tha ttheeffect of one bad data in s only noticeable in theneighbour-hood of the bad data location (the size of theneighbourhood de-pends onhe local redundancy). For example, for bad data inP3-13, wp3-13, p3-13 = 4.3 IO-1 whereas Wp3-13, p5 = 1.1 IO4;so bad data in P3-13 and P5 are noninteracting. The limited spreadingof bad data allows the reduction of the multiple noninteracting baddata problem to several one bad data problems (sec. 7) . The m ultipleinteracting bad data problem has to be solved separately (sec. 8).

3. EXAMPLE CASES STUDIED 5 . DETECTIONT h e numerical results to follow are based on the 25 bus networkshown in Fig. 3 . T h e basic me ter config uration indicated in Fig. 3 waschosen using a minimal cost criterion which assumes that the com- Consider J( ) as defined by (A l) with z=Z. Intuitively, it is tobe expected hat bad data or structural errors should cause an unex-

Table I: Effect of bad data in P3-13 on p3-13,$eore tical: (A1 1) with v = a (vz=O)simulated: (2.2) with -= ~J IxzBad dataizease ICase

a in 4W theoreticallsimulatedtheoreticallsimulated

22.8 22.1 35.2 35.845.6 44.8 70.5 71.9

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pectedly large value of J Q . This concept is formalized in (B1) interms of the J(i>t est. As indicated in Appendix B , it is possible tocomp ute P,, the false alarm probability and Pd, he detection proba-bility. Fig. 4 llustrates he type of results which can be obtained.Fo r case 1 a single bad data can be detec ted with 90 confidence pro-vided th e bad data is at least 2 0 MW. If the bad da ta is not removed,the error in P3-13 is then -1 1 MW. f this error is too big, the meteringconfigurationhas to be improved or a larger false alarm probabilityused. T h e probability calculations underlying Fig. 4 are based on thedistribution of J(@ in the presence of bad data, (A27). Monte Carlosimulations using the onlinear imulation verified these formulae.T h e linearized analysis metho ds for o ne bad data point apply to themultiple (N) bad data case by using 2.2) with b = C % a i .Instead of using J(&) it is just a s reasonable t o test the magnitud es ofthe individual residuals. This leads to the rr-test of (B2)and therr -test of (B4). Results such as Fig. 4 can be developed using thebound s of (B3). Such tudies and henonlinear imulations) haveshown that for a single bad d ata, the gq-test is usually superior to theJ(i>t est for arge networks.

Ni= 1

l .O 1 MMW

10 20 a[MW]30-Fig. 4. Performance of J (3 ) test and effect of undetecte d bad dataof size a nP3-13Pd: probability of detectionPe: false alarm probabilityError: error in estimate of P3-13 when bad data is not re-moved (A1 1 .

T h e following statements comparehe three detectionests:1 on-line implemeTtation is simple and fast.2) m-test requires off-line calculation of Q in (AI 2 ) while3 ) %-test is more effective than -g-t est in all cases.4) =-test is more effective than J(&>test for a single baddata point (for arge networks).5) J(g)-test is sometimes be tter than gq-te st nd vice ersaformultiple bad data (interacting or noninteracting) or

structure errors.

J ( i> tes t and-r- tes t do not .

These statemen ts lead to he recommendation t o implement:eitherJG Ftes t and -r-te st or J(g>test and 3- te st . A detection is in-dicated if eithe r test of a pair fails; i.e. if either J($)-test or m -te stdetects trouble. The J(f) -- pair is preferable if can be calculated.If sparcity echniquesandcareful programming are used then as avery rough rule of thumb, time to compute _ is less than or equalto the time to doone WLS estimate.

6. DECOMPOSITION OF NETWORKT h e limited spreading of bad data makes it possible to de-compose the network in to a contaminated part nd a healthypart by inspection of their nd r (sec. 5 . If this is done, on-

line state estimation can continue to update the system state in atracking mod e or thehealthy part even if no identification is c-complished . dentification echniques may be used jus t on he con-taminated part plus the boundary buses to the healthy part.7. IDENTIFICATION

7.1. Single or Multiple Noninteracting Bad DataFor the ordered residual search approach the weighted (normal-ized) residuals rm) f (A13) are put into descending order of m ag

nitudeand he measurementzicorresponding to the largest residualGax r )s removed first. T h e result of the subsequent WLS esti-mation with the reduced measurem ent vector Zr is then passed throughthe detection schemes of sec. 5. If bad data is still detected a furtherestimation is carried out with the second largest residual removed,etc. For one very large bad datapoint A16)states hat he largestnormalized residual rEax directly indicates the bad data location.

In thegrouped residual search the p largest residuals in Ior 3 are removed simultaneously and hen pu t back one after heotherunt il bad data is detected. It is particularly uited for iden-tification of multiple noninteracting bad da ta. Grouped residual searchmay requirea new Hypothesizemodel (Fig. 2) when the removalof residuals leads to asituation where some of the states cannot beestimated.

2 4 6 8 10 12 14

dS/drgradient

4

2

Fig. 5. Various estimation criteria2. Quadratic-straight1. Quadratic4. Quadratic-Square root3. Multiple segment5. Quadratic


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An alternate o residual search is the use of nonquadratic errorcriteria (see Fig. 5 and Appendix C). The main numerical problem withthenonquadratic error unction is thepote ntial difficulty of manylocal minima. In orde r to get a good starting value the result of a WLSestimation is used. Table I1 shows the normalized residuals= using aquadraticsquare root error functionfor bad data in P3-13.Table 11: Thehree largest normalized residuals for bad data inP3-13 of case 2 .

Measurement3.18.20.184-21 War] 5 0 .

2.95.523-13 [

40 MW0 MW=10 MWQ2 War 2.81 2.89.82

In agreement with he results of Fig. 4 bad data of less than 20MWcannot be detected. Larger errors result in a clear identification ofthe bad data point. The other nonquadratic error functions of Fig. 5have shown a similar behaviour.Residual search and nonquadratic criteria are actually closelyrelated concepts. Consider the quadratic-constant criteria of Fig. 5where the break point hl between he quadratic and constant part islowered afte r each trial estimation until he bad data is identified.Thiscorresponds to ordered residual search. In he grouped residualsearch the break point X1 is increased after e ach trial stimation.7.2. Multiple Interacting Bad Data

Multiple interacting bad datapoints may occur on occasionssuch as when all themeasure ments in one switchyard are trans-mitte d over the same channel, w hen several power flow me ters use acommon potential transformer, etc.Furthermore,due to he ran-dom nature of bad data they may occur in two adjacent buses.There are many reasonable ways to extend the residual searchlogics of sec. 7 for the multiple nteracting bad data case. Unf ortu-nately hebestdirection to follow is not obvious and it appearsthat there will always be cases where an extensive (almost exhaustive)search will be required. Th e difficulties of this search will e re-duced if thenetwork is decompos ed (sec. 6 into healthyandcon-taminated areas.Nonquadratic rror riteria are effective when 1) he local re-dunda ncy Ek is large enough and 2) henumbe r of nteracting baddata is not oo big. Some esults are summarized in Table 111 forcase 1 with bad data in P3-13 and P3-14 corresponding to 20 MWeach. For WLS the residuals are smeared while for all formsofthenonquadraticcriteria, the bad data location is obvious. In situa-tions such as these thenonquadraticcriteria have adefinite advan-tage over residual search procedures since only one onquadraticiterative sequence is requiredandsophisticated search logics do nothave to be developed. It is not necessary to decompose he network(although it canbe done if desired).

Nonquadratic criteria are not effective when the local redundancyek is too small or when the numbe r of interacting bad data points s toolarge as e.g. when bad d ata on all line flows at bus 3 are generated bysetting measured quantities qual to zero. It is not clear whethe rsome sophisticated residual search would be better.

7.3. Structural errorsThe most reliable approach to identification of str uctur al errorsis to use a search logic (based on the same philosophy as in residualsearch for bad data). This however can be difficult to autom ate anddemands much computer time. Network decomposition is almostmandatory.A istinction between bad data b parameterrrors p, andstructure errors 2 was made in Section 2 because they represent dif-ferent physical phenomena. However, frommathematical point

of view, (and also) may be interp reted as bad dat a in E. Forexample:a tructuralerror c can arise when a line is assumed to be inservice although it is actually disconnected atboth ends.Thiscan be viewed as bad data in the line admittance; i.e. in2.This kind of generalization te nds o obscure the mp ortan t physicaldifferences, is not entirelysatisfactory rom an intellectual point ofview and has some practical shortcomings. However it provides a use-ful engineering approach tohetructure errordentificationproblem. To be mo re precise, consider he estimation of both andfrom 7 .1)

Bad data identification logics can now be applied directly, where baddata in the measurementq is of interest.7.4. Discussions

Th e following discussions are based partly on solid theory, partlyon the numerical studies (only a few of which w ere actually presentedin this paper) and partly on personal judgement. They shouldbe viewedas guidelines, not as absolute truths.Some of the o ption s in the choice of the identification logic aresummarized in .Fig. 6 in term s of th ree levels; Level I: Attempt oidentify bad data; Level 11: Decompose the network; Level 111: Usemore sophisticated echniques to handle bad dataand/or tructuralerrors. T he orders of Levels I and I1 can be exch anged. Each level hasseveral options, including the option o do nothing. The minimumcomp uter cost solution is to choose to do nothing at all levels sothat theonlyaction aken after a detection is to warn the ystemoperator. A Level 111 capability requires the largest compu ter capacity(a Level I11 solution involving real-time display-input and an oper atorcontrolled search is a promising approach).Fig. 7 provides a comparison between t he use of residual searchand nonquadratic estimation. If a nonquadratic estimator approach isused the choice of criteria is not straightforward. In general terms, thequadratic-constant curve worksbest (when it works) but is leastreliable (most sensitive to iterative olution difficulties) while thequadratic-straight line is the most reliable but least effective. As a com-promise o ne of the middle curves (quadratic-square root or quad raticmultiple segment) is to be preferred. T he use of a normalized non-quad ratic error criteria (see Appendix C) provides bette r results butrequires the computation of E.

Table 111: Welghted and normalized residuals for five different estimation criteria and interacting bad data (case 1Measurements Quadra t i c -u l t i p l euad ra t i c - Quad ra t i c -LS s t r a i g h t f l a tegmentq u a r e o o tr w rN rw rW rNW rNW rN3-13i 3 - 1 42- 71-163-1

3.00.50

0.18 0.20.10 0.11.15.27.47 1.68.49 2.85 0.25.46.21.40.29 0 . 5 3.55 2.83.85.201.71 2.03.75 2.02.83 .11 .57 .97.04 3.51 9.720.980.10 11.42.88 11.17.96.03.48 7.349.75 14.680.055.09.574.28.43 8.16

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e s t a t e es timation1A

Remove b ad d a t aa n d / o rc o r r e cm o d e l s t r u c t u r e

Fig. 6. Option s in design of identification procedure.

Fne ba d da taN mul t ip lenonin-e r a c t i n g b a d d a t ac t i n g b a d d a t a

Number of quadra t i c e s t ima t ions whicheneededusing a sea rchWeighted re- Normalized res is i d u a l s d u a l s rNd e t e c t i o n

y : l a r g e: s m a l lt h r e s h o l d t h r e s h o l dd e t e c t i o n

s e v e r a l onee v e r a ls e v e r a l N one t o N depen-e v e r a l N ding on log icVery large . A l l poss ib lecombina t ionsmay have t o b e s tu die d

Performanceofone non-q u a d r a t i c e s t i m a t o r

E x c e l l e n t f o c a l e d u n d a n c yi s highenough

S a t i s f a c t o r y f1 local redundancy i s high2) number of b a dd a t asmal l

Notes: 1. As a very rough rule of thum b: time fo r one nonqu adratic estimate = ( I to 2 ) time for one WLS estimate.2. High detec tion threshold 7 mplies only large bad d ata detected .Fig. 7 . Comparison of residual search with nonquadratic estimation for bad data detection .

8. CONCLUSIONSThe residual sensitivity matrix W or WN of (A9) or (A 1 9 is apowerful tool or analyzing bad datawithou t resorting to multiple,nonlinearMonte Carlo simulations. Study of hismatrix hows he

limited spreading of bad data, the important distinction between inter-acting and noninteracting bad data, and that local redundancy ek is moreimportant than redundancy r) in evaluating a m eter configuration.Using the W or WN matrix, and an assumption of normal measure-ment errors, t is possible t o comp ute the false alarm probability andprobability of d etection (or boun ds there on). This provides a designtool for choosing a metering configuration.The choice of which detection tests to implement is relativelystraightforward. T h e choice of identification logic involves a radeoffbetween computer timeandperformance. dentification by residualsearch and by nonquadratic criteria are relatedconcepts.Both haveadvantages and disadvantages.Appendix A W L S Estimation and Linearized Analysis.

This appendix summarizes the mathem atical backgroun d of WLSestimation and its error analysis as used for state estimation in electricpower systems. The WLS estimate & is based on the criterionm

JP[z-l(x)ITR-[z-h(x)]=iil(q)i 2 ( A I )

ri = zi - hi(5). &satisfies the optimality condition

where H = d&/d& denotes the Jacobian matrix. T h e nonlinear equation(A2 ) is solved iteratively [Ref s. 1-31 using an algorithm such as

where denotes he WLS estimate of the k: th iteration and thechoice of affects the convergence properties (B=HT -18 is oftenused).Complete discussion on thechoice of E use of Sparcity Pro-gramming etc. is beyond the scope of this paper.With (2.2) it follows from (A21 that-T E-[h(5)+~-5(i)] 2. (A4)Define the state estimation error as

The rest of he equation s of Appendix A and B are based on linear-ization obtained by assuming& is small and sett ing &( x) =h (s) +B x.From (A4)


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and for v=v the covariance matrix of the estimation error d x isE[& hT] I . (A71

Similarly, for the residual vector- -2 i = h_@) +v-h_@ - w (A814 1-E E T 1 ( A 9

with the residual sensitivity matrix W given byand for x= the residual covariance matrix

E[- iT]&=s-_HT 5. (A101Finally, the difference between the true and estimatedmeasurementis givenby

= h c g -h_& = cw,-gv. (A1 1)Let the diagonal matrix _ denote- = diag (A12)Defme the weighted residual and the normalized residual as

Then the inequality holds-k2 . For v=yz the covariances areElrwgw a V, = JE L - T -1: JR I and (A141

where the diagonal elements of UN are all equal to one. Defrne thenormalized residual sensitivity matrix VN bys=flw (AIS)Let WN,jk denote he W element of the j:th column and thek:th row. Then from (A10) and (A15)

so that from the positive semidefinite nature of Cr

both become zero mean, unit variance Gaussian; i s . N 0 , l ) .(2.2) is given byFor one bad data in the i:th measurement of z the error f

v = + :?a.- (A191Substituting (A1 9) in (A8) the (normalized) residuals and r~ take theform

=Ez weia and &=WJVZ + a. (A20)J(3)n (A1 7) becomes

where (if is normal) he irst term is x2- distributed, the secondterm is normal distributed,and the hird erm is a constan t. As Kbecomes large I(&) n (A21) approachesanormaldistributionwithmean I. and variance 02.I I

where Wi s the (i,i):th element of the residual sensitivity matrix W.Thus, the standardized variables 51 and 32 of (A18) become

Appendix B Detection TheoryThe detection of bad dataor tructural errors can be viewed

as an hypothesis testing problemwith two hypothesis Ho and H1where Homo bad dataor tructural errorspresentH1 :Ho is not rue.Let Pe denote the false alarm probability, i s . Pe is theprobabilityof rejecting H,, when Ho is actually rue. Let Pd de-note he probability of dete ction; i.e. Pd is theprobability of ac-cepting H1 when H 1 is true.Threeme thod s of testing these hypo-thesis will be summarized.1 . The I(&)-test isAccept H, i f 5 1 < y ( o r 5 2 < y )

Rejec t H ( accep t H1) otherwise (B1)where c1 and 52 are given by (A1 8) and 7 is the detection thresholdlevel. The choice of 7 determines P,. Fo r example, when yz is normal,K is large, and Ho is true, then 51 (52) is N(0, l) so ~ 1 . 6 5orrespondsto Pe = 0.05. In a similar fashion it is possible to compute Pd for agiven 7 nd a using (A23).2. Th e 1_m-test s

Accept Ho i f I&,k l < y I k = l . . . m (B2)Rejec t H ( accep t H1) otherwise .When l, is normal and Ho is true, then from (A14), %,k is N(0,1)which provides a basis for choosing the hreshold 7 . For a given 7(and a), t is awkward to comp ute e xact values of P,, (and Pd). How-ever if Pe,k is probability l k , k l > 7 when Ho is trueandPd,k isprobabil ity l k , k l> 7 when H 1 is true then

pe 5 p e l k ; pd 2 Pd,k (B3)3. The rw-test is

Accept HO i f I;W,k] < y I k = l . . .Ill (B4)Rejec t H ( accep t HI) otherwiseIf the same value of 7 used in (B2) is used in (B4), he w -te st ismore sensitive, since du e to (A13): -k 219.Appendix C NonquadraticCriteria

For a nonquadratic cost function (A l) is replaced by

where the vector p(r) is a function of the residual r such that heerror criteria shown in Fig. 5 are obtained. The optimality ondi-tion is obtained via chain ruledJ-dJ d e df: = 2 ( & p ) g g=o (C2)dx de dy dx

where

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ing the optimality cond ition (C2) asH E S Q = ~

the similarity with (A2 ). he nonlinear equation (C4) isT - 1 (C4)-

T T - 1 - - T -1E E 51 G , [ ,+,- ,]=E S e (C5)e unctio nspi of e and the diagonal elements

of th e diagonal matrix G for the criteria shown in Fig. .of the reakpo ints are chosen by engineering judge ment,. . .5 for @ , @ and @ . In @ usually X1=2.5. . .. .6.5 and X3=9.5. . .12.

An important variation of (C l) is the normalized nonquadraticnction which is the same as (C l), (C2) except that the break-i are different for each residual or determined by@ f. Thus, for example, thecondition lri/uil < h is replaced byJL;~ I< X .D - References

References and ome alte rnat e approache s are now discussed.pti c bu t hopefully serve asguide to the literature. Apologies are m ade to authors whose workThe use of J f), -J J and lu for detection and identification was

1-3. The X2-nature of J(&) is discussed in Ref. 4 .5 discusses the use of J(i().The use of nonquadratic criteria for power system bad data de-dentification is introdu ced n Ref. 6 (which used the

7 considers the use of quadratic-with varying breakp oint which decreases ex-withhe numb er of terations. T h e technique used in8 apparentlyorresponds to the quadratic-constantriteria.of these attempts t o , handle bad data, the theory ofestimation Ref. 9) has been used in m athematical tatistics

come rom Ref.9. This work is presentlylo), using non-A stimulating discussion with Huber ndin thisA bridge between robust estimation and thepower ystems11 and 12.An excellent study on the effect of parametererrors p on is

The struc tura l error identification approach is taken from Ref. 14.References15and16 discuss bad data det ectio n and identifica-based on a deterministic load flow, using the redund ant informa-to id entify bad d ata. On e difficulty is the choice of an accurateThe ombined patial ndime orrelation pproach of Refs.

17, 18 assumes that a sensor which has failed at time k+l will show aifferent reading from the previous estimated measure-at time k. For any large difference, an autom atic search logic iso check oth er d ifferences which are spatially correlated withmeasurement in questionspatiallyorrelated is probablyto in teracting) to see if the sudden change is caused byoperating condition or bad data. This metho d appearsin detecting multiple interacting bad data, structuraland a singlebad data point caused by a meter which driftsThehypothe sis testing approach of Ref. 19 is based on the as-Ewitha scalar u2 i.e. all variances are multiplied by the

0 2 This seems to be a mathem atical artifice which doesIn practice, however, the

Table CI Nonquadratic estimation functionsand gradients for i = 1 2, ..., mQ Q u a d r a t i c p i = ri 1 = 1@Quadratic-straight

lri/uilsA:pi = ri gi = 1

@Multiple segmentlri/uilLXl: p i = r i gi = 1

-1/2gl=[ 2 k I - 1 1

gi = 0@ Quadratic-square root

/r./uilLA:pi= ri1 gi = 1

@Quadratic constant

gi = 1i

3 3 5


8/9

logics of Ref. 19 are closely related t o th e Je) nd gq tests, the maindifference being in the extra mathem atical complex ities resulting fromthe introductio n of 02 (see also Ref. 20).The network of Fig. 3 is taken from Ref. 2 1.Th e present paper is a condensed version of Ref. 22 which con-t a i n s many m ore details.The linearized analysis of Append ix A is straightforward and canbe found in m any places. Ref. 23 contains discussions on the generalapproach and also on the hypothesis testing concep ts of Appendix B.The X2-nature (and limit distributions) of J g) (when a = 0) for linearmodels an be found in many standard tatistical tex ts (see alsoRef. 23).Methods to solve (A2) for i (and other approaches) are discussedin many of the above references and elsewhere. Ref. 24 can be viewedas a compan ion paper to the present paper which discussed the overallstatic tate estimation roblem in more etail.Ref. 24 uses theterminology of this paper.

REFERENCES(1) F. C. Schweppe and J . Wildes, Power system static-state estima-tion Part I: Exact model, IEEE Trans., PAS-89, pp. 120-125,1970.(2) F. C. Schweppe h d D. B. Rom , Power system static-state esti-mation - Part 11: Approximatemodel, IEEE Trans., PAS-89,(3) F.C. Schweppe, Power system static-state estimation - Pa rt 111:(4) R. D. Masiello andF. C. ChWeDDe. Pow er svstem track ing

pp. 125-130, 1970.Implementation, IEEE Trans., PAS-89, pp. 130-135, 1970.. , state estimation. PSEG Report EE-D ept. M.I.T.-1970, see also

5) . F. DODZO. 0. A. Klitin and L. S. VanSlyck. State Calcula-IEEE Trans.,PAS-90, pp. 1025-1033, 1971.. _t ion of Power Systems from Line Flow Measurements, Par t II,IEEE Trans.,PAS-91, pp. 145-151, 1972.(6) H. M. Merrill and F. C.Schwep pe, Bad data suppression inpower system static-statestimation, IEEE Trans., PAS-90,(7) E. Handschin. H. Glavitsch and J. Kohlas. Estimation undpp. 2718-2725, 1971.. ,Entdeckung Lhle chter Messdaten, EDV in der Elektrizitatswirt-

Wien, 11. und 12. April, 1973.schaft, Seminar des Oesterreichischen ProduktiviMtszentrm,(8) B. Poretta and R . S . Dhillon, Performance evaluation of stateestimation from line flow measurements on Ontario Hydro powersystem, IEEE Winter Power Meeting, T 73 086-6, 1973.(9) D. F. Andrews et al., Robust estimates of location , PrincetonUniversity Press, 1972.(10) P. J. Huber nd F. Hampel, Private comm unication, Swiss

(1 1) J. Kohlas, On bad data suppression in estimation, IEEE Trans.,Federal Inst. of Technology, Zurich; 1973.(12) E. Handschin and J. Kohlas, Statistical data processing for(13) T. A. Stuart and C. J. Herget, A sensitivity analysis of weightedpower system op eration, Proc. 4th PSCC, Grenoble 1972.least-squares stat e estima tion for pow er ystems. IEEE WinterPower Meeting 1 973, T 73 085-8.(14) H. M. Merrill and F. C. Schweppe, On-line system model er rorcorrection, IEEE Winter Power Meeting 1973,C 73 106-2.(15) G. W. Stagg, J . F. Dopazo, 0. A. Klitin and L. S. VanSlyck,Techniques for real-time monitoring of power system opera-

tion, IEEE Trans., PAS-89, pp. 545-555, 1970.(16) . Ariatti et al., Methods for electric ower system stateestimation, Proc. 4th PSCC, Grenoble 1972.(17) A. S. Debs, R. E. Larson and L. S . Hajdu, On-line sequential(18) E. Handschin (editor), Real-time ontrol of elecmc owerstate estimation, Proc. 4th PSCC, Grenoble 1972.(19) J. F. Dopazo, 0. A. Klitin nd A. M. Sasson, State estima-systems, Chap. 2, Elesevier, Amsterdam, 1972.tion or power ystems: detection and dentification of grossmeasurem ent errors, Paper X-2, PICA Conference Minneapolis,1913.(20) F. C. Schweppeand E.Handschin, Discussion of the paper byJ . F. Dopazo et al.: State estimation for power ystems: de-tection nd dentification .of gross measurem ent rrors, pre-sented at PICA conferen ce 973,o be published in IEEE(21) R. G . Gungor, N . F. Tsang and B. Webb, A technique forTrans., PAS.optimizing real and reactive powerchedules, IEEE Trans.,

AC-17, pp. 827-828, 1972.

PAS-90,pp. 1781-1790, 1971.

(22) E.Handschin, F. C. Schweppe, J. Kohlas and A. Fiech ter, Baddata analysis in electric power systems, Brown Boveri ResearchReport 1973.(23)F. C. Schwep pe, Uncertain dyn amic ystem s, Prentice Hall,Englewood Cliffs, N. Y. 1 973.(24) F. C. Schweppeand E. Handschin,Static state estimation inelectric ower systems, t o be published in Special issue ofIEEE Proceed ing devoted to Power Systems and Computers.

DiscussionA. C. Sullivan, G. H.Couch, and J . A. Dembecki (Electricity Com-mission of New So uth Wales, Australia): The authors are to be congrat-ulated for their description of some alternative metho ds for detectingand identifying bad data in power system estimation.We in the Electricity Commission of New South Wales have beenactively assessing the feasibility of applying state estimation techniquesin our system and feel qualified to com men t on his paper. Our ob-servations are, in large, of a general nature.We have experienced similar effects using non qu adr atic criteriaas the authors[ 11. In particular we have found it desirable t o performone or two iterations with weighted least squares WLS) before intro-ducing, say, the quadratic-square root criterion. This has he effectof helping to speed convergence, avoiding spurious minima andallowing more flexible selection of penalty function breakpoints.

The authors discussion of n etwork decomposition, and their em-phasis of the requirem ent fo r local redundancy are most interesting. Inmate of states electrically remo te from that error would indicate thatfact, the relatively sm all effect of one measurem ent err or on the esti-state estimation is a task which is ideal for application of decomposi-tion techniques. This is, however, clearly conditional on the edundancycondition s in each locality.found convenient to inhibit the non quad ratic penalty function withIn applying state estimation to the N.S.W. system[ 11 it has beenvoltage measurement or where busbar real and reactive power in-respect to selected measurements. For example, if there is only onejections are know n to be zero, such measurements should no t be sub-ject to this form of bad d ata suppression. I n general, care should beexercised when t here are interacting measurem ents having widely dif-ferent standard errors.The residual search procedures described in the p ape r, particularlythe rn-test, are a result of the fact that the largest residuals from aWLS estimate do not necessarily correspondwith the bad data. Thenonquadratic criteria attempt to avoid the necessity of making severalestimates t o eliminate bad d ata. It would seem that, provided availablemodifiedpenalty function, should not simply be a function of thecomputing ime is sufficient, the matrix G , used to implement theresiduals r and measu rement errors R, but also of the residual sensitivitymatrix W. Both he sensitivity of the rn-test and he p erforman ce ofthenonq uadra tic criterion may be then used to greater advantage.Depending on the algorithm chosen for estimation, thestate covari-ance matrix may be available in triangular factorised form from whichthe residual covariance may be co mp uted with little additional effort.In practice it would be well to remember, that in the final out-come, it is the system operator who, once alerted, is perhaps the mostcomp etent identifier of bad data. With this qualification, the detectionphase must be regarded as having prime importance, with precise iden-tification being subordinate. How ever, any nformation which can begiven to the operator concerning the probable locality of the bad datawould be of the greatest assistance.

REFERENCE[ 1 G. H. Couch, A.C. Sullivan, and J. A. Dembecki, A StateEstimatorOriented to a Semi-Isolated 5 GW Power System,IEEE 1974 Summer Power Meeting, paper num ber C 74 346-3.

Manuscript received July 25, 1974.

Hyde M . Merrill (American Electric Power Service Corp., New York,N. Y.):This very informa tive paper consists of a survey and impor tantextensions of previous work in addition to work reported here for thefirst time. It is required reading for any one interested in power systemManuscript received August 5, 1974.

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because the bad d ata/model rrorproblems are sothis application.One point tha t I wish the authors had stressed more than they didh = Q he largest normalizeddual flags the bad d ata poin t. This ends additional confidence todual search metho ds.In equation (CS), which is part of the descriptionof the non-HTGTE-IGWJ can be replaced withe s impler @E-l or wi th any other gain matr ix suitable for WLSon. Further more, since the right hand side of (C5) involves Gp only as a product 2 = Q, most of the complicated erms inr instance, for curve @ , si = ri or= s i g n ( r i ) / m . s im il ar ca ncelatio ns o cc ur o r h eother non-This means tha t he practicaldifferencebetweenon WLS and nonquadraticia is very mino r:

WLS: solve B(kk+l-&k) = HTB-inonquadratic: solve B (gk+ l - jk ) = _HTB-Is,5 is a simple func tion ofz.When nonqu adratic criteria were fv st considered, it was with thethat an estim ator could be designed that would operate directlyraw data to give good answers. WLS estimators must converge to aand identify bad data, throw itout, and re-that the nonq uadratic estimator would weightout fter a detection and dentification process. The

This example involved two 100 bad data points. The residualsgiven in Table I11 seems to indicate that this hope may be@ ,@ , and @ were all close to 100. In ef-this means that the nonqu adratic estimato rs subtracted 100 (themeasurement error) from each of the bad measurements in thean estimate 2 to the measurements p . Removingbad m easurements altogether would not have given a better . Istypical of the results the authors obtained, and if so, do they reallynonqu adratic estimators should be used merely to identify6? Or do they feel that with non-estimators the detection-identification-removal steps may no

residual with 5 This is fine f or the algorithm of refs. 1 4 , whichThe app lication o f non qua drat ic criteria given involves replacingon I, ut this quantity does not appear explicitly in5. Dr. R. D. Masiello suggested inconversation ecently that henonquadraticconcept could be ap-One possible way of doing this5 : the WLS AEP algorithm computesgo, El, . . . that converges to an estimate of g, the bus&+I is fou nd by solving

(BTDB)Ek+l = BTD( mk-A&g), (a )k is a func tion of the ctual system measurements ndThe test for onvergence is applied by computing

PEk+ 1 = Ek+ 1 - Ek.+I C= Q, onvergence has occurred.now be given. From equation (1 1) of ref. 5 ,A theoretically and comp utationally equivalent restatem ent of the

BEk =Xck -

and (BTDB)Ek = BTD(lrck- @g). (b )Computing &+I directly by subtracting (b ) from (a) gives

(BT ?B@k+l = BT ?(Xmk-Xck)(BTPB)&k+l = gT ?Gkl&-S:k). (C)

The equivalent AEP algorithm is: comp ute &+I from (c). If it is = 0convergence has occurred. In any event, &+I = Ek + +I . Finally,since -k k =1 he residue,(BTpB)AEk+ = BTpB21*. (dl

Applying the nonqu adratic concept is now simple: replace I* in (d)withz*, where 5 is calculated as n the present paper.Do the authors feel this exercise is legitimate? O r do they havemator? Or do they feel it c annot be done?a better way of applying the nonquadratic criteria to the AEP esti-

E. Handschin, F. C. Schweppe, J . Kohlas, and A. Fiechter: We appre-ciate very much the efforts of he discussers in commenting and ex-panding on a few points of our paper. This contribu tions are a wel-come addition.The important result mentioned by Dr. M e d l , that or =the largest normalized residual eaxoincides with the bad data loca-tion is based on the condition that only one bad data point is present.lating the normalized residuals. T h i s must be emphasized because theFor identificationsufficient comp uter capacity is required for calcu-computation of the normalized residuals qq nvolves the calculation ofthe covariance matrix & of the residuals.As shown in Table CI, hecalculation of the produ ct G p leadsto thecancelation of many erms.Thus, there is no significant dif-ference between programming a WLS and aNonquadraticestimator.Furthermore, he use of nonqu adratic estimators yield resu lts closeprocessing. Indeed, the results in Table I11 indicate the equivalenceto thoseobtained by a WLS estimator with bad data removed beforebetween a WLS estimator with bad d ata removed and a nonquadraticestimator. However, there are twopoin ts o be stressed. First, non-quadratic riteria give good esults provided the normalized costfunctions are used; i.e., the break poi nts Xi in Fig. 5 are calculated asmentioned at the end of Appendix C. From a computational point ofview this approach requires a considerable amou nt of extra calculation.The results in Table 111 are obtained with normalized nonquadraticerror riteria.Second, rom convergence point of view it is ad-vantageous to start with WLS estimation.performance is excellent if local redundancy is high and the numberThe nonquadraticcriteriaproposed are nonconvex. Thus, heirof bad data is mall. Otherwise the nonquadratic approach may leadto unstablesolutions. It has been found hat the four stepapproachof Fig. 2 is safer; thu s special dete ctio n and identification steps arenecessary in order to be applicable in a wide variety of cases.The use of a nonqu adratic criterion with the line-only metho dseems to be a useful approach. It has not yet been numerically tested.

Manuscript received November 14, 1974.

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analysis for power system state estimation

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