Decision Theory Applied to Bad Data Identification in Power System State Estimation

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DECISION THEORY APPLIED TO BAD DATA IDENTIFICATION IN POWER SYSTEM STATE ESTIMATION L. Mili*. Th. Van Cutsem and M. Ribbens-Pavella DI'jlllrllllflll ()f Elnl riml ElIgillPerillg, Cl/il'I'nil.\' ()j Li
.)11 rl -Tillllflll ,

B28,

Abstract. This work proposes a method to identify bad measurements in electric power systems, whene v er they are detected. It is based on the hypothesis testing theory in conjunction with estimates of measurement errors obtained through a linear estimator, New results additional to those presented in a previous work (Mili et al. , 1984) are set forth. In particular , it is shown that the proposed esti mator is optimal, Moreover , an identification strategy under a bounded ~ risk is explored. It is found to combine reliability and computational efficiency and hence to provide an appropriate on-line identification algorithm. Keywords, State estimation; power systems; bad data identification; bad data detection; decision theory.

sive measurement eliminations followed by state reestimations , the measurements being processed either one-by - one or by g roups, The other class comprises the non-quadratic c riteria where the suspected mea surements are rejected during the very state estimation converg ence. But while both of these classes general l y provide good results in the case of single BD , they become ineffective as soon as multip l e and "interacting " BD appear. This is mainly attri butable to the relative l y low measurement redundan cy currently available in power systems (it generally varies between 1 ,5 and 2.5) , and to the resul ting dependence of a residual on many BD, especial l y those corresponding to electrically c l ose measurements ( " interacting" BD).

NOTATION

N,B. W-l.th bOme ObV-<'OU6 exc ept,,{,oYL6, toweJtcct6e-l.ta.U.c WteM -<'l1cUcate vectoM, capda.£. da.Uc and cap-l.tat Glte.ek teUe.M deno-Ce matJt,{.cVl, P [x::

A1 :

the probability that a random variable X assumes values less than or equal to A . Na : the va l ue of a standard normal random variable X such that p[x:5Nal=a 2 X 0" N (fJ ,0 ) indicates that X is a normal random 2 variable with mean J1 and variance 0 E[xl : expectat i on of the vector x. ID (D diagonal matrix) = diag (1Dii) , I : identity matrix .

Contrary to the above methods , the Hypothesis Testing Identification (HTI ) uses variables Wh06e. mean

INTRODUCTION

va.£.ue6 depend only 01'1 thUtt COltlt e6poncUng mect6U11.emen t s. These variables are the estimates of the measure-

State estimation software is at the heart of the security monitoring of a modern electric control center. Its basic role is to generate a complete I coherent and reliable data base from a set of mea surements, even in presence of grossly erroneous ones , whatever their origin (instrument failure, data communication error, calibration bias I nonzero sampling time under major transients, etc,) . To accomplish its task, a state estimator must be able to c le ar the set of measurements from such bad data (BD); this in turn implies its ability to detect and to identify BD whenever they occur (Schweppe et al., 1970) .

ment errors obtained through a linear estimator. Moreover, HTI assesses the quality of each measurement on the bas i s of a hypothesis testing applied -<.nd-<.v-<'duatty to each measurement error estimate. This paper reports new results of the HTI method, which was first presented and developed in Mili et al. (1984). In particular, the optimality of the proposed linear estimator of the measurement errors i s formula ted and discussed, A detai l ed description of the bounded (3 risk identification algorithm is also given. Finally, an example using typical data currently encountered in power systems illustrates the proposed method.

Unl i ke the detection which is based on a sound foundation given by the classica l testing theory applied to the residuals (normali zed or weighted ones) and to the quadratic cost function , the identification did not receive until recently a conceptually per se and practically satisfactory solution. Instead, the proposed identification procedures were merely extensions of the detection step (Handschin et al., 1975); indeed, they were relying on the mea surement residuals, declaring "suspected" those measurements whose residual moduli are lar ger than a given threshold. Two broad classes of methodolo gies were thus proposed (Mili et al., 1985). The one comprises the " identification by eliminat ion " procedures and consists of performing succes-

STATIC STATE ESTIMATION IN POWER SYSTEMS A power system state estimator aims at finding an estimate i of the true state X which best fits the measurements z related to x through the nonlinear model z = h (x) + e (1) where Z is the m- dimensiona l measurement vector, generally consi sti ng of active/reactive line power flows and nodal power injections as well as some voltage magnitudes; X is the n-dimensional state vector consisting of N vol tage magni tudes and (N-l ) phase angles (referred to a reference node); e is the m - dimensional measurement error vector; its i-th component is :

* On leave from "Societe Tunisienne de l'Electricite et du Gaz" (STEG) , Tunis, Tunisia.

945

L. !'.Iili. Th. \ '
9-Hi

CutSt'1ll

and M. Ribbens-l'a,t'll
- a normal noise N(O,Oi) if the corresponding measurement is valid, - an unknown deterministic quantity otherwise.

where the weighting matrix P rocal of cov (~) I -1 P = ( cov (a.,) ) =

has to be the recip-

Moreover, we shall denote by 1) = m/ n the global measurement redundancy, and by R = diag (Oi) the covariance matrix of the measurement noise vector.

in order the estimate to be of minimum variance. This estimate is ca lled "optimal" in the sequel.

The weighted least squares (WLS) estimate is the value of x which minimizes the quadratic index 1 J(x) = [z- h (X)]T R- [z-h(x)] (2)

Let us consider the particularly interesti ng case p = s , corresponding to a redundancy equal to one (p = s) ( Xiang Nian-de et al., 1982) ; estimator (11) then becomes (13 )

The m - dimensional residual vecto r is by defin ition = Z - h( x)

l'

(3)

Linearizing ( 3 ) yi e lds

We

l'

where

'" is the W

and H is the

(4 )

(m x m) re s idual sen s itivity matrix I - H (HT R- 1 H) -1 HT R- 1 (5) (m x n)

jacobian matrix

The matrix W has the following remarkable properties : r ank (W) = m - n = k In t he absence of BD , the mea sur ement resi dua l vec tor is distributed p'\,

N (O ,WR)

I n pra ctice, it is convenient to introduce t he norma li zed or the weigh ted residual respectively given by

.J (WR) i i

neceooM!f a.nd '~115 6-ic-ie.nt condilioll 05 the. ew te.nc e. 06 POE -to that ma.t4-ix Wss to be. ~e.911~, -i.e. that .the s me.MMemel1,t-6 to be -6-imu.Ua.neol1-6R.y ~edw1da.lU. Optimality of POE

(6)

H = o h/ox

Because this estima tor realizes no filtering and assumes that the observation vector r s i s perfect, we call i t "POE" I the "Perfect Obs e rvations Estimator" (Mili et al. , 1984). Let us at once obs erve that the

(7)

ri/oi OPTIMAL ESTIMATION OF MEASUREMENT ERRORS

From the above co nsidera tions one could expect that the quality of th e e s timate s given by the optima l estima tor (eqs . (11) and ( 12» would improve , i.e. the filt ering of the observation noise vec tor dp would increase, when the number of re dundant observations (p - s) increases. In other words, for a given se t of se l ected measurements , one could think that the best filtering would be obtained when the corr esponding er r or s e s are esti mated by mean S of the maximum a va ilab le redundant Obser vat i ons , namely the p = k independent residuals. According to the same reasoning, one could think that POE is not the most "accurate" estimator. Nevertheless, it is pos sible to show (Mi li, 1984) Che f ollowing r emarkable

e

THEORE,\{ : The opuma.l e.,~ uma.to,'t 9-ive.n by ecp . (11) a.nd (12) ~educ e-6 to .the P~6ec.t Obo~va..t-ton6 E6UmatM 9~v en by: e = ri~ ~ l's (13) s

I n this sec ti on we inve st i gate t he measurement e rr or estimators of the type of Fisher . Basic Mathematical Model On the basis of eq. (4) and knowledge of vector 1', we inte nd to determine es timates fo r s of e s some "-6ete.c.te d" measurements s; these are measurements chos en amo ng the suspected ones (see " Introduction" and "Measurement Se lection" below) . For the reasons given hereafter, the upper bound o f s is smax = m - n = k . In the following, we shall also be using subscript t (t for "true") whe r e

e

t=m-s

.

Since rank W m - n = k , it is possible to select p (p:: k) among the m relationships contained in (4)

:

(8)

in such a way that the (p x m) submatrix Wpm is of fu ll rank, i.e. rank Wpm = P . Further, let us partition e and Wpm in (8) as follows (9)

where Relationship (9) may be regarded as the particular model of a linear system whe re : Pp es

~s P

is is is is

the the the the

(p (s (p (p

x x x x

l) 1) s) 1)

observation vector, vector to be estimated, observation matrix, observation noise vector such that

E[~] = "'pt E[ e t ] = 0

a nd

Measurement Error Estimators

cov

(a.,)

= Wpt Rt W[,t (la)

We attempt to estimate e s from l"p . Since the corresponding model (9) is linear, the "best li near un biased" estimate of e s is given by the well-known expression (Sorenson , 1980 ) ~ -1 WT P ps 1'p (11) e s = (psPWps )

rl

In fact , this sur pri sing theorem is i n herent in the idempotence property of matrix W. Its analytical proof may be found in Mili (1984). It may also be shown through the following intuitive reasoning. Assume tha t the optimal estimator would achi eve an additional filtering of e t to that p rovided by the s ystem state estimator (namely add itio nal filtering of d p = f/p t et) , This would imply that the quality o f the electrical s tate would be be tter when correcting the e rroneous measurements via s ra ther than when e liminating them. This additional filtering would be provided from valid information contained i n the sele cted bad measurements . Obviousl y , this reasoning lead s to a n a bsurd statement.

e

Main Properties of POE POE has s eve ral remarkable pro?erties which are summariz ed he reaf ter (Mili et al., 1984) 1) the : r ro rs on s equal the errors on h s (xc) , where hs (xc) d enotes the estimated sub vector of h(xc ) corresponding t o Che case where the s measurements have been elimi nated ( c stand s for "correctedO) i 2) deducing s f r om z s amoun ts t o eliminat i n g these s measurements. Indeed , the residuals l's of these corrected measurements ar e found to b e zero; 3) th e accuracy of s incre ases when the number s of selec ted measureme n ts de creases; 4) the a ccuracy of s incr e ases when the accuracy of the remaining measurements increases.

e

e

e

e

Estimati on Error Ana lysis

e

The expr ess ion of the estimation e rror lies = e s - s c an b e ded uced through suitable partitioning of (4), the consideration of its s part (1 4 ) and further the sub s titution o f the la t ter into (13) . 1 Note that cov (a.,) when p > t .

may be singular, for e xample

'1-17

Bad 1)ata Id en tification in Po\\'er S\'stem This yields e s = W~! (Wsse s + Wste t ) = e s + ;1~!Wste t (15) Equation (15) leads to the following conclusions, (i) If all t measurements are valid , E[ "' t]= O =E[Oesl e s]= O =E[es le s ] = -" s

(16)

In this case, unlike the residuals , the mean value of s is equal to the true measurement error 2 As for the covariance of Oes' it can be shown that (Milietal., 1984):

e

cov (Oe

s

le s ) 6 E[Oe

s

1

(W-

ss

o! s le sl

-1)H

s

(17) 5

W- 1 (26) ss We observe that the variance of eSi is equal to the sum of the variances corresponding respectively to the true error and to the accuracy of POE.

r

where

=

Rvn~k. In the case where POE is biased, the probability densities (21) and (24) are no longer true since is a funct i on of all those BD which have s not been se l ected (i.e. those which are included in the list t ) .

e

Making Decisions

where I s is the (s x s) identity matrix. (ii) If there are BD among the t measurements,

UndeJt .the hypo tlte.~.u.. Ho ' let us define the normal ized error estimate as :

In this case, POE is biased.

Its

eN =..;;::r e s i - th

eS1'

_

e Ni =

THEORY OF TESTING APPL I ED TO MEASUREMENT ERRORS

The measur ement error estimates will be used as the variables of concern for identifying BD . Because of the random character of these variables, the theory of testing appears quite appropriate to accomplish this task. Three main steps have to be processed : (i) formulation of the hypotheses to be tested; (ii) determination of the probability densities of the decision variables under the various hypotheses, viz. of the measurement error estimates; (iii) definition of the decision rules based on a tradeoff between the a and {3 risks. The Hypotheses to be Tested They apply to each measurement .tl1di.v.tduate.y, and express as follows

Ho

the measurement is valid; the measurement is false.

:ta/ze.H M a whote.

e

(i) When the s measurements are wrong , the true error e s is an unknown variable. In this case, ( 15) yields ( 19) E[e s ] = E[e s ] = e s and (20) cov (Cs) = (W~~ - Is ) Rs Hence - 1 (21) s '\t N (e s ' (Wss - Is) ·"s)

e

measurements are valid, (13) yields - 1

~

and

E[ e s ] = Ws s E[r s ] = 0 .- 1

~

T

(22)

. -1T

cov (e s ) = '';ss E[rsr s ] (Wss )

or , since

E[l'sr~ J

= WssRs' ~

Hence

cov (e s ) =

6s

",-I

ss

R

s

'\, N (0, "'~! Rs )

=

Note that if

Ho

is true ,

i = 1 , ... , s

(28)

e Ni '\, N (0,1)

The hypothesis testing is performed by (i) choosing a level of significance (the ex risk) and thereby an " identificat i on threshold " (N _<:!.) i of the doub l e 1 tailed test and (ii) deciding 2 that Ho

is true if

Ho

is false

leNi l < (N _<.l.) i ' i= 1 , ... , s 1 2 is true) otherwise.

(29)

(HI

Each decision is taken with a certain ex risk of declaring false a measurement which is valid, or a certain {3 risk of declaring valid a measurement which is false; the value of (N 1 _cx)i materializes the tradeoff be tween these two 2 error probabilities. Let Pi denote the "identification pro.::. babili!y" which is the probability of accepting HI when HI is actually true; Pi is the complement of {3 (Pi = 1-13) an d characterizes t he power of the identification test.

Let us bility rupted {3 risk

explore how ev o lves the identification probaof the i-th measurement, assumed to be cor by a gross error es i . The expression of the is (3 = I - Pi = P [ le Ni (N _'!) i] (30) _

In what follows , we shall assume that POE is unbiased, i.e. that the t measurements are cleared from BD (E [ e t] = 0 ) . The decision making is based o n the knowledge of the probability densities of corresponding to the two possible kinds of meas surements : wrong and valid.

s

_

Jvar (esi)

I :::

The Probability Densities of Measurement Error Estimates

(ii) When the

(27)

Identif;cation Probability Analysis

Note that these hypotheses differ totally from those used in the detection test which concerns the pre sence (HI) or the absence (Ho) of BD among :the. mea~lL~emefl:t.6

(W~! Rs)

D = diag

where

component is

(23)

_1 2

Note that unde:: hypothesis HI ' eNi is no longer N(O,I) . Let ~Ni denote the normalized error estimate when H1 is true : e si - e si

~Ni =

Then

1-

p [

- X -le 1

0i J with

(31 )

°i J r ii -l

'I

S1:::

rii - 1

~ Ni :::

X1 - l eS1 · I

Oi~ 11

]

( 32)

~

Eq. (32) allows computing the ident i fication proba bility Pi in terms of ex , r ii and of the weighted error eWi = esi/Oi . Observe that the identification probability Pi incr e ases when : - the ex risk increases , i.e. the threshold (Nl_~) i decreases; - the magnitude of the measurement error increases; - the value of r ii decreases, i . e. the accuracy of eSi increases , see fig.1; - the number s of selected measurements decreases; this property derives from property 3 of "Main properties of POE " .

(24)

This result can also be inferred from (15). In par ticular, for the i - th measurement, it comes

The validity of this assertion depends upon the va lidity of the linearized model (4).

R~n~tk. As already mentioned , if some BD are not selected , POE becomes biased and the hypothesis testing looses its power. Next section provides a means to circumvent this difficulty.

L. \[ili . Th. \"an CutSCIIl and \1. Ribbcns·l'a\clla A Reliable Identification Strategy Pi 1,0

11

0. 8

r\ ~ l'-..

\

::3

0,

0.

a Fig.

w

r---

A BD identification strategy which naturally comes to mind is the one built on the classical decision theory app lied to the measurement error estimates as outlined in "Making Dec isions". This theo r y sug gests indeed to choose a small 0: risk and to compare the normalized error estimates t o a fixed thresho ld N1_~ whatever the iJ risk. However, because the latter may be large for inaccurate mea surement error est i mates , the test may fail to identify some BD.

\

0. 6

le~ ,I,lo0

1'\

The quest ion of concern here is "how to identify the BD actua lly present in a first list of select ed measurements" and only them.

~

t---

~

~ fwo

200

--600

~i

1. Identification probability

Pi vs. fii for various magnitudes of the weighted

measurement error

esi eWi = 0-:-

with

~

HYPOTHESIS TESTING IDENTIFICATION ALGORITHM In this section , we establish an identification pro cedure based on a selection criterion able to distinguish the suspected from the valid measurements. The adequacy of this cr iter ion is of great concern , since the very validity of the measurement error model depends on it. The TWO contradictory Aspects The question of how to draw up the list of selected measurements has two contradictory aspects . Indeed , on one hand , one shou ld select as many as possib l e measure ments in order to include all BD, t h~s avoiding b iase dness of POE (k = m- n is the maximum number of selected measurements). On the o ther hand, the analysis of the power of the identification test argues for the opposite thesis, since the smaller the number of selected measurements , the higher the probability Pi (see "Identification Probabili ty Analysis"). Obvious ly, the optimal select i o n consists of the BD only , since this provides the best esti mation of the measurement errors with the largest identification probability. The above contradiction can be evaded by proceeding to successive refinements of the list o f suspected

This difficulty can be circumvented by perfor ming successive cycles of detection - identi ficat i on - el imination-sta te reestimation until the detect io n test becomes negative; at each cycle a group of BD are ide ntified. Another more efficient way t o avoid the above difficulty is provided by a str ategy res u lting from modificati ons -r efinements of the previous one. In essence, this new strategy consists of : ( i) decreasing the threshold (N1 -~)i in such a way that the iJ risk is fixed for a given gros s error, i.e. in suspecting all those measure ments whose corresponding measurement error es timates are not accurate enough; (ii ) perfor ming successive r efinements of the list of suspected measurements so as to remove the mea surements already decla red valid by the hypothesis test. The process s t ops when all the selected mea surements are labe ll ed fa lse. The eli minat ion (at each cycl e ) of some valid measurements previously suspected and the concomitant increase in the power of the identification te s t allow decreas in g the num ber of se le c te d measurements and make the list conve r ge towards the one containing the BD on ly . This str ate gy is further developed and discussed hereafter. Identification Test With Bounded

Risk" below .

Let us determine the relationship between t he thres ho l d (N1 _~) i and the iJ risk. From (32) , it results that

iJ::::

P [ ~Ni ~

(N1 -9; ) i v ' r : - l e Wi l 2

and there fr om that leWil

Mea surement Selection

When the only available information is that provided by the state estimator, a first evaluation of the degree of the measurements susp icion may rely on the magnitude of their corresponding residuals (nor malized preferably to the weighted ones). In this case , the measure ments ar e selec ted (either one-by one or by groups) according to the decreasing mag nitudes of their residuals. But till wh ich bound? This question may be answered via a J - test. A com putationally convenien t way to carry out this tes t consists of using the correction formula derived by Ma Zh i-qi ang (1981) • • T n- 1 • ( 33) J(xc ) = J(x) - 1"s"s e s where J (x) and J (Xc) are the values of the qua dratic cost funct i on co rresponding respect i ve l y to m and (m-s) measurements. Note that J(x c ) is chi-squared with (k -s ) degrees of freedom. This test informs about the existence of BD among the non selected measurements provided that these lat t e r are redundant . But it becomes ineffective as soon as a measurement stops being redundant , i . e. as soon as it becomes critical in the cour se of the elimination process. This has to be kept i n mind when deriving an identification algorithm.

Risk

The leading idea is to perform a hypothesis test that guarantees a high identification p r obab ility for all gross errors large r than a given va l ue .

measurements; this is described in "Ident i fication

Test with Bounded iJ

iJ

(34)

Jfii - 1 +NiJ~l

(35)

~

Discussion. (i) If we choose a smal l iJ risk (for example 1 %) for a given we i ghted gross error eWi = l e Si I / Gi ' the threshold (N1 _'¥) i comput ed through (35) dec rea ses and hence the 0: risk increases when f ii increases . Roughly speaking , this mea ns that the probability of r eject ing a measurement i n creases when the accuracy of its e rror est i ma te de creases. Note that because of the appr ox imatio n in volved in eq. (35), it may happen that for large fii' (N 1 -" )i becomes nega tive; i n th i s case we shall set (N 1_~ ) i = 0 ; the probability of rejection is then 1 .

!

I

(i i ) Conve rse ly , for a decreasing r ii ' i.e . for an increa sing accuracy of esi ' the va l ue of (N1 _q )i may become too large. In this case , we pena li ze 2 the identification of BD of lower magnitude. For example, fig. 2 shows that for (N 1-q ) i = 11 the iJ risk is only of 1 % f o:: eWi = 30 ; 2 but it runs up to 99 % when eWi ! = 20 ! It is there fo r e advisable to choose an upper bound to the value of (N1-~ )i (a good va l ue is 3) by fixing a small 0: ris k" for t he accurate measurement error estimates. Thus, ~~e decision of declaring val id a se l ected measurement is made with a reasonable probability.

i

I

I

Bad Data Identification in Power System EXAMPLE

I ~i =5 I 1.0

/ 31

The simulations

20

/

•

0, 2

o

on

the

MW/MVar . ~

la

Fig. 2 .

been performed

- for zero injection pseudo-measurements : a::;: 0.3

./

0.0

have

well-known IEEE 118-node system. 472 measurements are taken on this network leading to a redundancy 17 ~ 2.01 . The following standard deviations have been used : - for power measurements : a ~ 1.7 MW/MVar at 132 kV and a ~ 5 MW/MVar at 345 kV except for the lines 9-10, 9-8 and 68-65 where a ~ 10.8 MW/MVar ; - for voltage measurements : a ~ 0.005 p . u. ;

1 I I I 1 1 ~ l/ V / ewd~IO

0, 5

0,

Network Features

V

If

o.a

949

A remote telemetry unit failure at node 65 comprising 7 erroneous measurements has been simulated. Table 1 summarizes their characteristics. These BD

are of the following types : - IN (injection) or FL (fl o w)

i3 risk vs. the threshold for l'ii ~ 5

of

P/ Q

(ac t ive /

reactive power);

-

Ivl

(voltage magnitude).

Identification Algorithm TABLE 1

Let us summarize the main steps of the identifica-

tion algorithm with bounded i3

risk.

Step 7. Order the measurements according to decreasing values of

IrNi l

or of

IrWil

.

Step 2. Add to the list s

of selected measurements one (or a group of ) redundant measurement(s) corresponding to the largest IrNi l (s)

St ep 3. Memorize the c

measurements which may not be selected because of observability constraints : they would become critical if the s measurements were eliminated. These measurements must be suspected.

Step 4. Estimate the errors of the through eq.

s

J(X c ) simulating the measurements. If the J (xc) test is positive, return to step 2 . Otherwise go to s

Step 6. Choose a small i3

risk and its corresponding Ni3 ~ b for a given weighted measurement error i eWi I "" a . Choose also a small a risk and its corresponding (NI-~)max' (N1_~)i

selected measurement as follows:

a+b~

Let if

vi ~ (NI-~)max

if

0 < vi < (N1-~)max

if

vi

~

0

take

for each

2

(N1_~) i ~ (N1-~)max

take

(NI_~)

take

(NI_~)

i

~

i ~ vi

0

Step 8. Perform the identification test and conclude that the i-th measurement is : - valid i f I"si l S; (NI_~)i _ suspect otherwise.

Step 9. I f some of the

measurements are declared

valid, perform a new selection of the suspected ones

along with those among the now b e s el ec ted. Estimate Otherwise, i.e.

c

measurements which can

es

and return to step 7.

Step 70. if some of the c been selected, perform one or a group of them previously s selected selected. Estimate the step 7.

Step 77. Eliminate the

measurements have never a substitution by selecting along with those among the measurements which now can be new vector s and return to

e

s measurements and carry out a state estimation. The measurements labelled false are the eliminated ones along with those which have bec ome c ri t ical.

!e Wi !

317.3 - 78. 5 lOB. 7 124.4 211.0 133.5

5.0 5.0 1. 7 1.7 5.0 5.0

53.5 15.7 53.9 73.2 42.2 25.7

- 0.055

0.005

11.0

500.0 - 40 .0

FLP FLQ FLP FLQ INP INQ

65-64 65-64 65-66 65-66 65 65

182. 7 3B.5 8.7 64.4 391.0 63.5

0 -- 100. 60.0

! V!

65

1. 005

0.95

180. 0 - 70 .0

-

The detection test performed after a first state estimation, indicated the presence of BD: J(x) 6330.6 whereas X~.Ol ~302 . 1 . On the other hand, 66 suspected measurements have their IrNil larger than 3. The selection of 22 measurements chosen by decreasing I rNi I makes the J (Xc) - test negative (J (Xc) ~ 180.5 is less than X~.Ol ~ 277.5) . This test guarantees (with a certain i3 risk) the validity of the redundant measurements among the non-selected pected measurements (namely INP49 and INP67) which have not been selected because of observability constraints. Three cycles have been required; the results are

reported in Table 11. The thresholds (NI_~)i are computed through eq. (35) with i3 ~ 1 % corresponding to Ni3 ~ -2.32 for I eWi I ~ 20 . In the first selection, we observe that 8 measurements including

2

s

°i

ones. However it does not inform about the two sus-

i = 1, ... ,s

y'1'U

esi =

zi - hi (xl

Detection and Identification Results

Step 5. compute the value of

Step 7. compute the threshold

"Measured lL value zi

measurements

(1 3 ).

elimination of the

BAD DATA

Actua 1 va 1ue hi (x l

the BD have been suspected. Note that four inaccurate measurement error estimates have their (N1-~)i less than 1. That renders the INP65 suspected 1 whereas it would not · have been identified if the threshold was fixed to 3 In the second selection,

10 measurements are re-

tained including the two previously suspected and not yet selected ones. Note that the increase in the estimate accuracy makes the thresholds equal to 3. Finally, the third selection comprises only the BD which are all identified.

L. l\lili. Th. Van Cutsem and M. Ribbens-Pa\'ella

9,,0

TA8LE 2 Sel ected measurements

esi

65-64 65-66 66-49 65-66 67-66 66-49 116 65 48-49 68-69 67-62 56-59 62-67 65-64 57 68-69 64-61 63-59 38-35 65

IVI 65 FLP 68-65

317.30 - 124.40 - 0.40 -108.70 1. 26 - 2.58 - 5.70 - 211.00 - 1. 58 1. 50 0.15 - 1. 72 - 1. 33 - 78.50 - 0.57 2.20 3.44 - 0.26 1. 30 - 133.50

318.79 - 121. 79 3.12 - 108.04 0.33 - 0.14 - 36.96 - 178.72 1. 90 0.41 1. 96 - 4.42 - 3.00 - 78.20 0.12 2.29 4.65 - 1. 37 2.25 - 130.57

5_58 7.56 13.00 25.19 4.73 4 . 80 41. 07 62.71 7.33 3.77 5.03 2.52 4.92 6.03 5.92 2.86 3.25 2.54 3.01 13.94

- 0.055

- 0.054 - 37.33

- 9.29

eN;

SECOND

57.09 - 16.11

-----0:24 - 4.29

3.00 0.13

----0:26

:-z:ss

0:00 2.44

0.11 0.39 - 1. 75 - 0.61 - 12.97

3.00 3.00 3.00 3.00 3. 00

---o.B2

2.5I

0.80 1.43 - 0.54 0.75 - 9.37

3.00 3.00 3.00 3.00 ~

0.005

- 10.24

40.07

- 0.93

~ 3.00

SELECTI

oN

I I~Q

317.38 - 125.10 - 109.14 - 8.02 - 212.13 - 80.01 - 137.91

5.52 4.26 5.28 11. 30 14.56 5.45 9.89

I VI

65

- 0.055

- 0.055

0.005

- 10.73

IN? INP

49 67

6 . 04 2.66

0.36 - 1.28

2.17 - 3 . 41

T H I RD

0.00

3.00

- 0.03 - 0.90

317.30 - 124.40 - 108.70 - 5.70 - 211.00 - 78.50 - 133.50 0.60 - 0.90

3.00

2.24

Q.37

---0:07

65-64 65-66 65-66 116 65 65-64 65

FLP FLQ FLP INP INP FLQ

(N1_~) ;

SELECT ION

F I RS T FLP FLQ FLP FLP FLP FLQ INP INP FLP FLP FLP FLP FLP FLQ INP FLQ FLQ FLQ FLP INQ

Jvar (e s ;)

€si

57.51 - 29.36

-:zo:b7 - 0.71 - 14.57 - 14.67 - 13.95

3.00

3.00 3.00 3.00 3.00

3.00 3.00

~ 3.00 3.00

S E L ECTION

FLP FLQ FLP INP FLQ INQ

65-64 65-66 65-66 65 65-64 65

317.30 - 124.40 - 108.70 - 211.00 - 78.50 - 133.50

317.02 - 124.81 - 108.95 -217.98 - 80.10 - 137.58

5.51 4.23 4 . 87 10.48 5.45 9.87

I VI

65

- 0.055

- 0.055

0.005

57.56 - 29.54 - 22.35 - 20.79

.:...li...M

3. 00

3.00 3.00 3.00

- 13 . 94

3.00 3.00

- 10.73

~

CONCLUSION The measurement error estimates provided throughan optimal linear estimator have several interesting properties. For these variables, the shortcoming of the BD interaction vanishes. Their use in conjunction with a hypothesis testing under bounded ~ risk allows elaborating appropriate identificationstrategies. Based on the latter, a practical algorithm is derived; its performances are found very satisfactory for the effective on-line BD icentification in power system state estimation.

REFERENCES Handschin, E., F.C. Schweppe, J. Kohlas, A. Fiechter (1975). Bad data analysis for power system state estimation. IEEE Trans. on PAS, Vol. PAS-94, ~, pp. 329-337. Ma Zhi-qiang (1981). Bad data reestimation-identification using residual sensitivity matrix. Proc. of the 7th PSCC Conf, Lausanne, July, pp. 10561060. Mili,L., Th. Van Cutsem, M. Ribbens-Pavella (1984). Hypothesis testing identification : A new method for bad data analysis in power system state estimation. IEEE Trans. on PAS, Vol. PAS-103, 11, pp. 3239-3252. Mili, L. (1984). Algorithmes fiables d'identification des fausses donnees par tests d'hypotheses. Internal report, Univ. of Liege, No. MBC/3.

Mili, L., Th. Van Cutsem, M. Ribbens-Pavella (1985). Bad data identification methodsin power system state estimation - A comparative study". Paper No. 85 WM 060-9. Presented at the IEEE PES-Winter Meeting, New York, Feb. 3-8. Schweppe, F.C., J. Wildes, D.B. Rom (1970). Power system static state estimation. Parts I , l l , Ill. IEEE Trans. on PAS, Vol. PAS-89, 1, pp. 120-135. Sorenson, H.W. (1980). Parameter esti~ation : principles and problems. Marcel Dekker, Inc. Xiang Nian-de, Wang Shi-Y~ng, Yu Er-keng (1982). A new approach for detection and identification of multiple bad data in power system state estimation. IEEE Trans. on PAS, Vol. PAS-101, ~, pp. 454-462 . Ya-lun Chou (1975). Statistical analysis. Holt, Rinehart and Winston, 2nd. ed.