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2010 IREP Symposium- Bulk Power System Dynamics and Control – VIII (IREP), August 1-6, 2010, Buzios, RJ, Brazil

Bad Data Analysis Using the Composed Measurements Errors for Power System State Estimation

Bretas, N.G. Bretas, A.S.

Elect. Eng. Dept., University of Sao Paulo Elect Eng.&Computer Dept., UFRGS

Abstract-In this paper, a topological/geometrical based approach is used to define the undetectability index (UI), which provides the position of a measurement related to the range space of the Jacobean matrix of the power system. The higher the value of this index, for a measurement, the closer it will be to the range space of that matrix, that is, the error in measurements with high UI is not reflected in their residuals, thus masking a possible gross error those measurements might have. Using the UI of a measurement, the possible gross error the state estimation process might mask is recovered; then the total gross error of that measurement is composed and used to the gross error detection and identification test. The gross error processing turns out to be very simple to implement, requiring only a few adaptations to the existing state estimation software. The IEEE-14 bus system is used to validate the proposed gross error detection and identification test.

Index Terms—State Estimation, Orthogonal Projections, Gross Errors Analysis, Recovering Errors, Undetectability Index.

I. INTRODUCTION HE ability to detect and identify gross errors is one of the most important attributes of the state estimation process in power systems. This characteristic to deal with gross errors makes the results of the state estimation process preferable if compared to the SCADA raw data [1]. In order to mitigate the influence of gross errors on the state estimation results, some robust estimators have been introduced in power systems [2]. One of the first robust state estimators applied to power systems was the Weighted Least Squares (WLS) Estimator endowed with the largest normalized residual test [3] for gross error detection and identification. However, this combination is not robust in the presence of single and multiple non-interacting gross errors [2]. Also, this estimator is not able to reliably identify multiple interacting gross errors, especially when the errors are conforming and/or occur in Leverage Points, which are highly influential measurements that “attract” the state estimation solution towards them [4].

Other alternative estimators, which are more robust than the WLS Estimator, have been proposed. The Weighted Least Absolute Value Estimator (WLAV), for example, can deal better with multiple gross errors, but it is prone to fail in the presence of a single gross error at a Leverage Point [2]. The Least Median of Squares Estimator (LMS) is another estimator alternative [2], [4]. It is a member of the family of estimators known collectively as high-breakdown point estimators. It is inherently resistant to outliers in Leverage Points and can handle multiple interacting gross errors, even when they are conforming. However it requires excessive computing time for on-line applications [1]. The idea of exploring geometry to detect gross errors in power systems is not new. Based on a geometric interpretation of the residual estimation for single gross errors, a method for the detection/identification of multiple gross errors was developed in [5]. The authors claim that their method is able to determine whether the residual vector lies in a subspace defined by the suspect measurements (suspect in terms of having gross errors) and whether any portion of that subspace is orthogonal to the residual vector. They also claim that their proposition is able to find the suspect measurements of the measurement set; however, applications of their proposition are not known in a real environment. In this paper, using the WLS Estimator, and based on the concepts of references [10, 11], more insights related to the detectability of gross errors in power system state estimation using geometrical approaches are provided. This is achieved decomposing the measurement error into two parts: the undetectable part and the detectable one. The ratio between the norms of those quantities, the undetectability index (UI), provides a measure of how difficult it is to detect errors in those measurements. As a consequence the UI gives a more comprehensive picture of the problem of gross error detection in power system state estimation than the critical measurements and leverage points concepts. Moreover, based on the UI index, a technique to recover the error not reflected in the state estimation residual, that is, the masked error is proposed. Even more, a new measurement gross error detection and identification test is proposed. In this paper, instead of the classical normalized measurement residual amplitude, the corresponding normalized composed

T

measurement residual amplitude is used in the gross error detection and identification test.

II. II. BACKGROUND [10-11]

Consider a power system with n buses and m measurements. The power system is modeled, for state estimation purposes, as a set of nonlinear algebraic equations that is:

( )z h x e= + , (1) where z (mx1) is the measurement vector, x (Nx1) is the state vector, h: RN → Rm (m > N) is a continuously nonlinear differentiable function, “e” is the (mx1) measurement error vector with zero mean and Gaussian probability distribution, and N=2n-1 is the number of unknown state variables to be estimated. Since number m of measurements is higher than number N, a common solution to estimate the states variables is the well-known WLS method, which searches for state x, minimizing the functional ( ) ( ( )) ( ( ))TJ x z h x W z h x= − − , where W is a symmetric and positive definite real matrix. In power system state estimation, the weight matrix W is usually chosen as the inverse of the measurement covariance matrix. Functional J is a norm in the measurement vector space Rm that is induced by the inner product <u,v>=uTWv, that is:

2( ) ( ) ( ), ( )WJ x z h x z h x z h x= − = − − =≺

= ( ) ( )Tz h x W z h x⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦− − . (2)

Let x be the estimated state, that is, the solution of the aforementioned minimization problem, and define the estimated measurement vector as ˆˆ ( )z h x= . The residual vector is defined as the difference between z and z , that is, ˆr z z= − . The linearization of (1), at a certain operating point x* (z*), yields:

z H x eΔ = Δ + , (3) where x

hH ∂∂= is the Jacobean of h calculated at x*.

If system (2) is observable, that is, rank(H)=N [6], then the vector space of measurements Rm can be decomposed into a direct sum of two vector subspaces, that is,

( )( ) ( )mR H H ⊥= ℜ ⊕ ℜ in which the range of H, ℜ(H), is an N-dimensional vector subspace into Rm and ℜ(H)┴ is its orthogonal complement, that is, if ( )u H∈ℜ and

( )v H ⊥∈ℜ , then <u,v>=uTWv=0. In the linear state estimation formulation, (3), the state estimation solution can be interpreted as a projection of the measurement vector mismatch Δz onto the ℜ(H). Let P be the linear operator which projects vector Δz onto ℜ(H), that is, z P zΔ = Δ and ˆr z z= Δ − Δ be the residual vector for the linearized model. The projection

operator P, which minimizes the norm J, is the one that projects Δz orthogonally onto ℜ(H) in the sense of the inner product <u,v> =uTWv, that is, the vector ˆz H xΔ = Δ is orthogonal to the residual vector. More precisely:

( ) ( )ˆ ˆˆ, 0Tz r H x W z H xΔ = Δ Δ − Δ = . (4) Solving this equation for xΔ , one obtains:

( ) 1ˆ T Tx H WH H W z

−Δ = Δ .

As ˆz H xΔ = Δ , the projection matrix P will be the idempotent matrix:

( ) 1T TP H H WH H W−

= . (5)

Obs.: If W is a diagonal matrix given by W=cI, where c>0 is a real number, and I is the identity matrix, then P=PT and P is said to be an orthogonal projection. In power system literature, matrix P is usually called Hat matrix and it is also known as K matrix [4]. The residual vector is found to be:

ˆ ( )r z z z P z I P z= Δ − Δ = Δ − Δ = − Δ , (6) where the idempotent matrix (I-P) is the operator that projects Δz onto ℜ(H)┴. Matrix (I-P) is given by

( ) ( ) 1T TI P I H H WH H W−

− = − , (7)

and is usually called residual sensitivity matrix; it is also known as S matrix [4] in power system literature. Figure 1 illustrates operator P acting on vector Δz.

III. UNDETECTABLE ERRORS AND UNDETECTABILITY INDEX (UI)

PROPOSITION In this section, the decomposition of the measurement vector space into a direct sum of ℜ(H) and ℜ(H)┴ will be used to decompose the measurement error vector e into two components: the detectable and the undetectable one. For that purpose, let xtrue be the vector of the true unknown states and define ( )true truez h x= . Consider the state estimation linearized model,

Fig. 1- Operator P acting on the vector Δz: a Geometric interpretation

Δz2

Δz1

Δz3

Δx1

Δx2 Δz*=P Δz Δx*

Δz

where *true truex x xΔ = − ; x* is the estimated state for

the operating point z * , and true truez H xΔ = Δ ; assuming the available Jacobean matrix H is close to the one obtained with measurements without gross errors, then ztrue is close to ℜ(H) and *true truez z zΔ ≈ − . The measurement error vector is given by

true truee z z z z= − ≈ Δ − Δ , and can be written as ( )e Pe I P e= + − .

Denominating Ue and De as PeeU =: , (8)

ePIeD )(: −= , (9) respectively, the undetectable and detectable components of “e”, one has DU eee += . It is easy to see that eU ∈ ℜ(H) while eD ∈ ℜ(H) ┴ and as a consequence the measurement error vector e can be composed by:

2 2 2D UW W W

e e e= + . (10)

The next proposition shows that the undetectable part of the error does not contribute to the residual vector. Proposition 1: The residual vector, r, of the WLS state estimation is not affected at all by the undetectable component eU of the measurement error vector. Proof: The measurement vector mismatch can always be written as:

true true U Dz z e z e eΔ ≈ Δ + ≈ Δ + + . As a consequence:

DUtrue

true

ePIePIzPIePIzPIzPIr

)()()()()()(

−+−+Δ−≈≈−+Δ−≈Δ−=

. (11)

Since ( )truez HΔ ∈ ℜ and )(HeU ℜ∈ , one

has ( ) 0trueI P z− Δ = and also 0)( =− UePI . Consequently, equation (11) becomes: ( ) ( ) Dr I P z I P e= − Δ ≈ − De≈ ■ The previous proposition shows that the difficulty of detecting gross errors in some measurements is due to the fact that their errors have a significant undetectable component when compared to the detectable one. As a consequence, their residuals are small even when those measurements are contaminated with gross errors. The largest normalized residual test is often employed to verify the existence of gross errors in the measurement set. If the random error vector e has a normal distribution with expected value equal to zero, then it is well known that the normalized residual ri

N of the i-th measurement is a random variable with normal distribution, expected value equal to zero, and variance equal to one. Given a snapshot of measurements, one can calculate the normalized residual of each measurement and verify the existence of gross errors. More precisely, if the largest normalized residual ri

N is larger than a threshold value we

can affirm, with a certain degree of confidence, the existence of gross error in the i-th measurement. However, the operator (I-P) is not invertible and, as a consequence, the backward implication is not true, that is, small normalized residuals do not imply small errors in the measurements due to the presence of undetectable components of the errors. More precisely, the residual does not change if an extra error that lies in the vector space ℜ(H) is added to the measurement vector. In other words, when the largest normalized residual ri

N is not larger than a threshold value, one can affirm, with a certain degree of confidence, that only the detectable components of the errors in all the available measurements are small. As a consequence, the largest normalized residual test can often fail in the detection of gross errors for the measurements that have a large undetectable error component. However, if additional information is provided, it is possible to obtain more about the relationship between residual and the components of the errors in the measurements.

In what follows an index, UI, based on the detectable and undetectable component of the measurement error vector will be proposed. Also an algorithm to calculate the UI for each available measurement will be presented.

For that purpose, suppose the existence of a single error in the i-th measurement, that is, i ie bδ=

with [0 1 0]i

Tiδ = … … , and using equations (8) and

(9) define the corresponding undetectable )( iU bPei

δ=

and detectable ))(( iD bPIei

δ−= components of that error.

With that in mind, the following definition of error undetectability index for the i-th measurement is proposed:

i

i

U Wi

D W

eUI

e= . (12)

A critical measurement is the limit case of a measurement with high UI, that is, it belongs to the range space of the Jacobean matrix, has an infinite UI index, and its error is totally masked, and as a consequence, in these cases, the state estimation results will have a low quality. Remark 1: As can be seen from the equations for the detectable and undetectable error vectors of the ith

measurement, the UIi does not depend on the error magnitude b but only on the projection matrix P as well as on the measurement topology.

III.A. Algorithm to compute the measurement UI

(i). For the ith measurement, suppose the existence of a known error vector given by i ie bδ= .

(ii). Compute the undetectable (eUi) and detectable (eDi) components of the error vector ei as previously defined. With these vectors in hands, compute the ith

measurement undetectability index using (12). (iii). Perform steps one and two of this algorithm for all available measurements. This algorithm uses the estimated state vector obtained from the available measurement set. Using (12), which defines the measurement UI, it is easy to understand that a measurement with UI equal to one will present the detectable error vector amplitude just equal to the corresponding undetectable component. That is, the measurement residual obtained from the state estimation does not show the total measurement error amplitude. III.B. Recovering of the masked errors and gross error detection and identification test For the purpose of estimating the measurement total error amplitude, the following algorithm is proposed:

(i) Process the state estimation and compute the residual vector r , (r i= eDi);

(ii) Using the UI index of each measurement compute the undetectable error vector eUi , that is:

.ii DiU eUIe = ; (iii) With those two vectors in hands, and

knowing they are orthogonal to each other, compute the ith measurement composed normalized error (CNE) amplitude, using for that purpose the following equation :

22222 )1( ||||+=||||+||=|||||| DiiUiiDi eUIeee

= (1+UIi2)ri

2 (13)

where ir is the ith measurement estimated residual. (iv) Then apply the gross error detection and

identification test (HTI), as in the Appendix, but using the composed normalized measurements error (CNE), as in (13), instead of the normalized measurements residual.

(v) Once a gross error was detected and identified perform the measurement correction, using the estimated measurement error, and then again estimate the power system states.

Remark 2: The understanding of this paper’s authors is that Schweppe et al. [7]-[9], and many others, have wrongly applied the classical Statistical Hypothesis Test (HTI) in power system for gross error detection and identification. The reason is that the HTI requires the non existence of masking effect in the estimation process and

in power system state estimation the masking effect exists in a significant way.

IV. NUMERICAL TESTS For the numerical tests, the IEEE -14 bus system will be used (Figure 2). The parameters of this system are available in www.ee.washington.edu/research/pstca/. The power measurements values (zif) used in the tests were obtained from a load flow solutions to which normally distributed noises were added. The measurements noise was assumed to have zero mean and

a Standard Deviation, “σ”, given by*3

lfpr zσ = ; with

pr, the meter precision, equal to 3%. Scenario-I: In this measurement scenario the measurement set shown in Table I will be used.

Fig. 2 – IEEE-14-bus system.

Remark 3: For the next Tables the following nomenclature is used: Ii – injection measurement at bus “i”; Fi-j – flow measurement from bus “i” to bus “j”; Vi – voltage magnitude measurement at bus “i”; UI(A) and UI(R) – the UI for the active and reactive power measurements respectively. In Table I, using the paper proposition, the UI for each measurement was calculated, as well as the measurements gross error detection level by means of the largest normalized residual test, considering a threshold value of 3λ = (see the Appendix).

The measurement detection level was obtained increasing, step by step, the gross error of that measurement until that measurement gross error is detected. As seen in that table, there is a close relation between the measurement detection level and the corresponding UI. Measurements with small UIs, in general below one, have gross errors detected and identified, with the normalized gross error detection level close to 3.0δ (average values for thirty different cases).

Table I - Measurements Scenario: Gross Error Detection Level.

Active Power

UI(A) DetecLevel ( δ )

React Power UI (R)

Detec Level ( δ )

I:3 0.942 0.63 3.5 0.061 1.475 5.9 I:8 0.000 --- --- 0.0176 0.72 3.8 I:9 0.295 0.11 4.4 -0.166 0.96 4.2

I:10 0.090 3.58 10.7 -0.058 2.60 7.7 I:11 0.035 2.33 7.80 -0.018 2.10 6.7 I:13 0.135 4.38 11.7 -0.058 3.92 11.7.7 I:14 0.149 1.05 4.8 -0.050 1.12 5.1

F:1-2 1.569 0.76 3.9 -0.204 1.41 5.2 F:2-1 1.526 0.74 3.7 0.277 0.86 3.9 F:1-5 0.755 0.39 3.4 0.039 4.38 12.4 F:5-1 0.727 0.39 3.2 0.022 4.56 13.6 F:3-4 0.233 2.29 7.8 0.045 1.01 4.4 F:4-5 0.612 0.96 4.4 0.158 0.89 4.2 F:5-4 0.617 0.98 4.6 -0.142 1.07 4.5 F:4-7 0.281 0.90 4.0 -0.097 0.35 3.7 F:4-9 0.161 0.43 3.2 0.040 4.80 14.4 F:9-4 0.161 0.43 3.4 0.017 0.86 4.2

F:11-6 0.073 0.67 3.7 -0.034 0.66 3.7 F:6-12 0.078 7.47 22.7 0.025 10.15 31.5 F:7-8 0.000 --- --- -0.172 0.69 3.9 F:8-7 0.000 --- --- 0.176 0.72 3.8 F:7-9 0.281 1.01 4.2 0.058 2.79 10.1

F:11-10 0.038 1.74 5.9 0.016 1.82 5.8 F:13-14 0.056 1.17 5.1 0.017 1.42 5.9

Magnitude UI Detection Level V:1 1.060 0.24 3.2 V:3 1.010 0.25 3.2 V:8 1.090 0.23 3.2

On the other hand, measurements with high UI require very high gross errors in order it can be detected and identified, that is, the masked error due to the estimating process is high. As a consequence, this Table results show that the largest normalized residual test, as performed today, as an identification procedure of measurement with gross error, may fail significantly when the gross errors are in those measurements with high UI. As can be seen at Table I, the masking measurement error in power system state estimation is a significant fact, that is, the measurement gross error detection level is related to the measurement UI; or, in another words, it is related to the measurement masked gross error. With that in mind, in the next test, Table II, gross error of 10δ is added, in an isolated way, to each measurement;

then the classical gross error detection identification test is performed and the corresponding normalized residual computed. Also the corresponding measurement CNE will be computed, using (13).

Table II – Measurements Scenario –I: Measurements Gross Errors Composition

UI (a)

rN )(δ

CNE(a) ( δ )

UI (r)

rN )(δ

CNE(r) ( δ )

I:3 0.56 8.52 10.08 1.75 5.00 10.0 I:8 --- --- --- 1.75 5.00 10.0 I:9 1.09 6.82 10.1 0.96 7.21 10.0

I:10 3.58 2.82 10.51 0.96 7.21 10.0 I:11 2.33 3.90 9.89 2.04 4.32 9.84 I:13 4.38 2.64 11.8 3.92 2.83 11.4 I:14 1.05 6.62 9.62 1.12 6.29 9.45

F:1-2 0.76 7.90 9.93 1.42 5.80 10.1 F:2-1 0.74 8.06 10.1 0.87 7.65 10.1 F:1-5 0.40 9.18 9.89 4.35 2.48 11.1 F:5-1 0.39 9.41 10.1 4.55 2.23 10.4 F:3-4 2.29 3.91 9.80 1.013 6.97 9.93 F:4-5 0.97 7.14 9.97 0.890 7.38 9.90 F:5-4 0.98 7.11 10.0 1.07 6.77 9.93 F:4-7 0.89 7.48 10.0 0.35 8.95 9.51 F:4-9 0.42 9.25 10,0 4.78 2.11 10.3 F:9-4 0.42 9.25 10.0 0.88 7.38 9.88

F:11-6 0.67 8.25 9.94 0.5 8.26 9.90 F:6-12 7.48 1.31 9.92 10.15 0.57 5.86 F:7-8 --- --- --- 0.68 8.07 9.79 F:8-7 --- --- --- 0.71 8.04 9.90 F:7-9 1.00 7.10 10.0 2.79 2.99 8.89

F:11-10 1.74 5.08 10.2 1.82 5.02 10.4 F:13-14 1.16 6.22 9.56 1.42 5.37 9.35

UI rN CNE V:1 0.24 9.68 9.95 V:3 0.25 9.64 9.95 V:8 0.22 9.72 9.97

V:13 0.24 9.67 9.96 The UI indexes were obtained using the estimated states.

As can be seen in Table II, for the measurements with low UI, their normalized residual and the corresponding CNE are close to each other; however for measurements with high UI they are quite different. As a consequence, for those measurements with high UI, the classical measurement gross error detection, identification, and amplitude correction will fail in a significant way. They will require first the measurement masked error estimation, then the composition of the total measurement error as proposed in paper through equation (13). Table II shows, in a clear way, that when the step of recovering the gross error is performed and the measurement gross error is composed, the HTI procedure will be very reliable. In the paper tests it was observed, as expected, that when the measurement redundancy level increases the measurements undetectability indexes decreases in general. The reason is that the increase in the

measurement redundancy level means that more information is given and then less masking effect will happen. Remark 4: The numerical tests have shown that further studies must be made when gross errors occur in redundant measurements. Suppose for example a case of a measurement critical pair. Since that measurement pair have the same normalized residual, when their errors are composed, the measurement with larger UI will have larger CNE. As a consequence applying the largest normalized error test, when the error is on the measurement with larger UI, the conclusion of gross error on that measurement will work correctly. However when the error is on the measurement of lower UI, some extra work in order to lead to correct gross error identification test must be made.

V. CONCLUSIONS In this paper, more insights related to detectability of gross errors in power system state estimation, using geometrical approaches has been provided. This is achieved decomposing the measurement error vector into two components: the undetectable part and the detectable one. The ratio between the w-norm of those quantities, the undetectability index, gives the distance of a measurement from the range space of the Jacobean matrix. In other words, the UI is a measure of how difficult it is to detect errors in those measurements. According to the paper’s results, it is possible to conclude that the higher a UI of a measurement the higher the masked error. Based on the UI index, a way to recover the masked errors, resulting from the measurement residual estimating process has been proposed. The total measurement residual is then composed. Also the proposed measurement gross error detection and identification test, using the measurement composed gross error, as proposed in this paper, works very well. This results show the gross error test as used in the existing literature is not correct. It does not consider the masking effect existing in the power system state estimation Using the paper’s approach, we are working on a multiple gross errors detection/identification test in power system state estimation.

REFERENCES [1] D. M. Falcão and M. A. Arias, “State estimation and observability analysis based on echelon forms of the linearized measurement models”, IEEE Transactions on Power Systems, vol. 9, n° 2, p. 979-987, 1994. [2] A. Monticelli, “Electric power system state estimation”, in Proc. of the IEEE, vol. 88, n02, pp. 262-271, 2000. [3] E. Handschin, F.C. Schweppe, J. Kohlas and A. Fiechter, “Bad data analysis for power system state estimation”, IEEE Trans. on Power Apparatus and Systems, vol-Pas-94, n° 2, pp.329-337, 1975. [4] A. Abur and A. G. Expósito, “Power system state estimation: Theory and implementation”. Marcel & Dekker Publishers, Nova York, EUA, 2004.

[5] K. A. Clements and P. W. Davis, “Multiple bad data detectability and identifiability: a geometric approach”, IEEE Trans. on Power Delivery, vol. 1, n° 3, pp. 355–360, 1986. [6] N.G. Bretas, "Network observability: theory and algorithms based on triangular factorization and path graph concepts", IEE Proceedings, Generation, Transmission and Distribution, vol. 143, n°1, pp. 123-128, 1996. [7] F. C. Schweppe; J. Wildes, “Power system static-state estimation, part I: Exact model”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-89,N°1, p. 120-125, Jan., 1970. [8] F. C. Schweppe; B. R. Douglas, “Power system static-state estimation, part II: Approximate model”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-89,N°1, p. 125-130, Jan., 1970. [9] F. C. Schweppe, “Power system static-state estimation, part III: Implementation”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-89,N° 1, p. 130-135, Jan., 1970. [10] N.G. Bretas, J.B.A. London Jr., L.F.C. Alberto, and R.A.S. Benedito, “Geometrical Approaches for Gross Errors Analysis in Power System State Estimation”. PowerTech09, Bucharest, June 2009. [11] N.G. Bretas, J.B.A. London Jr., L.F.C. Alberto, and R.A.S. Benedito, “Geometrical Approaches on Masked Gross Errors for Power Systems State Estimation”. PES-GM09, Calgary, July 2009. [12]

APPENDIX: DETECTION AND IDENTIFICATION OF MEASUREMENTS GROSS ERRORS

Given a measurement vector z, the estimated state x will depend on the projection operator P, which depends only on the inner product choice (W) and matrix H. Matrix “W” is given by the inverse of the measurement covariance matrix. After finding x , it is interesting to verify the existence of gross errors in the measurements. Therefore a routine for error detection is required. At this point only statistic concepts are needed. Assuming that measurement errors have a normal distribution, it is easy to show that index )ˆ(xJ , i.e., the function to be minimized in (2), has a Chi-square distribution (χ2) with (m-n) degrees of freedom. Choosing a probability “1-α” of false alarm and being “α” the significance level of the test, a number “C” is obtained (via Chi-square distribution table for 2

,1m n αχ − − )

such that, in the presence of gross errors )ˆ(xJ > C. Another way to detect the presence of gross errors is by the largest normalized residual test. Based on the same assumption about measurement errors, the vector of residuals r is normalized and subjected to a validation

test: ( ) λσ ≤= kkrkr

r

N )()( (threshold value),

where Nkr )( is the

largest, among all ( ) miir N ,,1, …= ; ( ) ( )kkkr ,Ω=σ is the Standard Deviation of the kth component of the residuals vector, and Ω is the residual covariance matrix given by 1 1

.( )T TW H H WH H− −= −Ω

If λ>Nkr )( , then the measurement with gross error is detected and the kth measurement will be the one with gross error (usually 3λ = [4]).

BIOGRAPHIES Bretas, N.G. received the Ph.D. degree from University of Missouri, Columbia, USA, in 1981. He became Full Professor of the University of Sao Paulo, Brazil, in 1989. His work has been primarily concerned with power system analysis, transient stability using direct methods, as well as power system state estimation. Dr. Bretas has also interest in energy restoration using ANN and GA for distribution systems. A. S. Bretas (M’ 1998) received the Ph.D. degree from Virginia Tech, Blacksburg, in 2001. Currently he is an associate professor of the Federal University of Rio Grande do Sul. His main areas of interest are power systems protection, restoration and analysis. E-mail: abretas@ece.ufrgs.br