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Abstract- In this paper, a topological and geometrical based approach is used to define an index, undetectability index (UI), which provides the distance of a measurement from 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. A two bus system is used to show how this geometrical view of gross error analysis works. The classical three bus system used in many papers of the state estimation field is used as an example of a small application of the proposed methodology.

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

I. GENERAL VIEW 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

This work was supported by FAPESP, Scientific Foundation of the State of São Paulo, Brazil and CNPq.

The first author is with the University of São Paulo, São Carlos, SP, Brazil (email: ngbretas@sc.usp.br)

The second author is with the Federal University of Rio Grande do Sul (UFRGS), Porto Alegre, RS, Brazil (e-mail: abretas@ece.ufrgs.br).

The third author is a Ph.D. student with the University of São Paulo, São Carlos, SP, Brazil (e-mail: saulopiereti@usp.br)

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]. In this paper, using the WLS estimator, and based on the concepts of references [6]-[7], 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 components: the undetectable component 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 measurement error amplitude is used in the gross error detection and identification test.

II. BACKGROUND The background as presented below is as in [6-9]. Consider a power system with n buses and m measurements 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 Rm 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 the number m of measurements is higher than the number N, a common solution to estimate the states variables is the well-known WLS method, which searches for the state vector x, which minimizes the functional

Bretas,1N.G., Senior Member, IEEE, Bretas, A.S., Member, IEEE, Piereti,S.A.R.

T

Masked Errors in Power Systems State Estimation and Measurement Gross Errors detection and

Identification

Paper accepted for presentation at the 2011 IEEE Trondheim PowerTech

978-1-4244-8417-1/11/$26.00 ©2011

2

( ) ( ( )) ( ( ))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 that is induced by the inner product <u,v>=uTWv, that is:

=)(xJ ( ) ( )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 )ˆ(ˆ xhz = . The residual vector is defined as the difference between z and z , that is:

zzr ˆ−= . The linearization of (1), at a certain operating point x*, yields:

z H x eΔ = Δ + , (3) where x

hH ∂∂= is the Jacobian of h(.) calculated at x*,

**)( zzxhzz −=−=Δ is the measurement vector mismatch,

and *xxx −=Δ is the state vector mismatch. If (2) is observable, that is, rank(H)=N [8], then the vector space of measurements (mx1), 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 z onto the ℜ(H). Let P be the linear operator which projects vector z 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 z 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 (4) 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 as Hat matrix and it is also known as K matrix [4]. The residual vector in the linear form of (1) 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 an operator that projects

z onto ℜ(H) . Matrix (I-P) is given by

( ) ( ) 1T TI P I H H WH H W−

− = − , (7)

and it is usually called residual sensitivity matrix; and it is also known as S matrix [4] in power system literature.

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 linearized state estimation

model, where ∧−=Δ xxx truetrue

; and true truez H xΔ = Δ ; assuming that the available Jacobian 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) In what follows an index, UI, based on the detectable and undetectable component of the measurement error vector will be proposed in order to solve that difficulty. 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 ith measurement, that is: i ie bδ= ,

with [0 1 0]i

Tiδ = , and using (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 ith measurement is proposed:

i

i

U Wi

D W

eUI

e= (10)

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

Δz2

Δz1

Δz3

Δx1

Δx2 zΔ =P Δz xΔ

Δz

3

OBS.: A critical measurement is the limit case of a measurement with high UI, that is, it belongs to the range space of the Jacobian matrix, has an infinite UI index, and its error is totally masked. EXAMPLE 1: Consider the system of two buses (n = 1) connected through two lines, as shown in Fig. 2. Taking bus 2 as a reference of angle (δ2 = 0), the Jacobian matrix for that

system becomes .102

=H Considering that the standard

deviations of z1 and z2 are equal to 1.0 ( 1 2 1σ σ= = ), that is,

“W = I”, the projection matrix P will be =961.0192.0192.0038.0

P .

Let ( )10.175 0.175true truex δ= = be the vector of the unknown

states then [ ]0.35 1.75 Ttrue truez Hx= = the true unknown

measurement vector. Suppose the existence of a gross error of magnitude 9σz2 in measurement z2, that is: [ ]Tz 75.1035.0= . Solving the linear WLS state estimator, with W I= , one obtains:

Fig. 2: Two-bus system for the example. - means measurement place

=−

=765.1765.1

,346.0

73.1 Nrr (they belong to a critical set) and

ˆ ˆ ˆ( ) ( ) ( ) 3.11TJ x z Hx W z Hx= − − = . The threshold value 5.02C = is obtained via chi-square distribution table

for 2,1m n αχ − − (see Appendix).

As a consequence, the hypothesis of error existence is erroneously rejected. This situation is depicted in Fig. 3. In order to understand the reason for that wrong decision, let us compute the UI for those measurements, whose values are shown at Table I.

TABLE I : UI – EXAMPLE 1 AND 2 z1 z2

UI 0.1997 5.0073

As shown in Table I, the UI of measurement z2 is very high, while the UI of measurement z1 is very low. REMARK 2: as seen in Fig. 3, the measurement z2 is closer to the range space of H than z1. As a consequence the measurement z2 behaves closer to a critical measurement, in terms of masking error, than z1; in this way, to detect a gross error in z2, using the standard techniques, that error has to be very large. EXAMPLE 2: Using the same system considered in Example 1 (Fig. 2), but now with a gross error of magnitude 9σz1 in measurement z1, that is, [ ]9.35 1.75 Tz = , and solving the linear WLS state estimation one obtains:

Fig. 3: Gross error in measurement z2 (Example 1) zwtge- measurement z without gross error

=−

=825.8825.8

,73.1

65.8 Nrr (as a consequence they belong to

a critical set) and 88.77)( =xJ (see Fig.4 bellow). In this case the error is detected ( 3=λ and 5.024C = ) because it occurred in measurement z1, with low UI.

Fig. 4: Gross error in measurement z1 (Example 2) zwtge- measurement z without gross error

Recovering of the Masked Errors and Gross Error Detection and Identification Test For the purpose of estimating the measurement total error, the following algorithm is proposed:

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

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

.iDiiU 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 :

( ) ( ) 2222222 11 iiDiiUiiDi rUIeUIeee +=||||+=||||+||=||||||

(11)

where ir is the ith measurement estimated residual.

j0.1

z2

j0.5 z1

4

(iv) Then apply the gross error detection and identification test (HTI), as in the Appendix, but using the CNE as in (11), instead of the normalized measurements residual. The degree of freedom for the composed error is now m instead of (m-N) of the measurement residual.

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

IV. APPLICATION In what follow a small power network will be presented in

order to better explain a paper propositions in gross errors detection and identification but using previous proposed solution. The measurement values used in this application were obtained from a load flow solution (zlf) to which normally distributed noises will be 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%.

Using the measurement set as indicated in Fig.4, with values as in Table II, for each of those measurements, once at a time, error is increased at steps of 0.1 until the gross error is flagged using the confidence degree of 95% that at least one measurement has its rN equal or superior to three has occurred. For each of those measurements it is also computed: the measurement CNEi. Also the added error at that time is available at that table.

In what follows the corresponding sequence of computation for the active measurement I(A):1 will be described (The other cases will be just repetition). After performing the state estimation analysis, the r1

N (3.04) for that measurement is obtained; the UI1=0.971 as well as the CNE1=4.34 is obtained.

In all the cases the measurement with gross error was correctly identified. It should be observed that in Table II the added gross error is a little bit different from the corresponding CNEi, the reason being that random errors has been previously added to the measurements After correcting that measurement identified as the one containing error, and performing another state estimation, no gross error has been flagged anymore. One should see that as much innovation a measurement has, closer it will be the measurements quantities ri

N, CNEi, as well as the added error.

1

2 G

30.10+j1.00

0.01+j0.10 0.10+j1.00

Active power injection measurementActive power flow measurement

Legend:Reactive power injection measurementReactive power flow measurement

Load GeneratorsG Fig. 4: The Three-bus-system

TABLE II

RESULTS CORRESPONDING TO THE THREE -BUS SYSTEM: FIG. 5

Meas.

II

rN Det.leve

l

Added Error (σ )

CNE

I(A):1 -1.6234

0.97 3.04 3.80 4.24

I(A):2 2.5962

0.53 3.04 3.90 3.44

F(A):2-1 2.0332

0.63 3.02 4.20 3.58

F(A):1-3 0.3661

1.31 3.06 2.80 5.07

F(A):2-3 0.5629

0.58 3.06 3.10 3.54

I(R):1 0.3726

1.43 3.01 4.40 5.22

I(R):3 0.4047

0.67 3. 02 3.20 3.64

F(R):1-2 0.3743

1.39 3.04 5.20 5.21

F(R):3-1 0.1446

1.03 3.03 3.90 4.35

F(R):3-2 0.2583

0.68 3.07 3.60 3.71

From the Table II one should observe: (i) In the added error

column it is not included the measurement random error; (ii).Although the gross error detection test worked well, column of the rNs, the measurement error at that time is quite different from the rN as shown at the column of the CNEs; (iii) Also, the measurements threshold values are different from the corresponding rNs, but very close to the CNEs. With the increase in the information degree, all those quantities will be closer and closer to each other.

V. CONCLUSIONS In this paper, more insights related to detectability of gross errors in power system state estimation, using of geometrical approaches has been provided. This is achieved decomposing the measurement error vector into two components: the undetectable component 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 Jacobian matrix. In other words, the UI is a measure of how difficult it is to detect errors in those measurements. Based on the UI index, a way to recover the masked errors, resulting from the measurement residual estimating process has been proposed. Once that error is recovered the total measurement error in then composed.

5

The paper 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.

ACKNOWLEDGMENTS The authors would like to acknowledge FAPESP, Scientific

Foundation of the State of São Paulo, Brazil, for the financial support given to this research. They also would like to acknowledge the graduate student Raphael A. S. Benedito, from the Elect. Eng. Dep. of EESC, USP, for the simulations performed.

REFERENCES [1] D. M. Falcão and M. A. Arias, “State estimation and observability analysis based on echelon forms of the linearized measurement models,” IEEE Trans. on Power Systems, vol. 9, no. 2, pp. 979-987, 1994. [2] A. Monticelli, “Electric power system state estimation,” Proc. of the IEEE, vol. 88, no. 2, 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, no. 2, pp.329-337, 1975. [4] A. Abur and A. G. Expósito, “Power system state estimation: Theory and implementation.” Marcel & Dekker Publishers, Nova York, USA, 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, no. 3, pp. 355–360, 1986. [6] 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,” in Proc. PowerTech09, Bucharest, June 2009. [7] 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,” in Proc. PES-GM09, Calgary, July 2009. [8] N.G. Bretas, S.A. Piereti, “The Innovation Concept in Bad Data Analysis Using the Composed Measurements Errors for Power System State Estimation”, in Proc. PES-GM10, Minneapolis, July 2010. [9] N.G. Bretas, London Jr., J.B.A., “Recovering of Masked Errors in Power Systems State Estimation”, in Proc. PESGM2010 Minneapolis, July 2010. [10] N.G. Bretas, "Network observability: theory and algorithms based on triangular factorization and path graph concepts," IEE Proc., Generation, Transmission and Distribution, vol. 143, no. 1, pp. 123-128, 1996.

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 and power system state estimation.

Bretas, A. S. (M’98) received the Ph.D. degree from Virginia Tech, Blacksburg, USA, in 2001. Currently, he is an Associate Professor of the Federal University of Rio Grande do Sul (UFRGS), Porto Alegre, Brazil. His main areas of research interest include power system protection, analysis and restoration. Piereti, S.A.R. is a Ph.D. student at the Elect. Eng. Dept., EESC-University of São Paulo, Brazil.