Convergence Property of the Measurement Gross Error ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013

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Convergence Property of the Measurement Gross Error Correction in Power System State Estimation, Using Geometrical Background N. G. Bretas, Life Senior Member, IEEE, A. S. Bretas, Senior Member, IEEE, and Andre C. P. Martins

Abstract—In this paper it is shown that once the measurements with gross errors are detected and identified, the geometrical approach to recover gross errors in power system state estimation, as we have previously proposed, is a convergent process. For the purpose of correcting the measurement gross errors, the measurement residuals are computed, and then the measurement gross errors are composed. For the detection and identification of the measurements with gross errors, the composed measurement error in the normalized form is used. The measurement magnitude corrections otherwise are performed using the composed normalized error . To support the thesis that, after the detection and identification of the measurements containing gross errors, the measurement correction is a convergent procedure, the generalization of the largest normalized error test is also provided. A two-bus power network is used to show in a didactic way the behavior of the gross error correction. The IEEE 14-bus system and the 45-bus equivalent of Brazil south are used to perform the tests for the multiple measurement gross errors case. Many gross error scenarios with different redundancy levels, for multiple gross error situations, have been tested. The measurement correction is made one at a time.

Detectable component of the measurement error.

Index Terms—Composing errors, gross errors analysis, orthogonal projections, recovering errors, state estimation.

Number of measurements.

Weight matrix. Measurement vector mismatch. State vector. Estimated state vector. Operating point. Estimated state

with

.

State vector mismatch. Jacobian matrix of

.

Number of unknown state variables. Global redundancy level. Objective function.

NOMENCLATURE

Continuous nonlinear differentiable function. Projection matrix.

Measurement vector. Measurement vector

Non-detectable component of the measurement error.

without gross error.

Identity matrix. Residual sensitivity matrix.

Unknown true measurement vector. Estimated value of .

Real number.

Measurement error vector.

Range of

.

Orthogonal complement of Manuscript received July 02, 2012; revised August 30, 2012 and March 04, 2013; accepted April 20, 2013. Date of publication May 15, 2013; date of current version October 17, 2013. This work was supported by FAPESP, State of Sao Paulo, Science Foundation and CNPq. The authors would like to acknowledge FAPESP, a research foundation of the state of Sao Paulo, Brazil, and CNPq, a Brazilian national research foundation, for the financial support given to this research. Paper no. TPWRS-00667-2012. N. G. Bretas is with the Department of Electrical Engineering, E.E.S.C.-University of São Paulo, São Paulo 13560-480, Brazil (e-mail: [email protected]. br). A.S. Bretas is with the Electrical and Computer Engineering Department of UFRGS, Brazil (e-mail: [email protected]). A. C. P. Martins is with the Department of Electrical Engineering, State University of São Paulo, São Paulo, Brazil (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2013.2260360 0885-8950 © 2013 IEEE

.

Normalized residual of the th measurement. Standard deviation. Added gross error measured in . Meter precision. Measurement obtained from the load flow. Active power injection at bus . Active flow measurement from bus to . Reactive flow measurement from bus to . Measurement voltage magnitude at bus .

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where is the measurement vector, is the state variable vector, is a continuously nonlinear differentiable function, is the measurement error vector assumed with zero mean and Gaussian probability distribution, and is the number of unknown state variables to be estimated. stands for the real number space. A classical technique to estimate the states variables of this well-known problem is the WLS formulation, which looks for the state vector that minimizes the functional:

Residual covariance matrix. Significance level of the gross error test. Composed measurement error. Composed normalized measurement error.

I. INTRODUCTION

T

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], [9]. One of the first 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 estimator is not able to reliably identify multiple interacting gross errors, especially when they are conforming and/or occur in leverage points [4], which are highly influential measurements that “attract” the state estimation solution [5], [6]. Other alternative estimators have been proposed. The weighted least absolute value estimator (WLAV), for instance, 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], [4]. Using the PMUs, [10]–[12] have proposed an appropriate placement of those units in order to enhance the bad data detection and identification, always requiring enough redundancy level and eliminating the leverage measurements. In those papers using the measurement residual approach the measurement gross errors are detected and identified. A natural consequence of all these proposed papers is that they use the residual as metric for the gross error analysis and [16] shows that this is not correct. More recently, using topological and geometric approaches Bretas et al. [13]–[20] have proposed methodologies to compose the measurement error and then correct the measurement magnitudes for those measurements identified as containing a gross error. In this paper it is shown that the correction of the measurement detected as having gross error, using the measurement , is a convergent process. It is also shown that with the generalization of the measurement largest normalized error test the previous process works correctly even for the case of multiple gross errors case.

where is a symmetric and positive definite real matrix. In power system state estimation, the weight matrix is usually chosen as the inverse of the measurement covariance matrix. Functional is a norm in the measurement vector space induced by the inner product , that is

(2) being the square of the norm of weighted by . with Let be the estimated state, that is, the solution of the aforementioned minimization problem; then the estimated measurement vector will be given by . The residual vector is defined as the difference between and , that is, . The linearization of (1), at a certain operating point , yields (3) , , and the Jacobian with of calculated at . If the system described by (3) is observable, that is, , then the vector space of measurements can be decomposed into a direct sum of two vector subspaces, i.e., , in which the range space of is an N-dimensional vector subspace into and is its orthogonal complement, that is, if and , then . In the linear state estimation formulation, the solution of (3) can be understood as a projection of the measurement vector mismatch onto . Let be the linear operator which projects vector onto , that is, . The projection operator which minimizes the norm is the one that projects orthogonally onto in the sense of the inner product , that is, the vector is orthogonal to the residual vector. More precisely (4)

II. THEORETICAL BACKGROUND The material presented below constitutes in its essence the background contained in [2] and [21]. Consider a power system with buses and measurements modeled, for state estimation purposes, as a set of nonlinear algebraic equations, that is

Solving this equation for , one obtains so that, iteratively, the nonlinear state estimation problem as formulated by (1) and (2) is solved. As , the projection matrix will be the idempotent matrix:

(1)

(5)

BRETAS et al.: CONVERGENCE PROPERTY OF THE MEASUREMENT GROSS ERROR CORRECTION

Fig. 1. Operator P acting on vector

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Remark 1: It can be seen, directly from (7), that the detectable component of the measurement error is nothing else than the measurement residual. Remark 2: One should be aware that the orthogonal concept, in a high dimension space, is a more general concept than it is the ninety degrees for a bi-dimensional space, that is: two vectors and are orthogonal to each other if 1) Innovation Index and the Measurement Error Recovery (summary): As presented at [16]–[18] the degree of innovation contained in a measurement corresponds to the new information carry out by it, and can be calculated by

: Geometric view.

1) Obs: If the weight matrix is a diagonal matrix given by , where is a real number and is the identity matrix, then . In power system literature, matrix is usually called Hat matrix and is also known as matrix [13]. On the other hand the matrix is usually called residual sensitivity matrix and known as matrix [19]. Fig. 1 illustrates operator acting on vector , that is, in any iteration step of the estimation, the operator is projecting a generic , that is, onto the Jacobian range space, the space of the estimated measurements. In that figure one can see the measurement residual as the vector corresponding to the difference between and . The residual is orthogonal to the Jacobian range space as can be seen in Fig. 1. One can be aware that and estimated are quantities in different directions and for each measurement those directions may be varying. III. UNDETECTABLE ERRORS AND INNOVATION INDEX (II) PROPOSITION

(9) As shown in [18]–[20], the th composed measurement error can be estimated by

(10) with II the Innovation Index, that is, the new information a measurement contains related to the other measurements of the measurement set. From (10) one can normalize the composed measurement error, using the measurement , which gives (11) One could also normalize the residual and then compose the error, that is (12)

In this section a summary [16]–[20] of the masked effect of the gross error in the state estimation, as was previously proposed [21], will be presented. The linear formulation of the state estimation (3) will be used to decompose the measurement vector space into a direct sum of and , that is, to decompose the measurement error vector into two parts: the detectable and the undetectable components. For that purpose, assuming the available Jacobian matrix is not far from the one obtained with the true states, and then is close to and, as a consequence, . The vector of the measurement error is then given by , and can be directly written, that is: . Denominating and (6) (7) respectively, the undetectable and detectable components of the measurement error, and also that: . It is easy to see that the vector while and as a consequence they are orthogonal to each other and the following relation between them holds: (8)

In our papers related to this subject [16]–[20] and [21] this quantity is called by composed normalized error. It has been shown also that numerically the measurement and are equal quantities, although they pertain to different sub-spaces, consequently with completely different properties. The residuals pertain to the residual sub-space, with degrees of freedom while the composed measurement errors pertain to the measurement sub-spaces, then with degrees of freedom. The measurement identified as having error is corrected by its corresponding . The reason is that since in the state estimation process the measurements (dimension ) are used to estimate the state variables (dimension N), and then the measurement residuals are calculated (dimension ), that is, in a lower dimension space; as a consequence the computation is more accurate than the computation, the first performed on the residual sub-space and the other obtained going from the residual sub-space to the measurement sub-space and then calculated, but in a lager dimension sub-pace. A natural question then appears: is the correction of the measurement with error, using the measurement , a convergent process? In what follows, a logical of the sequence of corrections which are made in the measurements detected with error is presented to show that the procedure is a convergent process.

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Fig. 2. Two-bus network used in the simulations.

Fig. 4. Active Power x Residual.

Fig. 3. Active Power x Phase Angle (degree).

IV. CORRECTION OF A MEASUREMENT CONTAINING GROSS ERROR USING THE MEASUREMENT CNE AS A CONVERGENT PROCEDURE 1) Single Gross Error Cases: In what follows the significant cases of the gross error in the measurement set will be analyzed: 1) The measurement set contains only perfect measurements: When estimating the network states, the measurement residuals will be zeros, and no error will be detected at all. The explanation for this conclusion is because the measurements must be in the Jacobian range space, since they are correct measurements, like the measurement of a load flow, and as a consequence the measurement residuals must be zeros. See Fig. 2 for a representation of a case of a two-bus and one line power network, with the active and reactive power flows as measurements, the bus voltage magnitudes as fixed values, and the phase angle between them being the state variable to be estimated. Simulation Description: The starting operating point uses the perfect measurements (point in the middle of the green ball); then errors will be first added to the active power measurement and the hull state estimation processed is performed obtaining the state variable (phase angle) between the two buses. The same procedure is repeated but subtracting error to the active power measurement. To the reactive measurement, only random errors with zero mean will be added as errors. Fig. 3 below shows the plane behavior and Fig. 4 shows the corresponding behavior. Obs: All the quantities in Figs. 2–5 are in p.u. For the specific operating point of active power measurement and reactive power ) the measurement residual ( equal to 12.016) and the

Fig. 5. Measurement GE correction.

estimated will be equal to 13.020. As a consequence the corrected active power flow is obtained, as can be seen in Fig. 5. In that figure it is also shown the true active power flow (without gross error); and as can be seen the corrected measurement value and the true measurement value are very close to each other. Description of Fig. 5 below: With and the phase angle between the two buses is estimated, the measurement residuals are computed, and the Jacobian is computed and linearized at that point. The is then obtained. Since the measurement with gross error and its residual are available, and since the measurement residual and are orthogonal to each other, the measurement is obtained. As a consequence the measurement correction is directly calculated, since the measurement standard deviation is known. Obs: The previous figure shows that the similar graphic of measurement correction which we have presented in our previous papers related to this subject deserves some improvement. In those papers we have presented the Jacobian range space as passing through the origin, and that is a very particular case where the measurement to be corrected is close to zero. The situation of perfect measurements can be identified in the Fig. 3 as corresponding to the graphic inflexion point, that is, the point where the power crosses the sinusoidal curve (ball


in green). In that point the Jacobian range space will be a vertical line. That point can be also identified in Fig. 4, the point where the measurement error (residual) becomes zero, that is, the phase angle theta equal to fifteen degrees. 2) The measurements contain random errors: When estimating the network state variables, the measurement residuals will be different from zero. However gross error should not be detected and, as a consequence no measurement correction will be performed. 3) The measurements contain, beyond the random errors one of them will contain gross error. a) The error will be increasing the measurement magnitude: Applying the gross error detection and identification test the measurement with error has been identified. Before going to explain the geometrical representation for this situation, one should be aware that the WLS works with the idea of average values. As a consequence, the estimated value of such a measurement with error (increased magnitude), will cause a correction so that the measurement magnitude will be decreased, that is, its residual will be positive (measured estimated ). In Fig. 3 that situation will be characterized for the angles which are larger than fifteen degrees. b) The error is one that decreases the measurement magnitude. The logical for the explanation for this situation is the same as the previous case but the correction will be increasing the measurement magnitude. That situation will be characterized for angles smaller than fifteen degrees in the two previous figures. 4) Gross Error Correction as Previously Proposed is a Convergent Process Suppose that all the measurements of the measurement set contain random errors, except one of them containing beyond that random error a gross error. Applying the gross error detection and identification test, the measurement with error will be identified and, as a consequence, the correction to the wrong measurement will be performed using (12) and as explained by (10). Three different situations may occur: a) the measurement correction will bring that measurement to inside the chosen hypersurface which cause no gross error detection anymore (less than the chosen gross error threshold value, generally three measurement standard deviation). Then the gross error analysis will finish; b) the performed gross error correction is not enough so that the next gross error detection test will still detect error in that measurement. In this case new gross error correction will be required using the same previous methodology, until gross error in the measurement is not detected anymore; c) the performed correction in the measurement with error is an overcorrection. In what follows a logical of thinking demonstrates that the overcorrection cannot happen. Suppose two equal sets of measurements, (1) and (2), of a network having the same measurement values, except one of the measurements, having different gross errors. If a gross error analysis is performed, using those two measurement sets, the measurement with larger gross error will have a larger correction to be made.

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Proof: Since the weight least square state estimation is being used, and since it works in accordance with the average principle, once a gross error has been detected the measurement with higher gross error must have a larger estimated value. As a consequence, it is more apart from its true measurement value than the measurement with a lower gross error. As a consequence its normalized residual will be also larger [20], then it will have also a larger correction to be made. q. e. d. This previous property can be seen in the previous Figs. 3and 4. In case that the measurement set contains multiple gross errors, first we will present the multiple gross error analysis approach and then how to make the measurement corrections. 2) Multiple Measurement Gross Errors Cases In case the measurement set has more than one measurement with gross error, the properties of the largest normalized gross error theorem are required. That theorem has been proposed in Bretas [20] and it will be summarized below. Theorem: The Largest Normalized Error Test as a Generalization of the Largest Normalized Residual Test: Assuming all the measurements of a measurement set with limited random errors, (inside a threshold value), and adding gross error only to one of the measurements, the measurement to which gross error was added will have the largest increment of error among all the measurements. Obs: In [20] it was proved that numerically the measurement error in its normalized form and the normalized residuals are equal, although they pertain to different subspaces, that is, the measurement error pertaining to the measurement sub-space, with degrees of freedom, and the measurement residual pertaining to the residual sub-space, of dimension , and, as a consequence, with degrees of freedom. Proof: The measurement residual for the situation in which the measurements have random errors is (13) Let us suppose now that gross error is added only to the measurement and as a consequence for this new measurement: , and all the other measurements staying with the same values. The nomenclature new stands for: new measurement, but with error. Following the same standard of demonstration of the classical largest normalized residual test one obtains: , where is the th column of the residual covariance matrix , as in the largest normalized residual test of [2]. As a consequence, and following the same standard of proof of the classical largest normalized residual test, one will obtain: , that is, , . q.e.d. Obs: With the Largest Normalized Error Theorem as previously presented, one is sure, within the uncertainty degree as chosen in the gross error hypothesis test, that once error in

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TABLE I MULTIPLE GROSS ERROR TEST USING IEEE 14-BUS SYSTEM: , IDENTIFICATION THROUGH THE LARGEST THE J(X) WITH AND CORRECTION USING THE

Fig. 6. IEEE 14-bus system.

the measurement set is detected the measurement with largest will have the error. Obs: With the largest normalized error test property, as presented before, the multiple gross error case reduce to a case of single gross error since the measurement with the largest error will have the gross error. In what follows some tests with the IEEE 14-bus network, for multiple gross error situation will be presented. The correction procedure as proposed will be tested for those multiple gross errors cases.

TABLE II MULTIPLE GROSS ERROR TEST USING IEEE 14-BUS SYSTEM: , IDENTIFICATION THROUGH THE LARGEST THE J(X) BUT WITH THE . AND CORRECTION USING THE

V. NUMERICAL TESTS The IEEE 14-bus system (Fig. 6) will be used for the paper tests. To obtain the parameters of IEEE 14-bus system, see [23]. The measurement values used in the tests were obtained from a load flow solution to which normally distributed noise was added. The measurement noise was assumed to have zero mean and a standard deviation given by ; where is the meter precision, equal to 3%. For Tables I–IV, the following nomenclature is used: -injection measurement at bus a; -flow measurement from bus to bus ; and —voltage magnitude measurement at bus . The capital letters and at those measurements stand for active and reactive power, respectively. 1) Measurement Scenario: The measurement set to be used on the multiple gross errors to be simulated is available at http:// www.sel.eesc.usp.br/LACOSEP. For this measurement scenario three different measurements were randomly chosen and random errors between five and fifteen standard deviations of magnitude were added to or subtracted from those measurements. In what follows some of the performed simulations will be described. The detection test will be: . It can be observed in the two previous simulations that for those measurements with larger II, the measurement and the added GE are closer to each other than the cases in which the measurements have a lower II.

The detection test will be: . The detection test will be: . Again using another measurement scenario and redundancy level, it is observed the improvement of the for those measurements with larger II. In all simulated cases the error was always detected and the measurements with error corrected and identified. In most of the cases one could identify the wrong measurements at once but it


TABLE III MULTIPLE GROSS ERROR TEST USING IEEE 14-BUS SYSTEM: , IDENTIFICATION THROUGH THE LARGEST THE J(X) BUT WITH THE . AND CORRECTION USING THE

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observed that as much redundancy level is used for the measurements more reliable the hull process of gross error analysis will be. VI. CONCLUSIONS

TABLE IV MULTIPLE GROSS ERROR TEST USING IEEE 14-BUS SYSTEM: , IDENTIFICATION THROUGH THE LARGEST , THE J(X) WITH AND CORRECTION, USING THE

This paper has shown that the proposed methodology where the measurements identified with error is corrected using the measurement is a convergent process. In this paper even multiple gross errors were simulated to show the gross error correction adequacy. A two-bus network has been used to show in detail all the measurement gross error correction steps. Also the measurement residuals were shown every time a gross error correction was performed. The IEEE 14-bus system has been used for multiple gross errors cases and for many measurement gross errors scenarios; the tests results has shown that the measurement with gross errors were always corrected, detected and identified, and after that the measurement with errors had, in an accurate way, their magnitudes recovered. The paper results also suggest that in the gross error analysis, the identification of the measurements with error and their error correction should be made one at a time, and not all in just one step. Although not presented in the paper, it has been observed that when the system size increases the measurement gross errors correction can be performed at once. The explanation for that may be due to the fact that in the case of large systems the interaction among the system events is significantly diminished; then the multiple gross error simulations will behave as in the case for single gross error in the small power systems. APPENDIX A. Detection and Identification of Gross Errors

is more reliable to identify the wrong measurements one at a time. Although it was presented three different gross errors scenarios with different redundancy level, many others were simulated and all them resulting in the correct detection and identification of the measurements containing gross errors. Also it was

Given a measurement vector , the estimated state will depend on the projection operator , which depends only on the inner product choice and matrix . Matrix is given by the inverse of the measurement covariance matrix. After finding , 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 the measurement errors have a normal distribution, it is easy to show that the index , i.e., the function to be minimized in (2), has a Chi-square distribution with degrees of freedom. Choosing a probability “ ” of false alarm and being “ ” the significance level of the test, a number “ ” is obtained (via Chi-square distribution table for ) such that, in the presence of gross errors, . Another way to detect the presence of gross errors is by the largest normalized residuals test. Based on the same assumption about measurement errors, the vector of residuals is normalized and subjected to a validation test: (threshold value), where is the largest among all , ;

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is the standard deviation of the th component of the residuals vector, and is the residual covariance matrix given by . If , the measurement with gross error is detected and the th measurement will be the one with gross error (usually [14]). B. Equations of the Quantities Appearing in the Geometrical State Estimation Measurement Innovation Index : Using the definition of Innovation Index it will be obtained:

with being the error magnitude. With the measurement innovation index as given above and using (11) and (12), the measurement CMEs and CNEs are directly obtained. 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. Power Syst., vol. 9, no. 2, pp. 979–987, May 1994. [2] A. Monticelli, “Electric power system state estimation,” Proc. IEEE, vol. 88, no. 2, pp. 262–271, Feb. 2000. [3] E. Handschin, F. C. Schweppe, J. Kohlas, and A. Fiechter, “Bad data analysis for power system state estimation,” IEEE Trans. Power App. Syst., vol. PAS-94, no. 2, pp. 329–337, 1975. [4] M. G. Cheniae, L. Mili, and P. J. Rousseeuw, “Identification of multiple interacting bad data via power system decomposition,” IEEE Trans. Power Syst., vol. 11, no. 3, pp. 1555–1563, Aug. 1996. [5] L. Mili, M. Cheniae, N. Vichare, and P. Rousseau, “Robust state estimation based on projection statistics,” IEEE Trans. Power Syst., vol. 11, no. 2, pp. 112–1118, May 1996.

[6] M. K. Çelik and A. Abur, “A robust WLAV state estimator using transformation,” IEEE Trans. Power Syst., vol. 7, no. 1, pp. 106–113, Feb. 1992. [7] P. J. Rousseau, “Least median of squares regression,” J. Amer. Statist. Assoc., vol. 79, pp. 871–880, 1984. [8] L. Mili, V. Phaniraj, and P. J. Rousseeuuw, “Least median of squares estimation in power systems,” IEEE Trans. Power Syst., vol. 6, no. 2, pp. 511–523, May 1991. [9] K. A. Clements and P. W. Davis, “Multiple bad data detectability and identifiably: A geometric approach,” IEEE Trans. Power Del., vol. 1, no. 3, pp. 355–360, Jul. 1986. [10] J. Zhu and A. Abur, “Bad data identification when using phasor measurements,” in Proc. IEEE Power Tech Conf., Laussane, Switzerland, 2007, pp. 1676–1681. [11] J. Chen and A. Abur, “Placement of PMUs to enable bad data detection in State estimation,” IEEE Trans. Power Syst., vol. 21, no. 4, pp. 1608–1615, Nov. 2006. [12] J. Chen and A. Abur, “Improved bad data processing via strategic placement of PMUs,” in Proc. IEEE Power Eng. Society General Meeting 2005, 2005, pp. 509–513. [13] J. B. A. London, Jr, L. F. C. Alberto, and N. G. Bretas, “Analysis of measurement set qualitative characteristics for state estimation purposes,” IET Gen., Transm., Distrib., vol. 1, pp. 39–45, 2007. [14] 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, Romania, Jun. 2009. [15] 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 system state estimation,” in Proc. IEEE Power Eng. Society General Meeting 09, Calgary, AB, Canada, Jul. 2009. [16] N. G. Bretas and S. A. Piereti, “The innovation concept in bad data analysis using the composed measurement errors for power system state estimation,” in Proc. IEEE Power Eng. Society General Meeting 2010, Minneapolis, MN, USA, Jul. 2010. [17] N. G. Bretas and J. B. A. London Jr., “Recovering of masked errors in power systems state estimation,” in Proc. IEEE Power Eng. Society General Meeting 2010, Minneapolis, MN, USA, 2010. [18] N. G. Bretas and A. S. Bretas, “Innovation concept for measurement gross error detection and identification in power system state estimation,” IET Gen., Transm., Distrib., vol. 5, pp. 603–608, Jun. 2011. [19] N. G. Bretas and A. S. Bretas, “Bad data analysis using the computed measurement errors for power system state estimation,” in Proc. IEEE Power Eng. Society General Meeting 2010, Minneapolis, MN, USA, 2010. [20] N. G. Bretas and A. Martins, “Statistical characterization of the measurement composed error and the innovation concept for gross error composition, detection, and identification in power system state estimation,” in Proc. IEEE Power Eng. Society General Meeting 2011, Detroit, MI, USA, 2011. [21] N. G. Bretas, S. A. Piereti, A. S. Bretas, and A. C. P. Martins, “A geometrical view for multiple gross error detection, identification, and correction in power system state estimation,” IEEE Trans. Power Syst., accepted for publication. [22] A. Abur and A. G. Expósito, Power System State Estimation: Theory and Implementation . Nova York, EUA: Marcel & Dekker, 2004. [23] Data for the test system. [Online]. Available: http://www.ee.washington.edu/research/pstca/. N. G. Bretas has main research interests that are concerned with power system analysis, transient stability using direct methods, power system state estimation, and energy restoration for distribution systems.

A. S. Bretas received the Ph.D. degree from VPI&SU, Blacksburg, VA, USA. Currently he is an Associate Professor at UFRGS, Porto Alegre, Brazil. His research interests are power systems state estimation, protection, and restoration.

Andre C. P. Martins received the Ph.D. degree from the University of São Paulo, Brazil, in 2005. Currently he is an Associate Professor at UNESP, Bauru, Brazil. His research interests are power systems state estimation and analysis.