Pires Mili Constrained Robust State Estimation 2014

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IEEE TRANSACTIONS ON POWER SYSTEMS 1

Constrained Robust Estimation of Power SystemState Variables and Transformer Tap Positions

Under Erroneous Zero-InjectionsRobson C. Pires, Member, IEEE, Lamine Mili, Senior Member, IEEE, and Flavio A. Becon Lemos, Member, IEEE

AbstractThis paper presents an equally constrained robustestimator of both the state and the transformer tap positions ofa power system able to withstand all types of outliers, includingbad leverage points and erroneous zero-injections. The statisticalrobustness of the estimator stems from the application of theSchweppe-type Huber GM estimator (SHGM) while its numericalrobustness originates from the use of an orthogonal iterativelyre-weighted least-squares algorithm together with the Van Loansmethod for processing the equality constraints. The good per-formance of the new estimator, termed EC-SHGM estimator forshort, is demonstrated on a small test system and on the BrazilianSouthern power system with increasing size ranging from 139buses to 1916 buses. It is shown that it exhibits superior conver-gence properties in all tested cases while the WLS method maysuffer from numerical instabilities or even divergence problemswhen large weights are assigned to zero power injections modelingfalse information.

Index TermsConfident zero bus injection and transformer tappositions estimates, robust bad data processing of measurementsand zero power injection.

I. INTRODUCTION

C ONSTRAINED power system state estimation aims atbuilding a reliable data base for power systems applica-tions such as static and dynamic security analysis and energymarket pricing system, to name a few. If the zero injections havetoo large residuals upon convergence of the state estimation al-gorithm, unacceptable errors in the power flow solutions mayresult [1], which will invalidate the contingency analysis andincrease the vulnerability of the power system to catastrophicfailures.

Manuscript received March 18, 2013; revised June 20, 2013; accepted July30, 2013. This work was supported in part by the Brazilian South State Elec-tric Utility (CEEE) under contract CEEE/2003-No. 9920524 and the NationalScience Foundation under the prime contract NSF EFRI-0835879, and the Sub-award Agreement CR-19806-477991. Paper no. TPWRS-00298-2013.R. C. Pires is with the Power and Energy SystemEngineeringGroupGESis/

ISEE at Federal University of Itajub, Minas Gerais (MG), Brazil (e-mail:[email protected]).L. Mili is with the Bradley Department of Electrical and Computer Engi-

neering, Northern Virginia Center, Virginia Tech, Falls Church, VA 22043 USA(e-mail: [email protected]).F. A. B. Lemos is with the Department of Electrical Engineering at Federal

University of Rio Grande do SulUFRS, Porto AlegreRio Grande do Sul(RS), 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.2013.2284734

In power system state estimation, two methods have beenproposed to satisfy the equality constraints. These are 1) as-signing large weights to the zero-injections, which are treatedas pseudo-measurements, and 2) formulating the problem asa constrained optimization problem, which is usually solvedvia the Lagrangian method [2]. Unfortunately, both approachessuffer from convergence problems in the event where anequality constraint is modeling incorrect information [3], forexample a wrong status of a load circuit breaker assumed to beopen whereas it is closed.It turns out that all the literature dealing with constrained

power system state estimation is assuming perfect zero injectioninformation when dealing with solution methods. Specifically,many papers concentrate either on modeling issues [4], [5] and[6] or on addressing numerical ill-conditioning problems whentoo large measurement weights are being assigned [7][10] and[11]. A few papers focus on improving the statistical robustnessof the constrained state estimation to non-leverage outliers inthe measurements using statistical tests on the residuals [2], [4],[12][14] and [15] or on the Lagrangian multipliers [16], [17].Therefore, all the proposed methods suffer from convergenceproblems under erroneous zero injection or bad leverage points.Recall that a leverage point is a measurement whose projectionon the space spanned by the row vectors of the Jacobian matrixis distant from the bulk of the measurements projections [18].By contrast, the equality constrained state estimation method

proposed here, which is termed hereafter EC-SHGM for short,is statistically and numerically robust to three types of outliers,namely vertical outliers, bad leverage points, and erroneouszero injections. Its statistical robustness stems from the use ofthe Schweppe-type Huber GM-estimator [18], the so-calledSHGM, along with a new techniques inspired by [6] to copewith erroneous injections. As for its numerical robustness, itoriginates from the use of an orthogonal iteratively reweightedleast-squares (OIRLS) algorithm using Givens rotations [19]and [20] together with the application of the generalizedsingular value decomposition method initiated by Van Loan[21] to deal with the equality constraints (see the Appendix).In summary, the proposed OIRLS algorithm that implementsthe EC-SHGM estimator exhibits the following interestingproperties:1) It implements the Van Loans method [21] and the orthog-onal transformations described in Golub [22] in a unifiedmanner;

2) It assigns, during the first iteration steps, weights of thesame order of magnitudes to both zero-injections and

0885-8950 2013 IEEE

Lamine MiliTypewritten Text


Lamine MiliTypewritten TextVOL. 29, NO. 3, MAY 2014



2 IEEE TRANSACTIONS ON POWER SYSTEMS

regular measurements. Then upon convergence of thealgorithm, it forces the residuals of all the zero-injec-tions flagged as correct data by the EC-SHGM to verysmall values by applying the Van Loans method. By thisway, ill-conditioning problems of the Jacobian matrix areprevented while incorrect zero-injections together witherroneous measurements are being suppressed.

This paper is organized as follows. Section II presents theEC-SHGM problem formulation. Section III presents com-putational issues regarding the algorithm implementation. InSection IV, numerical results are presented and discussed.Finally, conclusions are outlined in Section V.

II. PROBLEM FORMULATION

In this section, the EC-SHGM estimator of a power systemstate and transformer tap positions is developed and solvedusing the OIRLS algorithm for numerical robustness togetherwith the iterative refinement weighting method to cope witherroneous zero injections.

A. Equality Constrained State Estimation Problem

For an power system with on-loadtap changer transformers, the extended vector has

entries. It is related to the -dimensionalvector containing tele- and pseudo-measurements through

, where is an -dimensional vector-valuednonlinear function and is an -dimensional vector assumedto contain independent random variables with zero mean andknown covariance matrix, . The vectoris estimated by using an EC-SHGM estimator that minimizes

an objective function given by

(1)

subject to

(2)

Here is the Huber function defined as

forfor

(3)

and its first derivative with respect to , , is expressed as

forfor (4)

In the previous equations, is a standardized residual definedas , where is the th entry of the residualvector, and . The latteris a weight that is either equal to one up to a given threshold, ,or to a decreasing function of the squared projection statistic,

, associated with the th-measurement. The reader is re-ferred to [18] for a description of an efficient procedure for cal-culating PS. Note that it is the that make the SHGM robustagainst bad leverage points. Interestingly, they are calculated of-fline from the Jacobianmatrix assessed at the flat voltage profile;

they need to be updated only if the measurement configurationchanges or the topology is modified.In (2), is an -dimensional vector-valued nonlinear

function and is an -dimensional vector containing con-stant real and reactive power injections to be exactly satisfied(i.e., zero values for the zero injections). In our approach, theequality constraints do not remain the same throughout theiterations. Indeed, an equality constraint is suppressed or notdepending upon whether it is identified as outliers or not via theEC-SHGM weight function, Specifically,if the -function is equal to one during the initial iterations ofthe OIRLS algorithm, the associated constraint is deemed to bevalid and, therefore, is enforced in the final iterative step viathe Van Loans method [21], while if it is smaller than one, theassociated pseudo-measurement will be downweighted. Thismethod is described next.

B. Unconstrained SE Solution/Outer Loop

In the outer loop, the set of equality constraints are handled asany other measurements, i.e., the corresponding weights are ofthe same order of magnitude as those assigned to the measure-ments. Therefore, it is an unconstrained SHGM-estimator thatis being solved. It is a root of the necessary condition of opti-matility derived from (1), , which is expressed as

(5)

The foregoing equation is solved using the iterativelyreweighted least-squares (IRLS) algorithm because the latteris less prone to numerical problems than Newtons method[18], [23], [24]. This algorithm is derived by dividing andmultiplying by in (5) and using the definition ofyielding

(6)

which, after algebraic simplifications, reduces to

(7)

Now, putting (7) in a matrix form, we get

(8)

where

(9)

Let us replace in (8) by its first Taylor series expansionabout expressed as

(10)

which gives

(11)


PIRES et al.: CONSTRAINED ROBUST ESTIMATION OF POWER SYSTEM STATE VARIABLES 3

or

(12)

where This is equivalent to obtaining thesolution for a linear system of redundant equations defined as

(13)

where

(14)

and

(15)

Because the weight matrix in (14) and (15) changesthroughout the iterations, the matrix in (13) must be factoredat each iteration. This is the price to pay for suppressing outliersduring the updates of via . The iterationsare stopped when

(16)

where is the infinity norm.In order to gain numerical robustness, we now apply Givens

rotations as described in [22]. To this end, we redefine the linearset of equations given by (13) as follows:

(17)

Let us partition (17) into two subsets of equations, one associ-ated with the measurements and the other one associated withthe zero-injections flagged as valid, yielding

(18)

Here, and are matrices of dimension and, respectively, while and are vectors partition of

dimension and , respectively. Also,and are expressed as

(19)

where the parameter is a weight factor to be assigned toand .

Now, the augmented Jacobian matrix of the OIRLS algorithmis given by

(20)

which, once factored by applying the Givens rotations algorithmdescribed in [25] or [26], results in

(21)

The OIRLS outer loop solution is obtained by applying backsubstitution to the upper part of (21), yielding

(22)

Here, is the unitary upper triangular matrix that stems fromGivens rotations applied to defined in (20) as proposed by[25]. By contrast, the algorithm proposed in [26] leads to anon-unitary upper triangular matrix, As for , it is the cor-responding updated vector in the right-hand-side of (18), oncethe same orthogonal transformations are applied in (20).In the solution of (13), the set of non-valid equality con-

straints are identified as those associated with small diagonalweights of the matrix given in (15), say smaller than 0.099.Note that this outlier identification must be employed over allthe iterations of the OIRLS outer loop solution while assignedto the equality constraints is given the same value as the weightsassigned to the voltage measurements, which is . Bythis way, only the equality constraints flagged as valid are pro-cessed through the iterative refinement weighting method that isdescribed next.

C. Equality Constrained SE Solution/Inner Loop

The iterative refinement weighting method is proposed in [21]for solvingWLS estimator subject to equality constraints. How-ever, it is here adapted for solving the EC-SHGM estimatorwhile using Van Loans algorithm. When using an iterative re-finement to reach a solution, the augmented Jacobian matrix ex-pressed in (20) does not require to be refactored. Therefore, onlythe right-hand-side of the linear system of equations to be solvedin this step is updated. Consequently, the sequence of Givens ro-tations are stored instead of the orthogonal matrix .The iterative refinement weighting method aims at min-

imizing the Euclidean norm of residual vector partitioncorresponding to the equality constraints. Although the min-imization process is performed through an iterative scheme,the algorithm presents fast convergence rate at each outer-loopiteration. Once (22) is solved while the convergence criteriongiven by (16) is not satisfied, the inner loop starts to applythe Van Loans Method[21]. To minimize the zero-injectiondeviations provided by the iterative refinement weighting algo-rithm , the value previously assigned to the parameter shouldbe replaced by another one, since it is lower than the highestrecommended value, i.e., , where is processoraccuracy [27]. For instance, in our work is pickedin double precision calculations. Essentially, this algorithm canbe summarized as follows:1) After the inner loop counter has been initialized, i.e.,

, set equal to and update the parameter .2) Check the convergence criterion given by

(23)

where the tolerance can be the same one used in the outerloop and where and are the Euclidean and Infinity



norms, respectively. If the criterion is satisfied, go to step7. Otherwise, go to step 3.

3) Calculate4) While applying Givens rotations, solve for thatsatisfies

(24)

which is the solution of

(25)

where is an orthonormal matrix that stemsfrom the factorization of the Jacobian matrix defined in(18) as: , and is the residual vectorrelated to the non-violated equality constraints.

5) Calculate6) Increment inner loop counter: and return tostep 2.

7) Increment outer loop counter: and performnext step.

8) Update as and return to the begin-ning of the iterative outer loop.

III. COMPUTATIONAL ISSUES

A. Ordering Schemes

When solving the EC-SHGM, the Jacobian matrix has to befactored at each iteration, except in the inner loop. Techniquesemployed for compact storage and row and column orderingschemes are essential for an enhanced computational perfor-mance of Givens method [26]. The row ordering proposedin [28] aims at minimizing the intermediary fill-in while thecolumn ordering [29] aims at minimizing the fill-in in theunitary upper triangular matrix, that is obtained fromthe orthogonal Givens rotations [25]. Additionally, it is wellproven that the matrix has the same pattern and entries asthe matrix that stems from the Cholesky decomposition whenboth matrices result from the WLS solution [24]. This is themain reason for applying the above ordering schemes on thegain matrix, , so it can be symbolically factored. Thesestrategies decrease the computing time during the numericalfactorization of the augmented Jacobian Matrix [30], as it isshown in (20).

B. Iteration Strategy for Convergence

The iterative process implementing the EC-SHGM estimatorrequires a starting point that is not too far from a reliable solu-tion. For instance, if the iterative solution begins from the flatstart condition, then most of the corresponding residual mag-nitudes may be greater than the break-even-point, . Thus, therelated measurements will be strongly downweighted. The rec-ommended strategy is to perform the first iteration using a WLSestimator, and then switch to the EC-SHGM estimator. This pro-cedure takes 3 to 5 iterations to attain convergence. However,special care should be taken with the weights assigned to the

pseudo-tap-position measurements placed on an on-load-tap-changer (OLTC) transformer whose tap position is modeled asan unknown parameter. This important issue is discussed next.

C. Estimating the Tap Position of the OLTC Transformers

The OLTC model implemented in this work follows that pre-sented in [31]. Regarding the choice of the number and theplacement of the measurements for successfully estimating thetap positions of OLTC transformers, the reader is referred to[32]. To enhance the measurement redundancy and improve theconvergence rate of the algorithm, phasor measurement units(PMUs) may be used [33]. As for the iterative procedure, theimplementation of the following heuristics is recommended:1) Take as an initial value the nominal tap position for eachOLTC. This procedure allows us to reduce the magnitudeof the residuals of the associated voltage and active andreactive power flow measurements;

2) To avoid instability of the iterative process, place pseudo-measurements on the tap positions to be estimated, set themequal to the nominal values and pick their standard devia-tions equal to one-third of the tap range.

IV. NUMERICAL RESULTS

The proposed methodology is implemented in a program de-veloped in C++ Builder Version 6.0. It makes use of an ob-ject-oriented framework [34][36]. Besides the demonstrationperformed on an example system depicted in Fig. 1, the OIRLSalgorithm is executed on a real-life system of about 150 buses.Also, its performance is evaluated on the Brazilian Southernpower systems with increasing size of 340, 730 and 1916 buses.In all the simulations carried out on these systems, standard de-viations of 0.1% and 1% are respectively assigned to the volt-ages and the power measurements. To model the uncertainty inthe measurements, Gaussian errors with zero mean and givenvariances are added to the corresponding true values obtainedfrom load flow calculations. Gross measurements and erroneouszero injections are included in the measurement set by replacingthe associated good values with large values.

A. EC-SHGM Estimator Applied to an Example System

The one-line diagram of an example system, including themeasurement configuration identified by means of bulletsplaced at each measurement point, is shown in Fig. 1, whilethe corresponding network parameters in pu are presented inTable IIn this example, two OLTC transformers are included, whose

models are given in [31]. Therefore, there are 9 states variablesand 2 tap positions to be estimated. There are also four equalityconstraints associated with zero power injections (active/reac-tive) on Buses #3 and 4. Two bad data are included in the mea-surement set. Firstly, a reactive power injection is set to zero atBus #3 whereas in the field it has a non-zero value, which re-sults in one topological error. Secondly, a gross error is addedby changing the sign of the reactive power injection measure-ment at Bus #5. This case simulates an inversion of the cur-rent winding polarity following a measurement calibration pro-cedure.



Fig. 1. Online diagram of the 5-bus system.

TABLE IPARAMETER VALUES OF THE 5-BUS SYSTEM

TABLE IIESTIMATION RESULTS FOR THE 5-BUS SYSTEM

Table II presents the estimated variables throughout the iter-ations, except for the first one where the WLS iterative solutionbegins from the flat state condition and the tap setting state vari-able starts from the nominal position. The OIRLS outer/innerloop counter, viz. , are indicated in the headings of Table II.Table III presents measured and estimated values along with

the weighting factors and the chi-squared indices at eachstep of the iterative process. All underlined measurements areleverage points. As observed, both bad data, and , which

are bad leverage points, are correctly identified and suppressed.Regardless whether the reactive power injection at Bus #3 iswrongly set to zero or the reactive power injection at Bus #5 isincorrectly assigned a negative value, the state estimator solu-tion points out that there is actually a shunt capacitor connectedto Bus #3, and that Bus #5 is actually supplying the networkwith reactive power. These bad data are severely downweightedduring the iterative solution; they are the main cause for thelarge number of iterations performed in the inner loop. Usually,if only true information is modeled as equality constraints, theOIRLS outer/inner loop algorithms require on average 3/1 itera-tions to converge. Moreover, the deviations of the active powerinjections at Buses #3 and 4 and on the reactive power injectionat Bus #4 are reduced to very small values. These results showthe effectiveness of EC-SHGM estimator.Note that for the estimation results shown in Table II, the

tap positions are modeled as unknown variables while thoseshown in Table III, they are modeled as pseudo-measurements.In both cases, the standard-deviation assigned to them complieswith heuristic #1 and 2 and their weights remain unchangedthroughout the iterations.

B. EC-SHGM Estimator Applied to Power Systems NetworksTable IV summarizes the monitoring features of large-scale

power system networks used for evaluating the EC-SHGM es-timator.In this work, a network of about 150 buses is used as a bench-

mark system. The corresponding historical data are snapshots ofstored digital and analog measurements taken from the networkevery minute from January 2003 to July 2007. For instance, inTable IV the data of this system correspond to the snapshot takenat 04:00 PM on July 4, 2007. The other 340-, 730-, and 1916-bussystems are used to evaluate the computational performance ofthe implemented algorithm.The benchmark system consists of two interconnected asyn-

chronous subsystems belonging to neighboring countries, onesubsystem stretches in Argentina and Uruguay and operates at50 Hz and the other one stretches in Brazil and operates at 60Hz. Unlike other applications, power flows over back-to-backDC links are modeled as equality constraints of power injec-tions because DC link primary control mode is a constant powercontrol [37]. Specifically, the interconnections between the Ar-gentinian and the Uruguayan grid and between the Argentinianand the Brazilian grid are made through a frequency-converterstation with a maximum scheduled interchanged power around50 MW. As for the interconnection between the Brazilianand the Uruguayan grids, it is made through the Livramentofrequency-converter station with a maximum scheduled inter-changed power around 70 MW. Power interchanged via theseconverters are used to meet local load seasonally.The performance of the EC-SHGM estimator was also evalu-

ated on a real-life power system of about 150 buses of an indus-trial district located in the city of Porto Alegre in Rio Grande doSul, a Brazilian Southern State. Fig. 2 displays the measured andestimated values of the voltage magnitude and real and reactivepower at Bus #1258 over 288 snapshots recorded every 5 minfrom 00:00 AM to 11:55 PM on July 4, 2007. The infinite normsof the residual vectors for , , and are 0.01356, 0.00988 and



TABLE IIIMETERED VALUES AND ESTIMATES FOR THE 5-BUS SYSTEM

TABLE IVCHARACTERISTICS OF THE BRAZILIAN POWER SYSTEMS

, respectively. All these values are within stan-dard deviation of the residuals.Table V shows various performance indices of the OIRLS

algorithm evaluated for the Brazilian 159-, 340-, 730-, and1916-bus power systems. These indices are the numbers ofiterations of the outer and inner loop along with the averageand the maximum values of the active and reactive power

TABLE VPERFORMANCE INDICES FOR THE BRAZILIAN POWER SYSTEMS

injection absolute deviations,and , at selected buses. The bus where the maximumdeviation has occurred is also indicated. The largest deviationvalue showed in the table is the reactive power injection atSTA-ESUL 230-kV bus. It was induced by a gross error onSTA 525-kV voltage, which is a neighboring bus where anequality constraint is imposed as zero reactive power injection(AL2 230-kV) whereas in the field, actually, an equivalentshunt reactor compensator of 77.1 MVAr (3 25 MVAr) is



Fig. 2. Measured (in blue) and estimated (in red) values recorded over one day of (a) the voltage, (b) the real power injection, (c) the reactive power injection atBus #1258 of the power system of the industrial district located in the city of Porto Alegre/Rio Grande Do Sul.

connected.The main reason for the large number of iterationsrequired to reach the solution is the existence of a large numberof gross errors to be suppressed. Note that if an unconstrainedSHGM is carried out, the convergence is not attained.

Finally, Table VI presents the computing times of the OIRLSalgorithm for the four test systems. Each of these times is aver-aged over different hardware platforms. Also, the table indicatesoverhead times for up and down allocated memory. As it can be



TABLE VINUMBER OF ITERATIONS OF THE OUTER/INNER LOOP AND COMPUTING TIMES IN SECONDS

observed, the computing times of the OIRLS algorithm that im-plements an EC-SHGM estimator are compatible with real-timeapplications, even for very large systems. Notice that it may alsobe applied in study mode with additional features that requiresheavier computing times.

V. CONCLUSION

A reliable orthogonal iterative algorithm for solving a robustequality-constrained state estimator has been proposed. It es-timates power system state variables and transformer tap po-sitions under erroneous zero-power injections. Simulations onlarge-scale systems showed that the proposed method has theability to suppress gross errors corrupting both measurementsand zero injections, be they in position of leverage or not. More-over, it presents high convergences rates with low number ofiterations and small computing times. These features can befurther enhanced if a fast decoupled OIRLS version is imple-mented.

APPENDIXPRINCIPLES OF THE VAN LOANS METHOD

One way to satisfy the equality constraints in power systemstate estimation is to assign large weights to the zero-injections,which are treated as pseudo-measurements. An approximatedsolution for this problem can be reached by solving the fol-lowing WLS unconstrained problem:

(26)

where and are defined as in (18) and and are expressedas in (19).Applying a generalized singular value decomposition as sug-

gested in [21], we get

(27)

where ,and As shown in [27], the exact solutionto the WLS problem given by (1) and (2) is expressed as

(28)

while the solution using WLS estimator in the problem ex-pressed in (26) is given by

(29)

and when Obviously, dependson the value assigned to .To get small errors for zero-power injections, must be as-

signed large values. Following the recommendations made in[26], in this applications is set around while themeasurement weights are set around . Note that forother parameter values, numerical instabilities may occur.

ACKNOWLEDGMENT

Prof. R.C. Pires gratefully acknowledges the contributionsof N. Dallocchio for his work on the original version of theVDTap program that provides estimates of both the state vari-ables and the transformer tap positions of a power system. Thisprogram was developed as part of Dallocchios Bachelor andMS research work [36] carried out when he was enrolled in theUNIFEI Electrical Engineering program at ISEE/GESis.

REFERENCES[1] C. Gonzlez-Prez and B. F. Wollenberg, Analysis of massive

measurement loss in large-scale power system state estimation, IEEETrans. Power Syst., vol. 16, no. 4, pp. 825832, Nov. 2001.

[2] A.Monticelli, Testing equality constraint hypotheses in weighted leastsquares state estimators, IEEE Trans. Power Syst., vol. 15, no. 3, pp.950954, Aug. 2000.

[3] A. Monticelli, State Estimation in Electric Power SystemsA Gener-alized Approach. New York, NY, USA: Springer, 1999.



[4] A. Abur and M.K. elik, Least absolute value state estimation withequality and inequality constraints, IEEE Trans. Power Syst., vol. 8,no. 2, pp. 680686, May 1993.

[5] K. A. Clements, P. Davis, and K. Frey, Treatment of inequality con-straints in power system state estimation, IEEE Trans. Power Syst.,vol. 10, no. 2, pp. 567574, May 1995.

[6] A. S. Costa and J. Gouvea, A constrained orthogonal state estimatorfor external system modeling, Elect. Power Energy Syst., vol. 22, pp.555562, 2000.

[7] K. A. Clements, G. W. Woodzell, and R. C. Burchett, A new methodfor solving equality-constrained power system static-state estimation,IEEE Trans. Power Syst., vol. 5, no. 4, pp. 12601266, Nov. 1990.

[8] R. Nucera and M. Gilles, A blocked sparse matrix formulation for thesolution of equality-constrained state estimation, IEEE Trans. PowerSyst., vol. 6, no. 1, pp. 214224, Feb. 1991.

[9] E. Kliokys and N. Singh, Minimum correction method for enforcinglimits and equality constraints in state estimation based on orthog-onal transformations, IEEE Trans. Power Syst., vol. 15, no. 4, pp.12811286, Nov. 2000.

[10] G. N. Korres, A robust method for equality constrained state estima-tion, IEEE Trans. Power Syst., vol. 17, no. 2, pp. 305314, May 2002.

[11] G. N. Korres and T. A. Alexopoulos, A constrained ordering forsolving the equality constrained state estimation, IEEE Trans. PowerSyst., vol. 27, no. 4, pp. 19982005, Nov. 2012.

[12] F. Wu, W.-H. E. Liu, L. Holten, A. Gjelsvik, and S. Aam, Observ-ability analysis and bad data processing for state estimation usingHachtels augmented matrix method, IEEE Trans. Power Syst., vol.3, no. 2, pp. 604611, May 1988.

[13] F. Wu, W.-H. E. Liu, and S.-M. Lun, Observability analysis and baddata processing for state estimation with equality constraints, IEEETrans. Power Syst., vol. 3, no. 2, pp. 541548, May 1988.

[14] H. Singh, F. Alvarado, and W. L. e. Al, Constrained LAV state esti-mation using penalty functions, IEEE Trans. Power Syst., vol. 12, no.1, pp. 383388, Feb. 1997.

[15] R. Jabr, Power system Huber m-estimation with equality and in-equality constraints, Elect. Power Syst. Res., vol. 74, pp. 239246,2005.

[16] K. Clements andA. Simes-Costa, Topology error identification usingnormalized Lagrange multipliers, IEEE Trans. Power Syst., vol. 13,no. 2, pp. 347353, May 1998.

[17] A. Gjelsvik, The significance of the Lagrangemultipliers inWLS stateestimation with equality constraints, in Proc. 11th Power Syst. Com-putation Conf.PSCC, 1993, pp. 619625.

[18] L. Mili, M. Cheniae, N. Vichare, and P. Rousseeuw, Robust state es-timation based on projection statistics, IEEE Trans. Power Syst., vol.11, no. 2, pp. 11181127, May 1996.

[19] R. Pires, A. Simes-Costa, and L. Mili, Iteratively reweighted least-squares state estimation through Givens rotations, IEEE Trans. PowerSyst., vol. 14, no. 4, pp. 14991505, Nov. 1999.

[20] R. Pires and A. Simes-Costa, Fast decoupled IRLS state estimationthrough Givens rotations, in Proc. 13th PSCC, Trondheim, Norway,Jul. 1999, pp. 434440.

[21] V. Loan, On the method of weighting for equality constrained least-squares problems, SIAM J. Numer. Anal., vol. 22, no. 5, pp. 851864,1985.

[22] G. Golub, Numerical methods for solving linear least-squares prob-lems, Numer. Math., no. 7, pp. 206216, 1965.

[23] P. Holland and R. Welsch, Robust regression using iterativelyreweighted least- squares, Commun. Statist., vol. A6, no. 9, pp.813827, 1977.

[24] A. Bjork, Methods for Sparse Linear Least-Squares Problems, ser.Sparse Matrix Computation. New York, NY, USA: Academic, 1975.

[25] W. Gentleman, Least squares computations by Givens transforma-tions without square roots, J. Math. Applicat., vol. 12, pp. 329336,1974.

[26] N. Vempati, I. Slutsker, and W. Tinney, Enhancements to Givens ro-tations for power system state estimation, IEEE Trans. Power Syst.,vol. 6, no. 2, pp. 842849, May 1991.

[27] C. L. Lawson and R. J. Hanson, Solving least squares problems,SIAMClassics Appl. Math., vol. 15, 1995.

[28] W. Tinney and J. Walker, Direct solution of sparse network equationsby optimally ordered triangular factorization, Proc. IEEE, vol. 55, pp.18011809, 1967.

[29] A. George and M. Heath, Solution of linear least squares problemsusing Givens rotations, Linear Algebra, vol. 34, pp. 6983, 1980.

[30] A. Pandian, K. Parthasarathy, and S. Soman, Towards faster Givensrotations based power system state estimator, IEEE Trans. PowerSyst., vol. 14, no. 3, pp. 837843, Aug. 1999.

[31] L. Barboza, H. Zrn, and R. Salgado, Load tap change transformer: Amodeling reminder, IEEE Power Eng. Rev., pp. 5152, 2001.

[32] G. N. Korres, P. J. Katsikas, and G. C. Contaxis, Transformer tapsetting observability in state estimation, IEEE Trans. Power Syst., vol.19, no. 2, pp. 699706, May 2004.

[33] M. Shiroie and S. Hosseini, Observability and estimation of trans-former tap setting with minimal PMU placement, in Proc. IEEEPower and Energy Soc. General MeetingConv. and Delivery ofElect. Energy in the 21st Century, 2008, pp. 14.

[34] J. Rumbaugh, M. Blaha, and W. P. E. Al, Object-Oriented Modelingand Design. Englewood Cliffs, NJ, USA: Prentice Hall, 1991.

[35] S. Soman, S. Khaparde, and S. Pandit, Computational Methods forLarge Sparse Power Systems: An Object Oriented Approach. NewYork, NY, USA: Springer, 2001.

[36] N. Dallocchio, Modelagem Orientada a Objetos de Estimadores deEstados Robustos Aplicados Superviso da Segurana Operativa deRredes Eltricas, Masters thesis, Universidade Federal de ItajubUNIFEI, Setembro, MG, Brazil, 2007.

[37] P. Kundur, Power System Stability and Control. New York, NY,USA: McGraw-Hill Education, 1993.

Robson C. Pires (S98A99M02) received theB.Sc. degree in 1983, the M.Sc. degree in 1989,and the D.Sc. degree in 1998, all in electricalengineering, from Fluminense Federal University(UFF)RJ; Federal University of Itajub (UNIFEI)MG, and Federal University of Santa Catarina(UFSC) SC, respectively, all in Brazil.In 1996 (January to July), he did part of his

graduate program at The Bradley Department ofElectrical and Computer Engineering VTech,Blacksburg, VA, USA. Since 1987 he has been

with the Power System and Energy Institute (ISEE) at UNIFEI, where he iscurrently an Associate Professor. His research interests include electromagneticcompatibility (EMC) issues, analysis and control of large power systems androbust state estimation network application.

Lamine Mili (SM90) received the electrical engi-neering diploma from EPFL, Lausanne, Switzerland,in 1976, and the Ph.D. degree from the University ofLiege, Liege, Belgium, in 1987.He is a Professor of electrical and computer

engineering at Virginia Tech, Blacksburg, VA, USA.His research interests include robust statistics, powersystem dynamics and control and risk managementof critical infrastructures. He has five years ofindustrial experience with the electric power utility,STEG, where he worked as an engineer in the

planning department and the Test and Metering Laboratory from 1976 until1981. He is co-founder and co-editor of the International Journal of CriticalInfrastructures, http://www.inderscience.com/jhome.php?jcode=ijcis.

Flavio A. Becon Lemos (S94M01) received theB.Sc. degree in electrical engineering in 1988 fromthe Federal University of Santa Maria UFSM, andthe M.Sc. degree in 1994 and the Ph.D. degree in2000 in electrical engineering from the Federal Uni-versity of Santa Catarina UFSC, respectively.During 19961997, he was a fellow research

at Brunel University, U.K. He was with the StateElectrical Utility in Porto Alegre-RS, from 1988 to1992. Currently, he is an Adjunct Professor in theDepartment of Electrical Engineering at Federal

University of Rio Grande do Sul UFRS, Porto Alegre, Brazil. His mainresearch interests include power systems dynamics, voltage coordinated controland stability, and power system operation.

Pires Mili Constrained Robust State Estimation 2014

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