This letter introduces a robust generalized state estimator which is able to detect and reject gross measurement errors, parameter errors, and topology errors simultaneously. The solution is based on finding a consistent estimate which minimizes the total number of hypothesized gross errors. The problem is formulated as a mixed integer nonlinear program. A small-scale ac estimation example is given which illustrates some of the properties of the method

Irving, MR

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1886 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 4, NOVEMBER 2009Robust Algorithm for Generalized State EstimationM. R. IrvingAbstract—This letter introduces a robust generalized state esti-mator which is able to detect and reject gross measurement errors,parameter errors, and topology errors simultaneously. The solu-tion is based on finding a consistent estimate which minimizes thetotal number of hypothesized gross errors. The problem is formu-lated as a mixed integer nonlinear program. A small-scale ac esti-mation example is given which illustrates some of the properties ofthe method.Index Terms—Generalized state estimation, mathematical pro-gramming, robust estimation.I. INTRODUCTIONG ENERALIZED state estimation algorithms [1]–[3] op-erate with busbar-level models, so that topology errors(e.g., incorrect circuit-breaker status) can be handled by grosserror processing. By extending a previous robust estimator ap-proach [4], [5], a new algorithm is introduced which is capableof simultaneously processing gross measurement errors, grossparameter errors, and topology errors. Test results on a smallexample network illustrate the capabilities of this method.II. PROBLEM FORMULATIONIn addition to conventional transmission line models, a gen-eralized estimator also models the flow of active and reactivepower on “links”, where a link represents a zero-impedance con-nection (such as a closed circuit-breaker). Flows on links andadditional voltage magnitudes and phase angles (at their con-nection points) are included as states of the system. The non-linear ac loadflow equations then become(1)(2)where, active/reactive injection at bus ;, active/reactive line flow, bus to ;, active/reactive link flow, bus to ;over connected buses.(3)(4)where, bus voltage magnitudes;, bus voltage phase angles;, line conductance and susceptance parameters;Manuscript received January 21, 2009; revised March 31, 2009. First pub-lished August 25, 2009; current version published October 21, 2009. Paper no.PESL-00007-2009.The author is with the Brunel Institute of Power Systems, Brunel University,Uxbridge UB8 3PH, U.K.Digital Object Identifier 10.1109/TPWRS.2009.2030116, shunt conductance and susceptance at bus ;, ac loadflow functions.Following the principle of adopting an “uncertainty range” foreach measured quantity [4], and also introducing uncertainty inthe line parameters, we can define a set of uncertainty inequali-ties:(5)wheremeasurement (or parameter) ;estimated value;lower tolerance;upper tolerance;0/1 binary variable;arbitrarily large positive scalar value.The set of uncertainty inequalities include all available mea-surements of , , , , , , , , plus toler-ance ranges for , , , and . The tolerance ranges aredefined according to the familiar engineering practice of speci-fying a quantity as accurate to within plus or minus a tolerance.If all the measurements and parameters are within tolerance, itwill be possible to find a solution which satisfies all the aboveinequalities with . However, if some measurements, orparameters, have unexpectedly large errors, it will be necessaryto use to “switch off” the corresponding uncertaintyconstraint. The scalar value must be chosen to be large incomparison with possible gross errors, in order to eliminate anyinfluence on the solution from “switched off” constraints.Topology errors can be handled in a similar manner. A linkindicated as closed (between bus and bus ) is represented bythe following inequalities:(6)(7)These constraints are “complementary”, since the same vari-able is used in both (6) and (7). If the link is actually closed,should be zero, constraining the voltages at each end of the linkto be equal and switching off the flow constraints. If the link isactually open, should be found to be 1, constraining the linkflows to zero and switching off the voltage constraints. A link in-dicated as open can be represented by similar inequalities, withand interchanged.The robust estimation criteria used is to select a state estimatethat minimizes the total number of measurements, parameters,and link states that are considered out-of-tolerance, i.e.,(8)0885-8950/$26.00 © 2009 IEEEAuthorized licensed use limited to: Brunel University. Downloaded on November 25, 2009 at 11:49 from IEEE Xplore.  Restrictions apply. IRVING: ROBUST ALGORITHM FOR GENERALIZED STATE ESTIMATION 1887Fig. 1. Example network.This results in a mixed integer nonlinear program in the binaryvariables and real variables , , , , , , , ,, and . In the case that a link has no status indicationavailable, inequalities (6) and (7) can be used with the corre-sponding value omitted from the objective function.III. RESULTS AND ANALYSISThe problem can be readily solved using standard mathe-matical programming software, such as the MINLP algorithmof Fletcher and Leyffer, available via the NEOS web-service[6]. The proposed method is therefore related to the branch andbound method of Monticelli [1]. The robustness of the pro-posed method can be illustrated using the small example net-work shown in Fig. 1. An accurate loadflow solution was ob-tained, with the link between bus 2 and bus 3 closed, and withcorrect network parameters as shown in the second column ofTable I. Measurement and parameter uncertainty ranges werethen defined, as shown in columns 3 and 4 of Table I. Three si-multaneous gross errors were introduced, on , , and thelink status (which was erroneously indicated as “open”). The so-lution of the mixed integer program gave the estimates shown incolumn 5 of Table I. The three gross errors have been detectedand corrected. The minimized cost function has a value of 3, asthe binary variables associated with the gross errors have valuesof one, with all other binary variables remaining at zero.A disadvantage of the proposed method is that little noisefiltering is provided, as the estimator will use the full rangeof tolerance available to find an estimate that minimizes thenumber of hypothesized gross errors. This problem could becounteracted by introducing a second stage of noise filteringusing weighted least squares, once the gross errors have beeneliminated. Mixed integer nonlinear programming is a compu-tationally demanding task. This limits the online application ofthe method to small areas of the network where multiple grosserrors are suspected and where sufficient local redundancy isavailable.TABLE IUNCERTAINTY RANGES AND ESTIMATES FOR EXAMPLE NETWORKIV. CONCLUSIONSA robust generalized state estimator has been presented. Theuse of binary decision variables throughout (to accept or rejectmeasurements, parameter values, and topological states) allowsall three types of gross errors to be judged on a comparablebasis. Simultaneous gross errors in measurements, parameters,and topology can be identified and corrected. The computationaleffort of solving the required mathematical program probablylimits the practical application of the method to relatively smallnetworks, or sections of the network, at present. However, it ishoped that future advances in numerical algorithms and com-puter hardware will widen the scope of application in the future.REFERENCES[1] A. Monticelli, State Estimation in Electric Power Systems–A General-ized Approach. Norwell, MA: Kluwer, 1999.[2] H. Singh and F. L. Alvarado, “Network topology determination usingleast absolute value state estimation,” IEEE Trans. Power Syst., vol. 10,no. 3, pp. 1159–1165, Aug. 1995.[3] E. M. Lourenco, A. S. Costa, and K. A. Clements, “Bayesian-basedhypothesis testing for topology error identification in generalized stateestimation,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 1206–1215,May 2004.[4] M. R. Irving, “Robust state estimation using mixed integer program-ming,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1519–1520, Aug.2008.[5] S. Gastoni, G. Granelli, and M. Montagna, “Robust state estimationprocedure based on the maximum agreement between measurements,”IEEE Trans. Power Syst., vol. 19, no. 4, pp. 2038–2043, Nov. 2004.[6] NEOS, NEOS Server for Optimisation, 2008. [Online]. Available:http://neos.mcs.anl.gov/neos.Authorized licensed use limited to: Brunel University. Downloaded on November 25, 2009 at 11:49 from IEEE Xplore.  Restrictions apply. 

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