1808 IEEE TRANSACTIONS ON SMART GRID, VOL. 5, NO. 4, JULY 2014 LAV Based Robust State Estimation for Systems Measured by PMUs Murat Göl, Student Member, IEEE, and Ali Abur, Fellow, IEEE Abstract—The weighted least squares (WLS) estimator is com- monly employed to solve the state estimation problem in today’s power systems, which are primarily measured by SCADA mea- surements, including bus power injection, branch power flow and bus voltage magnitude measurements. Despite being widely used, WLS estimator remains to be non-robust, i.e., it fails in the pres- ence of bad measurements. The so called least absolute value (LAV) estimator is more robust, but is not widely used due to its higher computational cost. LAV estimator has the desirable property of automatic bad-data rejection provided that the measurement set does not include any “leverage measurements.” There will be two important advantages to using LAV estimator when systems are observed by phasor measurements: 1) if the measurement set con- sists of only phasors, the leveraging effect of measurements can be easily eliminated by strategic scaling; 2) computational perfor- mance of LAV estimator will become competitive with that of WLS due to the linearity of the estimation problem for phasor mea- surements. This paper demonstrates these advantages and argues that the LAV estimator will be a statistically robust and computa- tionally competitive estimator for those power systems that will be measured entirely by PMUs. Index Terms—Least absolute value, phasor measurement units, state estimation. I. INTRODUCTION T ODAY’S POWER systems are measured by conventional measurements, including bus voltage magnitude, branch power flow, and bus injection measurements. State estima- tion problem for those systems is solved iteratively by using weighted least squares estimator (WLS), which is a widely used and well-investigated method. Despite being iterative, WLS estimator is quite fast due to the efficient sparse matrix methods used in its implementation. This is, however, true only for the main solution engine. As well known, WLS estimator is non-robust and will be biased even in the presence of a single bad measurement. Hence, the solution is customarily followed by a bad data processor whose function is to detect, identify, and correct any existing bad data. This is commonly accomplished by the largest normalized residual test. In this Manuscript received May 19, 2013; revised October 01, 2013, December 03, 2013; accepted January 19, 2014. Date of publication April 28, 2014; date of current version June 18, 2014. This work made use of Engineering Research Center Shared Facilities supported by the Engineering Research Center Program of the National Science Foundation and the Department of Energy under NSF Award Number EEC-1041877 and the CURENT Industry Partnership Program. Paper no. TSG-00406-2013. The authors are with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 USA (e-mail: muratgol@ece.neu. edu; abur@ece.neu.edu). 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/TSG.2014.2302213 test, the main bottleneck is the computation of the residual covariance matrix, which requires calculation of a subset of the elements in the inverse of the sparse gain matrix. Even when highly efficient sparse inverse method [1]–[3] is employed, the computational complexity grows approximately proportional to the number of measurements. A practical alternative which avoids this post-processing stage is the use of re-weighted least squares method where measurement weights are modified based on their respective residuals during the iterative solution [4]–[7]. Despite its simplicity, this approach may lead to biased solutions especially when multiple interacting bad data are present. Breakdown point of an estimator can be simply defined as the smallest amount of contamination (number of gross errors) that can cause an estimator to give an arbitrarily incorrect solution [8]. Estimators with high breakdown points have been investi- gated and developed by researchers in the past couple of decades [9], [10]. Some of these have also been applied to power system state estimation [11]–[14]. Among these robust estimators, the Least Absolute Value (LAV) estimator was shown to have de- sirable properties where its implementation can be made com- putationally efficient by taking advantage of power system’s properties [15]–[17]. However, LAV estimator remains vulner- able against the so-called leverage measurements [12], [18]. This shortcoming along with the added computational burden brought on by the linear programming (or interior point) based problem formulation have so far made widespread implementa- tion of LAV estimators non-viable. Here the bad data refers to measurements with gross errors. Measurement error is defined as the difference between the mea- sured and the true value of the measurement. Note that true value of a measurement and therefore its measurement error cannot be known. In power system state estimation a measurement may be considered an outlier either because of its wrong value (it may contain a gross error) or because of the very large or very small entries (compared to the rest of the entries) of the mea- surement jacobian in the row corresponding to that measure- ment. The latter type of outlier will likely be a leverage mea- surement (this is what is referred as an outlier in this paper). Note that a leverage measurement may or may not carry bad data. Leverage point is an observation (or measurement), which lies away from the rest of the measurements in the measurement space. In the special case of power system state estimation, a leverage point (or measurement) will have distinctly different values in the row of the measurement jacobian corresponding to this measurement. There are several ways to identify leverage measurements which are well documented [10]–[12]. As evident from the large number of publications on phasor measurements, their optimal deployment and utilization for a 1949-3053 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.