458 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 Combined State Estimation and Measurement Calibration Shan Zhong, Student Member, IEEE, and Ali Abur, Fellow, IEEE Abstract—A measurement calibration method is described in this paper. The proposed method identifies calibration models for the uncalibrated measurements and estimates the calibration model parameters along with the system states. The permanent nature of calibration errors allows their estimation by using multiple scans of measurements. The parametric models of the measurements can be estimated by reformulating the conventional system state estimation problem and incorporating calibration models. The paper also addresses the issues of network and parameter observability and provides simulation results for cases involving the IEEE 14-bus test system. Index Terms—Parameter estimation, power system state estima- tion, remote measurement calibration. I. INTRODUCTION M EASUREMENTS that are transmitted to the control centers are processed by the state estimators in order to determine the state of the system during normal operation. Measurements may not all be correct, due to errors introduced by the transducers, telecommunication medium, etc. These er- rors may have both random as well as systematic components, depending upon the error source. State estimators are designed to filter the random component of the errors in the telemetered quantities. Most state estima- tion formulations, including the popular weighted least squares (WLS) method, are developed based on the assumption that the measurements contain only random errors with zero mean and known variance. Exceptions to the WLS estimator that are more robust against gross errors also exist but are not widely imple- mented due to their computational complexity. Any existing sys- tematic errors are, therefore, filtered either by post WLS esti- mation methods or via alternative robust estimation methods. Unfortunately, performances of all of these methods are lim- ited by the measurement redundancy. The number of bad data that can be handled by these methods cannot exceed an upper limit, which is dictated by the local measurement redundancy and configuration. Hence, in order to eliminate all gross errors, those measurements with only random errors should constitute the majority of the overall measurement set. Any calibration er- rors in these measurements should be eliminated before using them in state estimation. Measurement calibration at the substa- tion is a labor-intensive and costly process, not to mention the fact that it has to be repeated at regular intervals during the year. Manuscript received January 29, 2004. This work was supported in part by the NSF/PSERC. Paper no. TPWRS-00670-2003. The authors are with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail: zhshan@ee.tamu.edu; abur@ee.tamu.edu). Digital Object Identifier 10.1109/TPWRS.2004.841237 The alternative, which is referred to as “soft” calibration, has been proposed, and various ways of calibration that can be re- motely conducted in the control centers are discussed in [1]–[4]. In [1], the authors utilize the measurement residuals to do a linear regression between the estimated/measured measurement pairs. The procedure that is described in [2] and [3] is executed at individual substations by taking advantage of the redundancy provided by multiply-measured quantities. A calibration model is assumed, and its parameters are estimated using these mul- tiply-measured quantities. A system-wide calibration approach, which relies on a set of essentially reliable measurements, is proposed in [4], but implementation and performance details are not provided. In this paper, measurement calibration is considered as a pa- rameter estimation problem, where the users choose the cali- bration models for the measurements of interest. An excellent review of parameter estimation methods used in power system applications is given in [5]. The commonly used method of aug- menting the state vector with the unknown parameters of in- terest is applied to the measurement calibration problem, and preliminary results are given in [6]. The main idea is to relate the true and measured values by parametric equations and esti- mate these parameters simultaneously with the system states by using a modified state estimation program. In order to filter the random noise and provide the needed redundancy, the proposed technique can be implemented offline utilizing several recorded measurement scans. This paper will further elaborate on this ap- proach and also address the related issues, such as the determi- nation of the suspect measurement set, verification of the cali- bration results, and observability analysis. The paper is organized such that Section II introduces the gen- eral formulation of the proposed method. Section III discusses the implementation of the method for a specific case, where the measurement calibration model is chosen as a quadratic function. Section IV mainly addresses the observability issue. Results of testing the performance of the proposed method using simulated data for different-size IEEE systems are shown in Section V. II. FORMULATION OF THE PROBLEM Power system state estimation is formulated based on the measurement equations given below: (1) where measurement vector of dimension ; nonlinear function relating the error-free measure- ments to the system states; 0885-8950/$20.00 © 2005 IEEE