Weighted Least Square State Estimation Research Articles

Fast decoupled state estimation including thyristor controlled series compensator devices

This paper extends the weighted-least-square state estimation (WLS-SE) formulation to enable the solution of networks with embedded thyristor-controlled series compensation (TCSC). In the proposed methodology, the TCSCs reactances are treated as new state variables along with the complex voltage at the network nodes. Information regarding the specified active power flow through transmission lines connected to TCSCs and net power injection at their connecting nodes are strategically modeled as new active and reactive pseudo-measurements to be included into both: the linear and the fast-decoupled WLS-SE problem. Applications of the proposed extension in connection with the exact solution of a TCSC embedded power flow algorithm are presented and discussed. The IEEE 14- and 118-bus test systems are employed to illustrate and evaluate the proposed approach. The results point out that the new tool is effective to accurately estimate the system states and the control parameters of the TCSCs under distinct operating conditions.

A new algorithm for improving the numerical stability of power system state estimation

State estimation (SE) is based on an iterative process for solving weighted least squares (WLS) via the so‐called normal equations (NE). This process is prone to numerical instability and may lead to an ill‐conditioned state estimator in many cases. Although several methods have been proposed in the past to deal with the ill‐conditioning problem in high‐voltage transmission systems, the SE stability of low‐voltage distribution systems remains a challenge due to fewer measurements and high R/X ratios. This paper highlights the main reasons for the ill‐conditioning problem and focuses on the impact of the R/X ratio of the distribution systems. NE‐based SE uses a comparatively simple algorithm with much lower storage size. Hence, this paper proposes a regularized version of the WLS state estimator to solve the problem of ill‐conditioning using an adjustable regularization parameter. Simulation results on U.K. 18‐bus and radial 33‐bus distribution systems show improved performance in terms of reducing the computational complexity, measurements redundancy, the impact of high R/X ratios and improving the accuracy and numerical stability of the SE solution. © 2018 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Hybrid State Estimation in Complex Variables

Power system state estimation is classically formulated as a weighted least squares (WLS) optimization problem over the real domain, with the state variables being the voltage magnitudes and angles or the voltage real and imaginary parts. Although the voltages in ac networks are complex variables, a solution in the complex domain has not been previously reported; this is because a real function of complex variables is nonanalytic in its arguments, i.e., the Taylor series expansion in its arguments alone does not exist. By making use of the Wirtinger calculus, this paper presents a new implementation of the WLS state estimation problem in complex variables. The partial derivatives employed in the method give rise to a structure that is very simple to implement. The method can straightforwardly handle both legacy and phasor measurements; particularly, the processing of phasor current measurements does not require any special provisions, unlike previously reported hybrid state estimator implementations over the real domain. Numerical results are reported on networks with up to 9241 nodes, and they demonstrate that the accuracy of the complex variable implementation is competitive with that of the real variable constrained hybrid state estimator, while being significantly faster.

A Mixed WLS Power System State Estimation Method Integrating a Wide-Area Measurement System and SCADA Technology

To address the issue that the phasor measurement units (PMUs) of wide area measurement system (WAMS) are not sufficient for static state estimation in most existing power systems, this paper proposes a mixed power system weighted least squares (WLS) state estimation method integrating a wide-area measurement system and supervisory control and data acquisition (SCADA) technology. The hybrid calculation model is established by incorporating phasor measurements (including the node voltage phasors and branch current phasors) and the results of the traditional state estimator in a post-processing estimator. The performance assessment is discussed through setting up mathematical models of the distribution network. Based on PMU placement optimization and bias analysis, the effectiveness of the proposed method was proved to be accurate and reliable by simulations of different cases. Furthermore, emulating calculation shows this method greatly improves the accuracy and stability of the state estimation solution, compared with the traditional WLS state estimation.

Multiple gross errors detection, identification and correction in three-phase distribution systems WLS state estimation: A per-phase measurement error approach

This paper presents an analytical methodology for multiple measurement gross error detection, identification and correction for three-phase distribution systems weighted least square (WLS) state estimation. Initially, a three-phase WLS state estimator, considering the inherent distribution systems characteristics, as unbalance operation and different load types, is presented. Extended formulations for Jacobian matrix elements calculation, relating to different load models are proposed. Detection, identification and correction of gross errors are performed considering hypothesis testing in a per-phase component approach. Gross error detection is made through a Chi-square (χ2) Hypothesis Testing (HT) applied to the phase composed measurement error (CME). Composed errors are estimated with measurements’ phase innovation index (II). Gross error identification is made through the Largest Normalized Error Test property. Gross error correction is made considering the phase composed normalized error (CNE). A method to correct pseudo measurements identified as having gross errors is proposed. Validation is made with the IEEE 123 bus test system. Comparative tests with the state of art are presented. Test results are encouraging, indicating potential aspects for real-life applications.

Performance Assessment of Linear State Estimators Using Synchrophasor Measurements

This paper aims to assess the performance of linear state estimation (SE) processes of power systems relying on synchrophasor measurements. The performance assessment is conducted with respect to two different families of SE algorithms, i.e., static ones represented by weighted least squares (WLS) and recursive ones represented by Kalman filter (KF). To this end, this paper firstly recalls the analytical formulation of linear WLS state estimator (LWLS-SE) and Discrete KF state estimator (DKF-SE). We formally quantify the differences in the performance of the two algorithms. The validation of this result, together with the comprehensive performance evaluation of the considered state estimators, is carried out using two case studies, representing distribution (IEEE 123-bus test feeder) and transmission (IEEE 39-bus test system) networks. As a further contribution, this paper validates the correctness of the most common process model adopted in DKF-SE of power systems.

Detecting False Data Injection Attacks in AC State Estimation

Estimating power system states accurately is crucial to the reliable operation of power grids. Traditional weighted least square (WLS) state estimation methods face the rising threat of cyber-attacks, such as false data injection attacks, which can pass the bad data detection process in WLS state estimation. In this paper, we propose a new detection method to detect false data injection attacks by tracking the dynamics of measurement variations. The Kullback–Leibler distance (KLD) is used to calculate the distance between two probability distributions derived from measurement variations. When false data are injected into the power systems, the probability distributions of the measurement variations will deviate from the historical data, thus leading to a larger KLD. The proposed method is tested on IEEE 14 bus system using load data from the New York independent system operator with different attack scenarios. We have also tested our method on false data injection attacks that replace current measurement data with historical measurement data. Test results show that the proposed approach can accurately detect most of the attacks.

RETRACTED: A Robust WLS Power System State Estimation Method Integrating a Wide-Area Measurement System and SCADA Technology

With the development of modern society, the scale of the power system is rapidly increased accordingly, and the framework and mode of running of power systems are trending towards more complexity. It is nowadays much more important for the dispatchers to know exactly the state parameters of the power network through state estimation. This paper proposes a robust power system WLS state estimation method integrating a wide-area measurement system (WAMS) and SCADA technology, incorporating phasor measurements and the results of the traditional state estimator in a post-processing estimator, which greatly reduces the scale of the non-linear estimation problem as well as the number of iterations and the processing time per iteration. This paper firstly analyzes the wide-area state estimation model in detail, then according to the issue that least squares does not account for bad data and outliers, the paper proposes a robust weighted least squares (WLS) method that combines a robust estimation principle with least squares by equivalent weight. The performance assessment is discussed through setting up mathematical models of the distribution network. The effectiveness of the proposed method was proved to be accurate and reliable by simulations and experiments.

The Effect of Variable Weights in a WLS State Estimator Considering Instrument Transformer Uncertainties

The state estimator constitutes the cornerstone of the supervisory control and data acquisition system since it provides the power system operating situation in consecutive time intervals. Furthermore, the output of the state estimator is used by other tools responsible for the monitoring and control of the power system. Therefore, there is a need for the power system state estimator to be as accurate and reliable as possible. The main source of uncertainty that may deteriorate the accuracy of a weighted least squares (WLS) state estimator, provided that the network parameters are perfectly known, is the uncertainty that is encompassed in the measurements. It is well known that the measurement chain is not ideal and this information is passed to the state estimator through the measurement weights. In this paper, the effect of the measurement weights, which are calculated by considering both the standard uncertainties associated with the measurement devices and the instrument transformers, on the WLS state estimator is examined.

A distributed implementation of multi-area power system state estimation on a cluster of computers

This paper presents an efficient weighted least squares (WLS) distributed algorithm for multi-area power system state estimation including measurements provided by the supervisory control and data acquisition (SCADA) system and the synchronized phasor measurement units (PMUs). Each area performs independently its own state estimation using local measurements and exchanges border information at a coordination level that computes the system-wide state. The basic criterion of partitioning a power system into several control areas is to have areas as equal in size as possible, so that the workload on each area processor is as balanced as possible, and interconnections between distinct areas be limited, as much as possible, to reduce the amount of inter-process communication necessary. A true distributed implementation on a cluster of computers running Linux is developed. The code is written in C and sparse linear algebra routines from the Portable, Extensible Toolkit for Scientific computation (PETSc) library are used. Communication among area processors and the coordinator processor is implemented by using the MPICH2 implementation of the message-passing interface (MPI) protocol. By using extensions MPICH-G2, ES-MPICH2, MVAPICH2 of MPICH2 protocol, a portable Web-based distributed multi-area state estimator can be developed based on geographically distributed and heterogeneous cluster of computers. The proposed distributed approach has the same accuracy and redundancy level as the centralized estimator and provides the same solution only from the strictly mathematical point-of-view, by assuming that the measurement arrival delays and skew errors are identical to those of the centralized state estimator. Numerical simulations with a 1180 bus system, partitioned into equally or unequally sized area networks, demonstrate the efficacy of the proposed implementation in comparison with a centralized WLS state estimation.

Optimal static state estimation using improved particle swarm optimization and gravitational search algorithm

In this paper, two novel evolutionary search techniques based on Improved Particle Swarm Optimization (IPSO) algorithm and Gravitational Search Algorithm (GSA), have been proposed to solve the static State Estimation (SE) problem as an optimization problem. The proposed methods are tested on five IEEE standard test systems along with two ill-conditioned test systems under different simulated conditions and the results are compared with the same of standard Weighted Least Square State Estimation (WLS-SE) technique, Particle Swarm Optimization (PSO) based SE and Hybrid Particle Swarm Optimization Gravitational Search Algorithm (PSOGSA) based SE technique. The optimization performance and the statistical error analysis show the superiority of the proposed GSA based SE technique over the other two techniques.

Power system state estimation: Undetectable bad data

SUMMARY This paper proposes an index, called Undetectability Index (UI), to classify the measurements according to their characteristics of not reflecting their errors into the residuals of the weighted least squares state estimator (WLS) from a geometric analysis of this estimator. Gross errors in measurements with high UIs are very difficult to be detected by methods based on residual analysis, because the errors in those measurements are “masked”, i.e. they are not reflected in their residuals. The paper also presents a very simple algorithm to compute the UI of all available measurements in a metering system. This algorithm is very simple and uses routines already available in the existing WLS Estimator software. Two UI applications are investigated (i) to design more reliable metering systems for bad data processing, i.e. metering systems formed by measurements with UI lower than a pre-specified value and (ii) to estimate the measurement errors. Several simulation results (with IEEE-14 and 30 bus systems) have validated the propositions. Copyright © 2013 John Wiley & Sons, Ltd.

Optimal Static State Estimation Using hybrid Particle Swarm-Differential Evolution Based Optimization

In this paper, swarm optimization hybridized with differential evolution (PSO-DE) technique is proposed to solve static state estimation (SE) problem as a minimization problem. The proposed hybrid method is tested on IEEE 5-bus, 14-bus, 30-bus, 57-bus and 118-bus standard test systems along with 11-bus and 13-bus ill-conditioned test systems under different simulated conditions and the results are compared with the same, obtained using standard weighted least square state estimation (WLS-SE) technique and general particle swarm optimization (GPSO) based technique. The performance of the proposed optimization technique for SE, in terms of minimum value of the objective function and standard deviations of minimum values obtained in 100 runs, is found better as compared to the GPSO based technique. The statistical error analysis also shows the superiority of the proposed PSO-DE based technique over the other two techniques.

Investigation of WLS state estimation convergence under topology errors and load increment

State estimation plays an essential role in real-time power system monitoring, control, and automation. Utilising non-linear optimisation theory, it determines the optimal system states based on the network model and the gathered measurements from the system. Conventional weighted least square (WLS) state estimation algorithm may suffer divergence problem due to the occurrence of topology error. This paper studies the impact of the topology errors on the convergence property of WLS state estimation under load increment. Two types of load increment are utilised: one is the load increment of all load buses, and the other is a single load increment. The IEEE 118-bus system is used as the test case and simulation results are reported. In addition, phasor measurement unit (PMU) measurements including voltage magnitude and angle measurements are added to the measurement vector of state estimation to verify if they could solve the divergence problem and improve state estimation accuracy.

PMU Measurement Uncertainty Considerations in WLS State Estimation

A method to assign weights to the measurements obtained through phasor measurement units (PMUs) in a weighted least squares (WLS) state estimation is presented in this paper. The uncertainties for direct measurements are obtained from the manufacturer's specifications. For pseudo-measurements, the uncertainties are evaluated by using the classical uncertainty propagation theory. The propagation of measurement uncertainty as a function of line length and conductor type is also investigated. The lower and upper bounds of the estimated states considering the measurement uncertainties are found by using linear programming. The proposed method is applied on the IEEE 14-, 30-, 57-, and 118-bus test systems, and the state estimation results including the lower and upper bounds of the estimated states are presented.

Topology Identification, Bad Data Processing, and State Estimation Using Fuzzy Pattern Matching

A new method for topology identification, bad data detection and state estimation (SE) is proposed in this paper. The method utilizes fuzzy clustering and pattern matching for this purpose. The proposed approach is very efficient and accurate for simple SE and to aid in improving the accuracy and time performance for conventional weighted least squares state estimator (WLSE). In this method, a fuzzy pattern vector is generated based on the analog measurement vector (telemetry data) received. The topology identification and gross errors are detected from difference between the fuzzy pattern vector and the analog measurement data. The gross errors and topology errors in the measurement data are detected and corrected by using the fuzzy pattern vector which can directly be used as good measurement data for SE. The effectiveness of the proposed method is demonstrated on IEEE 14-bus, IEEE 57-bus, practical 75-bus Indian system and IEEE 118-bus systems. The SE results obtained by proposed method are also compared with ANN models reported in the literatures.

Auto Tuning of Measurement Weights in WLS State Estimation

This paper describes an approach for choosing and updating measurement weights used in weighted least squares (WLS) state estimation. Since the weights are related to the measurement error variances, sample variances are estimated using historical data from previous measurement scans and the corresponding WLS estimation results. The proposed approach can be implemented as a one-time estimation function for off-line execution or as a recursive function for updating the measurement weights on-line. Simulated measurement data and state estimation results are used to test and verify the accuracy of the proposed method. The proposed method can be integrated into an existing WLS state estimator as an added feature.

Decoupled State-Estimation in Energy Control Centres

Abstract STATE-ESTIMATION (SE) is the process of assigning a value to an unknown system state-variable based on measurements from that system according to some criteria. Usually, the process involves imperfect measurements inherent in the process and therefore estimating the system state is based on a statistical criteria that estimates the true value of the state-variable so as to minimize or maximize the selected criteria. A commonly used and familiar criteria is that of minimizing the sum of squares of the differences between the estimated and ‘true’ (i.e. measured) values. Power System Static State-Estimation (PSSE) is a proven technique for providing modern energy control centres with reliable real-time data bases for security monitoring and on-line control. The state-estimation processes real-time raw measurements to obtain optimal estimates of voltage magnitudes and angles at all buses, which determines the static state of the Power System (PS). The estimation problem of PSSE is usually formulated as a nonlinear least-square problem. Among state estimators based on the normal equations approach, the weighted least square (WLS) estimator provides reliable estimates but is slow, mainly due to its requirement for updating and factorization of the full-matrix in every iteration. The decoupled state-estimator requires less computation time per iteration and less storage requirement than WLS. The concept of p- δ, Q-V decoupling technique as applied in load flow studies is extended to state estimation technique also which reduces the storage requirement and also the time for convergence. The most common approach to the SE problem is to formulate it as a nonlinear problem usually solved by an iterative procedure based on successive linearization; in each iteration one has to solve a linear least-squares problem. Associated with WLS estimation is a set of redundant equations are to be solved to yield the solution. These equations are ‘Normal Equations’ and are solved using Cholesky factorization. The normal equations are very satisfactory with respect to preserving sparsity, but they may give rise to numerical ill-conditioning. Methods using orthogonal transformations, such as those of Householder have far better stability properties. The Method of Data Equations which uses the Householder orthogonal transformation is more efficient than the conventional method of normal equations. Also, it is less prone to computer round-off errors and is more numerically stable. The standard approach to the solution of the WLS state estimation in power system is the iterative normal equations method. Occassional ill-conditioning has been experienced with this method. Alternative approaches to state-estimation are based on ‘orthogonal transformation’. These methods have improved numerical behaviour, but generally their sparsity suffers when the measurement redundancy is high. In this paper the decoupled estimator based on transformation method is presented using different algorithms and the results agree with the method of normal equations. Measurements are transformed into new ‘measurements’ that are functions of the states and of the original measurements. In PSSE redundancy rate is an important aspect. Estimator obtained from various redundancy rates, by considering active and reactive redundancy seperately obtained by uniformly distributing the measurements over the system is presented. Line flows and bus injection measurements required for stateestimation studies, have been obtained from load-flow computations. Error (noise) terms (σ = 0.01 and 0.001) are added to these measurements. The error vector being simulated using a random number generator. Convergence tolerance of 0.001 PU and 0.001 are used for V and δ, respectively. Results obtained for various combinations of measurements; and with different order of processing these measurements have been presented. Measurements are classified as. Type 1, 2 : Power injections. Type 3, 4 : Power flows. Type 5 : Voltage magnitudes. Different cases analysed are. i) Case 1 : 1, 2, 3, 4 & 5 Type measurements. ii) Case 2 : 1, 3, 4 and 5 Type measurements. iii) Case 3 : 1, 2, 3 and 4 Type measurements. iv) Case 4 : 3 and 5 Type measurements. It is important that at least few voltage measurements (NBUS/ 10 to NBUS/5) must be included while estimating the states. When line-flows only are used as measurements, solution takes large number of iterations to converge. Comparative results obtained from ‘method of Normal equations’ and ‘Data equations’ have been presented. Illustrative examples considered are IEEE-14 bus system and 5-bus sytem.

A Hybrid State Estimator: Solving Normal Equations By Orthogonal Transformations

The standard approach to the solution of the weighted least square (WLS) state estimation in power system is the iterative normal equations method. Occasional ill-conditioning has been experienced with this method. Recently alternative solution approaches based on orthogonal transformations have been proposed. Sparsity generally suffers in these methods. In this paper, the network condition which causes ill- conditioning is studied. A hybrid approach which enjoys the same numerical robustness of the orthogonal methods with sparsity close to the standard WLS method is presented. The modification needed to implement the hybrid method on an existing standard WLS state estimation program is quite small.

Convergence analysis of weighted least-squares and fast decoupled weighted least-squares state estimation

An eigenvalue analysis technique is used to determine the convergence properties of weighted least-squares and fast decoupled weighted least-squares state estimation. Weighted least-squares state estimation has zero eigenvalues, and its convergence is insensitive to variations. Convergence of fast decoupled weighted-least squares state estimation is insensitive to measurement accuracy and operating-point variations and remains excellent for transmission-circuit reactance-to-resistance ratios as low as 4:1. Both algorithms converge rapidly in three-four iterations.