Measurement Jacobian Matrix Research Articles

Microgrid state and frequency estimation using Kalman filter: an approach considering an augmented measurement Jacobian matrix

Microgrid state and frequency estimation using Kalman filter: an approach considering an augmented measurement Jacobian matrix

Blind false data injection attacks in smart grids subject to measurement outliers

ABSTRACT False data injection attacks (FDIAs) can manipulate measurement data from Supervisory Control and Data Acquisition (SCADA) system and threat state estimation in smart grids. Blind FDIAs (BFDIAs) enhance traditional FDIAs, which eliminate the limitation of grasping measurement Jacobian matrix H in advance, but when there are outliers in measurement data, attack performance is degraded. In this paper, improved BFDIAs are proposed. In off-line phase, low-dimensional measurement matrix without outliers calculated by Linear Local Tangent Space Alignment algorithm (LLTSA) is sent into Continuous Deep Belief Network (CDBN) as training data to learn their probability distribution. In on-line phase, real-time low-dimensional measurement matrix with outliers are sent into the trained model as inputs, and outputs are reconstructed by the probability distribution in off-line phase, which eliminates the influence of outliers indirectly. Simulations are implemented on PJM 5-bus and IEEE 14-bus systems to verify the performance of proposed strategy compared with PCA-based BFDIAs.

On the Silent Perturbation of State Estimation in Smart Grid

In this article, we address the problem of integrity of the state estimation in presence of a novel bad data injection threat. We consider persistent adversaries who inject attack vectors “silently,” i.e., they drift the results of the state estimation gradually in multiple steps, with each step bypassing the bad data detection step of state estimation. Prior works on the topic have shown how an adversary can bypass the bad data detection by constructing attack vectors as linear combinations of the column vectors of the measurement Jacobian matrix. We show here that the attack surface is much broader than implied by this assumption. We demonstrate our attack strategy using realistic load patterns from NYISO in the IEEE 14-bus system. We also propose a detection method that uses the expected energy of normalized residues and a paired t -test, and we show its effectiveness. The generality of the proposed attack and detection strategy implies that it can be used in other cyber-physical systems involving state estimation based on linearized state perturbation.

A Greedy Algorithm for Observability Analysis

This letter presents a numerical algorithm for observability analysis. The proposed algorithm is based on a greedy strategy combined with a Gram-Schmidt orthonormalization procedure of the rows of the measurement Jacobian matrix. The algorithm outputs the system is observable with the available measurements, and if not, it looks for the best candidate measurements to restore observability. Also, an efficient calculation of the null space of the Jacobian matrix can be obtained. The algorithm is an improvement with respect to other existing algorithms that calculate a factorization of the Jacobian matrix. Throughout the process it is not necessary to invert matrices, nor select pivots and the errors caused when dividing by values close to zero are avoided. Four examples are given to illustrate the effectiveness of the proposed method.

Observability Analysis in Water Transport Networks: Algebraic Approach

AbstractState estimation (SE) techniques are being applied to different network systems in order to convert system measurements into real information about the network state. SE applications to water systems are relatively novel, but these techniques have been implemented in other fields for decades. In those applications, observability analysis (OA) is required prior to application of SE techniques with different purposes: (1) to identify redundant information, (2) to detect elements that make no contribution in the subsequent SE process, or (3) to identify observable islands. However, no discussion has been found in the pertinent literature regarding any interest in applying OA to water networks, with there being only a few basic applications. The aim of this paper is twofold: first, to present the implementation of a novel algebraic OA approach to water networks, which is based on the application of a Gauss elimination technique to the measurement Jacobian matrix, and to discuss and justify the interes...

State estimation and observability analysis for phasor measurement unit measured systems

The synchronised phasor measurement units (PMUs) can provide extremely accurate measurements that will improve the state estimation performance and observability. Although full observability based only on PMUs appears very impractical today, it will be the case in a few years when these devices become standard equipment at substations. This paper compares the performances of three state estimation formulations, in terms of the convergence properties and the variance in the estimated states. Phasor measurements, as well as zero injections, are incorporated and either rectangular or polar coordinates can be used to represent the bus voltage and branch current phasors measured by PMUs. Current phasors in polar coordinates may give rise to ill-conditioning problems, which are resolved by conversion into rectangular coordinates format, using the uncertainty propagation theory. The behaviour of the elements in the measurement Jacobian matrix, because of the current measurements, is investigated for possible ill-conditioning of the state estimator coefficient matrix. A non-iterative topological-numerical algorithm is proposed for observability checking and restoration in case of observability loss. The proposed algorithms are demonstrated with the IEEE 14-bus and IEEE 118-bus systems.

Observability Analysis Based on Echelon Form of a Reduced Dimensional Jacobian Matrix

This letter presents an efficient numerical method to determine the observable islands and additional measurements for placement, in an unobservable system. This is accomplished in a noniterative manner via the use of the Echelon form of a reduced order measurement Jacobian matrix, that is associated with a subset of the injection measurements and a subnetwork of the entire system. The proposed method is faster and easier to implement than existing approaches and gives more reliable results due to better conditioning of the processed matrices. The IEEE 14-bus system illustrates the steps of the proposed method.

Inclusion of PMU current phasor measurements in a power system state estimator

Three different methods of inclusion of current measurements by phasor measurement units (PMUs) in a power system state estimator is investigated. A comprehensive formulation of the hybrid state estimator incorporating conventional, as well as PMU measurements, is presented for each of the three methods. The behaviour of the elements because of the current measurements in the measurement Jacobian matrix is examined for any possible ill-conditioning of the state estimator gain matrix. The performances of the state estimators are compared in terms of the convergence properties and the variance in the estimated states. The IEEE 14-bus and IEEE 300-bus systems are used as test beds for the study.

A new method for redundancy analysis of measurements applied to three-phase state estimation

In this paper, a new method based on the properties of the Gram matrix is proposed to verify the redundancy of measurements in three-phase state estimation. The Gram matrix is constructed using rows of the measurement Jacobian matrix as vectors. The method can be used for measurement systems planning or updating of three-phase networks so that the overall system remain observable even under possible branch contingencies and loss of measurements. It is a fast and robust numerical method based on the Gram matrix factorization that is easy to implement. A small numerical example that shows the application of the method is presented.

On the Use of Gram Matrix in Observability Analysis

This letter presents some notes on the use of the gram matrix in observability analysis. This matrix is constructed considering the rows of the measurement Jacobian matrix as vectors, and it can be employed in observability analysis and restoration methods. The determination of nonredundant pseudo-measurements (normally injections pseudo-measurements) for merging observable islands into an observable (single) system is carried out analyzing the pivots of the gram matrix. The gram matrix can also be used to verify local redundancy, which is important in measurement system planning. Some numerical examples are used to illustrate these features. Others features of the gram matrix are under study.

A simple factorization-based observability analysis and meter placement method

A simple factorization-based observability analysis and meter placement method is presented in this paper. Measurement Jacobian matrix is extended to include non-existing branch flow measurements so that observable islands are identified simultaneously when extra candidate measurements are placed to make the network observable if initially the system is unobservable. The method is described in detail along with numerical examples, which illustrates its effectiveness.

Jacobian Matrix-Based Observability Analysis for State Estimation

In this paper, numerical observability analysis is restudied. Algorithms to determine observable islands and to decide a minimal set of pseudo-measurements to make the unobservable system observable are presented. The algorithms make direct use of the measurement Jacobian matrix. Gaussian elimination, which makes the whole process of observability analysis simple and effective, is the only computation required by the algorithms. Numerical examples are used to illustrate the proposed algorithms. Comparison of computation expense on the Texas system among the proposed algorithm and the existing algorithms is performed.

State Estimation Observability Based on the Null Space of the Measurement Jacobian Matrix

This letter describes a new technique for state estimation observability analysis. This technique is computationally efficient and based on calculating the null space of the measurement Jacobian matrix. The technique is illustrated through a clarifying example. Conclusions are duly drawn.

Observability analysis for state estimation using linear programming

The linear programming technique is used to analyse the observability of power system state estimation networks. This technique has the capability of identifying the observable islands of the system network. It can also identify the necessary branches or nodes needed for pseudo-measurement placement to recover the overall system observability. The technique manipulates only integer numbers. It uses the network connectivity matrix and the measurement Jacobian matrix as equality constraints. Computational aspects of the proposed technique are presented and results of several examples are reported.

A fast integer algorithm for observability analysis using network topology

An integer algorithm for determining the observability of a power system network with its linearized state estimator model is presented. By using some properties of the allocation of the flow and injection measurements and the network topology, the algorithm manipulations only integer numbers in the measurement Jacobian matrix. Theorems that determine the rank of the special kinds of matrices are proposed to speed up the observability analysis. Neither manipulation of real numbers nor a complicated combinatorial problem is involved in the proposed algorithm. Therefore, it is much simpler and the speed for observability analysis is much faster compared to previous methods. Moreover, no numerical problems are encountered. A detailed example of the IEEE 14-bus test system following the steps in the algorithm is presented and test results on the algorithm are reported. >

Modelling and state estimation of power plant steam turbines

The development of a robust state estimator for ill-conditioned systems based upon the Levenberg-Marquardt algorithm in connection with Givens orthogonal row processing technique is described. The estimator works in two stages: the square-root-free Givens transformation is applied to the measurement Jacobian matrix to reduces it to the product of a unit upper triangular matrix and a diagonal matrix, then the square-root-free Givens transformation is again used to transform a diagonal matrix whose diagonal element is equal to the Levenberg-Marquardt parameter. The Levenberg-Marquardt parameter is adjusted so that the reduction in quadratic cost function is downhill in successive iterations and rapid convergence of the estimator is ensured. An observability analysis algorithm is also developed which is simple and easy to implement. Both description and implementation of the square-root-free Givens transformation are presented. The estimator is applied to a 250 MW, double-flow, single reheat steam turbine which has seven feedwater heaters. Tests results show that the estimator performs well under a wide range of measurement errors and measurement redundancy ratios. A sample of estimated results for rated loading condition are presented. In the appendix, the turbine modelling procedure is discussed.

Structure of measurement Jacobian matrices for power systems

In steady-state modelling of power systems, the measurement equations together with the system states may be ordered so that the measurement Jacobian matrix H displays a particular block-sparsity structure, which will become more apparent as system size increases. The resulting partitioned form for H may be regarded as a simply specified decomposition of the measurement equations. The decomposition may then be exploited using a theorem which enables the rank of H to be determined from a set of rank tests on reduced-order matrices. This result has implications for the testing of numerical observability on power systems, particularly when the new rank tests are distributed within a multiprocessor computer system. A novel feature of this approach to the determination of numerical observability is that a degree of localisation of unobservable states becomes possible. The fundamental structure of the decomposition described for H is suitable for exploitation in other areas of steady-state power systems analysis such as state estimation.

Network Observability Analysis through Measurement Jacobian Matrix Reduction

A method of observability analysis for the state estimation in power systems is presented. The method is based on a symbolic reduction of the measurement Jacobian matrix without its actual computation. The algorithm is characterized as being extremely simple and fast. The results of the algorithm application in the Energy Management System (EMS) of a large electric utility are given.