An investigation of the significance of singular value decomposition in power system dynamics

Abstract

A theoretical explanation of the basics of singular value decomposition (SVD) is given to justify their use in applications in power system dynamics and control. The minimum singular value of a matrix not only specifies the rank of the matrix, it also gives a measure of distance of the matrix from the set of matrices having a rank less than its rank. This distance is used as a measure to compare the ability of inputs to control a mode. It is also used to compare the observability of a mode in a signal and to synthesize the optimum signal to maximize the minimum singular value of the observability matrix. The ideas are applied to a single-machine infinite busbar system with a static var controller (SVC). The system has two inputs and the SVD is used to compare the effectiveness of the outputs in controlling a mode. It is also used to determine the best output (signal) to have maximum observability of a mode in it.

Introduction

Oscillatory modes have been identified in recorded signals in power systems and in simulation studies of linear models. These modes may cause instabilities or unacceptable behavior at some operating conditions. These so-called ‘troublesome’ modes dictate an introduction of control action to improve the overall system response. Excitation controllers designed on the basis of linear models have been successfully implemented in the last decades. Controllability and observability are properties of linear models which are useful in the design and evaluation of controllers [1], [2]. A model that is not completely controllable must have some noncontrollable modes. An interesting question is to find a way to determine how difficult it is to control a mode from an input of the system. A mode may be easier to control from one input rather than another. The Popov–Belevitch–Hautus (PBH) test [2] specifies a rank test to indicate whether a mode is controllable or not. Numerical analysts paid a lot of attention to the problem of determining the rank of a matrix [1], [3], [4], [5]. It is generally accepted that the singular value decomposition is the most reliable, if not the most computationally economic, method for determining the rank of a given matrix.

The rank test is theoretically important but as far as application is concerned it does not lend itself to direct use. However, the singular value decomposition (SVD) offers not only a binary result on the question of controllability and observability of a given mode. It also gives an indication of how far is the matrix used to test the controllability of the mode from losing rank. The minimum singular matrix offers a measure of the distance of the matrix from matrices with a rank less than its rank. This measure is the one that turns out to be useful in comparing two inputs: which one makes the controllability matrix move away from loosing rank. A similar argument holds also for comparing outputs for the observability matrix. We have used the SVD in previous articles [6], [7], [8] in applications in power system dynamics. However in this article we give a theoretical explanation of the basics of using SVD in power system dynamics.

The rest of this article is arranged as follows. In Section 2 we give a brief description of models in power systems. The role of the notion of the rank of a matrix in specifying control properties of linearized model is emphasized. The controllability and observability of such models can be stated as rank tests on some input and output matrices. In Section 4 the well-established theorem of SVD is stated without proof. Three more theorems on applications of the SVD to power system models to obtain practical measures to quantify modal controllability and observability are given and proved. In Section 4 we apply the ideas to a model of a single machine connected to an infinite busbar taken from reference [9]. Conclusions are given in Section 5.