Smooth perturbation on a classical energy function for lossy power system stability analysis

The efforts to find Lyapunov functions for power systems with losses have been until now in vain. Despite that, engineers have been using approximated energy-like functions to obtain good estimates of the critical clearing time (CCT) in transient stability analysis of power systems. These approximated energy-like functions are not Lyapunov functions, and are usually obtained by an integration process followed by an approximation of the integration path. Therefore, the good CCT estimates obtained with these functions are not supported by a sound theory. Nevertheless, it is shown in this paper, for a particular approximated energy-like function, a theoretical approach to support these good estimates. The approximated energy-like function studied in this paper is well known in the literature, and was proposed by Athay et al. in the COA formulation. It is shown that this approximated energy-like function is neither a Lyapunov function in the usual sense, nor an extended Lyapunov function, when the transfer conductances are taken into account. In spite of that, a function attending the requirements of the extension of the Invariance Principle, that is, an extended Lyapunov function, can be obtained by smooth perturbations on that energy-like function. This perturbed function can be used to estimate the attraction area without approximations or conjectures. Indeed, the difference between the proposed extended Lyapunov function and the approximated energy-like function has the order of a smooth perturbation. This fact supports the good CCT estimates that have been obtained using these approximated energy-like functions, and encourages engineers to keep using them for CCT estimates.

I. Introduction

Over THE past three decades, a great effort has been devoted to the application of direct methods to power-system transient stability analysis in real time. In spite of that, some difficulties were not yet overcome. One issue is the nonexistence of Lyapunov functions for power systems with losses. Considerable efforts have been made in finding energy functions for power systems with losses. However, these efforts have been in vain. In this paper, the word “losses” is used to refer to transmission-line resistances in the network-preserving models, or to the transfer conductance which appears in the nodal admittance matrix as a consequence of network reduction.