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

The efforts to find Lyapunov functions for power systems with losses have been used approximated energy like functions to obtain good estimates of the attraction area of power systems. In this work, it is shown that the well known approximated energy-like function proposed by Athay et al. [Athay, T. et al., 1979] in the COA formulation is neither a Lyapunov function in the usual sense nor an extended function [Bretas, N. G. and Alberto, L. F. C.], if the transfer conductances are taken into account. In spite of that, it is shown 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 through the use of the extension of the invariance principle [Rodrigues H. M. et al., 2001]. The differences between the proposed extended Lyapunov function and the energy-like function of Athay et al. is of the order of the smooth perturbation. This suggest an explanation for the good critical clearing time estimates that have been obtained in the past using these approximated energy-like functions.