Algorithmic Construction of Lyapunov Functions for Power System Stability Analysis

We present a methodology for the algorithmic construction of Lyapunov functions for the transient stability analysis of classical power system models. The proposed methodology uses recent advances in the theory of positive polynomials, semidefinite programming, and sum of squares decomposition, which have been powerful tools for the analysis of systems with polynomial vector fields. In order to apply these techniques to power grid systems described by trigonometric nonlinearities we use an algebraic reformulation technique to recast the system's dynamics into a set of polynomial differential algebraic equations. We demonstrate the application of these techniques to the transient stability analysis of power systems by estimating the region of attraction of the stable operating point. An algorithm to compute the local stability Lyapunov function is described together with an optimization algorithm designed to improve this estimate.

I. Introduction

A traditional approach to transient stability analysis of power systems involves the numerical integration of the nonlinear differential equations describing the complicated interactions between its components. This method provides an accurate description of transient phenomena but its computational cost prevents time-domain simulations from providing real-time transient stability assessments and significantly constraints the number of cases which can be analyzed [1].