Estimating Relevant Portion of Stability Region using Lyapunov Approach and Sum of Squares

Traditional Lyapunov-based transient stability assessment approaches focus on identifying the stability region (SR) of the equilibrium point under study. When trying to estimate this region using Lyapunov functions, the shape of the final estimate is often limited by the degree of the function chosen - a limitation that results in conservativeness in the estimate of the SR. More conservative the estimate is in a particular region in state space, smaller is the estimate of the critical clearing time (CCT) for disturbances that drive the system towards that region. In order to reduce this conservativeness, we propose a methodology that uses the disturbance trajectory data to skew the shape of the final Lyapunov-based SR estimate. We exploit the advances made in the theory of sum of squares decomposition to algorithmically estimate this region. The effectiveness of this technique is demonstrated on a power systems classical model.

I. Introduction

The transient stability analysis of power systems traditionally relied on time domain simulations that were computationally intensive, thereby making online applications challenging. This led to the ongoing development of various direct methods [1] which start by defining a scalar function whose specific level set approximates the stability region (SR) for the post fault system configuration. The direct methods only required knowledge of the system state values at the time of fault clearing to predict the stability, thus significantly reducing the simulation time. Direct methods were of two types, namely, analytical energy function-based and Lyapunov-based. The techniques belonging to the first category start by characterizing the stability boundary and then defining an analytical energy function that is proved to exist under certain system related assumptions. This function is constrained to decrease along all the post-fault system trajectories and can take both positive and negative values. Using a unique level set of the energy function for estimating the SR (also known as the closest unstable equilibrium point approach) was seen to be extremely conservative in terms of the size of the final SR estimate relative to the actual SR size [1]. This led to the idea of relevant stability boundary and the controlling unstable equilibrium point (CUEP) approach where only the portion of the SR relevant to the particular disturbance was estimated using a particular level set of the energy function thus reducing the conservativeness. This chosen level set differed according to the disturbance being studied. While these approaches relied on a derived analytical energy function proven to satisfy the required criteria, there were numerous system related assumptions (such as transverse intersection of stable and unstable manifolds on the stability boundary, hyperbolicity and finiteness of equilibrium points, low transmission conductances, etc. [1]) that were impossible to verify and/or were found to be violated under various system conditions [2].