Numerical Computation of Critical System Recovery Parameter Values by Trajectory Sensitivity Maximization

Consider a particular finite-time disturbance applied to a system governed by ordinary differential equations and which possesses a stable equilibrium point. The recovery of the system from a disturbance is a function of the system parameter values. It is an important though challenging problem to identify the system parameter values, called critical parameter values, for which the system is just marginally unable to recover from a particular disturbance. Such critical parameter values correspond to cases where the system state, at the instant when the disturbance clears, is on the boundary of the region of attraction of the stable equilibrium point. The paper proposes novel algorithms for numerically computing critical parameter values, both for one and arbitrary dimensional parameter spaces. In the latter case, the algorithm computes the critical parameter values that are nearest to a given point in parameter space. The key idea underpinning the algorithms is that on the boundary of the region of attraction, the trajectory becomes infinitely sensitive to small changes in parameter value. Therefore, critical parameter values are found by varying parameters so as to maximize trajectory sensitivities. The algorithms are demonstrated using a fourth-order power system test case.

I. Introduction

Engineered systems experience disturbances which have the potential to disrupt desired operation. The ability of the system to recover from a particular finite-time disturbance, such as a fault on a certain transmission line in a power system, to a desired operating point depends on the system parameters. From a systems perspective, the disturbance can be thought of as a parameter dependent initial condition to the post-disturbance dynamical system (which itself is parameter dependent). It is an important and challenging problem to determine the parameter values for which the system is just marginally unable to recover from a particular disturbance, which we term critical parameter values. Solving this problem is of value for many applications, such as assessing fault vulnerability in power systems. This paper develops novel, theoretically motivated algorithms for efficient numerical computation of critical parameter values in both one and arbitrary dimensional parameter space.