Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems

TL;DR

This paper develops a method to assess the stability of stochastic differential-algebraic systems using Lyapunov exponents, with applications demonstrated in power systems engineering.

Contribution

It introduces a novel approach to stability analysis of SDAEs via Lyapunov exponents, including numerical methods and power system applications.

Findings

Numerical methods effectively compute Lyapunov exponents for SDAEs.

Application to power systems demonstrates practical utility.

Methods provide insights into system stability under stochastic influences.

Abstract

In this paper we discuss Stochastic Differential-Algebraic Equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-one SDAEs and their reformulation as ordinary stochastic differential equation (SDE). Via ergodic theory it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDE. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous $QR$ decomposition-based numerical methods are implemented to compute the fundamental solution matrix and to use it in the computation of the LEs. Important computational features of both methods are illustrated via numerical tests. Finally, the methods are applied to two applications from power systems engineering, including the single-machine infinite-bus (SMIB) power system model.