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

González-Zumba, Andrés; Fernández-de-Córdoba, Pedro; Cortés, Juan-Carlos; Mehrmann, Volker

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-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. 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 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.
Published in: Mathematics, 10.3390/math8091393, MDPI