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-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.
Highlights
Modeling the dynamic behavior of systems employing differential-algebraic equations is a mathematical representation paradigm that is widely used in many areas of science and engineering.On the other hand, the dynamics of systems perturbed by stochastic processes have been adequately modeled by stochastic differential equations (SDEs)
The SDE system (5) preserves the inherent dynamics of a strangeness-free stochastic differential-algebraic equations (SDAEs) system [22]. In this way, the algebraic equations have been removed from the system, but, whenever a numerical method is used for the numerical integration, one has to make sure that the algebraic equations are properly solved at each time step, so that the back-transformation to the original state variables can be performed
The results obtained from our numerical experiments illustrate the approximations of the corresponding Lyapunov exponents (LEs) converge to degenerate random variables, i.e., the LE can be interpreted as a deterministic value, since in the limit the variance of the approximations tends to zero
Summary
Modeling the dynamic behavior of systems employing differential-algebraic equations is a mathematical representation paradigm that is widely used in many areas of science and engineering. The need of a generalized concept that covers both DAEs and SDEs, and allows the modeling and analysis of constrained systems subjected to stochastic disturbances, has led to the formulation of stochastic differential-algebraic equations (SDAEs). We review and extend the main concepts of this approach for asymptotic stability assessment of differential-algebraic equations driven by Gaussian white noise and apply the technique in the setting of power systems. This paper follows the ideas exposed in [21], where these QR-based methods were extended to the stochastic case These concepts and computational techniques are used to assess the asymptotic stability of power systems affected by stochastic fluctuations.
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