Stochastic dynamics in power systems excited by discrete-continuous random disturbances

This paper presents a unified framework for analyzing power systems subjected to both discrete and continuous random disturbances—a critical gap in existing literature that typically treats these disturbances separately. Unlike conventional approaches that focus on either continuous stochastic processes or discrete switching events in isolation, our novel methodology simultaneously captures both types of uncertainties within an integrated Markovian jump framework. The stochastic model of multi-machine power systems is formulated as a high-dimensional hybrid system and transformed into a quasi-Hamiltonian system with Markovian jump processes. A pioneering two-step approximation method is developed that first converts the hybrid system into a weighted-average model, then reduces it to a one-dimensional averaged Itô equation representing system energy dynamics. The approximate analytical solution of the corresponding Fokker-Planck-Kolmogorov (FPK) equation provides stationary response estimates for the original hybrid systems. A Lyapunov exponent approach is employed for asymptotic stability analysis with probability one. The methodology is validated through comprehensive analysis of Kundur's 4-machine 2-area system, demonstrating superior computational efficiency and analytical insights compared to traditional Monte Carlo simulations.