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This paper discusses the stochastic evolutions of the state trajectories of a stochastic Single Machine Infinite Bus (SMIB) system. The dynamical equation of the SMIB system assumes the structure of a second-order non-linear differential equation. That is described as a power swing equation. After accounting for the stochasticity in the machine swing equation as well as accomplishing the phase space formulation, we are led to a vector stochastic differential equation. Importantly, the Itô stochasticity as well as the Stratonovich stochasticity coincide for the stochastic swing equation, since the Wang-Zakai correction term of the stochastic integral will vanish for the special case. Despite universality of the second-order non-linear machine swing equation in theoretical studies, e.g. power system dynamics, circuits and systems literature, the stochasticity of the machine swing equation is not analysed in a greater detail yet. In this paper, we analyse the stochasticity of the swing equation after utilizing the notion of the stochastic evolution of a scalar function of vector stochastic processes in combination with conditional expectation. Note that the stochastic state vector of the machine swing equation of this paper is a Markov process.

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In power systems dynamics and control literature, theoretical and practical aspects of the wind turbine-generator system have received considerable attentions. The evolution equation of the induction machine encompasses a system of three first-order differential equations coupled with two algebraic equations. After accounting for stochasticity in the wind speed, the wind turbine-generator system becomes a stochastic system. That is described by the standard and formal Itô stochastic differential equation. Note that the Itô process is a strong Markov process. The Itô stochasticity of the wind speed is attributed to the Markov modeling of atmospheric turbulence. The article utilizes the Fokker-Planck method, a mathematical stochastic method, to analyse the noise-influenced wind turbine-generator system by doing the following: (i) the authors develop the Fokker-Planck model for the stochastic power system problem considered here; (ii) the Fokker-Planck operator coupled with the Kolmogorov backward operator are exploited to accomplish the noise analysis from the estimation-theoretic viewpoint.

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