On the stochasticity of a machine swing equation using Itô differential

Abstract

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.

On the Stochastic Filtering Theory of a Power System Dynamics

“…That involves formal, systematic, unified and well-based mathematical method to achieve non-linear filtering of the stochastic system of the paper. This paper is different from our recent previous contri-bution [9] and the available results become a special case of the results of this paper. This paper 'respects' the process noise term as well as observation noise term of the filtering model of the SMIB system by accounting their contributions to the non-linear filtering equations of the SMIB system.…”

“…This paper 'respects' the process noise term as well as observation noise term of the filtering model of the SMIB system by accounting their contributions to the non-linear filtering equations of the SMIB system. On the other hand, the observation noise correction term is ignored in [9].…”

“…Finally, we achieve the non-linear filtering equations of the SMIB system, which are the con-sequence of the Kushner-Stratonovich equation and the exact conditional moment stochastic evolution equation. This paper is an extension of a recently published paper [9] in the following senses: (i) this paper revisits the stochastic swing equation in the Kushner-Stratonovich setting. On the other hand, the Fokker-Planck setting and multi-dimensional stochastic differential rules were the major ingredients to analyse the stochasticity of the SMIB system [9].…”

“…This paper is an extension of a recently published paper [9] in the following senses: (i) this paper revisits the stochastic swing equation in the Kushner-Stratonovich setting. On the other hand, the Fokker-Planck setting and multi-dimensional stochastic differential rules were the major ingredients to analyse the stochasticity of the SMIB system [9]. (ii) This paper accounts for the contributions of the process noise correction term as well as the observation noise correction term in the non-linear filtering of the SMIB system.…”

“…(ii) This paper accounts for the contributions of the process noise correction term as well as the observation noise correction term in the non-linear filtering of the SMIB system. On the other hand, the observation noise correction term is ignored in [9]. Notation Throughout the paper, the notation ⟨⟩ denotes the action of conditional expectation operator E on the Random Variable.…”