Mathematical Modeling of Hybrid Electrical Engineering Systems

Abstract

A large class of systems that have found application in various industries and households, electrified transportation facilities and energy sector has been classified as electrical engineering systems. Their characteristic feature is a combination of continuous and discontinuous modes of operation, which is reflected in the appearance of a relatively new term “hybrid systems”. A wide class of hybrid systems is pulsed DC converters operating in a pulse width modulation, which are non-linear systems with variable structure. Using various methods for linearization it is possible to obtain linear mathematical models that rather accurately simulate behavior of such systems. However, the presence in the mathematical models of exponential nonlinearities creates considerable difficulties in the implementation of digital hardware. The solution can be found while using an approximation of exponential functions by polynomials of the first order, that, however, violates the rigor accordance of the analytical model with characteristics of a real object. There are two practical approaches to synthesize algorithms for control of hybrid systems. The first approach is based on the representation of the whole system by a discrete model which is described by difference equations that makes it possible to synthesize discrete algorithms. The second approach is based on description of the system by differential equations. The equations describe synthesis of continuous algorithms and their further implementation in a digital computer included in the control loop system. The paper considers modeling of a hybrid electrical engineering system using differential equations. Neglecting the pulse duration, it has been proposed to describe behavior of vector components in phase coordinates of the hybrid system by stochastic differential equations containing generally non-linear differentiable random functions. A stochastic vector-matrix equation describing dynamics of the processes has been obtained in the paper. The equation contains both continuous and discrete components, which characterize an amplitude signal modulation. An equation for probability density of phase coordinate distribution in the system has been developed on the basis of a mathematical model for a hybrid system.