Calculation of transient processes in electrical circuits using Chebyshev polynomials

Abstract

Transient electromagnetic processes in electrical systems present a significant risk to machines and influence the reliability of relay protection; therefore, the investigation of these processes is important. Transient modeling in transformers is presently done by construction and solution of state equations for magnetoelectrical equivalent schemes that bring together the electric and magnetic circuits of these devices, taking into account the interaction with each other. Modeling of these processes can be quite long when using modern software. An aim of the given work is development of more fast-acting method of calculation of electromagnetic transients, using a polynomial approximation of the solution, and also development of scheme model of method that creates convenience for engineers in modeling. A method of calculation of transients in electric circuits is developed on the basis of expansion of the solution of state equations in series of orthogonal Chebyshev polynomials. A scheme interpretation of the developed method is offered. It is shown that, in the special equivalent scheme, Kirchhoff’s laws apply to images of currents and magnetic fluxes as vectors containing the values of coefficients of decomposition of these values by Chebyshev polynomials. The given method allows replacement of operations with instantaneous values of currents by operations with direct currents in the presented equivalent scheme. Transient calculation via the suggested method in a single-phase transformer processor is twice as fast as with well-known methods. The presented method is especially appropriate for transient calculation in magnetoelectrical circuits, since it allows the use of voltage sources controlled by the current derivative for easy calculation of the integrals of solutions.