Manuscript Information Authors

[1]	Tokhir Makhmudov, Power Engineering Faculty, Tashkent State Technical University, Tashkent, Uzbekistan.

Abstract

Approximately 20-25 years ago, new approaches to the research of automatic control systems based on matrix methods appeared in the literature. New matrix designs (zero divisors, canonizers) have been developed that make it possible to extend the range of solvable problems, including not only minimal-phase, but also non-minimal-phase systems. It is obvious that matrix methods of investigation of linear systems are promising directions for the development of analysis and synthesis of dynamic objects, including studies of the steady–state stability of complex electrical systems. The mathematical formulation of the problem of studying the steady–state stability of electric power systems boils down to the following. Since all processes in the elements of the automatic control system are described by differential equations, the stability analysis reduces to investigating the properties of the solution of linearized equations for small perturbations. When analyzing and synthesizing dynamic systems, it becomes necessary to solve matrix equations. Along with the known methods for solving matrix equations, the article gives a method called the canonization method. Advantages of this method is its analyticity, i.e. this method allows us to carry out analytical studies of the resulting matrix equations. Canonicalization is based on a modified Gauss algorithm, in which the computational procedure is minimized. Here it should be noted that in the electrical system, when perturbations occur, the loss of stability occurs as a result of the synchronous generator leaving the synchronism or in the general case of rotating machines. Static elements affect the stability of electrical systems only by their parameters, which are usually assumed to be constant or slowly changing. Therefore, determining the conditions for the output from synchronism of a particular synchronous generator or their grostaups (stations) in a complex electrical system is the main task. The technology of embedding systems is an effective method for studying the steady–state stability (small oscillations) of an electrical system that makes it possible to determine all possible dynamic and structural properties of the linear matrix system under study.

Keywords

Electrical System, Mathematical Models, Embedding Matrix, Promatrix, Transient Characteristic

Reference

[01]	V. N. Bukov Vlojeniye system. Analiticheskiy podhod k analizu I sintezu matrichnix system. [Embedding systems. Analytical approach to the analysis and synthesis of matrix systems], N. F. Bochkareva Publisher, Kaluga (in Russian), 2006.
[02]	V. Olshevsky, E. Tyrtyshnikov Matrix Methods: Theory, Algorithms and Applications World Scientific, World Scientific Publishing Co. Pie. Ltd., 2010.
[03]	K. R. Allaev, A. M. Mirzabaev Matrichnye metody analiza malyh kolebaniy elektricheskih system [Matrix methods for the analysis of small oscillations of electrical systems], Fan va texnologiya Publ., Tashkent (in Russian), 2016.
[04]	J. C. Das Transients in Electrical Systems: Analysis, Recognition, and Mitigation, The McGraw-Hill Companies, 2010.
[05]	Altuve Ferrer Hector J., and Edmund O. Schweitzer. Modern Solutions for Protection, Control, and Monitoring of Electric Power Systems. Pullman, Wash. (2350 NE Hopkins Court, Pullman, WA 99163 USA): Schweitzer Engineering Laboratories, 2010.
[06]	Stefan G Johansson, Gunnar Asplund, Erik Jansson, Roberto Rudervall, “Power System Stability Benefits With Vsc DC-Transmission Systems”, CIGRE conference, Paris, 2004, pp. 1-8.
[07]	B. M. Abdellatif, “Stability with respect to part of the variables of nonlinear Caputo fractional differential equations”, Mathematical Communications, 23, 2018, pp. 119-126.
[08]	A. Klos Mathematical Models of Electrical Network Systems: Theory and Applications - An Introduction, Springer International Publishing AG, 2017.
[09]	K. R. Allaev, A. M. Mirzabaev, T. F. Makhmudov, T. A. Makhkamov “Matrix Analysis of Steady–State Stability of Electric Power Systems”, AASCIT Communications, Vol. 2, Issue 3, 2015, pp. 74-81.
[10]	S. Kovalenko, A. Sauhats, I. Zicmane, A. Utans “New Methods and Approaches for Monitoring and Control of Complex Electrical Power Systems Stability”, IEEE 16th International Conference on Environment and Electrical Engineering (EEEIC 2016), Italy, Florence, 2016, pp. 270-275.