Èlektron. model. 2020, 42(5):82-96
The mathematical properties of nonlinear complex equations of steady-state modes of electrical systems as monoanalytic functions of many variables are investigated. The existing concepts of formal partial derivatives and the areolar derivative of polyanalytic functions are based on the assumption that complex variables are independent z and , at the same time, do not allow the existence of other complex variables. Therefore, the linearization of the system of nonlinear complex equations for the analysis of their physical stability is proposed to be performed using the pseudo-derivative of complex power, as a polygenic function of many variables with an infinite number of values. A possible approach to the construction of the limiting surface that bounds the region of physically stable modes is proposed. It is shown that splitting complex equations into two real equations is incorrect for the analysis of physical stability, since mathematical operations with real equations do not take into account the composition laws of a system of complex numbers.
steady state, monoanalytical functions of many complex variables, pseudo-derivative of polygenic function, hypercomplex numbers, physical stability.
- Kasner, E. (1936), “A complete characterization of the derivative of a polygenic function”, the Proceedings of the National Academy of Sciences, Vol. 22, pp. 172-177.
- Petrov A.M. (2006), Kvaternionnoye predstavleniye vikhrevykh dvizheniy [Quaternion representation of vortex motions, Kompaniya Sputnik+, Moscow, Russia.
- Klipkov S.I. (2019), “Quaternion Analysis of Electrical System Modes”, Elektronnoye modelirovaniye, Vol. 41, no. 6, pp. 15-35.
- Fedorovskiy, K.Yu. (2016), Approksimatsiya polianaliticheskimi mnogochlenami [Approximation by polyanalytic polynomials], IPM im. M.V. Keldysha, Moscow, Russia.
- Balk M.B. (1991), “Polyanalytic functions and their generalizations”, Itogi nauki i tekhn. Sovrem. probl. Matem. Fundam. Napravleniya, Vol. 85, pp. 187-246.
- Sekene, Y. and Yokojama, A. (1981), “Multisolutions for load flow problem of power System and their physical stability”, the Proceeding of 7th Power Syst. Comput. Conf, Lausanne, pp. 819-826.
- Klipkov, S.I. (2012), “The use of hypercomplex numerical systems for mathematical modeling of the limiting modes of electrical systems”, Reestratsiya, zberigannya i obrob. danykh, 14, no. 4, pp. 11-23.
- Klipkov, S.I. (2011), “On a new approach to the construction of hypercomplex number systems of rank two over the field of complex numbers”, Mat. Zhurn, Vol. 63, no. 1, pp. 130-139.
- Sin’kov, M.V., Boyarinova, Yu.Ye. and Kalinovskiy, Ya.A. (2010), Konechnomernyye giperkompleksnyye chislovyye sistemy. Osnovy teorii. Primeneniya [Finite-dimensional hypercomplex number systems. Foundations of the theory. Applications], Infodruk, Kiev, Ukraine.
- Klipkov,I. (2010), “Using a harmonic approach to the analysis of the limit modes of AC electrical systems”, Elektricheskiye seti & sistemy, no. 6, pp. 71-82.
- Klipkov, S.I. (2015), “Features of harmonic analysis of limiting modes of electrical systems”, Elektronnoye modelirovaniye, Vol. 37, no. 1, pp. 113-127.