# FEATURES OF THE ANALYSIS OF PHYSICAL STABILITY OF STEADY-STATE MODES OF AC ELECTRICAL SYSTEMS

S.I. Klipkov

Èlektron. model. 2020, 42(5):82-96
https://doi.org/10.15407/emodel.42.05.082

### ABSTRACT

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.

### KEYWORDS

steady state, monoanalytical functions of many complex variables, pseudo-derivative of polygenic function, hypercomplex numbers, physical stability.

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