An Application of the Hamilton–Ostrogradsky Principle to the Modeling of an Asymmetrically Loaded Three-Phase Power Line

Andriy Chaban, Marek Lis, Andrzej Szafraniec () and Vitaliy Levoniuk
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Andriy Chaban: Faculty of Transport, Electrical Engineering, and Computer Science, University of Technology and Humanities, 26-600 Radom, Poland
Marek Lis: Faculty of Electrical Engineering, Czestochowa University of Technology, 42-201 Czestochowa, Poland
Andrzej Szafraniec: Faculty of Transport, Electrical Engineering, and Computer Science, University of Technology and Humanities, 26-600 Radom, Poland
Vitaliy Levoniuk: Department of Electrical Systems, Lviv National Agrarian University, 80381 Dubliany, Ukraine

Energies, 2022, vol. 15, issue 21, 1-19

Abstract: This paper presents a mathematical model of an electric power system which consists of a three-phase power line with distributed parameters and an equivalent, unbalanced RLC load cooperating with the line. The above model was developed on the basis of the modified Hamilton–Ostrogradsky principle, which extends the classical Lagrangian by adding two more components: the energy of dissipative forces in the system and the work of external non-conservative forces. In the developed model, there are four types of energy and four types of linear energy density. On the basis of Hamilton’s principle, the extended action functional was formulated and then minimized. As a result, the extremal of the action functional was derived, which can be treated as a solution of the Euler–Lagrange equation for the subsystem with lumped parameters and the Euler–Poisson equation for the subsystem with distributed parameters. The derived system of differential equations describes the entire physical system and consists of ordinary differential equations and partial differential equations. Such a system can be regarded as a full mathematical model of a dynamic object based on interdisciplinary approaches. The partial derivatives in the derived differential state–space equations of the analyzed object are approximated by means of finite differences, and then these equations are integrated in the time coordinate using the Runge–Kutta method of the fourth order. The results of computer simulation of transient processes in the dynamic system are presented as graphs and then discussed.

Keywords: Hamilton–Ostrogradsky principle; extended Lagrangian; interdisciplinary modeling; electric power system; three-phase power supply line; telegrapher’s equations; distributed parameter system; unbalanced load (search for similar items in EconPapers)
JEL-codes: Q Q0 Q4 Q40 Q41 Q42 Q43 Q47 Q48 Q49 (search for similar items in EconPapers)
Date: 2022
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