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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Andriy Chaban & Marek Lis & Andrzej Szafraniec & Vitaliy Levoniuk, 2022. "An Application of the Hamilton–Ostrogradsky Principle to the Modeling of an Asymmetrically Loaded Three-Phase Power Line," Energies, MDPI, vol. 15(21), pages 1-19, November.
  • Handle: RePEc:gam:jeners:v:15:y:2022:i:21:p:8255-:d:963797
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    References listed on IDEAS

    as
    1. Mengran Zhou & Tianyu Hu & Kai Bian & Wenhao Lai & Feng Hu & Oumaima Hamrani & Ziwei Zhu, 2021. "Short-Term Electric Load Forecasting Based on Variational Mode Decomposition and Grey Wolf Optimization," Energies, MDPI, vol. 14(16), pages 1-17, August.
    2. Andriy Chaban & Marek Lis & Andrzej Szafraniec & Vitaliy Levoniuk, 2022. "Mathematical Modelling of Transient Processes in a Three Phase Electric Power System for a Single Phase Short-Circuit," Energies, MDPI, vol. 15(3), pages 1-16, February.
    3. Hui Zhang & Cunhua Pan & Yuanxin Wang & Min Xu & Fu Zhou & Xin Yang & Lou Zhu & Chao Zhao & Yangfan Song & Hongwei Chen, 2022. "Fault Diagnosis of Coal Mill Based on Kernel Extreme Learning Machine with Variational Model Feature Extraction," Energies, MDPI, vol. 15(15), pages 1-14, July.
    4. Adrian Pană & Alexandru Băloi & Florin Molnar-Matei, 2019. "Mathematical Explanations of a Paradox Observed in a HVAC (High Voltage Alternating Current) Untransposed Overhead Line," Energies, MDPI, vol. 12(4), pages 1-17, February.
    5. Paolo Casoli & Carlo Maria Vescovini & Fabio Scolari & Massimo Rundo, 2022. "Theoretical Analysis of Active Flow Ripple Control in Positive Displacement Pumps," Energies, MDPI, vol. 15(13), pages 1-22, June.
    6. Yingjie Chen & Jianhong Fu & Tianshou Ma & Anping Tong & Zhaoxue Guo & Xudong Wang, 2018. "Numerical Modeling of Dynamic Behavior and Steering Ability of a Bottom Hole Assembly with a Bent-Housing Positive Displacement Motor Under Rotary Drilling Conditions," Energies, MDPI, vol. 11(10), pages 1-23, September.
    7. Sourav Pramanik & Sohel Anwar, 2020. "Look Ahead Based Control Strategy for Hydro-Static Drive Wind Turbine Using Dynamic Programming," Energies, MDPI, vol. 13(20), pages 1-22, October.
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