# Науковий вісник НГУ

## Mathematical modeling of free voltage oscillations on transformer windings taking into account winds mutual induction under surge overvoltage

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Authors:

M.S. Seheda, Dr. Sci. (Tech.), Professor, Lviv Polytechnic National University, Head of the Department of Electric Stations, Lviv, Ukraine

Ye.V. Cheremnykh, Dr. Sci. (Phys.-Math.), Professor, Lviv Polytechnic National University, Professor of the Department of Higher Mathematics, Lviv, Ukraine

T.A. Mazur, Lviv Politechnic National University, assistant Lecturer of the Department of Electric Stations, Lviv, Ukraine

Abstract:

Purpose. To create a mathematical model for study of wave processes in the transformer windings with mutual induction between their coils and the solution of integral-differential equations in partial derivatives.

Methodology. The mathematical model of the transformer has been obtained on the basis of the equivalent circuit of its winding per unit length along its axis with considering of the leakage currents between coils relatively to the ground and mutual induction between their coils. For solving the integral-differential equations the method of separation of variables has been proposed. The initial conditions have been determined based on the algorithm of calculating the steady state mode of electric circuits. Boundary conditions have been determined using the solution of equations describing the beginning and the ending of the equivalent circuit.

Findings. The proposed mathematical model allows us to analyze the wave processes in the transformer windings taking into account all parameters of the equivalent circuit. During the study of wave processes in the transformer windings it is necessary to consider the ratio between time interval of electromagnetic wave propagation along the entire length of the winding and time interval, during which the current and voltage take considerable changes with respect to the their full change during the process under research. The voltage oscillations in turn arise with frequency corresponding both to the entire winding and the self frequency of the winding turn. Free oscillations are characterized by high frequencies and die out completely.These oscillations depend on inductive and capacitive coupling in the winding.

Originality. We suggest the new approach to the formation of the mathematical model which takes into account the distributed parameters and makes it possible to model the surge overvoltage taking into account all parameters of the equivalent scheme and solution of the integral-differential equations in partial derivatives using the method of separation of variables.

Practical value. Reliability of the transformers insulation during the surge overvoltage is very important and depends on its correct coordination. The insulation of the high-voltage transformers must ensure their accident-free operation both during the long-term action of the applied voltage and the short-time external and internal overvoltage. Development of the mathematical models for the wave processes study in the power transformers with consideration of these factors is an important and actual problem because the insulation of the transformer windings has a significant influence on its size, cost and weight, increases the reactance, thereby reducing its bandwidth.

References:

1. Математичне моделювання в електроенергетиці: Підручник/ О.В. Кириленко, М.С. Сегеда, О.Ф. Буткевич, Т.А. мазур– Львів, 2010. – 608 с.

Kyrylenko, O.V., Seheda, M.S., Butkevych, O.F. and Mazur, T.A. (2010), Matematycne modeliuvannia v elektroenerhetytsi [Mathematical Modeling In Electric Power Engineering], Lviv, Uraine.

2. Сегеда М.С. Математичне моделювання хвильових процесів у трансформаторах / Сегеда М.С. // Техн. електродинаміка. – 2002. – №3. – С. 47–49.

Seheda, M.S. (2002), “Mathematical Modeling of Wave Processes in Transformers”, Tekhnicha elektrodynamika, no.3, pp. 47–49.

3. Seheda, M.S. and Mazur, T.A. (2008), “modeling of electromagnetic processes in transformers with considering of distributed parameters”, Proc. of the 6th Sci. and Tech. Conf. “Sieci elektroenergetyczne w przemysle i energetyce”, SIECI 2008, Szklarska Poreba, Poland, pp. 415–424.

4. Seheda, M.S.,Mazur, T.A. and Chutora, I. (2011), “Analysis of overvoltages in transformer windings with considering of distributed parameters”, Proc. of the 12th International Workshop “Computational Problems of Elektrical Engineering”, CPEE’2011, Kostryna, pp. 47.

5. Seheda, M.S., Cheremnykh, Y. and Mazur, T.A. (2012), “Mathematical modeling of freevoltage oscillations in the transformer windings during pulse overvoltages”, Proc. of the 7th Sci. and Tech. Conf.“Sieci elektroenergetyczne w przemysle i energetyce”, SIECI 2012, September 19–21, Szklarska Poreba, Poland, ISBN: 978-83-935801-0-1.

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ISSN (print) 2071-2227,
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Journal was registered by Ministry of Justice of Ukraine.
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