Solution of nonlinear stiff differential equations for a three-phase no-load transformer using a Runge–Kutta implicit method

• Autorzy: Baron Bernard , Kolasińska-Płuska Joanna , Łukaniszyn Marian , Spałek Dariusz , Kraszewski Tomasz

Identyfikatory

DOI

10.24425/aee.2022.142125

• Języki publikacji: EN

Abstrakty

EN

The paper presents an approach to differential equation solutions for the stiff problem. The method of using the classic transformer model to study nonlinear steady states and to determine the current pulses appearing when the transformer is turned on is given. Moreover, the stiffness of nonlinear ordinary differential state equations has to be considered. This paper compares Runge–Kutta implicit methods for the solution of this stiff problem.

Słowa kluczowe

EN

circuit model of a three-phase transformer; Runge-Kutta implicit methods; stiff nonlinear ordinary diferential equations

• Wydawca: Polish Academy of Sciences, Committee on Electrical Engineering
• Czasopismo: Archives of Electrical Engineering
• Rocznik: 2022
• Tom: Vol. 71, nr 4
• Strony: 1081--1106
• Opis fizyczny: Bibliogr. 23 poz., rys., tab., wz.

Twórcy

autor

Baron Bernard

• Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology Prószkowska 76, 45-758 Opole, Poland

• https://orcid.org/0000-0002-4466-4111

autor

Kolasińska-Płuska Joanna

• Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology Prószkowska 76, 45-758 Opole, Poland

• https://orcid.org/0000-0001-6165-3468

autor

Łukaniszyn Marian

• Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology Prószkowska 76, 45-758 Opole, Poland

• https://orcid.org/0000-0003-2816-1917

autor

Spałek Dariusz

• Institute of Electrotechnics and Informatics, Silesian University of Technology 10 Akademicka St., 44-100 Gliwice, Poland

• https://orcid.org/0000-0002-0237-2583

autor

Kraszewski Tomasz

• Research and Development Center GLOKOR Sp. z o.o. Górnych Wałów 27A St., 44-100 Gliwice, Poland

• https://orcid.org/0000-0003-2208-8536

Bibliografia

• [1] Baron B., Kolańska-Płuska J., Numerical methods of solving ordinary differential equations in C#, Politechnika Opolska Publisher (in Polish), ISBN 978 83 65235 30 5 (2015).
• [2] Baron B., Kolańska-Płuska J., Waindok A., Kraszewski T., Kawala-Sterniuk A., Application of Runge–Kutta implicit methods for solving stiff non-linear differential equations of a single-phase transformer model in the no-load state. Innovation Management and information Technology impact on Global Economy in the Era of Pandemic, Proceedings of the 37th International Business Information Management Association Conference (IBIMA), ISBN-13: 978-1-4200-0760-2 (Ebook-PDF), pp. 8071–8087, https://ibima.org/accepted-paper/application-of-runge-kuttas-implicit-methods-for-solving-stiff-non-linear-differential-equations-of-a-single-phase-transformer-model-in-the-no-load-state/ (2021).
• [3] Baron B., Kolańska-Płuska J., Kraszewski T., Application of Runge–Kutta implicit methods to solve the rigid differential equations of a single-phase idle transformer model, Politechnika Poznańska, Zeszyty Naukowe nr 100 (in Polish), Electrical Engineering (2019), DOI: 10.21008/j.1897-0737.2019.100.0007.
• [4] Dos Passos W., Numerical Methods Algorithms and Tools in C#, CRC Press, Taylor & Francis Group LLC, Boca Raton London New York, ISBN-13: 978-1-4200-0760-2, Ebook-PDF (2010), https://www.routledge.com/Numerical-Methods-Algorithms-and-Tools-in-C/Passos/p/book/9781439859070.
• [5] Dekker K., Verwer J.G., Stability of Runge–Kutta methods for stiff nonlinear differential equations, Elsevier Science Publishers B.V., North-Holland Amsterdam-New York – Oxford (1984), DOI: 10.1002/zamm.19870670128.
• [6] Hairer E., Wanner G., Solving Ordinary Differential Equations II: Stiff and Differential-algebraic Problems, 2nd revised ed., Springer, Berlin (2010) https://rd.springer.com/content/pdf/bfm:978-3-642-05221-7/1.pdf.
• [7] de Swart J.J.B., Soederlind G., On the construction of error estimators for implicit Runge–Kutta methods, Journal of Computational and Applied Mathematics 86, pp. 347–358 (1997), DOI: 10.1016/S0377-0427(97)00166-0.
• [8] Dombek G., Nadolny Z., Liquid kind, temperature, moisture, and ageing as an operating parameters conditioning reliability of transformer cooling system, Eksploatacja i Niezawodność – Maintenance and Reliability, vol. 18, no. 3, pp. 413–417 (2018), DOI: 10.17531/ein.2016.3.13.
• [9] Gutten M., Korenciak D., Kucera M., Sebok M., Opielak M., Zukowski P., Koltunowicz T., Maintenance diagnostics of transformers considering the influence of short-circuit currents during operation, Eksploatacja i Niezawodność – Maintenance and Reliability, vol. 19, no. 3, pp. 459–466 (2017), DOI: 10.17531/ein.2017.3.17.
• [10] Horiszny J., Research of leakage magnetic field in deenergized transformer, Compel – The international journal for computation and mathematics in electrical and electronic engineering, vol. 37, pp. 1657–1667 (2018), DOI: 10.1108/compel-01-2018-0040.
• [11] Horiszny J., Analysis and reduction of transformer inrush current, Gdansk University of Technology, monograph 159 (in Polish) (2016).
• [12] Pandey S.B., Lin C., Estimation for a life model of transformer insulation under combined electrical and thermal stress, IEEE Transactions on Reliability, vol. 41, no. 3, pp. 466–468 (1992), DOI: 10.1109/24.159823.
• [13] Nicolet A., Delince F., Implicit Runge–Kutta methods for transient magnetic field computation, IEEE Trans. Magn., vol. 32, no. 3, pp. 1405–1408 (1996), DOI: 10.1109/20.497510.
• [14] Noda T., Takenaka K., Inoue T., Numerical integration by the 2-stage diagonally implicit Runge–Kutta method for electromagnetic transient simulations, IEEE Trans. Power Del., vol. 24, no. 1, pp. 390–399 (2009), DOI: 10.1109/TPWRD.2008.923397.
• [15] Pries J., Hoffmann H., State Algorithms for Nonlinear Time-Periodic Magnetic Diffusion Problems Using Diagonally Implicit Runge–Kutta Methods, Magnetics IEEE Transactions on, vol. 51, no. 4, pp. 1–12 (2015), DOI: 10.1109/TMAG.2014.2344005.
• [16] Rosser J.B., A Runge–Kutta for all seasons, SIAM Rev., vol. 9, pp. 417–452 (1967), DOI: 10.1137/1009069.
• [17] Wang X., Weile D.S., Implicit Runge–Kutta methods for the discretization of time domain integral equations, IEEE Trans. Antennas Propag., vol. 59, no. 12, pp. 4651–4663 (2011), DOI: 10.1109/TAP.2011.2165469.
• [18] Noda T., Takenaka K., Inoue T., Numerical Integration by the 2-Stage Diagonally Implicit Runge–Kutta Method for Electromagnetic Transient Simulations, IEEE Transactions on Power Delivery, vol. 24, no. 1 (2009), DOI: 10.1109/TPWRD.2008.923397.
• [19] Gołębiowski M., Mazur D., Measurement and calculation of 3-column 15-winding autotransformer, Archives of Electrical Engineering, vol. 60, no. 3, pp. 223–230 (2011), DOI: 10.2478/v10171-011-0021-8.
• [20] Chraygane M., El Ghazal N., Fadel M., Bahani B., Belhaiba A., Ferfra M., Bassoui M., Improved modeling of new three-phase high voltage transformer with magnetic shunts, Archives of Electrical Engineering, vol. 64, no. 1, pp. 157–172 (2015), DOI: 10.1515/aee-2015-0014.
• [21] Bavendiek G., Leuning N., Müller F., Schauerte B., Thul A., Hameyer K., Magnetic anisotropy under arbitrary excitation in finite element models, Archives of Electrical Engineering, vol. 68, no. 2, pp. 455–466 (2019), DOI: 10.24425/aee.2019.128280.
• [22] Spałek D., Two relations for generalized discrete Fourier transform coefficients, Bulletin of the Polish Academy of Sciences: Technical Sciences, vol. 66, no. 3, pp. 275–281 (2018), DOI: 10.24425/123433.
• [23] Spałek D., Nonlinear magnetic circuit – self inductance definitions, passivity and waveforms distortion, Bulletin of the Polish Academy of Sciences, Technical Sciences (2022), DOI: 10.24425/bpasts.2022.141003.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).