AN ADJUSTING APPROACH TO THE DETERMINATION OF THE PERMEABILITY FUNCTIONAL DEPENDENCIES OF ANISOTROPIC COLD-ROLLED ELECTROTECHNICAL STEELS
DOI:
https://doi.org/10.15588/1607-6761-2018-2-1Keywords:
magnetization characteristics, anisotropy, permeability, spline interpolation, functional basis, nonlinear regression method, electromagnetic fieldAbstract
Purpose. To develop an effective approach for the determination of mathematical functional relationships for the high-precision description of the nonlinear dependences of the permeability on the magnetic flux density, taking into account the anisotropy of modern cold-rolled electrotechnical steel, which are used in the production of new series of power transformers.
Methodology. The researches were carried out using interpolation methods, approximation, regression analysis, mathematical physics, electromagnetic field theory.
Findings. The methods of functional conception of nonlinear characteristics of magnetization of ferromagnetic materials, including cold-rolled anisotropic electrotechnical steels, are researched. An adjustment approach is proposed to increase the dimension of the initial data, based on the spline interpolation method on the Hermite polynomial basis for the digital data of the magnetization characteristics. This excludes an increase of the errors in determining the permeabilities in the sections between the nodal values of the magnetic flux density. For a mathematical description of the dependences of the permeability on the magnetic flux density, a special functional basis with Gaussian functions and additional error functions has been developed, that ensures the continuity of derivatives of various orders at fixed angular values between the rolling direction of anisotropic electrical steel and magnetic flux. The method of nonlinear regression in the structure of function package of the Mathcad software for the high-precision determination of regression coefficient vectors in functional descriptions of permeabilities is applied. With the help of verification and validation of the results, which adjusting nonlinear regression with computational data using the generic method and with experimental data, a significant decrease in the relative errors that do not exceed 1.62% for fixed angles between the direction of rolling of anisotropic electrical steel and magnetic flux is obtained. The application of the new regression equations substantially improves the conditions for the field simulation of electromagnetic fields in the open-circuited mode of the power transformer in the Comsol Multiphysics software, and ensures the stability of iterative computing processes.
Originality. A new approach to increasing the dimension of input arrays, based on spline interpolation of digital data arrays of magnetization characteristics for fixed angles between the directions of rolling of the electrical steel and magnetic flux is implemented. For nonlinear regression equations, a new functional basis with Gaussian functions and additional error functions for mathematical descriptions with continuous derivatives for the dependences of the permeabilities of anisotropic cold-rolled electrotechnical steels on the magnetic flux density with high accuracy in the range of angles variation between the rolling directions and magnetic flux from 0º to 90º.
Practical value. The approaches and techniques for the basis of Gaussian functions and additional error functions proposed in this paper make it possible to significantly improve the accuracy of determining the nonlinear characteristics of anisotropic electrotechnical steels. Also to reduce the relative errors to 1.62% when the angles between directions of rolling of the electrotechnical steels and magnetic flux vary from 0° to 90°.
References
[1] Tikhomirov, P. M. (1986). Raschet transformatorov [Calculation of transformers]. M.: Energoatomizdat, 528. (in Russian).
[2] Kulkarni, S.V., Khaparde, S.A. (2004). Transformer Engineering, Design and Practice, New York: Marcel Dekker, 478. (in English).
[3] Tang Qi, Guo S., Wang Z. (2015). Magnetic flux distribution in power transformer core with mitred joints. Journal of Applied Physics, 117, 17, 17D522-1-17D522-4. DOI: 10.1063/1.4919119.
[4] Rashtchi, V., Rahimpour, E., Rezapour, E. M. (2011). Parameter identification of transformer detailed model based on chaos optimisation algorithm. IET Electric Power Applications, 5, 2, 238-246. DOI: 10.1049/iet-epa.2010.0147.
[5] Paikov, I.A., Tikhonov, А.I. (2015). Analiz modeley dla elektromagnitnogo rascheta silovykh transformatorov [Analysis of models for the electromagnetic calculation of power transformers]. Vestnik IGEU, 3, 38-43. (in Russian).
[6] Jazebi, S., de León, F., Farazmand, A., Deswal, D. (2013). Dual Reversible Transformer Model for the Calculation of Low-Frequency Transients. IEEE Transactions on Power Delivery, 28, 4, 2509-2517. DOI: 10.1109/TPWRD.2013.2268857.
[7] Bul', O.B. (2006). Metody rascheta magnitnykh system electrichescikh apparatov. Programma ANSYS. Uchebnoe posobie dla studentov vuzov [Methods for calculating the magnetic systems of electrical apparatus. ANSYS program Training manual for university students]. М.: Academiya, 288. (in Russian).
[8] Butarev I. Ju., Potapov, L.A. (2011). Comsol multiphysics: Modelirovanie electromehanicheckikh ustroystv [Comsol multiphysics: Modeling of electromechanical devices]. Bryansk: Izdatel'stvo Bryanskogo gosudarstvennogo tehnicheskogo universiteta, 113. ISBN 978-5-89838-520-0. (in Russian).
[9] Bul', O.B. (2005). Metody rascheta magnitnyh system electrichescih apparatov. Magnitnye cepi, polya I programma FEMM. [Methods for calculating the magnetic systems of electrical apparatus. Magnetic circuits, fields and the FEMM program]. Uchebnoe posobie dla vuzov po special'nosti «Electricheskie i electronnye apparaty» napravleniya «Electrotehnika, electromehanika i electrotehnologiya», Мoscow, Akademiya, 336 (in Russian).
[10] Molotilov, B.V., Mironov, L.V., Petrenko, A.G., and et al. (1989). Kholodnokatanyye elektrotekhnicheskiye stali [Cold-rolled electrical steel: Reference], Moscow, Metallurgiya, 167 (in Russian).
[11] Demidenko, E.Z. (1981). Lineynaya i nelineynaya regresiya [Linear and nonlinear regression]. М.: Finansy i statistika, 304. (in Russian).
[12] Korol' E.G. (2007). Analiz metodov modelirovaniya magnitnyh harakteristik elektromagnitov dla kompensacii magnitnogo polya electrooborudovaniy [Analysis of methods for modeling the magnetic characteristics of electromagnets for compensation of the magnetic field of electrical equipment]. Elektrotehnica i Electromehanika, 2, 31-34. (in Russian)
[13] Bessonov L.A. (1948). Electricheskie cepi so stal'ju [Electric circuits with steel]. М.: Gosenergoizdat, 344. (in Russian).
[14] Demirchan K.S. (1974). Modelirovanie magnitnykh poley [Modeling of magnetic fields]. Leningrad: Energiya, 288. (in Russian).
[15] Pentegov, I.V., Krasnozhon, A.V. (2006). Universal'naya approksimaciya krivykh namagnichevaniya elektrotehnicheskikh staley [Universal approximation of magnetization curves for electrical steel]. Elektrotehnica i Electromekhanika, 1, 66-70. (in Russian).
[16] Yarymbash, D., Yarymbash, S., Divchuk, T., Kylymnik, I. (2016). Osoblyvosti vyznachennya parametriv korotkoho zamykannya sylovykh transformatoriv zasobamy polʹovoho modelyuvannya [Determination features of the power transformer short circuit parameters through field modeling]. Electrical Engineering And Power Engineering, 1, 12-17. doi: http://dx.doi.org/10.15588/1607-6761-2016-1-2 (in Ukrainian).
[17] Yarymbash, D., Yarymbash, S., Divchuk, T., Kylymnik, I. (2016). Osoblyvosti rozpodilennya mahnitnykh potokiv u rezhymi nerobochoho khodu sylovykh transformatoriv [The features of magnetic flux distribution of the idling mode of the power transformers]. Electrical Engineering And Power Engineering, 2, 5-12. doi: http://dx.doi.org/10.15588/1607-6761-2016-2-1 (in Ukrainian).
[18] Divchuk, T., Yarymbash, D., Yarymbash, S., Kylymnyk, I., Kotsur, M., Bezverkhnia, Y. (2018). Podkhod k opredeleniyu tokov kholostogo khoda silovykh trekhfaznykh transformatorov s ploskimi sterzh-nevymi magnitnymi sistemami [Approach to determination of no load current of three-phase power transformers with plane rods magnetic systems]. Electrical Engineering And Power Engineering, 2, 56-66. doi: http://dx.doi.org/10.15588/1607-6761-2017-2-6 (in Russian).
[19] Graph2Digit: www.plsoft.su
[20] Pan'kiv, V.I., Tankevich, E.M., Lutchin, M.M. (2014). Aproksymaciya kharakterystyk namagnichuvannya transformatoriv strumu [Approximation of magnetization characteristics of current transformers]. Praci Instytutu electrodynamiky,37, 82-90. (in Ukrainian).
[21] Malyar, V., Malyar, A., Grechin, D. (2004). Aproksymaciya kharakterystyk namagnichuvannya elektrotekhnicnyh staley [Approximation of magnetization characteristics of electrotechnical steels]. Teoretychna electrotekhnika, 57, 78-85. (in Ukrainian).
[22] Malyar, V.S, Dobushovs'ka, I.A. (2010). Aproksymaciya kharakterystyk namagnichuvannya elektrotekhnicnyh staley splaynamy drugogo poryadku [Approximation of magnetization characteristics of electrotechnical steels by splines of second order]. Elektroenergetychni ta electromekhanichni systemy, 671, 67-72. (in Ukrainian).
[23] Zhermen-Lakur, P., Shenen, P., Kosnar, M., Gardan, I., Rober, F., Rober, I., Vitomski, P., Kastel'zho, P. (1988). Matematika i SAPR: v 2-h kn. Kn. 1. Perevod s franc. М.: Mir, 204. ISBN 5-03-000417-3, OCR. (in Russian).
[24] Korn, G., Korn, T. (1977). Spravochnik po matematike (dla nauchnyh rabotnikov i inzhenerov) [Manual on mathematics (for scientists and engineers)]. М.: Nauka, 830. (in Russian).
[25] (2013). Algorytmizaciya ta programuvannya. MathCAD. [Algorithmization and programming]. Pidruchnyk. L'viv: Vydavnyctvo L'vivs'koi politehniky, 364. (in Ukrainian).
[26] Yarymbash, D., Kotsur, M., Yarymbash, S., & Kotsur, I. (2017). Osobennosti opredeleniya parametrov skhemy zameshcheniya asinkhronnogo dvigatelya dlya rezhima korotkogo zamykaniya [Features of parameter determination of the induction motor substitution circuit for short-circuit mode]. Electrical Engineering And Power Engineering, 1, 24-30. doi: http://dx.doi.org/10.15588/1607-6761-2017-1-4 (in Russian).
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