The Poynting Vector and the New Theory of a Transformer. Part 11. Three-Phase Three-Core Transformers without a Neutral Wire
Abstract
A topological equivalent circuit for a three-phase three-core transformer reflecting the spatial structure of its magnetic system is developed. Owing to this approach, it became possible to represent the magnetic fluxes of the magnetic circuit’s all main sections and the apertures for each of three phases directly in the circuit in the absence of the windings’ neutral wires. The circuit is constructed by stitching together the anatomical circuit models of single-phase transformers obtained in the previous parts with taking into account the relationships between the fluxes at the junctions of the phase zones in iron. Its validity is confirmed by the rigor nature of the physical and mathematical relations for idealized transformers with infinite magnetic permeability of iron and simplified magnetic field patterns, which corresponds to the generally accepted approach with neglecting the magnetization currents. The difference lies in the fact that the developed model takes into account the heterogeneity of magnetization in different parts of the magnetic circuit with allocating more than 30 sections in the iron and apertures. The transition to the model of a real three-core transformer is carried out by adding four nonlinear transverse magnetization branches in each extreme phase zone and eight branches in the central phase zone to the idealized equivalent circuit. It is shown that in cases of winding connections without neutral wires, there is no flux of the Poynting vector in interphase zones in any unbalanced mode. In this case, the problems connected with the occurrence of fluxes exceeding the no-load fluxes under the conditions of symmetric and asymmetric short circuits, as well as the occurrence of buckling fluxes in these modes in the region outside the transformer iron, are solved.
References
1. Электродинамическая стойкость трансформаторов и реакторов при коротких замыканиях/Под ред. А.И. Лурье. М.: Знак, 2005, 520 с.
2. Силовые трансформаторы. Справочная книга/Под ред. С.Д. Лизунова и А.К. Лоханина. М.: Энергоиздат, 2004. 616 с.
3. Electromagnetic transient-type transformer models for geomagnetically-induced current (GIC) studies. Palo Alto (CA): EPRI; 2013. 3002000832.
4. Ларин В.С. Вопросы трансформаторостроения на 44-й сессии СИГРЭ. — Электричество, 2013, № 5, с. 51-63.
5. Chen X, Venkata S.S. A three-phase three-winding core-type transformer model for low-frequency transient studies. — IEEE Trans Power Deliv 1997; 12(2):775—82. 0885-8977/97/$10.00.
6. Martinez JA, Walling R, Mork BA, Martin-Arnedo J, Durbak D. Parameter determination for modeling system transients. Part III: Transformers. — IEEE Trans Power Deliv 2005;20(3):2051—62 [Электрон. ресурс] https://doi.org/10.1109/TPWRD.2005.848752 (дата обращения 05.03.2020).
7. Chiesa N, Mork B.A, I liiiidiiicn H.K. Transformer model for inrush current calculations: Simulations, measurements and sensitivity analysis. — IEEE Trans Power Deliv 2010;25(4):2599—608 [Электрон. ресурс] https://doi.org/10.1109/TPWRD.2010.2045518 (дата обращения 05.03.2020).
8. Mork B.A, Gonzalez F, Ishchenko D, Stuehm D.L, Mitra J. Hybrid transformer model for transient simulation—Part I: Development and parameters. — IEEE Trans Power Deliv 2007;22(1):248—55 [Электрон. ресурс] https://doi.org/10.1109/ TPWRD.2006.883000 (дата обращения 05.03.2020).
9. Hrnidalen H.K, Chiesa N, Avendaсo A, Mork B.A. Developments in the hybrid transformer model — Core modeling and optimization. In: Int. conf. power systems transients, Delft, the Netherlands; 2011, 122 р.
10. Leon F., Gomez P., Martinez-Velasco, Rioual M. Transformers in Power System Transients: Parameter Determination. Ed. Boca Raton, FL: CRC, 2009, ch. 4, pp.177—250.
11. Шакиров М.А. Вектор Пойнтинга и новая теория трансформаторов. Часть 2. — Электричество, 2014, № 10, с. 53—65.
12. Шакиров М.А. Вектор Пойнтинга и новая теория трансформаторов. Часть 4. «Анатомия» трансформатора. — Электричество, 2017, № 3, с. 37—49.
13. Шакиров М.А. Вектор Пойнтинга и новая теория трансформаторов. Часть 10. Стержневые трансформаторы. — Электричество, 2020, № 3 , с. 39—50.
14. Шакиров М.А., ТкачукА.А. Ф-инвариантные поверхности в обмотках броневого двухобмоточного трансформатора. — Изв. ПГУПС, 2018, вып. 4, с. 643—659.
15. Шакиров М.А., Ткачук А.А. Универсальные характеристики магнитного потока в броневом трансформаторе. — Изв. НТЦ единой энергетической системы, 2019, № 2 (81), с. 113—128.
16. Васютинский С.В. Вопросы теории и расчета трансформаторов. М.: Энергия, Ленинградское отделение, 1970, 432 с.
17. Сергеенков Б.Н., Киселев В.М., Акимова Н.А. Электрические машины. Трансформаторы/Под ред. И.П. Копылова. М.: Высшая школа, 1989, 352 с.
#
1. Elektrodinamicheskaya stoykost’ transformatorov i reaktorov pri korotkikh zamykaniyakh/Pod red. A.I. Lur’ye (Electrodynamic Resistance of Transformers and Reactors at Short Circuits/Ed. A.I. Lurie). M.: Znak, 2005, 520 p.
2. Silovyye transformatory. Spravochnaya kniga/Pod red. S.D.Lizunova i A.K. Lokhanina (Power transformers. Reference book/Ed. S.D. Lizunov and A.K. Lokhanin). M.: Energoizdat, 2004. 616 p.
3. Electromagnetic transient-type transformer models for geomagnetically-induced current (GIC) studies. Palo Alto (CA): EPRI; 2013. 3002000832.
4. Larin V.S. Elektrichestvo — in Russ. (Electricity), 2013, No. 5, pp. 51—63.
5. Chen X, Venkata S.S. A three-phase three-winding core-type transformer model for low-frequency transient studies. — IEEE Trans Power Deliv 1997; 12(2):775—82. 0885-8977/97/$10.00.
6. Martinez JA, Walling R, Mork BA, Martin-Arnedo J, Durbak D. Parameter determination for modeling system transients. Part III: Transformers. — IEEE Trans Power Deliv 2005;20(3):2051—62 [Elektron. resource] https://doi.org/10.1109/TPWRD.2005.848752 (Date of appeal 05.03.2020).
7. Chiesa N, Mork B.A, Hrnidalen H.K. Transformer model for inrush current calculations: Simulations, measurements and sensitivity analysis. IEEE Trans Power Deliv 2010;25(4):2599—608 [Elektron. resource] https://doi.org/10.1109/TPWRD.2010.2045518 (Date of appeal 05.03.2020).
8. Mork B.A, Gonzalez F, Ishchenko D, Stuehm D.L, Mitra J. Hybrid transformer model for transient simulation—Part I: Development and parameters. — IEEE Trans Power Deliv 2007;22(1):248—55 [Elektron. resource] https://doi.org/10.1109/ TPWRD.2006.883000 (Date of appeal 05.03.2020).
9. Hrnidalen H.K, Chiesa N, Avendaw A, Mork B.A. Developments in the hybrid transformer model — Core modeling and optimization. In: Int. conf. power systems transients, Delft, the Netherlands; 2011, 122 r.
10. Leon F., Gomez P., Martinez-Velasco, Rioual M. Transformers in Power System Transients: Parameter Determination. Ed. Boca Raton, FL: CRC, 2009, ch.4, pp.177—250.
11. Shakirov M.A. Elektrichestvo — in Russ. (Electricity), 2014, No. 10, pp. 53—65.
12. Shakirov M.A. Elektrichestvo — in Russ. (Electricity), 2017, No. 3, pp. 37—49.
13. Shakirov M.A. Elektrichestvo — in Russ. (Electricity), 2020, No. 3, pp. 39—50.
14. Shakirov M.A., Tkachuk A.A. Izv. PGUPS — in Russ. (News of the St. Petersburg State University of Railway Engineering), 2018, iss. 4, pp. 643—659.
15. Shakirov M.A., Tkachuk A.A. Izv. NTTs edinoy energeticheskoy sistemy — in Russ. (News of the Scientific-Technical Center of the Unified Energy System), 2019, No. 2 (81), pp. 113—128.
16. Vasyutinskiy S.V. Voprosy teorii i rascheta transformatorov (Questions of theory and calculation of transformers). M.: Energiya, Leningradskoye otdeleniye, 1970, 432 p.
17. Sergeyenkov B.N., Kiselev V.M., Akimova N.A. Elektricheskiye mashiny. Transformatory/Pod red. I.P. Kopylova (Electric cars. Transformers/Ed. I.P. Kopylov). M.: Vysshaya shkola, 1 989, 352 p.
