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<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="ru"><front><journal-meta><journal-id journal-id-type="publisher-id">probener</journal-id><journal-title-group><journal-title xml:lang="ru">Известия высших учебных заведений. ПРОБЛЕМЫ ЭНЕРГЕТИКИ</journal-title><trans-title-group xml:lang="en"><trans-title>Power engineering: research, equipment, technology</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">1998-9903</issn><issn pub-type="epub">2658-5456</issn><publisher><publisher-name>Kazan State Power Engineering  University</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.30724/1998-9903-2024-26-6-81-93</article-id><article-id custom-type="elpub" pub-id-type="custom">probener-3224</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>ЭЛЕКТРОТЕХНИЧЕСКИЕ КОМПЛЕКСЫ И СИСТЕМЫ</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>ELECTROTECHNICAL COMPLEXES AND SYSTEMS</subject></subj-group></article-categories><title-group><article-title>Выбор аналитического метода расчета индуктивности рассеяния при оптимизации конструкции высоковольтного испытательного трансформатора</article-title><trans-title-group xml:lang="en"><trans-title>Selecting analytical method of leakage inductance calculation for optimizing high-voltage test transformer design</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-7204-5449</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Сахно</surname><given-names>Л. И.</given-names></name><name name-style="western" xml:lang="en"><surname>Sakhno</surname><given-names>L. I.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Сахно Людмила Ивановна – д-р техн. наук, профессор Высшей школы высоковольтной энергетики  </p><p> г. Санкт-Петербург </p></bio><bio xml:lang="en"><p> Liudmila I. Sakhno </p><p> St. Petersburg </p></bio><email xlink:type="simple">lsahno2010@yandex.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Парамонов</surname><given-names>Е. Д.</given-names></name><name name-style="western" xml:lang="en"><surname>Paramonov</surname><given-names>E. D.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Парамонов Евгений Дмитриевич – аспирант Высшей школы высоковольтной энергетики </p><p> г. Санкт-Петербург </p></bio><bio xml:lang="en"><p> Evgenii D. Paramonov </p><p> St. Petersburg </p></bio><email xlink:type="simple">paramon.j.e.k@gmail.com</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Сахно</surname><given-names>О. И.</given-names></name><name name-style="western" xml:lang="en"><surname>Sakhno</surname><given-names>O. I.</given-names></name></name-alternatives><bio xml:lang="ru"><p> Сахно Ольга Ивановна – канд. техн. наук, доцент кафедры «Высшая математика» </p><p> г. Санкт-Петербург </p></bio><bio xml:lang="en"><p> Olga I. Sakhno </p><p> St. Petersburg </p></bio><email xlink:type="simple">olga.sakhnoj@yandex.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Кочеткова</surname><given-names>Е. Ю.</given-names></name><name name-style="western" xml:lang="en"><surname>Kochetkova</surname><given-names>E. Y.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Кочеткова Елена Юрьевна – канд. техн. наук, доцент Высшей школы высоковольтной энергетики </p><p> г. Санкт-Петербург </p></bio><bio xml:lang="en"><p> Elena Y. Kochetkova </p><p> St. Petersburg </p></bio><email xlink:type="simple">k_ey@mail.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Миневич</surname><given-names>Т. Г.</given-names></name><name name-style="western" xml:lang="en"><surname>Minevich</surname><given-names>T. G.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Миневич Татьяна Геннадьевна – канд. техн. наук, доцент Высшей школы высоковольтной энергетики </p><p> г. Санкт-Петербург </p></bio><bio xml:lang="en"><p> Tatyana G. Minevich </p><p> St. Petersburg </p></bio><email xlink:type="simple">tm_21@rambler.ru</email><xref ref-type="aff" rid="aff-1"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru"><institution>Санкт-Петербургский Политехнический университет Петра Великого</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Peter the Great St.Petersburg Polytechnic University</institution><country>Russian Federation</country></aff></aff-alternatives><pub-date pub-type="collection"><year>2024</year></pub-date><pub-date pub-type="epub"><day>13</day><month>01</month><year>2025</year></pub-date><volume>26</volume><issue>6</issue><fpage>81</fpage><lpage>93</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Сахно Л.И., Парамонов Е.Д., Сахно О.И., Кочеткова Е.Ю., Миневич Т.Г., 2024</copyright-statement><copyright-year>2024</copyright-year><copyright-holder xml:lang="ru">Сахно Л.И., Парамонов Е.Д., Сахно О.И., Кочеткова Е.Ю., Миневич Т.Г.</copyright-holder><copyright-holder xml:lang="en">Sakhno L.I., Paramonov E.D., Sakhno O.I., Kochetkova E.Y., Minevich T.G.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://www.energyret.ru/jour/article/view/3224">https://www.energyret.ru/jour/article/view/3224</self-uri><abstract><p>АКТУАЛЬНОЙ ЗАДАЧЕЙ проектирования трансформатора является оптимизация его конструкции, поскольку она позволяет создать конкурентоспособный трансформатор. Одной из целевых функций в алгоритмах оптимизации высоковольтных испытательных трансформаторов является отношение индуктивного сопротивления рассеяния трансформатора, приведенного к вторичной стороне, к емкостному сопротивлению нагрузки. Так как целевые функции в процедуре оптимизации надо находить многократно, для расчета индуктивности рассеяния целесообразно использовать приближенные аналитические методы расчета. ЦЕЛЬЮ статьи является анализ погрешности традиционного аналитического метода расчета индуктивности рассеяния трансформатора, разработанного для силовых трансформаторов с прямоугольным осевым сечением обмоток, и разработка уточненного аналитического метода, в котором учтена основная конструктивная особенность высоковольтных испытательных трансформаторов – трапецеидальная форма сечения вторичной обмотки. В уточненном методе учтено изменение количества витков в радиальном направлении при сохранении основных допущений традиционного метода. МЕТОДЫ. Исследование погрешности проводится путем сравнения результатов расчета с использованием указанных методов с результатами численного расчета, которые приняты за точные. РЕЗУЛЬТАТЫ. Показано, что погрешность расчета с использованием предложенного в статье метода существенно уменьшена по сравнению с погрешностью метода, разработанного для силовых трансформаторов с прямоугольным сечением обмоток. Определены диапазоны изменения относительных геометрических параметров базового трансформатора, в которых погрешность этого метода расчета не превышает 10 %. В качестве базового трансформатора выбран трансформатор ТГИ 50/100, номинальная мощность которого составляет 5 кВА. Разработанный в статье аналитический метод рекомендуется для использования в алгоритмах оптимизации высоковольтных испытательных трансформаторов.</p></abstract><trans-abstract xml:lang="en"><p>AN ACTUAL PROBLEM in transformer design is to optimize its design, since it allows you to create a competitive transformer. One of the objective functions in optimization algorithms for high-voltage test transformers is the ratio of the leakage inductive reactance of the transformer, reduced to the secondary side, to the load capacitance. Since the objective functions in the optimization procedure must be found many times, it is advisable to use approximate analytical calculation methods to calculate the leakage inductance. THE PURPOSE of the article is to analyze the error of the traditional analytical method for calculating the leakage inductance of power transformers with a rectangular axial cross-sections of the windings, and to develop a refined analytical method that takes into account the main design feature of high-voltage test transformers, which consists in the trapezoidal shape of the secondary winding cross-section. The refined method takes into account the change in the number of turns in the radial direction while maintaining the basic assumptions of the traditional method. METHODS. The error is studied by comparing the calculation results using these methods with the results of a numerical calculation of leakage inductance based on a 3D magnetostatic field, which are accepted as accurate. RESULTS. It is shown that the calculation error using the method proposed in the article is significantly reduced compared to the error of the method developed for power transformers with rectangular winding cross-sections. The ranges of changes in the relative geometric parameters of the base transformer have been determined, in which the error of this calculation method does not exceed 10%. The TGI 50/100 transformer, whose rated power is 5 kVA, was chosen as the base transformer. The developed analytical method is recommended for high-voltage test transformers optimization algorithms.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>высоковольтный испытательный трансформатор</kwd><kwd>индуктивность рассеяния</kwd><kwd>численные методы</kwd><kwd>методы оптимизации</kwd><kwd>магнитостатическое поле</kwd><kwd>плоскопараллельное поле</kwd></kwd-group><kwd-group xml:lang="en"><kwd>high voltage test transformer</kwd><kwd>leakage inductance</kwd><kwd>numerical methods</kwd><kwd>optimization methods</kwd><kwd>magnetostatic field</kwd><kwd>plane-parallel field</kwd></kwd-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017</mixed-citation><mixed-citation xml:lang="en">K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," in IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182-197, April 2002, doi: 10.1109/4235.996017</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Г.Н. Петров. Электрические машины. Часть первая. Введение. Трансформаторы. М: Энергия. 1974, 240 с.</mixed-citation><mixed-citation xml:lang="en">G.N. Petrov. Elektricheskie mashiny. Chast' pervaya. Vvedenie. Transformatory. M: Energiya. 1974, 240 s.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Бутырин П.А., Алпатов М.Е. Аналитическая теория трансформаторов М: Национальный исследовательский университет МЭИ. 2019. C. 112</mixed-citation><mixed-citation xml:lang="en">Butyrin P.A., Alpatov M.E. Analiticheskaya teoriya transformatorov M: Nacional'nyj issledovatel'skij universitet MEI. 2019. C. 112</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Ouyang, W. G. Hurley and M. A. E. Andersen, "Improved Analysis and Modeling of Leakage Inductance for Planar Transformers," in IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 7, no. 4, pp. 2225-2231, Dec. 2019, doi: 10.1109/JESTPE.2018.2871968.</mixed-citation><mixed-citation xml:lang="en">Ouyang, W. G. Hurley and M. A. E. Andersen, "Improved Analysis and Modeling of Leakage Inductance for Planar Transformers," in IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 7, no. 4, pp. 2225-2231, Dec. 2019, doi: 10.1109/JESTPE.2018.2871968.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">L. M. R. Oliveira and A. J. M. Cardoso, "Leakage Inductances Calculation for Power Transformers Interturn Fault Studies," in IEEE Transactions on Power Delivery, vol. 30, no. 3, pp. 1213-1220, June 2015, doi: 10.1109/TPWRD.2014.2371877.</mixed-citation><mixed-citation xml:lang="en">L. M. R. Oliveira and A. J. M. Cardoso, "Leakage Inductances Calculation for Power Transformers Interturn Fault Studies," in IEEE Transactions on Power Delivery, vol. 30, no. 3, pp. 1213-1220, June 2015, doi: 10.1109/TPWRD.2014.2371877.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Dawood K., Isik F., Kömürgöz Kırış G. Comparison of Analytical Method and Different Finite Element Models for the Calculation of Leakage Inductance in Zigzag Transformers, ELEKTRONIKA IR ELEKTROTECHNIKA, cilt.28, sa.1, pp.16-22, 2022.</mixed-citation><mixed-citation xml:lang="en">Dawood K., Isik F., Kömürgöz Kırış G. Comparison of Analytical Method and Different Finite Element Models for the Calculation of Leakage Inductance in Zigzag Transformers, ELEKTRONIKA IR ELEKTROTECHNIKA, cilt.28, sa.1, pp.16-22, 2022.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">A. Sharma and J. W. Kimball, " Evaluation of Transformer Leakage Inductance Using Magnetic Image Method,” in IEEE Transactions on Magnetics, vol. 57, no. 11, pp. 1-12, Nov. 2021, Art no. 8401912, doi: 10.1109/TMAG.2021.3111479.</mixed-citation><mixed-citation xml:lang="en">A. Sharma and J. W. Kimball, " Evaluation of Transformer Leakage Inductance Using Magnetic Image Method,” in IEEE Transactions on Magnetics, vol. 57, no. 11, pp. 1-12, Nov. 2021, Art no. 8401912, doi: 10.1109/TMAG.2021.3111479.</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">M. Lambert, F. Sirois, M. Martinez-Duro and J. Mahseredjian, "Analytical Calculation of Leakage Inductance for Low-Frequency Transformer Modeling," in IEEE Transactions on Power Delivery, vol. 28, no. 1, pp. 507-515, Jan. 2013, doi: 10.1109/TPWRD.2012.2225451.</mixed-citation><mixed-citation xml:lang="en">M. Lambert, F. Sirois, M. Martinez-Duro and J. Mahseredjian, "Analytical Calculation of Leakage Inductance for Low-Frequency Transformer Modeling," in IEEE Transactions on Power Delivery, vol. 28, no. 1, pp. 507-515, Jan. 2013, doi: 10.1109/TPWRD.2012.2225451.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Hernandez, F. de Leon and P. Gomez, "Design Formulas for the Leakage Inductance of Toroidal Distribution Transformers," in IEEE Transactions on Power Delivery, vol. 26, no. 4, pp. 2197-2204, Oct. 2011, doi: 10.1109/TPWRD.2011.2157536.</mixed-citation><mixed-citation xml:lang="en">Hernandez, F. de Leon and P. Gomez, "Design Formulas for the Leakage Inductance of Toroidal Distribution Transformers," in IEEE Transactions on Power Delivery, vol. 26, no. 4, pp. 2197-2204, Oct. 2011, doi: 10.1109/TPWRD.2011.2157536.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">M. Lambert, F. Sirois, M. Martinez-Duro and J. Mahseredjian, "Analytical Calculation of Leakage Inductance for Low-Frequency Transformer Modeling," in IEEE Transactions on Power Delivery, vol. 28, no. 1, pp. 507-515, Jan. 2013, doi: 10.1109/TPWRD.2012.2225451.</mixed-citation><mixed-citation xml:lang="en">M. Lambert, F. Sirois, M. Martinez-Duro and J. Mahseredjian, "Analytical Calculation of Leakage Inductance for Low-Frequency Transformer Modeling," in IEEE Transactions on Power Delivery, vol. 28, no. 1, pp. 507-515, Jan. 2013, doi: 10.1109/TPWRD.2012.2225451.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">R. Torchio, "A Volume PEEC Formulation Based on the Cell Method for Electromagnetic Problems From Low to High Frequency," in IEEE Transactions on Antennas and Propagation, vol. 67, no. 12, pp. 7452-7465, Dec. 2019, doi: 10.1109/TAP.2019.2927789</mixed-citation><mixed-citation xml:lang="en">R. Torchio, "A Volume PEEC Formulation Based on the Cell Method for Electromagnetic Problems From Low to High Frequency," in IEEE Transactions on Antennas and Propagation, vol. 67, no. 12, pp. 7452-7465, Dec. 2019, doi: 10.1109/TAP.2019.2927789</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Жуйков А.В., Кубаткин М.А., Ларин В.С., Матвеев Д.А., Никулов И.И., Хренов С.И. К определению индуктивностей рассеяния обмоток трансформаторов // Электротехника. 2019. №8. c. 46- 53.</mixed-citation><mixed-citation xml:lang="en">Zhujkov A.V., Kubatkin M.A., Larin V.S., Matveev D.A., Nikulov I.I., Hrenov S.I. K opredeleniyu induktivnostej rasseyaniya obmotok transformatorov // Elektrotekhnika. 2019. №8. c. 46-53.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">M. D’Antonio, S. Chakraborty and A. Khaligh, "Planar Transformer With Asymmetric Integrated Leakage Inductance Using Horizontal Air Gap," in IEEE Transactions on Power Electronics, vol. 36, no. 12, pp. 14014-14028, Dec. 2021, doi: 10.1109/TPEL.2021.3089606.</mixed-citation><mixed-citation xml:lang="en">M. D’Antonio, S. Chakraborty and A. Khaligh, "Planar Transformer With Asymmetric Integrated Leakage Inductance Using Horizontal Air Gap," in IEEE Transactions on Power Electronics, vol. 36, no. 12, pp. 14014-14028, Dec. 2021, doi: 10.1109/TPEL.2021.3089606.</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Бутырин, Павел А., Семен Д. Дубицкий, и Николай В. Коровкин. "Численное моделирование электромагнитных полей: мультифизические задачи, инструментарий и обучение." Электричество 6 (2019): с.51-58.</mixed-citation><mixed-citation xml:lang="en">Butyrin, Pavel A., Semen D. Dubickij, i Nikolaj V. Korovkin. \"Chislennoe modelirovanie elektromagnitnyh polej: mul'tifizicheskie zadachi, instrumentarij i obuchenie.\" Elektrichestvo 6 (2019): s.51- 58.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">COMSOL Multiphysics User’s Guide, p.563. 16. E. D. Paramonov, I. Sakhno Liudmila, I. Sakhno Olga and E. Y. Kochetkova, "Analysis of Methods for Calculating Leakage Inductance of a High-Voltage Test Transformer," 2024 Conference of Young Researchers in Electrical and Electronic Engineering (ElCon), Saint Petersburg, Russian Federation, 2024, pp. 680-684, doi: 10.1109/ElCon61730.2024.10468519.</mixed-citation><mixed-citation xml:lang="en">COMSOL Multiphysics User’s Guide, p.563.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">E. D. Paramonov, I. Sakhno Liudmila, I. Sakhno Olga and E. Y. Kochetkova, "Analysis of Methods for Calculating Leakage Inductance of a High-Voltage Test Transformer," 2024 Conference of Young Researchers in Electrical and Electronic Engineering (ElCon), Saint Petersburg, Russian Federation, 2024, pp. 680-684, doi: 10.1109/ElCon61730.2024.10468519.</mixed-citation><mixed-citation xml:lang="en">E. D. Paramonov, I. Sakhno Liudmila, I. Sakhno Olga and E. Y. Kochetkova, "Analysis of Methods for Calculating Leakage Inductance of a High-Voltage Test Transformer," 2024 Conference of Young Researchers in Electrical and Electronic Engineering (ElCon), Saint Petersburg, Russian Federation, 2024, pp. 680-684, doi: 10.1109/ElCon61730.2024.10468519.</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
