Mathematical Modeling of the Skin Effect in ACSR Wires
Keywords:
power line, resistance, skin effect, higher harmonic components, power loss
Abstract
In overhead power lines made of aluminum conductor steel reinforced (ACSR) wires, part of active power loss is due to the higher harmonic current components. This is manifested in that the current sinusoidal waveform is distorted, and the current is displaced to the conductor surface with increasing the frequency (a skin effect). At the same time, increasing the frequency in electrical and autonomous power supply systems (for example, up to 400 or 800 Hz) is advisable for the possibility to achieve significantly smaller weight and overall dimensions of the equipment, especially for mobile facilities. The article presents mathematical models and the results of finite element modeling carried out taking into account the influence of temperature on the resistance growth coefficients caused by the skin effect. A comparative analysis was carried out, which has shown the possibility of using an analytical method for determining the resistance of ACSR wires with taking into account the higher harmonic current components. Two methods for obtaining analytical characteristics of the skin effect coefficients in the power line wires are considered. It is shown that a stranded wire can be considered as an "equivalent" homogeneous (single core) wire if the steel content does not exceed 25 % of the wire total volume.
References
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8. ГОСТ 32144–2013. Электрическая энергия. Совместимость технических средств электромагнитная. Нормы качества электрической энергии в системах электроснабжения общего назначения. М.: Стандартинформ, 2014, 16 с.
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10. Zhang X.-P., Yan Z. Energy Quality: A Definition. – IEEE Open Access Journal of Power and Energy, 2020, vol. 7, pp. 430–440, DOI: 10.1109/OAJPE.2020.3029767.
11. Product data Bulletin No. 8803PD9402. Causes and Effects of Variable Frequency Drives Relative to the IEEE 519-1992 Standard. Raleigh, U.S.A.: SQUARE D, 1994, 8 p.
12. Харлов Н.Н. и др. Энергетическое обследование несинусоидальных режимов многопроводных линий электропередачи. – Электричество, 2011, № 12, с. 12–15.
13. Боровиков В.С., Харлов Н.Н., Акимжанов Т.Б. О необходимости включения добавочных потерь от высших гармоник тока в технологические потери при передаче электрической энергии. – Известия Томского политехнического университета, 2013, т. 322, № 4, с. 91–93.
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15. Коверникова Л.И. Оценка дополнительных потерь активной мощности в воздушных ЛЭП 110 кВ при несинусоидальных режимах. – Электричество, 2024, № 6, с.16–26.
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17. Payne A. Skin Effect, Proximity Effect and the Resistance of Circular and Rectangular Conductors, 2021, 41 p.
18. Ramo S., Whinnery J.R., Van Duzer T. Fields and Waves in Communication Electronics. Third Edition. Hoboken: John Wiley & Sons, 1994, 864 p.
19. Карташев И.И., Зуев Э.Н. Качество электроэнергии в системах электроснабжения. Способы его контроля и обеспечения. М.: Изд-во МЭИ, 2001, 120 с.
20. Жежеленко И.В. Высшие гармоники в системах электроснабжения промпредприятий. М.: Энергоатомиздат, 1994, 272 с.
21. Манусов В.З., Хрипков В.В., Фролова В.В. Сравнительный анализ математических моделей для определения коэффициента увеличения активного сопротивления проводников от высших гармоник. – Научные проблемы транспорта Сибири и Дальнего Востока, 2018, № 1, с. 184–188.
22. ГОСТ Р МЭК 60287-1-1-2022. Кабели электрические. Расчет номинальной токовой нагрузки. М.: Российский институт стандартизации, 2022, 26 с.
23. Kennelly A.E., Laws F.A., Pierce P.H. Experimental Researches on Skin Effects in Conductors. – Transactions of the American Institute of Electrical Engineers, 1915, vol. 34, No. 2, pp. 1953–2021, DOI: 10.1109/T-AIEE.1915.4765283.
24. Monteiro J.H.A. et al. Simplified Skin-Effect Formulation for Power Transmission Lines. – IET Science, Measurement & Technology, 2014, vol. 8, No. 2, pp. 47–53, DOI: 10.1049/iet-smt.2013.0072.
25. Сухичев М.И., Скочко Е.М. К вопросу о необходимости учета скин-эффекта для сталеалюминиевых проводов. – Промышленная энергетика, 2023, № 1, с. 2–8.
26. Рукавишников В.А., Мосолапов А.О. Весовой векторный метод конечных элементов для одной задачи электромагнетизма с сильной сингулярностью. – Доклады академии наук, 2013, т. 449, № 2, с. 144–148.
#
1. Shandrygin D.A. et al. Vestnik Irkutskogo gosudarstvennogo tehnicheskogo universiteta – in Russ. (Bulletin of Irkutsk State Technical University), 2020, vol. 24, No. 2, pp. 396–407.
2. Ashraf N. et al. Power Quality Analysis of the Output Voltage of AC Voltage and Frequency Controllers Realized with Various Voltage Control Techniques. – Applied Sciences, 2021, vol. 11, iss. 2, DOI: 10.3390/app11020538.
3. Dutta N., Kaliannan P., Subramaniam U. Experimental Analysis of PQ Parameter Estimation of VFD Drives. – IOP Conference Series: Materials Science and Engineering, 2020, vol. 937, DOI: 10.1088/1757-899X/937/1/012042.
4. Jyothi R. et al. IoT Application for Real-Time Condition Monitoring of voltage Source Inverter Driven Induction Motor. – Innovative Data Communication Technologies and Application, 2021, vol. 59, pp. 97–105, DOI: 10.1007/978-981-15-9651-3_8.
5. Hu H. et al. Overview of Harmonic and Resonance in Railway Electrification Systems. – IEEE Transactions on Industry Applications, 2018, vol. 54, No. 5, pp. 5227–5245, DOI: 10.1109/TIA.2018.2813967.
6. Levačić G., Župan A., Čurin M. An Overview of Harmonics in Power Transmission Networks. – First International Colloquium on Smart Grid Metrology (SmaGriMet), 2018, DOI: 10.23919/SMAG-RIMET.2018.8369828.
7. Arrilaga Dzh., Bredli D., Bodzher P. Garmoniki v elektri-cheskih sistemah (Power Systems Harmonics). M.: Energoatomizdat, 1990, 320 p.
8. GOST 32144-2013. Elektricheskaya energiya. Sovmestimost’ tehnicheskih sredstv elektromagnitnaya. Normy kachestva elektricheskoy energii v sistemah elektrosnabzheniya obshchego naznache-niya (Electrical Energy. Electromagnetic Compatibility of Technical Means. Standards of Quality of Electric Energy in General Purpose Power Supply Systems). M.: Standartinform, 2014, 16 p.
9. Silvério E.T., Macedo Junior J.R. Measuring and Modeling the Skin Effect for Harmonic Power Flow Studies. – Energies, 2023, vol. 16, DOI: 10.3390/en16237913.
10. Zhang X.-P., Yan Z. Energy Quality: A Definition. – IEEE Open Access Journal of Power and Energy, 2020, vol. 7, pp. 430–440, DOI: 10.1109/OAJPE.2020.3029767.
11. Product data Bulletin No. 8803PD9402. Causes and Effects of Variable Frequency Drives Relative to the IEEE 519-1992 Standard. Raleigh, U.S.A.: SQUARE D, 1994, 8 p.
12. Harlov N.N. et al. Elektrichestvo – in Russ. (Electricity), 2011, No. 12, pp. 12–15.
13. Borovikov V.S., Harlov N.N., Akimzhanov T.B. Izvestiya Tomskogo politehnicheskogo universiteta – in Russ. (Proceedings of Tomsk Polytechnic University), 2013, vol. 322, No. 4, pp. 91–93.
14. Borovikov V.S. et al. Opyt korporativnogo obsledovaniya elektricheskih setey 110 kV Sibiri: monografiya (The Experience of Corporate Inspection of 110 kV Electric Networks in Siberia: a Monograph). Tomsk: Izd-vo Tomskogo politehnicheskogo universiteta, 2010, 228 p.
15. Kovernikova L.I. Elektrichestvo – in Russ. (Electricity), 2024, No. 6, p.16–26.
16. Demirchyan K.S. et al. Teoreticheskie osnovy elektrotehniki (Theoretical Foundations of Electrical Engineering). Vol. 3. SPb.: Piter, 2003, 377 p.
17. Payne A. Skin Effect, Proximity Effect and the Resistance of Circular and Rectangular Conductors, 2021, 41 p.
18. Ramo S., Whinnery J.R., Van Duzer T. Fields and Waves in Communication Electronics. Third Edition. Hoboken: John Wiley & Sons, 1994, 864 p.
19. Kartashev I.I., Zuev E.N. Kachestvo elektroenergii v sistemah elektrosnabzheniya. Sposoby ego kontrolya i obespecheniya (The Qua-lity of Electricity in Power Supply Systems. Ways to Control and Ensure it). M.: Izd-vo MEI, 2001, 120 p.
20. Zhezhelenko I.V. Vysshie garmoniki v sistemah elektrosnab-zheniya prompredpriyatiy (Higher Harmonics in Industrial Power Supply Systems). M.: Energoatomizdat, 1994, 272 p.
21. Manusov V.Z., Hripkov V.V., Frolova V.V. Nauchnye prob-lemy transporta Sibiri i Dal’nego Vostoka – in Russ. (Scientific Problems of Transport in Siberia and the Far East), 2018, No. 1, pp. 184–188.
22. GOST R IEC 60287-1-1-2022. Kabeli elektricheskie. Raschet nominal'noi tokovoi nagruzki (Electric Cables. Calculation of the Current Rating) M.: Rossiyskiy institut standartizatsii, 2022, 26 p
23. Kennelly A.E., Laws F.A., Pierce P.H. Experimental Resear-ches on Skin Effects in Conductors. – Transactions of the American Institute of Electrical Engineers, 1915, vol. 34, No. 2, pp. 1953–2021, DOI: 10.1109/T-AIEE.1915.4765283.
24. Monteiro J.H.A. et al. Simplified Skin-Effect Formulation for Power Transmission Lines. – IET Science, Measurement & Technology, 2014, vol. 8, No. 2, pp. 47–53, DOI: 10.1049/iet-smt.2013.0072.
25. Suhichev M.I., Skochko E.M. Promyshlennaya energetika – in Russ. (Industrial Energy), 2023, No. 1, pp. 2–8.
26. Rukavishnikov V.A., Mosolapov A.O. Doklady akademii nauk – in Russ. (Reports of the Academy of Sciences), 2013, vol. 449, No. 2, pp. 144–148
2. Ashraf N. et al. Power Quality Analysis of the Output Voltage of AC Voltage and Frequency Controllers Realized with Various Voltage Control Techniques. – Applied Sciences, 2021, vol. 11, iss. 2, DOI: 10.3390/app11020538.
3. Dutta N., Kaliannan P., Subramaniam U. Experimental Analysis of PQ Parameter Estimation of VFD Drives. – IOP Conference Series: Materials Science and Engineering, 2020, vol. 937, DOI: 10.1088/1757-899X/937/1/012042.
4. Jyothi R. et al. IoT Application for Real-Time Condition Monitoring of Voltage Source Inverter Driven Induction Motor. – Innovative Data Communication Technologies and Application, 2021, vol. 59, pp. 97–105, DOI: 10.1007/978-981-15-9651-3_8.
5. Hu H. et al. Overview of Harmonic and Resonance in Railway Electrification Systems. – IEEE Transactions on Industry Applications, 2018, vol. 54, No. 5, pp. 5227–5245, DOI: 10.1109/TIA.2018.2813967.
6. Levačić G., Župan A., Čurin M. An Overview of Harmonics in Power Transmission Networks. – First International Colloquium on Smart Grid Metrology (SmaGriMet), 2018, DOI: 10.23919/SMAGRIMET.2018.8369828.
7. Аррилага Дж., Брэдли Д., Боджер П. Гармоники в электрических системах / Пер. с англ. М.: Энергоатомиздат, 1990, 320 с.
8. ГОСТ 32144–2013. Электрическая энергия. Совместимость технических средств электромагнитная. Нормы качества электрической энергии в системах электроснабжения общего назначения. М.: Стандартинформ, 2014, 16 с.
9. Silvério E.T., Macedo Junior J.R. Measuring and Modeling the Skin Effect for Harmonic Power Flow Studies. – Energies, 2023, vol. 16, DOI: 10.3390/en16237913.
10. Zhang X.-P., Yan Z. Energy Quality: A Definition. – IEEE Open Access Journal of Power and Energy, 2020, vol. 7, pp. 430–440, DOI: 10.1109/OAJPE.2020.3029767.
11. Product data Bulletin No. 8803PD9402. Causes and Effects of Variable Frequency Drives Relative to the IEEE 519-1992 Standard. Raleigh, U.S.A.: SQUARE D, 1994, 8 p.
12. Харлов Н.Н. и др. Энергетическое обследование несинусоидальных режимов многопроводных линий электропередачи. – Электричество, 2011, № 12, с. 12–15.
13. Боровиков В.С., Харлов Н.Н., Акимжанов Т.Б. О необходимости включения добавочных потерь от высших гармоник тока в технологические потери при передаче электрической энергии. – Известия Томского политехнического университета, 2013, т. 322, № 4, с. 91–93.
14. Боровиков В.С. и др. Опыт корпоративного обследования электрических сетей 110 кВ Сибири: монография. Томск: Изд-во Томского политехнического университета, 2010, 228 с.
15. Коверникова Л.И. Оценка дополнительных потерь активной мощности в воздушных ЛЭП 110 кВ при несинусоидальных режимах. – Электричество, 2024, № 6, с.16–26.
16. Демирчян К.С. и др. Теоретические основы электротехники. Т. 3. СПб.: Питер, 2003, 377 с.
17. Payne A. Skin Effect, Proximity Effect and the Resistance of Circular and Rectangular Conductors, 2021, 41 p.
18. Ramo S., Whinnery J.R., Van Duzer T. Fields and Waves in Communication Electronics. Third Edition. Hoboken: John Wiley & Sons, 1994, 864 p.
19. Карташев И.И., Зуев Э.Н. Качество электроэнергии в системах электроснабжения. Способы его контроля и обеспечения. М.: Изд-во МЭИ, 2001, 120 с.
20. Жежеленко И.В. Высшие гармоники в системах электроснабжения промпредприятий. М.: Энергоатомиздат, 1994, 272 с.
21. Манусов В.З., Хрипков В.В., Фролова В.В. Сравнительный анализ математических моделей для определения коэффициента увеличения активного сопротивления проводников от высших гармоник. – Научные проблемы транспорта Сибири и Дальнего Востока, 2018, № 1, с. 184–188.
22. ГОСТ Р МЭК 60287-1-1-2022. Кабели электрические. Расчет номинальной токовой нагрузки. М.: Российский институт стандартизации, 2022, 26 с.
23. Kennelly A.E., Laws F.A., Pierce P.H. Experimental Researches on Skin Effects in Conductors. – Transactions of the American Institute of Electrical Engineers, 1915, vol. 34, No. 2, pp. 1953–2021, DOI: 10.1109/T-AIEE.1915.4765283.
24. Monteiro J.H.A. et al. Simplified Skin-Effect Formulation for Power Transmission Lines. – IET Science, Measurement & Technology, 2014, vol. 8, No. 2, pp. 47–53, DOI: 10.1049/iet-smt.2013.0072.
25. Сухичев М.И., Скочко Е.М. К вопросу о необходимости учета скин-эффекта для сталеалюминиевых проводов. – Промышленная энергетика, 2023, № 1, с. 2–8.
26. Рукавишников В.А., Мосолапов А.О. Весовой векторный метод конечных элементов для одной задачи электромагнетизма с сильной сингулярностью. – Доклады академии наук, 2013, т. 449, № 2, с. 144–148.
#
1. Shandrygin D.A. et al. Vestnik Irkutskogo gosudarstvennogo tehnicheskogo universiteta – in Russ. (Bulletin of Irkutsk State Technical University), 2020, vol. 24, No. 2, pp. 396–407.
2. Ashraf N. et al. Power Quality Analysis of the Output Voltage of AC Voltage and Frequency Controllers Realized with Various Voltage Control Techniques. – Applied Sciences, 2021, vol. 11, iss. 2, DOI: 10.3390/app11020538.
3. Dutta N., Kaliannan P., Subramaniam U. Experimental Analysis of PQ Parameter Estimation of VFD Drives. – IOP Conference Series: Materials Science and Engineering, 2020, vol. 937, DOI: 10.1088/1757-899X/937/1/012042.
4. Jyothi R. et al. IoT Application for Real-Time Condition Monitoring of voltage Source Inverter Driven Induction Motor. – Innovative Data Communication Technologies and Application, 2021, vol. 59, pp. 97–105, DOI: 10.1007/978-981-15-9651-3_8.
5. Hu H. et al. Overview of Harmonic and Resonance in Railway Electrification Systems. – IEEE Transactions on Industry Applications, 2018, vol. 54, No. 5, pp. 5227–5245, DOI: 10.1109/TIA.2018.2813967.
6. Levačić G., Župan A., Čurin M. An Overview of Harmonics in Power Transmission Networks. – First International Colloquium on Smart Grid Metrology (SmaGriMet), 2018, DOI: 10.23919/SMAG-RIMET.2018.8369828.
7. Arrilaga Dzh., Bredli D., Bodzher P. Garmoniki v elektri-cheskih sistemah (Power Systems Harmonics). M.: Energoatomizdat, 1990, 320 p.
8. GOST 32144-2013. Elektricheskaya energiya. Sovmestimost’ tehnicheskih sredstv elektromagnitnaya. Normy kachestva elektricheskoy energii v sistemah elektrosnabzheniya obshchego naznache-niya (Electrical Energy. Electromagnetic Compatibility of Technical Means. Standards of Quality of Electric Energy in General Purpose Power Supply Systems). M.: Standartinform, 2014, 16 p.
9. Silvério E.T., Macedo Junior J.R. Measuring and Modeling the Skin Effect for Harmonic Power Flow Studies. – Energies, 2023, vol. 16, DOI: 10.3390/en16237913.
10. Zhang X.-P., Yan Z. Energy Quality: A Definition. – IEEE Open Access Journal of Power and Energy, 2020, vol. 7, pp. 430–440, DOI: 10.1109/OAJPE.2020.3029767.
11. Product data Bulletin No. 8803PD9402. Causes and Effects of Variable Frequency Drives Relative to the IEEE 519-1992 Standard. Raleigh, U.S.A.: SQUARE D, 1994, 8 p.
12. Harlov N.N. et al. Elektrichestvo – in Russ. (Electricity), 2011, No. 12, pp. 12–15.
13. Borovikov V.S., Harlov N.N., Akimzhanov T.B. Izvestiya Tomskogo politehnicheskogo universiteta – in Russ. (Proceedings of Tomsk Polytechnic University), 2013, vol. 322, No. 4, pp. 91–93.
14. Borovikov V.S. et al. Opyt korporativnogo obsledovaniya elektricheskih setey 110 kV Sibiri: monografiya (The Experience of Corporate Inspection of 110 kV Electric Networks in Siberia: a Monograph). Tomsk: Izd-vo Tomskogo politehnicheskogo universiteta, 2010, 228 p.
15. Kovernikova L.I. Elektrichestvo – in Russ. (Electricity), 2024, No. 6, p.16–26.
16. Demirchyan K.S. et al. Teoreticheskie osnovy elektrotehniki (Theoretical Foundations of Electrical Engineering). Vol. 3. SPb.: Piter, 2003, 377 p.
17. Payne A. Skin Effect, Proximity Effect and the Resistance of Circular and Rectangular Conductors, 2021, 41 p.
18. Ramo S., Whinnery J.R., Van Duzer T. Fields and Waves in Communication Electronics. Third Edition. Hoboken: John Wiley & Sons, 1994, 864 p.
19. Kartashev I.I., Zuev E.N. Kachestvo elektroenergii v sistemah elektrosnabzheniya. Sposoby ego kontrolya i obespecheniya (The Qua-lity of Electricity in Power Supply Systems. Ways to Control and Ensure it). M.: Izd-vo MEI, 2001, 120 p.
20. Zhezhelenko I.V. Vysshie garmoniki v sistemah elektrosnab-zheniya prompredpriyatiy (Higher Harmonics in Industrial Power Supply Systems). M.: Energoatomizdat, 1994, 272 p.
21. Manusov V.Z., Hripkov V.V., Frolova V.V. Nauchnye prob-lemy transporta Sibiri i Dal’nego Vostoka – in Russ. (Scientific Problems of Transport in Siberia and the Far East), 2018, No. 1, pp. 184–188.
22. GOST R IEC 60287-1-1-2022. Kabeli elektricheskie. Raschet nominal'noi tokovoi nagruzki (Electric Cables. Calculation of the Current Rating) M.: Rossiyskiy institut standartizatsii, 2022, 26 p
23. Kennelly A.E., Laws F.A., Pierce P.H. Experimental Resear-ches on Skin Effects in Conductors. – Transactions of the American Institute of Electrical Engineers, 1915, vol. 34, No. 2, pp. 1953–2021, DOI: 10.1109/T-AIEE.1915.4765283.
24. Monteiro J.H.A. et al. Simplified Skin-Effect Formulation for Power Transmission Lines. – IET Science, Measurement & Technology, 2014, vol. 8, No. 2, pp. 47–53, DOI: 10.1049/iet-smt.2013.0072.
25. Suhichev M.I., Skochko E.M. Promyshlennaya energetika – in Russ. (Industrial Energy), 2023, No. 1, pp. 2–8.
26. Rukavishnikov V.A., Mosolapov A.O. Doklady akademii nauk – in Russ. (Reports of the Academy of Sciences), 2013, vol. 449, No. 2, pp. 144–148
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2025-01-30
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