On the Uniqueness of the Solution of Electric Power System Load Flow Nonlinear Equation Systems in the Form of a Power Balance
Keywords:
electric power system, steady state, power balance, system of nonlinear equations, uniqueness of solution
Abstract
The article addresses the problem of finding the uniqueness of a solution to electric power system load flow nonlinear equation systems in the form of a power balance. A mathematical formulation of the problem is presented, and expressions for determining power losses in transmitting power along power lines are given. It is pointed out that the stated problem does not have a unique solution, and to obtain it, a special criterion has to be formulated that would reflect the physical essence of the power flow distribution process in electric power systems. Based on D.K. Maxwell’s theorem “about the distribution of currents in a DC electrical circuit”, a theorem “of determining the uniqueness of the solution of nonlinear equation systems for arbitrary (AC and DC) electric power systems” has been formulated along with the uniqueness criterion: the minimum modulus of the total power loss values. A discrete optimization problem is stated using the formulated uniqueness criterion. In the experimental part of the article, the multiplicity of solutions is considered using the example of a four-node electric power system. It has been revealed that, when directly solving a system of nonlinear steady-state equations, eighteen solutions are obtained. Based on the formulated uniqueness criterion, a physically realizable solution is determined.
References
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14. Park S. et al. Uniqueness of Power Flow Solutions Using Monotonicity and Network Topology. – IEEE Transactions on Control of Network Systems, 2021, 8 (1), DOI: 10.1109/TCNS.2020.3027783.
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17. Lisboa A.C. et al. A Fast Power Flow Method for Radial Networks with Linear Storage and no Matrix Inversions. – International Journal of Electrical Power & Energy Systems, 2014, 63(1), pp. 901–907, DOI:10.1016/j.ijepes.2014.07.002.
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22. Максвелл Дж.К. Трактат об электричестве и магнетизме, т. 1. М.: Наука, 1986, 416 с.
#
1. Idel'chik V.I. Raschety i optimizatsiya rezhimov elektricheskih setey (Calculations and Optimization of Modes of Electric Networks). M.: Energoatomizdat, 1988, 287 p.
2. Ayuev B.I. et al. Vychislitel'nye modeli potokoraspredeleniya v elektricheskih sistemah (Computational Models of Flow Distribution in Electrical Systems). M.: Flinta: Nauka, 2008, 256 p.
3. Krumm L.A. Metody optimizatsii pri upravlenii elektroener-geticheskimi sistemami (Optimization Methods in the Management of Electric Power Systems). Novosibirsk: Nauka, 1981, 319 p.
4. Gurevich Yu.E., Libova L.E., Okin A.A. Raschyoty ustoychivosti i protivoavariynoy avtomatiki v energosistemah (Calculations of Stability and Emergency Automation in Power Systems). М.: Ener-goatomizdat, 1990, 390 p.
5. Milano F. Power System Modelling and Scripting. Springer, 2010, 578 p.
6. Kundur P. Power System Stability and Control. McGraw-Hill, Inc., 1994, 1176 p.
7. Anapol'skiy L.Yu. Izvestiya RAN. Energetika – in Russ. (News of the Russian Academy of Sciences. Energy Industry), 2006, No. 4, pp. 16–26.
8. ANARES-2010 [Electron. resource]. URL: https://isem.irk.ru/upload/iblock/f19/f19c24a44598ba28da299ae1f1cf42d3.pdf (Date of appeal 08.06.2024).
9. RastrWin3. User's Guide, 2020 [Electron. resource]. URL: https://www.rastrwin.ru/download/Files/RastrWin3_2020_10_05.pdf (Date of appeal 08.06.2024).
10. Fazylov H.F., Nasyrov T.H. Izvestiya Akademii nauk CCCP. Energetika i transport – in Russ. (Proceedings of the CCCP Academy of Sciences. Energy and Transport), 1971, No. 6, pp. 36–44.
11. Idel'chik V.I. Elektrichestvo – in Russ. (Electricity), 1983, No. 6, pp. 56–59.
12. Idel'chik V.I. Raschety ustanovivshihsya rezhimov elektr-icheskih sistem (Calculations of Steady-State Modes of Electrical Systems) / By ed. V.А. Venikov. М.: Energiya, 1977, 192 p.
13. Ilic M. Network Theoretic Conditions for Existence and Uniqueness of Steady State Solutions to Electric Power Circuits. – IEEE International Symposium on Circuits and Systems (ISCAS), 1992, DOI: 10.1109/ISCAS.1992.230611.
14. Park S. et al. Uniqueness of Power Flow Solutions Using Monotonicity and Network Topology. – IEEE Transactions on Control of Network Systems, 2021, 8 (1), DOI: 10.1109/TCNS.2020.3027783.
15. Zhang H. et al. Uniqueness of Power Flow Solutions Using Graph-theoretic Notions. – IEEE Transactions on Control of Network Systems, 2022, 9 (1), pp. 100–112, DOI: 10.1109/TCNS.2022.3145749.
16. Meisel J., Barnard R.D. Application of Fixed-Point Techniques to Load-Flow Studies. – IEEE Transactions on Power Apparatus and Systems, 1970, PAS-89 (9), pp. 136–140, DOI: 10.1109/TPAS.1970.292681.
17. Lisboa A.C. et al. A Fast Power Flow Method for Radial Networks with Linear Storage and no Matrix Inversions. – International Journal of Electrical Power & Energy Systems, 2014, 63(1), pp. 901–907, DOI:10.1016/j.ijepes.2014.07.002.
18. Bolognani S., Zampieri S. On the Existence and Linear Approximation of the Power Flow Solution in Power Distribution Networks. – IEEE Transactions on Power Systems, 2016, 31 (1), DOI: 10.1109/TPWRS.2015.2395452.
19. Yu S., Nguyen H.D., Turitsyn K.S. Simple Certificate of Solvability of Power Flow Equations for Distribution Systems. – IEEE Power & Energy Society General Meeting, 2015, DOI: 10.1109/PESGM.2015.7286371.
20. Wang C. et al. Explicit Conditions on Existence and Uniqueness of Load-Flow Solutions in Distribution Networks. – IEEE Transactions on Smart Grid, 2018, 9(2), DOI: 10.1109/TSG.2016.2572060.
21. Panteleev A.V., Yakimova A.S. Teoriya funktsiy kompleksnogo peremennogo i operatsionnoe ischislenie v primerah i zadachah (Theory of Complex Variable Functions and Operational Calculus in Examples and Problems). М.: Vysshaya shkola, 2001, 445 p.
22. Maxwell J.K. Traktat ob elektrichestve i magnetizme (Treatise on Electricity and Magnetism), vol. 1. М.: Nauka, 1986, 416 p.
2. Аюев Б.И. и др. Вычислительные модели потокораспределения в электрических системах. М.: Флинта: Наука, 2008, 256 с.
3. Крумм Л.А. Методы оптимизации при управлении электроэнергетическими системами. Новосибирск: Наука, 1981, 319 с.
4. Гуревич Ю.Е., Либова Л.Е., Окин А.А. Расчёты устойчивости и противоаварийной автоматики в энергосистемах. М.: Энергоатомиздат, 1990, 390 с.
5. Milano F. Power System Modelling and Scripting. Springer, 2010, 578 p.
6. Kundur P. Power System Stability and Control. McGraw-Hill, Inc., 1994, 1176 p.
7. Анапольский Л.Ю. Численно-аналитическое исследование электроэнергетических систем. – Известия РАН. Энергетика, 2006, № 4, с. 16–26.
8. Программно-вычислительный комплекс АНАРЭС-2010 [Электрон. ресурс]. URL: https://isem.irk.ru/upload/iblock/f19/f19c24a44598ba28da299ae1f1cf42d3.pdf (дата обращения 08.06.2024).
9. Программный комплекс «RastrWin3». Руководство пользователя, 2020 [Электрон. ресурс]. URL: https://www.rastrwin.ru/download/Files/RastrWin3_2020_10_05.pdf (дата обращения 08.06.2024).
10. Фазылов Х.Ф., Насыров Т.Х. Некоторые вопросы итерационного расчета установившихся режимов электрических систем. – Известия Академии наук CCCP. Энергетика и транспорт, 1971, № 6, c. 36–44.
11. Идельчик В.И. Пример анализа существования и единственности решения уравнений установившегося режима. – Электричество, 1983, № 6, с. 56–59.
12. Идельчик В.И. Расчеты установившихся режимов электрических систем / под ред. В.А. Веникова. М.: Энергия, 1977, 192 с.
13. Ilic M. Network Theoretic Conditions for Existence and Uniqueness of Steady State Solutions to Electric Power Circuits. – IEEE International Symposium on Circuits and Systems (ISCAS), 1992, DOI: 10.1109/ISCAS.1992.230611.
14. Park S. et al. Uniqueness of Power Flow Solutions Using Monotonicity and Network Topology. – IEEE Transactions on Control of Network Systems, 2021, 8 (1), DOI: 10.1109/TCNS.2020.3027783.
15. Zhang H. et al. Uniqueness of Power Flow Solutions Using Graph-theoretic Notions. – IEEE Transactions on Control of Network Systems, 2022, 9 (1), pp. 100–112, DOI: 10.1109/TCNS.2022.3145749.
16. Meisel J., Barnard R.D. Application of Fixed-Point Techniques to Load-Flow Studies. – IEEE Transactions on Power Apparatus and Systems, 1970, PAS-89 (9), pp. 136–140, DOI: 10.1109/TPAS.1970.292681.
17. Lisboa A.C. et al. A Fast Power Flow Method for Radial Networks with Linear Storage and no Matrix Inversions. – International Journal of Electrical Power & Energy Systems, 2014, 63(1), pp. 901–907, DOI:10.1016/j.ijepes.2014.07.002.
18. Bolognani S., Zampieri S. On the Existence and Linear Approximation of the Power Flow Solution in Power Distribution Networks. – IEEE Transactions on Power Systems, 2016, 31 (1), DOI: 10.1109/TPWRS.2015.2395452.
19. Yu S., Nguyen H.D., Turitsyn K.S. Simple Certificate of Solvability of Power Flow Equations for Distribution Systems. – IEEE Power & Energy Society General Meeting, 2015, DOI: 10.1109/PESGM.2015.7286371.
20. Wang C. et al. Explicit Conditions on Existence and Uniqueness of Load-Flow Solutions in Distribution Networks. – IEEE Transactions on Smart Grid, 2018, 9(2), DOI: 10.1109/TSG.2016.2572060.
21. Пантелеев А.В., Якимова А.С. Теория функций комплексного переменного и операционное исчисление в примерах и задачах. М.: Высшая школа, 2001, 445 с.
22. Максвелл Дж.К. Трактат об электричестве и магнетизме, т. 1. М.: Наука, 1986, 416 с.
#
1. Idel'chik V.I. Raschety i optimizatsiya rezhimov elektricheskih setey (Calculations and Optimization of Modes of Electric Networks). M.: Energoatomizdat, 1988, 287 p.
2. Ayuev B.I. et al. Vychislitel'nye modeli potokoraspredeleniya v elektricheskih sistemah (Computational Models of Flow Distribution in Electrical Systems). M.: Flinta: Nauka, 2008, 256 p.
3. Krumm L.A. Metody optimizatsii pri upravlenii elektroener-geticheskimi sistemami (Optimization Methods in the Management of Electric Power Systems). Novosibirsk: Nauka, 1981, 319 p.
4. Gurevich Yu.E., Libova L.E., Okin A.A. Raschyoty ustoychivosti i protivoavariynoy avtomatiki v energosistemah (Calculations of Stability and Emergency Automation in Power Systems). М.: Ener-goatomizdat, 1990, 390 p.
5. Milano F. Power System Modelling and Scripting. Springer, 2010, 578 p.
6. Kundur P. Power System Stability and Control. McGraw-Hill, Inc., 1994, 1176 p.
7. Anapol'skiy L.Yu. Izvestiya RAN. Energetika – in Russ. (News of the Russian Academy of Sciences. Energy Industry), 2006, No. 4, pp. 16–26.
8. ANARES-2010 [Electron. resource]. URL: https://isem.irk.ru/upload/iblock/f19/f19c24a44598ba28da299ae1f1cf42d3.pdf (Date of appeal 08.06.2024).
9. RastrWin3. User's Guide, 2020 [Electron. resource]. URL: https://www.rastrwin.ru/download/Files/RastrWin3_2020_10_05.pdf (Date of appeal 08.06.2024).
10. Fazylov H.F., Nasyrov T.H. Izvestiya Akademii nauk CCCP. Energetika i transport – in Russ. (Proceedings of the CCCP Academy of Sciences. Energy and Transport), 1971, No. 6, pp. 36–44.
11. Idel'chik V.I. Elektrichestvo – in Russ. (Electricity), 1983, No. 6, pp. 56–59.
12. Idel'chik V.I. Raschety ustanovivshihsya rezhimov elektr-icheskih sistem (Calculations of Steady-State Modes of Electrical Systems) / By ed. V.А. Venikov. М.: Energiya, 1977, 192 p.
13. Ilic M. Network Theoretic Conditions for Existence and Uniqueness of Steady State Solutions to Electric Power Circuits. – IEEE International Symposium on Circuits and Systems (ISCAS), 1992, DOI: 10.1109/ISCAS.1992.230611.
14. Park S. et al. Uniqueness of Power Flow Solutions Using Monotonicity and Network Topology. – IEEE Transactions on Control of Network Systems, 2021, 8 (1), DOI: 10.1109/TCNS.2020.3027783.
15. Zhang H. et al. Uniqueness of Power Flow Solutions Using Graph-theoretic Notions. – IEEE Transactions on Control of Network Systems, 2022, 9 (1), pp. 100–112, DOI: 10.1109/TCNS.2022.3145749.
16. Meisel J., Barnard R.D. Application of Fixed-Point Techniques to Load-Flow Studies. – IEEE Transactions on Power Apparatus and Systems, 1970, PAS-89 (9), pp. 136–140, DOI: 10.1109/TPAS.1970.292681.
17. Lisboa A.C. et al. A Fast Power Flow Method for Radial Networks with Linear Storage and no Matrix Inversions. – International Journal of Electrical Power & Energy Systems, 2014, 63(1), pp. 901–907, DOI:10.1016/j.ijepes.2014.07.002.
18. Bolognani S., Zampieri S. On the Existence and Linear Approximation of the Power Flow Solution in Power Distribution Networks. – IEEE Transactions on Power Systems, 2016, 31 (1), DOI: 10.1109/TPWRS.2015.2395452.
19. Yu S., Nguyen H.D., Turitsyn K.S. Simple Certificate of Solvability of Power Flow Equations for Distribution Systems. – IEEE Power & Energy Society General Meeting, 2015, DOI: 10.1109/PESGM.2015.7286371.
20. Wang C. et al. Explicit Conditions on Existence and Uniqueness of Load-Flow Solutions in Distribution Networks. – IEEE Transactions on Smart Grid, 2018, 9(2), DOI: 10.1109/TSG.2016.2572060.
21. Panteleev A.V., Yakimova A.S. Teoriya funktsiy kompleksnogo peremennogo i operatsionnoe ischislenie v primerah i zadachah (Theory of Complex Variable Functions and Operational Calculus in Examples and Problems). М.: Vysshaya shkola, 2001, 445 p.
22. Maxwell J.K. Traktat ob elektrichestve i magnetizme (Treatise on Electricity and Magnetism), vol. 1. М.: Nauka, 1986, 416 p.
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2024-07-31
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