Determining the Maximum Apparent Power of Electric Power System Nodes when Representing the Load by a Quasi-Impedance
DOI:
https://doi.org/10.24160/0013-5380-2026-7-32-42Keywords:
electric power system, steady state mode, load quasi-impedance, voltage collapse, static stability, limit loadAbstract
The article considers the problem of determining full PU curves, including their sections both above and below the voltage collapse point, which is a crucial factor in analyzing the power system static stability margins. Existing methods based on defining loads in terms of active and reactive power (P-Q) do not make it possible to calculate the load flow modes in the low-voltage domain in a correct way. The article presents a new load representation method, using which it becomes possible to obtain PU curves for arbitrarily changing power factor for all possible steady-state conditions, including domains with voltages below the critical value. In the new approach, the node's energy consumption is represented by a quasi-impedance parameterized according to a specified law of change with active power. This opens the possibility to reformulate the load flow mode calculation problem and obtain a system of equations the solution of which corresponds to all physically feasible voltages, including a low-level one. The occurrence of two roots (two stable modes) for a single active power and the conditions for their existence is analyzed from the physical point of view. An example of a calculated full PU curve passing through the limit point and containing a section below the critical voltage is given. It is shown that specifying the load with a constant complex impedance (a Z-model) is a necessary and sufficient condition for the existence of a low-voltage solution.
References
1. Крупенев Д.С. О единственности решения систем нелинейных уравнений установившихся режимов электроэнергетических систем в форме баланса мощности. – Электричество, 2024, № 9, с. 37–44.
2. Venikov V.A. et al. Estimation of Electrical Power System Steady-State Stability in Load Flow Calculations. – IEEE Transactions on Power Apparatus and Systems, 1975, vol. 94, No. 3, pp. 1034–1041, DOI: 10.1109/T-PAS.1975.31937.
3. Karimi M. et al. Application of Newton-Based Load Flow Methods for Determining Steady-State Condition of Well and Ill-Conditioned Power Systems: A Review. – International Journal of Electrical Power & Energy Systems, 2019, vol. 113. pp. 298–309, DOI: 10.1016/j.ijepes.2019.05.055.
4. Ajjarapu V., Christy C. The Continuation Power Flow: A Tool for Steady State Voltage Stability Analysis. – IEEE Transactions on Power Systems, 1992, vol. 7, No. 1, pp. 416–423, DOI: 10.1109/59.141737.
5. Chiang H.D. et al. CPFLOW: A Practical Tool for Tracing Power System Steady-State Stationary Behavior Due to Load and Generation Variations. – IEEE Transactions on Power Systems, 1995, vol. 10, No. 2, pp. 623–634, DOI: 10.1109/59.387897.
6. Sheng H., Chiang H.D. CDFLOW: A Practical Tool for Tracing Stationary Behaviors of General Distribution Networks. – IEEE Transactions on Power Systems, 2014, vol. 29, No. 3, pp. 1365–1371, DOI: 10.1109/TPWRS.2013.2289917.
7. MATPOWER [Электрон. ресурс], URL: https://matpower.org/2024/05/17/matpower-8-0-released/ (дата обращения 14.11.2025).
8. Alves D.A. et al. Continuation Fast Decoupled Power Flow with Secant Predictor. – IEEE Transactions on Power Systems, 2003, vol. 18, No. 3, pp. 1078–1085, DOI: 10.1109/TPWRS.2003.814892.
9. Ghiocel S.G., Chow J.H. A Power Flow Method Using a New Bus Type for Computing Steady-State Voltage Stability Margins. – IEEE Transactions on Power Systems, 2014, vol. 29, No. 2, pp. 958–965, DOI: 10.1109/TPWRS.2013.2288157.
10. Liu J.H., Chu C.C. Wide-Area Measurement-Based Voltage Stability Indicators by Modified Coupled Single-Port Models. – IEEE Transactions on Power Systems, 2014, vol. 29, No. 2, pp. 756–764, DOI: 10.1109/TPWRS.2013.2284475.
11. Su H.Y., Liu C.W. Estimating the Voltage Stability Margin Using PMU Measurements. – IEEE Transactions on Power Systems, 2016, vol. 31, No. 4, pp. 3221–3229, DOI: 10.1109/TPWRS.2015.2477426.
12. Gómez-Quiles C. et al. Computation of Maximum Loading Points via the Factored Load Flow. – IEEE Transactions on Power Systems, 2016, vol. 31, No. 5, pp. 4128–4134, DOI: 10.1109/TPWRS. 2015.2505185.
13. Trias A., Marín J.L. The Holomorphic Embedding Loadflow Method for DC Power Systems and Nonlinear DC Circuits. – IEEE Transactions on Circuits and Systems I: Regular Papers, 2016, vol. 63, No. 2, pp. 322–333, DOI: 10.1109/TCSI.2015.2512723.
14. Qiu Y. et al. Global Approximation of Static Voltage Stability Region Boundaries Considering Generator Reactive Power Limits. – IEEE Transactions on Power Systems, 2018, vol. 33, No. 5, pp. 5682–5691, DOI: 10.1109/TPWRS.2018.2800722.
15. Ayuev B.I., Davydov V.V., Erokhin P.M. Fast and Reliable Method of Searching Power System Marginal States. – IEEE Transactions on Power Systems, 2016, vol. 31, No. 6, pp. 4525–4533, DOI: 10.1109/TPWRS.2016.2538299.
16. Ayuev B.I., Davydov V.V., Erokhin P.M. Models of Closest Marginal States of Power Systems in p-Norms. – IEEE Transactions on Power Systems, 2018, vol. 33, No. 2, pp. 1195–1208, DOI: 10.1109/TPWRS.2017.2719125.
17. Liu C. et al. Online Voltage Stability Assessment for Load Areas Based on the Holomorphic Embedding Method. – IEEE Transactions on Power Systems, 2018, vol. 33, No. 4, pp. 3720–3734, DOI: 10.1109/TPWRS.2017.2771384.
18. Wang B. et al. Multi-Stage Holomorphic Embedding Method for Calculating the Power-Voltage Curve. – IEEE Transactions on Power Systems, 2018, vol. 33, No. 1, pp. 1127–1129, DOI: 10.1109/TPWRS.2017.2711260.
19. Zhang W., Wang T., Chiang H.-D. A Novel FFHE-Inspired Method for Large Power System Static Stability Computation. – IEEE Transactions on Power Systems, 2022, vol. 37, No. 1, pp. 726–737, DOI: 10.1109/TPWRS.2021.3093236.
20. Воропай Н.И. и др. Спектральный и модальный методы в исследованиях устойчивости электроэнергетических систем и управлении ими. – Автоматика и телемеханика, 2020, № 10, с. 3–34.
21. Ali M. et al. Calculating Voltage Feasibility Boundaries for Power System Security Assessment. – International Journal of Electrical Power & Energy Systems, 2023, vol. 146, DOI: 10.1016/j.ijepes.2022.108739.
22. Грязина Е.Н., Балуев Д.Ю. Исследование допустимых и предельных режимов работы энергетических систем. – Автоматика и телемеханика, 2024, № 5, с. 112–128.
23. Данилов М.И., Романенко И.Г. Определение PV-кривых и пределов нагрузки шин электроэнергетических систем. – Электрические станции, 2025, № 11, с. 22–31.
24. Данилов М.И., Романенко И.Г. Об определении границы области, предшествующей предельным установившимся режимам электроэнергетических систем, методом анализа тропической геометрии уравнений балансов мощности. – Автоматика и телемеханика, 2024, № 1, с. 95–109.
25. Данилов М.И. и др. Сравнительный анализ предельной точности расчётов установившихся режимов электрических энергосистем современными вычислительными комплексами. – Электрические станции, 2024, № 3, с. 38–45.
26. Данилов М.И., Романенко И.Г. Математический метод контроля теплового состояния проводов воздушных линий питающих сетей. – Электрические станции, 2024, № 7, с. 11–19.
27. Данилов М.И., Романенко И.Г. Определение потоков мощности и температуры проводов электрической сети установившегося состояния энергосистемы. – Электрические станции, 2022, № 7, с. 25–37.
#
1. Krupenev D.S. Elektrichestvo – in Russ. (Electricity), 2024, No. 9, pp. 37–44.
2. Venikov V.A. et al. Estimation of Electrical Power System Steady-State Stability in Load Flow Calculations. – IEEE Transactions on Power Apparatus and Systems, 1975, vol. 94, No. 3, pp. 1034–1041, DOI: 10.1109/T-PAS.1975.31937.
3. Karimi M. et al. Application of Newton-Based Load Flow Methods for Determining Steady-State Condition of Well and Ill-Conditioned Power Systems: A Review. – International Journal of Electrical Power & Energy Systems, 2019, vol. 113. pp. 298–309, DOI: 10.1016/j.ijepes.2019.05.055.
4. Ajjarapu V., Christy C. The Continuation Power Flow: A Tool for Steady State Voltage Stability Analysis. – IEEE Transactions on Power Systems, 1992, vol. 7, No. 1, pp. 416–423, DOI: 10.1109/59.141737.
5. Chiang H.D. et al. CPFLOW: A Practical Tool for Tracing Power System Steady-State Stationary Behavior Due to Load and Generation Variations. – IEEE Transactions on Power Systems, 1995, vol. 10, No. 2, pp. 623–634, DOI: 10.1109/59.387897.
6. Sheng H., Chiang H.D. CDFLOW: A Practical Tool for Tracing Stationary Behaviors of General Distribution Networks. – IEEE Transactions on Power Systems, 2014, vol. 29, No. 3, pp. 1365–1371, DOI: 10.1109/TPWRS.2013.2289917.
7. MATPOWER [Electron. resource], URL: https://matpower.org/2024/05/17/matpower-8-0-released/ (Accessed on 14.11.2025).
8. Alves D.A. et al. Continuation Fast Decoupled Power Flow with Secant Predictor. – IEEE Transactions on Power Systems, 2003, vol. 18, No. 3, pp. 1078–1085, DOI: 10.1109/TPWRS.2003.814892.
9. Ghiocel S.G., Chow J.H. A Power Flow Method Using a New Bus Type for Computing Steady-State Voltage Stability Margins. – IEEE Transactions on Power Systems, 2014, vol. 29, No. 2, pp. 958–965, DOI: 10.1109/TPWRS.2013.2288157.
10. Liu J.H., Chu C.C. Wide-Area Measurement-Based Voltage Stability Indicators by Modified Coupled Single-Port Models. – IEEE Transactions on Power Systems, 2014, vol. 29, No. 2, pp. 756–764, DOI: 10.1109/TPWRS.2013.2284475.
11. Su H.Y., Liu C.W. Estimating the Voltage Stability Margin Using PMU Measurements. – IEEE Transactions on Power Systems, 2016, vol. 31, No. 4, pp. 3221–3229, DOI: 10.1109/TPWRS.2015.2477426.
12. Gómez-Quiles C. et al. Computation of Maximum Loading Points via the Factored Load Flow. – IEEE Transactions on Power Systems, 2016, vol. 31, No. 5, pp. 4128–4134, DOI: 10.1109/TPWRS. 2015.2505185.
13. Trias A., Marín J.L. The Holomorphic Embedding Loadflow Method for DC Power Systems and Nonlinear DC Circuits. – IEEE Transactions on Circuits and Systems I: Regular Papers, 2016, vol. 63, No. 2, pp. 322–333, DOI: 10.1109/TCSI.2015.2512723.
14. Qiu Y. et al. Global Approximation of Static Voltage Stability Region Boundaries Considering Generator Reactive Power Limits. – IEEE Transactions on Power Systems, 2018, vol. 33, No. 5, pp. 5682–5691, DOI: 10.1109/TPWRS.2018.2800722.
15. Ayuev B.I., Davydov V.V., Erokhin P.M. Fast and Reliable Method of Searching Power System Marginal States. – IEEE Transactions on Power Systems, 2016, vol. 31, No. 6, pp. 4525–4533, DOI: 10.1109/TPWRS.2016.2538299.
16. Ayuev B.I., Davydov V.V., Erokhin P.M. Models of Closest Marginal States of Power Systems in p-Norms. – IEEE Transactions on Power Systems, 2018, vol. 33, No. 2, pp. 1195–1208, DOI: 10.1109/TPWRS.2017.2719125.
17. Liu C. et al. Online Voltage Stability Assessment for Load Areas Based on the Holomorphic Embedding Method. – IEEE Transactions on Power Systems, 2018, vol. 33, No. 4, pp. 3720–3734, DOI: 10.1109/TPWRS.2017.2771384.
18. Wang B. et al. Multi-Stage Holomorphic Embedding Method for Calculating the Power-Voltage Curve. – IEEE Transactions on Power Systems, 2018, vol. 33, No. 1, pp. 1127–1129, DOI: 10.1109/TPWRS.2017.2711260.
19. Zhang W., Wang T., Chiang H.-D. A Novel FFHE-Inspired Method for Large Power System Static Stability Computation. – IEEE Transactions on Power Systems, 2022, vol. 37, No. 1, pp. 726–737, DOI: 10.1109/TPWRS.2021.3093236.
20. Voropay N.I. et al. Avtomatika i telemekhanika – in Russ. (Automation and Telemechanics), 2020, No. 10, pp. 3–34.
21. Ali M. et al. Calculating Voltage Feasibility Boundaries for Power System Security Assessment. – International Journal of Electrical Power & Energy Systems, 2023, vol. 146, DOI: 10.1016/j.ijepes.2022.108739.
22. Gryazina E.N., Baluev D.Yu. Avtomatika i telemekhanika – in Russ. (Automation and Telemechanics), 2024, No. 5, pp. 112–128.
23. Danilov M.I., Romanenko I.G. Elektricheskie stantsii – in Russ. (Electrical Stations), 2025, No. 11, pp. 22–31.
24. Danilov M.I., Romanenko I.G. Avtomatika i telemekhanika – in Russ. (Automation and Telemechanics), 2024, No. 1, pp. 95–109.
25. Danilov M.I. et al. Elektricheskie stantsii – in Russ. (Electrical Stations), 2024, No. 3, pp. 38–45.
26. Danilov M.I., Romanenko I.G. Elektricheskie stantsii – in Russ. (Electrical Stations), 2024, No. 7, pp. 11–19.
27. Danilov M.I., Romanenko I.G. Elektricheskie stantsii – in Russ. (Electrical Stations), 2022, No. 7, pp. 25–37
