Stability analysis of a hybrid power system with photovoltaic generation using the dq0-transformation method

This article presents a method for stability analysis of hybrid power systems with a high share of renewable energy sources, particularly photovoltaic generators, based on the dq0-transformation (Park’s transformation). The key feature of this approach is the unification of all system components – synchronous generators, network elements, and power inverters – into a single generalized reference frame rotating at synchronous speed. This enables the construction of a holistic dynamic model of complex power systems that maintains high accuracy across a wide frequency range while preserving time-invariance for steady-state analysis.
The relevance of this research stems from the critical need for advanced mathematical modeling methods in electric power systems undergoing structural transformations due to integration of renewable-based distributed generation. The extensive integration of stochastic generation sources fundamentally alters power system dynamics, creating challenges for traditional modeling approaches. Existing methods demonstrate limited effectiveness in analyzing transients under high penetration of intermittent generation. Consequently, developing a mathematical framework for modeling hybrid power system dynamics represents a significant scientific challenge essential for ensuring reliable operation and sustainable development of power systems. Traditional phase-coordinate models provide accuracy but are non-stationary, complicating stability analysis, while quasi-static models, though stationary, neglect high-frequency dynamics. The proposed method addresses this methodological gap through a unified approach to modeling heterogeneous power system components.
The work details the mathematical apparatus for transforming equations of main system components into a unified coordinate system. For passive network elements (inductances, capacitances, resistors), state equation transformation accounts for the transformation operator derivative. The synchronous generator is represented by a physical salient-pole machine model considering mutual influence of magnetic fields along d- and q-axes, winding active resistances, and rotor circuit dynamics. Described by six state variables, this model incorporates inductive parameters ignored in simplified models. The photovoltaic generator utilizes an inverter model with DC-link voltage control loop employing PI-controller and output capacitive storage. The procedure for integrating inverter output variables into the common coordinate system is demonstrated.
A crucial methodological aspect is the network model reduction procedure. We describe a node elimination algorithm for buses unconnected to generators or loads, reducing problem dimensionality and focusing analysis on key generator bus dynamics. Reduction is achieved by controlling corresponding bus input variables to zero their outputs, followed by state vector transformation and new dynamic model formation using matrix algebra methods like LU-decomposition. This procedure is particularly important for large power systems where full models may contain redundant research information.
The method was tested through numerical experiments using MATPOWER test networks, including a modified IEEE 14-bus system with partial conventional generator replacement by photovoltaic stations. Comparative analysis of quasi-static, phase-domain (abc), and dq0-models was performed, along with small-signal dynamics investigation through linearization and eigenvalue analysis. Particular attention was given to verifying model adequacy under various system parameter variation scenarios.
Initial four-bus network simulation confirmed identical steady-states across all three model types. Phase-domain and dq0-models showed complete transient response agreement, while the quasi-static model inadequately represented high-frequency electromagnetic transients. Computational efficiency assessment analyzed matrix sparsity and non-zero element counts, revealing comparable abc and dq0 model parameters, demonstrating dq0-approach practicality for complex system modeling without significant computational burden increase.
A hybrid system based on modified IEEE 14-bus network was investigated, with synchronous generators at buses 6 and 8 replaced by photovoltaic stations. A complete nonlinear dq0-coordinate system model was developed, incorporating two synchronous generators, two photovoltaic inverters, passive network, and infinite bus. Steady-state parameters were obtained from power flow solutions. This test case enables investigation of mutual influence between conventional and renewable generation under varying parameters.
Stability analysis employed individual component model linearization around operating points followed by general state-space model formation. Frequency response analysis (Bode plots) of state-space equations examined inter-machine connections from mechanical to electrical power, revealing significant mutual influence near 30 rad/s resonance frequency. Root locus analysis tracked dominant eigenvalue trajectories under external disturbances: stepped mechanical power increase and photovoltaic power reduction. Pole migration toward imaginary axis indicated reduced stability margin and response speed under loading. Photovoltaic inverter power variation proved less dynamically significant than synchronous generator power changes, attributable to lower rated power in this test case.
The results demonstrate how synchronous generator and photovoltaic inverter power variations affect system stability and dynamics, confirming the approach’s effectiveness for large-scale hybrid power system modeling. The developed methodology provides a formalized tool for determining maximum permissible renewable generation shares while maintaining static and dynamic stability constraints. These findings enable future research into coordinated control of heterogeneous generators in complex power systems.

1. Burman, A. M. Modeling of synchronous machines in the d-q-0 coordinate system for studying the dynamics of power systems / A. M. Burman, G. V. Rozhdestvensky // Elektrichestvo. – 2005. – No. 4. – Pp. 20-28.

2. Venikov, V. A. Theory of similarity and modeling applied to problems of electric power engineering / V. A. Venikov, V. V. Venikov. – Moscow: Higher School, 1984. – 439 p.

3. German-Galkin, S. G. Computer Modeling of Semiconductor Systems in MATLAB: Textbook / S. G. German-Galkin. – St. Petersburg: Korona-Vek, 2010. – 320 p.

4. Gurevich, Yu. E. Intelligent Power Supply Systems: From Smart Grid to Microgrid / Yu. E. Gurevich, A. V. Prokhorov. – Moscow: Infra-Engineering, 2016. – 288 p.

5. Zhabrikov, S. M. Modeling and Analysis of the Stability of Electric Power Systems with Distributed Generation / S. M. Zhabrikov, V. N. Kozlov // Izvestiya NTC of the Unified Energy System. – 2018. – No. 2 (55). – Pp. 15-25.

6. Kostyuk, V. N. Stability of power systems with powerful wind turbines / V. N. Kostyuk, A. S. Abramovich // Energetik. – 2017. – No. 5. – Pp. 35-39.

7. Lykin, A. V. Electric Systems and Networks: Modeling and Calculation of Stationary and Transient Modes: Textbook for Universities / A. V. Lykin. – Moscow: MEI Publishing House, 2010. – 476 p.

8. Rozhdestvensky, G. V. Modeling of Dynamic Modes in Electric Power Systems Using dq0-Transformation / G. V. Rozhdestvensky, A. M. Burman // Izvestiya vuzov. Elektromekhanika. – 2006. – No. 3. – Pp. 45-52.

9. Sergienko, A. B. Digital Modeling of Electric Power Systems / A. B. Sergienko, Yu. Ya. Kovalev. – St. Petersburg: Publishing house of the Polytechnic. University, 2012. – 312 p.

10. Sokolov, V. I. Modes of operation of power systems with wind power plants / V. I. Sokolov, P. A. Lokhanin // Industrial power engineering. - 2015. – No. 8. – Pp. 15-21.

11. Strakhov, S. V. Modeling of photovoltaic systems connected to the grid for the analysis of power systems operation modes / S. V. Strakhov, D. A. Sukhanov // Scientific works of the Far Eastern State Technical University. – 2019. – No. 45. – Pp. 112-120.

12. Strizhkov, D. A. Methods for analyzing the dynamic stability of electric power systems using dq0 models / D. A. Strizhkov, K. V. Suslov // Electric stations. - 2021. – No. 5. – Pp. 41-48.

13. Fedotov, A. S. Analysis of the stability of hybrid power supply systems with photovoltaic plants / A. S. Fedotov, M. V. Ivanova // Alternative energy and Ecology (ISJAEE). – 2020. – № 1-3. – Pp. 64-73.

14. Ulyanov, S. A. Electromagnetic transient processes in electrical systems / S. A. Ulyanov. – Moscow: Energiya, 1970. – 520 p.

15. Anderson, P. M. Power System Control and Stability / P. M. Anderson, A. A. Fouad. – 2nd ed. – Wiley-IEEE Press, 2002. – 364 p.

16. Bevrani, H. Robust Power System Frequency Control / H. Bevrani. – 2nd ed. – Springer, 2014. – 322 p.

17. Bialasiewicz, J. T. A DQ-Based Phase-Locked Loop for Unbalanced Line Voltages / J. T. Bialasiewicz // IEEE Power Electronics Letters. – 2004. – Vol. 2, no. 2. – Pp. 62-65.

18. Blaabjerg, F. Overview of Control and Grid Synchronization for Distributed Power Generation Systems / F. Blaabjerg, R. Teodorescu, M. Liserre, A. V. Timbus // IEEE Transactions on Industrial Electronics. – 2006. – Vol. 53, №. 5. – Pр. 1398-1409.

19. Bolognani, S. A Distributed Control Strategy for Reactive Power Compensation in Smart Microgrids / S. Bolognani, S. Zampieri // IEEE Transactions on Automatic Control. – 2013. – Vol. 58, no. 11. – Pp. 2818-2833.

20. Bose, B. K. Power Electronics and Motor Drives: Advances and Trends / B. K. Bose. – Academic Press, 2006. – 911 p.

21. Carrasco, J. M. Power-Electronic Systems for the Grid Integration of Renewable Energy Sources: A Survey / J. M. Carrasco, L. G. Franquelo, J. T. Bialasiewicz // IEEE Transactions on Industrial Electronics. – 2006. – Vol. 53, no. 4. – Pp. 1002-1016.

22. Ellis, A. Model Development and Validation: Generator and Power Plant / A. Ellis, E. Muljadi, J. J. Sanchez-Gasca // Renewable Energy Integration: Practical Management of Variability, Uncertainty, and Flexibility in Power Grids / ed. by L. E. Jones. – 2nd ed. – Amsterdam: Academic Press, 2014. – Pр. 187-201.

23. Hatziargyriou, N. Distributed Energy Resources: Integration Challenges and Solutions / N. Hatziargyriou // IEEE Power and Energy Magazine. – 2014. – Vol. 12, no. 2. – Pp. 49-61.

24. Katiraei, F. Microgrids Management / F. Katiraei, M. R. Iravani // IEEE Power and Energy Magazine. – 2008. – Vol. 6, no. 3. – Pp. 54-65.

25. Krause, P. C. Analysis of Electric Machinery and Drive Systems / P. C. Krause, O. Wasynczuk, S. D. Sudhoff, S. Pekarek. – 3rd ed. – Piscataway: IEEE Press, 2013. – 713 p.

26. Kundur, P. Power System Stability and Control / P. Kundur. – New York: McGraw-Hill, 1994. – 1176 p.

27. Levron Y., Belikov J., Baimel D. A Tutorial on Dynamics and Control of Power Systems with Distributed and Renewable Energy Sources Based on the DQ0 Transformation // Journal of Renewable and Sustainable Energy. – 2018. – Vol. 10, № 5. – 034302. – DOI: 10.1063/1.5049212.

28. Liserre, M. Stability of Photovoltaic and Wind Turbine Grid-Connected Inverters for a Large Set of Grid Impedance Values / M. Liserre, R. Teodorescu, F. Blaabjerg // IEEE Transactions on Power Electronics. – 2006. – Vol. 21, no. 1. – Pp. 263-272.

29. Milano, F. Power System Modelling and Scripting / F. Milano. – Springer, 2010. – 450 p.

30. Rocabert, J. Control of Power Converters in AC Microgrids / J. Rocabert, A. Luna, F. Blaabjerg // IEEE Transactions on Power Electronics. – 2012. – Vol. 27, no. 11. – Pp. 4734-4749.

31. Zimmerman, R. D. MATPOWER: SteadyState Operations, Planning, and Analysis Tools for Power Systems Research and Education / R. D. Zimmerman, C. E. Murillo-Sánchez, R. J. Thomas // IEEE Transactions on Power Systems. – 2011. – Vol. 26, №. 1. – Pр. 12-19.