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ISSN 2542-1905 (Online)

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Khramenkov V. A., Dmitrichev A. S., Nekorkin V. I. Threshold stability of the synchronous mode in a power grid with hub cluster topology. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 2, pp. 120-139. DOI: 10.18500/0869-6632-2020-28-2-120-139

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Threshold stability of the synchronous mode in a power grid with hub cluster topology

Khramenkov Vladislav Anatolevich, Institute of Applied Physics of the Russian Academy of Sciences
Dmitrichev Aleksej Sergeevich, Institute of Applied Physics of the Russian Academy of Sciences
Nekorkin Vladimir Isaakovich, Institute of Applied Physics of the Russian Academy of Sciences

The main purpose of this paper is to investigate the dynamics of the power grid model with hub cluster topology based on the Kuramoto equations with inertia. It is essential to study the stability of synchronous grid operation mode and to find conditions of its global stability. The conditions that ensure establishment of the synchronous mode instead of coexisting asynchronous ones are considered. Methods. In this paper we use numerical modelling of different grid operation modes. Also we use an approach based on the second Lyapunov method, which allows to give an estimate of the area of safe perturbations that do not violate the synchronous mode. Results. Various power grid operation modes and boundaries of their existence in the parameter space are considered. An approach allowing to estimate the magnitude of safe disturbances that do not violate the synchronous mode, is described. Conclusion. The paper considers a power grid model with hub cluster topology. For hub-clusters of three and four elements, their parameter spaces are partitioned into areas corresponding to different operation modes. In particular, parameter areas with global asymptotic stability of synchronous modes that is with trouble-free operations under any initial conditions has been identified. To characterize the modes of hub clusters outside the areas of global asymptotic stability, estimates of the areas of safe perturbations that do not violate the synchronous grid operation mode is given.


Acknowledgements. The work on the study of the dynamic regimes of hub clusters of three and four elements (Section 4) was performed as part of the state assignment of the IAP RAS, project No. 0035–2019–0011. The approach for estimating the magnitude of safe disturbances (Section 5) was developed with support from Russian Foundation for Basic Research (grants No. 18-29-10040, No. 18-02-00406).

  1. Grzybowski J.M.V., Macau E.E.N., Yoneyama T. Power-Grids as Complex Networks: Emerging Investigations into Robustness and Stability. Springer International Publishing, 2012, pp. 287–315.
  2. Pagani G.A., Aiello M. The power grid as a complex network: A survey. Physica A: Statistical Mechanics and its Applications, 2013, vol. 392, no. 11, pp. 2688–2700.
  3. Zhdanov P.S. Stability Issues for Electrical Systems. M.: Energy, 1979 (in Russian). 
  4. Okulovskaya T.Y., Pavlova M.V., Panikovskaya T.Yu., Smirnov V.A. Stability of Electrical Systems. Training manual. Yekaterinburg: UGTU, 2001 (in Russian).
  5. Hruschev Yu.V., Zapodovnikov K.I., Yushkov A.Yu. Electromechanical Transients in Electric Networks: Training Manual. Tomsk: Publishing house of Tomsk Polytechnic University, 2012 (in Russian).
  6. Narovlinskyi V.G. Modern Methods and Means of Preventing the Asynchronous Mode of the Electric Power System. M.: Energoatomizdat, 2004 (in Russian).
  7. Smirnov K.A. On criteria for steady state stability of power system. Elektrichestvo, 1978, no. 3, pp. 12–16 (in Russian).
  8. Venikov V.A., Tsukernik L.V. The development of methods for power systems stability studies. Elektrichestvo, 1978, no. 2, pp. 1–7 (in Russian).
  9. Rohden M., Sorge A., Timme M., Witthaut D. Self-organized synchronization in decentralized power grids. Physical Review Letters, 2012, vol. 109, no. 6, 064101.
  10. Witthaut D., Timme M. Braess’s paradox in oscillator networks, desynchronization and power outage. New Journal of Physics, 2012, 083036.
  11. Menck P.J., Heitzig J. How dead ends undermine power grid stability. Nature communications, 2014, vol. 5, p. 3969.
  12. Lozano S., Buzna L., Dıaz-Guilera A. Role of network topology in the synchronization of power systems. The European Physical Journal B., 2012, vol. 85, no. 7, p. 231.
  13. Motter A.E., Myers S.A., Anghel M., Nishikawa T. Spontaneous synchrony in power-grid networks. Nature Physics, 2013, vol. 9, pp. 191–197.
  14. Fortuna L., Frasca M., Fiore S.A. Analysis of the Italian power grid based on Kuramoto-like model. 5th International Conference on Physics and Control (PhysCon 2011). Leon, Spain. September 5–8, 2011.
  15. Filatrella, G., Nielsen, A.H., Pedersen, N.F. Analysis of a power grid using a Kuramoto-like model. The European Physical Journal B., 2008, vol. 61, no. 4, pp. 485–491.
  16. Dmitrichev A.S., Zakharov D.G., Nekorkin, V.I. Global stability of a synchronous regime in hub clusters of the power networks. Radiophysics and Quantum Electronics, 2017, vol. 60, no. 6, pp. 506–512.
  17. Arinushkin P.A., Anishchenko V.S. Analysis of synchronousmodes of coupled generators, stability of dynamic modes. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, no. 3, pp. 62–77 (in Russian).
  18. Belykh V.N., Bolotov M.I., Osipov G.V. Kuramoto phase model with inertia: bifurcations leading to the loss of synchrony and to the emergence of chaos. Modeling and Analysis of Information Systems, 2015, vol. 22, no. 5, pp. 595–608.
  19. Nishikawa T., Motter A.E. Comparative analysis of existing models for power-grid synchronization. New Journal of Physics, 2015, vol. 17, no. 1, 015012.
  20. Chang Y., Wang X., Xu D. Bifurcation analysis of a power system model with three machines and four buses. International Journal of Bifurcation and Chaos, 2016, vol. 26, no. 5, 1650082.
  21. Zhang W., Huang S., Mei S. et al. Exponential synchronization of the Kuramoto model with star topology. Proceedings of the 35th Chinese Control Conference, Chengdu, China, July 27–29, 2016.
  22. Zhang X., Papachristodoulou A. A real-time control framework for smart power networks with star topology. American Control Conference (ACC), Washington, DC, USA, June 17–19, 2013.
  23. Schiffer J., Efimov D., Ortega R. Almost global synchronization in radial multi-machine power systems. 57th IEEE Conference on Decision and Control (CDC 2018), Miami Beach, FL, United States, December 17–19, 2018. 
  24. Long Vu Th., Turitsyn K. Lyapunov functions family approach to transient stability assessment. IEEE Transactions on Power Systems, 2016, vol. 31, no. 2, pp. 1269–1277.
  25. Gorev A.A. Transients of Synchronous Machines. M.: State Energy Publishing House, 1950 (in Russian).
  26. Park R. Two-reaction theory of synchronous machines: Generalized method of analysis – part I. Transactions of the AIEE, 1929, vol. 48, pp. 716–730.
  27. Dorfler F., Bullo F. Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators. SIAM Journal on Control and Optimization, 2012, vol. 50, no. 3, pp. 1616–1642.
  28. Gray R.M. Toeplitz and circulant matrices: A review Foundations and Trends in Communications and Information Theory, 2006, vol. 2, no. 3, pp. 155–239.
  29. Strachov S.V. Wyman M.J. The present state of development and possibilities for practical application of Lyapunov’s second method in determining the transient stability of power systems. Elektrichestvo, 1977, no. 10, pp. 7–9 (in Russian).
  30. Barbashin E.A. Lyapunov’s Functions. M: Science, 1979 (in Russian).