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Collocation Method for First Passage Time Problem of Power Systems Subject to Stochastic Excitations

Research Article - Electrical Engineering

Abstract

Large penetration of renewable energies heavily threats the stable and reliable operation of power systems due to their randomness and intermittence characteristics. The first passage time problem is one of the critical issues in reliability assessment of new energy power systems. In this paper, we present and analyze the first passage time problem of power systems with stochastic excitation by collocation method. The power systems with stochastic excitations are modeled by stochastic differential equations. Then, the backward Kolmogorov equations and the generalized Pontryagin equations governing the conditional reliability function and the conditional moments of first passage time, respectively, are established based on the stochastic averaging method. The corresponding initial and boundary conditions are also provided. A numerical collocation method was proposed to solve the equations, and case studies were executed on a single-machine infinite-bus system under Gaussian excitation. Illustrations of the conditional reliability function and probability density functions for some cases are presented.

Keywords

SDEs First passage time Stochastic averaging method Backward Kolmogorov equation Generalized Pontryagin equation Collocation method 

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References

  1. 1.
    Kundur, P.: Power System Stability and Control. McGraw-Hill, New York (1994)Google Scholar
  2. 2.
    Kundur, P.; Paserba, J.; Ajjarapu, V.; Andersson, G.; Bose, A.; Canizares, C.; Hatziargyriou, N.; Hill, D.; Stankovic, A.; Taylor, C.; Van Cutsem, T.; Vittal, V.: Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions. IEEE Trans. Power Syst. 19(3), 1387–1401 (2004)CrossRefGoogle Scholar
  3. 3.
    Qiu, J.; Shahidehpour, M.; Schuss, Z.: Effect of small random perturbations on power systems dynamics and its reliability evaluation. IEEE Trans. Power Syst. 4(1), 197–204 (1989)CrossRefGoogle Scholar
  4. 4.
    Akhmatov, V.; Knudsen, H.: An aggregate model of a grid-connected, large-scale, offshore wind farm for power stability investigations–importance of windmill mechanical system. Int. J. Electr. Power Energy Syst. 24, 709–717 (2002)CrossRefGoogle Scholar
  5. 5.
    Wang, C.; Shi, L.B.; Yao, L.Z.; Wang, L.M.; Ni, Y.X.; Bazargan, M.: Modeling analysis in power system small signal stability considering uncertainty of wind generation. In: Proc. IEEE PES General Meeting, pp. 1–7 (2010)Google Scholar
  6. 6.
    Ghaffari, A.; Krstic, M.; Seshagiri, S.: Extremum seeking for wind and solar energy applications. Focus Dyn. Syst. Control 2, 13–21 (2014)Google Scholar
  7. 7.
    Zárate-Minano, R.; Anghel, M.; Milano, F.: Continuous wind speed models based on stochastic differential equations. Appl. Energy 104, 42–49 (2013)CrossRefGoogle Scholar
  8. 8.
    Li, G.; Yue, H.; Zhou, M.; Wei, J.: Probabilistic assessment of oscillatory stability margin of power systems incorporating wind farms. Int. J. Electr. Power Energy Syst. 58, 47–56 (2014)CrossRefGoogle Scholar
  9. 9.
    Parinya, P.; Sangswang, A.; Kirtikara, K.; Chenvidhya, D.; Naetiladdanon, S.; Limsakul, C.: A study of effects of stochastic wind power and load to power system stability using stochastic stability analysis method. Adv. Mater. Res. 931–932, 878–882 (2014)CrossRefGoogle Scholar
  10. 10.
    Yuan, B.; Zhou, M.; Li, G.Y.; Zhang, X.P.: Stochastic small-signal stability of power systems with wind power generation. IEEE Trans. Power Syst. 30(4), 1680–1689 (2015)CrossRefGoogle Scholar
  11. 11.
    Vergejo, H.; Kliemann, W.; Vargas, L.: Application of linear stability via Lyapunov exponents in high dimensional electrical power systems. Int. J. Electr. Power Energy Syst. 64, 1141–1146 (2015)CrossRefGoogle Scholar
  12. 12.
    Odun-Ayo, T.; Crow, M.: Structure-preserved power system transient stability using stochastic energy functions. IEEE Trans. Power Syst. 27(3), 1450–1458 (2012)CrossRefGoogle Scholar
  13. 13.
    Dhople, S.V.; Chen, Y.C.; Deville, L.; Dominguez-Garcia, A.D.: Analysis of power system dynamics subject to stochastic power injections. IEEE Trans. Circuits Syst. I Regul. Pap. 60(12), 3341–3353 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Milano, F.; Zárate-Miñano, R.: A systematic method to model power systems as stochastic differential algebraic equations. IEEE Trans. Power Syst. 28(4), 4537–4544 (2013)CrossRefGoogle Scholar
  15. 15.
    Zhang, J.; Ju, P.; Yu, Y.; Wu, F.: Responses and stability of power system under small Gauss type random excitation. Sci. China Technol. Sci. 55(7), 1873–1880 (2012)CrossRefGoogle Scholar
  16. 16.
    Zhou, M.; Yuan, B.; Zhang, X.P.; Li, G.Y.: Stochastic small signal stability analysis of wind power integrated power systems based on stochastic differential equations. Proc. CSEE 34(10), 1575–1582 (2014). (in Chinese)Google Scholar
  17. 17.
    Vergejo, H.; Vargas, L.; Kliemann, W.: Stability of linear stochastic systems via Lyapunov exponents and applications to power systems. J. Appl. Math. Comput. 218, 1021–1032 (2012)MathSciNetGoogle Scholar
  18. 18.
    Wang, K.; Crow, M.: The Fokker-Planck equation for power system stability probability density function evolution. IEEE Trans. Power Syst. 28(3), 2994–3001 (2013)CrossRefGoogle Scholar
  19. 19.
    Dong, Z.Y.; Zhao, J.H.; Hill, D.: Numerical simulation for stochastic transient stability assessment. IEEE Trans. Power Syst. 27(4), 1741–1749 (2012)CrossRefGoogle Scholar
  20. 20.
    Zhu, W.Q.: Nonlinear stochastic dynamics and control in Hamiltonian formulation. Appl. Mech. Rev. 59(4), 230–248 (2006).  https://doi.org/10.1115/1.2193137 MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chen, L.C.; Zhu, W.Q.: First passage failure of quasi non-integrable generalized Hamiltonian systems. Arch. Appl. Mech. 80(8), 883–893 (2010)CrossRefMATHGoogle Scholar
  22. 22.
    Nwankpa, C.; Shahidehpour, M.: A nonlinear stochastic model for small disturbance stability analysis of electric power systems. Int. J. Elect. Power Energy Syst. 13(3), 139–147 (1991)CrossRefGoogle Scholar
  23. 23.
    Nwankpa, C.; Shahidehpour, M.; Schuss, Z.: A stochastic approach to small signal stability analysis. IEEE Trans. Power Syst. 7(3), 1519–1528 (1992)CrossRefGoogle Scholar
  24. 24.
    Khasminskii, R.Z.: On the averaging principle for Itô stochastic differential equations. Kibernetka 3(4), 260–279 (1968). (in Russian)Google Scholar
  25. 25.
    Burden, R.L.; Faires, J.D.: Numerical Analysis, 7th edn. Brooks/Cole, Pacific Grove (2001)MATHGoogle Scholar
  26. 26.
    Li, R.: Numerical Solutions of Partial Differential Equations. Higher Education Press, Beijing (2005). (in Chinese)Google Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.School of Mathematical and Physical ScienceNorth China Electric Power UniversityBeijingPeople’s Republic of China
  2. 2.State Key Laboratory of Alternate Electrical Power System with Renewable Energy SourcesNorth China Electric Power UniversityBeijingPeople’s Republic of China

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