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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Kundur, P.: Power System Stability and Control. McGraw-Hill, New York (1994)
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)
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)
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)
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)
Ghaffari, A.; Krstic, M.; Seshagiri, S.: Extremum seeking for wind and solar energy applications. Focus Dyn. Syst. Control 2, 13–21 (2014)
Zárate-Minano, R.; Anghel, M.; Milano, F.: Continuous wind speed models based on stochastic differential equations. Appl. Energy 104, 42–49 (2013)
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)
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)
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)
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)
Odun-Ayo, T.; Crow, M.: Structure-preserved power system transient stability using stochastic energy functions. IEEE Trans. Power Syst. 27(3), 1450–1458 (2012)
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)
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)
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)
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)
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)
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)
Dong, Z.Y.; Zhao, J.H.; Hill, D.: Numerical simulation for stochastic transient stability assessment. IEEE Trans. Power Syst. 27(4), 1741–1749 (2012)
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
Chen, L.C.; Zhu, W.Q.: First passage failure of quasi non-integrable generalized Hamiltonian systems. Arch. Appl. Mech. 80(8), 883–893 (2010)
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)
Nwankpa, C.; Shahidehpour, M.; Schuss, Z.: A stochastic approach to small signal stability analysis. IEEE Trans. Power Syst. 7(3), 1519–1528 (1992)
Khasminskii, R.Z.: On the averaging principle for Itô stochastic differential equations. Kibernetka 3(4), 260–279 (1968). (in Russian)
Burden, R.L.; Faires, J.D.: Numerical Analysis, 7th edn. Brooks/Cole, Pacific Grove (2001)
Li, R.: Numerical Solutions of Partial Differential Equations. Higher Education Press, Beijing (2005). (in Chinese)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported in part by National Natural Science Foundation of China (Grant No.51207052), the Major Program of the National Natural Science Foundation of China (Grant No. 51190103), and the Fundamental Research Funds for the Central Universities(Grant No. 2014MS62).
Rights and permissions
About this article
Cite this article
Wei, J., Li, G. Collocation Method for First Passage Time Problem of Power Systems Subject to Stochastic Excitations. Arab J Sci Eng 44, 2205–2212 (2019). https://doi.org/10.1007/s13369-018-3361-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13369-018-3361-5