Nonlinear Dynamics

, Volume 86, Issue 1, pp 401–420 | Cite as

Fixed-time dynamic surface high-order sliding mode control for chaotic oscillation in power system

  • Junkang Ni
  • Ling Liu
  • Chongxin Liu
  • Xiaoyu Hu
  • Tianshi Shen
Original Paper


In this paper, a fixed-time dynamic surface high-order sliding mode control approach is presented for chaos suppression and voltage stabilization in three-bus power system via design of current source converter-based static synchronous compensator controller. The proposed control strategy constructs two high-order sliding mode surfaces to achieve control objective. By combining backstepping idea with dynamic surface control (DSC) technique, high-order sliding mode controller is designed and the inherent problem of “explosion of complexity” in backstepping design is avoided. Further, a new stability concept is introduced into DSC design to achieve semi-global uniform ultimate boundedness of the signals in high-order sliding mode system within finite time independent of initial condition. In addition, stability analysis is provided to show that the proposed control scheme can achieve semi-globally fixed-timely uniformly ultimately bounded stabilization. Finally, simulation results are given to demonstrate the effectiveness of the proposed control scheme and the superior performance over conventional DSC.


Chaos suppression Dynamic surface control (DSC) Fixed-time stability High-order sliding mode control Current source converter-based static synchronous compensator (CSC-STATCOM ) Power system 



This Project was supported by the National Natural Science Foundation of China (Grant Nos. 51177117 and 51307130) and the Creative Research Groups Fund of the National Natural Science Foundation of China (Grant No. 51221005).


  1. 1.
    Nayfeh, M.A., Hamdan, A.M.A., Nayfeh, A.H.: Chaos and instability in a power system—primary resonant case. Nonlinear Dyn. 1, 313–339 (1990)CrossRefGoogle Scholar
  2. 2.
    Ni, J.K., Liu, C.X., Liu, K., Pang, X.: Variable speed synergetic control for chaotic oscillation in power system. Nonlinear Dyn. 78, 681–690 (2014)CrossRefGoogle Scholar
  3. 3.
    Yu, Y.X., Jia, H.J., Li, P., Su, J.F.: Power system instability and chaos. Electr. Power Syst. Res. 65, 87–95 (2003)CrossRefGoogle Scholar
  4. 4.
    Wei, D.Q., Zhang, B., Qiu, D.Y., Luo, X.S.: Effect of noise on erosion of safe basin in power system. Nonlinear Dyn. 61, 477–482 (2010)CrossRefMATHGoogle Scholar
  5. 5.
    Yamashita, K., Joo, S.K., Li, J., Zhang, P., Liu, C.C.: Analysis, control and economic impact assessment of major blackout events. Eur. Trans. Electr. Power 18, 854–871 (2008)CrossRefGoogle Scholar
  6. 6.
    Dobson, I., Chiang, H.D.: Towards a theory of voltage collapse in electric power system. Syst. Control Lett. 13, 253–262 (1989)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chiang, H.D., Liu, C.W., Varaiya, P.P., Wu, F.F., Lauby, M.G.: Chaos in a simple power system. IEEE Trans. Power Syst. 8, 1407–1417 (1993)CrossRefGoogle Scholar
  8. 8.
    Wang, H.O., Abed, E.H., Hamdan, A.M.A.: Bifurcations, chaos, and crises in voltage collapse of a model power system. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41, 294–302 (1994)CrossRefGoogle Scholar
  9. 9.
    Budd, C.J., Wilson, J.P.: Bogdanov–Takens bifurcation points and Sil’nikov homoclinicity in a simple power-system model of voltage collapse. IEEE Trans. Circuits Syst I Fundam. Theory Appl. 49, 575–590 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Tan, C.W., Varghese, N., Varaiya, P., Wu, F.F.: Bifurcation, chaos and voltage collapse in power systems. Proc. IEEE 83, 1484–1496 (1995)CrossRefGoogle Scholar
  11. 11.
    Nayfeh, A.H., Harb, A.M., Chin, C.M.: Bifurcations in a power system model. Int. J. Bifurc. Chaos 6, 497–512 (1996)CrossRefMATHGoogle Scholar
  12. 12.
    Wang, X.Y., Song, J.M.: Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control. Commun. Nonlinear Sci. Numer. Simul. 14, 3351–3357 (2009)CrossRefMATHGoogle Scholar
  13. 13.
    Wei, D.Q., Luo, X.S.: Passivity-based adaptive control of chaotic oscillations in power system. Chaos Solitons Fractals 31, 665–671 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lin, D., Wang, X.Y., Nian, F.Z., Zhang, Y.L.: Dynamic fuzzy neural networks modeling and adaptive backstepping tracking control of uncertain chaotic systems. Neurocomputing 73, 2873–2881 (2010)CrossRefGoogle Scholar
  15. 15.
    Wei, D.Q., Luo, X.S., Qin, Y.H.: Controlling bifurcation in power system based on LaSalle invariant principle. Nonlinear Dyn. 63, 323–329 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zirkohi, M.M., Lin, T.C.: Interval type-2 fuzzy neural network indirect adaptive sliding mode control for an active suspension system. Nonlinear Dyn. 79, 513–526 (2015)CrossRefGoogle Scholar
  17. 17.
    Wang, X.Y., Zhang, X.P., Ma, C.: Modified projective synchronization of fractional-order chaotic systems via active sliding mode control. Nonlinear Dyn. 69, 511–517 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ni, J.K., Liu, C.X., Pang, X.: Fuzzy fast terminal sliding mode controller using an equivalent control for chaotic oscillation in power system. Acta Phys. Sin. 62, 190507 (2013) (in Chinese) Google Scholar
  19. 19.
    Saad, M.S., Hassouneh, M.A., Abed, E.H., Edris, A.: Delaying instability and voltage collapse in power systems using SVCs with washout filter-aided feedback, In: American Control Conference, Portland, OR, USA, pp. 4357–4362 (2005)Google Scholar
  20. 20.
    Ginarsa, I.M., Soeprijanto, A., Purnomo, M.H.: Controlling chaos and voltage collapse using an ANFIS-based composite controller-static var compensator in power systems. Int. J. Electr. Power Energy Syst. 46, 79–88 (2013)CrossRefGoogle Scholar
  21. 21.
    Chakrabortty, A., Scholtz, E.: Time-scale separation designs for performance recovery of power systems with unknown parameters and faults. IEEE Trans. Control Syst. Technol. 19, 382–390 (2010)CrossRefGoogle Scholar
  22. 22.
    Liaw, D.C., Chang, S.T., Huang, Y.H.: Voltage tracking design for electric power systems via SMC approach. In: 48th IEEE Conference on Decision and Control, Shanghai, pp. 7854–7859 (2009)Google Scholar
  23. 23.
    Li, S., Zhao, Q., Chen, C., Xu, Y.: A comparative study on voltage stability bifurcation control ability of SVC and STATCOM. In: China International Conference on Electricity Distribution, Shanghai, pp. 1–4 (2012)Google Scholar
  24. 24.
    Mithulananthan, N., Canizares, C.A., Reeve, J., Rogers, G.J.: Comparison of PSS, SVC, and STATCOM controllers for damping power system oscillations. IEEE Trans. Power Syst. 18, 786–792 (2003)CrossRefGoogle Scholar
  25. 25.
    Ye, Y., Kazerani, M., Quintana, V.H.: Current-source converter based STATCOM: modeling and control. IEEE Trans. Power Deliv. 20, 795–800 (2005)CrossRefGoogle Scholar
  26. 26.
    Griffo, A., Lauria, D.: Two-leg three-phase inverter control for STATCOM and SSSC applications. IEEE Trans. Power Deliv. 23, 361–370 (2008)CrossRefGoogle Scholar
  27. 27.
    Gupta, R., Ghosh, A.: Frequency-domain characterization of sliding mode control of an inverter used in DSTATCOM application. IEEE Trans. Circuits Syst. I Regul. Pap. 53, 662–676 (2006)CrossRefGoogle Scholar
  28. 28.
    Luo, A., Tang, C., Shuai, Z.K., Tang, J., Xu, X.Y., Chen, D.: Fuzzy-PI-based direct-output-voltage control strategy for the STATCOM used in utility distribution systems. IEEE Trans. Ind. Electron. 56, 2401–2411 (2009)CrossRefGoogle Scholar
  29. 29.
    Xu, Y., Li, F.X.: Adaptive PI control of STATCOM for voltage regulation. IEEE Trans. Power Deliv. 29, 1002–1011 (2014)CrossRefGoogle Scholar
  30. 30.
    Levant, A.: Sliding order and sliding accuracy in sliding mode control. Int. J. Control 58, 1247–1263 (1993)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Levant, A.: Chattering analysis. IEEE Trans. Autom. Control 55, 1380–1389 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Boiko, I., Fridman, L., Castellanos, M.I.: Analysis of second-order sliding-mode algorithms in the frequency domain. IEEE Trans. Autom. Control 49, 946–950 (2004)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Valenciaga, F., Puleston, P.F.: High-order sliding control for a wind energy conversion system based on a permanent magnet synchronous generator. IEEE Trans. Energy Convers. 23, 860–867 (2008)CrossRefGoogle Scholar
  34. 34.
    Bartolini, G., Pisano, A., Punta, E., Usai, E.: A survey of applications of second order sliding mode control to mechanical systems. Int. J. Control 76, 875–892 (2003)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Rubio-Astorga, G., Sanchez-Torres, J.D., Canedo, J., Loukianov, A.G.: High-order sliding mode block control of single-phase induction motor. IEEE Trans. Control Syst. Technol. 22, 1828–1836 (2014)CrossRefGoogle Scholar
  36. 36.
    Defoort, M., Nollet, F., Floquet, T., Perruquetti, W.: A third order sliding mode controller for a stepper motor. IEEE Trans. Ind. Electron. 56, 3337–3346 (2009)CrossRefGoogle Scholar
  37. 37.
    Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34, 379–384 (1998)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Polyakov, A., Poznyak, A.: Lyapunov function design for finite-time convergence analysis: twisting controller for second-order sliding mode realization. Automatica 45, 444–448 (2009)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Moreno, J.A., Osorio, M.: Strict Lyapunov functions for the supertwisting algorithm. IEEE Trans. Autom. Control 57, 1035–1040 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Bhat, S.P., Bernstein, D.S.: Geometric homogeneity with applications to finite-time stability. Math. Control Signals Syst. 17, 101–127 (2005)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41, 823–830 (2005)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Swaroop, D., Hedrick, J.K., Yip, P.P., Gerdes, J.C.: Dynamic surface control for a class of nonlinear systems. IEEE Trans. Autom. Control 45, 1893–1899 (2000)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Zhang, T.P., Ge, S.S.: Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form. Automatica 44, 1895–1903 (2008)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Wang, D., Huang, J.: Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form. IEEE Trans. Neural Netw. 16, 195–202 (2005)CrossRefGoogle Scholar
  45. 45.
    Xu, Y.Y., Tong, S.T., Li, Y.M.: Adaptive fuzzy fault-tolerant control of static var compensator based on dynamic surface control technique. Nonlinear Dyn. 73, 2013–2023 (2013)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Polyakov, A., Efimov, D., Perruquetti, W.: Finite-time and fixed-time stabilization: implicit Lyapunov function approach. Automatica 51, 332–340 (2015)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Autom. Control 57, 2106–2110 (2012)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Cruz-Zavala, E., Moreno, J.A., Fridman, L.M.: Uniform robust exact differentiator. IEEE Trans. Autom. Control 56, 2727–2733 (2011)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Zuo, Z.Y.: Nonsingular fixed-time consensus tracking for second-order multi-agent networks. Automatica 54, 305–309 (2015)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Zuo, Z.Y., Tie, L.: A new class of finite-time nonlinear consensus protocols for multi-agent systems. Int. J. Control 87, 363–370 (2014)Google Scholar
  51. 51.
    Zuo, Z.Y.: Non-singular fixed-time terminal sliding mode control of non-linear systems. IET Control Theory Appl. 9, 545–552 (2015)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Walve, K.: Modelling of power system components at severe disturbances. In: Proceedings of the International Conference on Large High Voltage Electric Systems (CIGRE), pp. 38–48, Paris, France, 27 Aug–4 Sept (1986)Google Scholar
  53. 53.
    Schauder, C., Mehta, H.: Vector analysis and control of advanced static var compensators. IEE Proc. C Gener. Transm. Distrib. 140, 299–306 (1993)CrossRefGoogle Scholar
  54. 54.
    Hardy, G., Littlewood, J., Polya, G.: Inequalities. Cambridge University Press, London (1951)Google Scholar
  55. 55.
    Xie, X., Duan, N., Zhao, C.: A combined homogeneous domination and sign function approach to output-feedback stabilization of stochastic high-order nonlinear systems. IEEE Trans. Autom. Control 59, 1303–1309 (2014)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determine Lyapunov exponents from a time-series. Phys. D 16, 285–317 (1985)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Ma, J., Wu, X.Y., Qin, H.X.: Realization of synchronization between hyperchaotic systems by using a scheme of intermittent linear coupling. Acta Phys. Sin. 62, 170502 (2013)Google Scholar
  58. 58.
    Wang, C.N., He, Y.J., Ma, J., Huang, L.: Parameters estimation, mixed synchronization, and antisynchronization in chaotic systems. Complexity 20, 64–73 (2014)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Ma, J., Li, F., Huang, L., Jin, W.Y.: Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system. Commun. Nonlinear. Sci Numer. Simul. 16, 3770–3785 (2011)CrossRefMATHGoogle Scholar
  60. 60.
    Ma, J., Xu, Y., Ren, G.D., Wang, C.N.: Prediction for breakup of spiral wave in a regular neuronal network. Nonlinear Dyn. 84, 497–509 (2016)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Song, X.L., Wang, C.N., Ma, J., Ren, G.D.: Collapse of ordered spatial pattern in neuronal network. Phys. A 451, 95–112 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical EngineeringXi’an Jiaotong UniversityXi’anPeople’s Republic of China

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