Chaos in a Fractional-Order Single-Machine Infinite-Bus Power System and Its Adaptive Backstepping Control

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DOI: 10.4236/ijmnta.2016.53013    536 Downloads   710 Views  

ABSTRACT

This paper has numerically studied the dynamical behaviors of a fractional-order single-machine infinite-bus (FOSMIB) power system. Periodic motions, period- doubling bifurcations and chaotic attractors are observed in the FOSMIB power system. The existence of chaotic behavior is affirmed by the positive largest Lyapunov exponent (LLE). Based on the fractional-order backstepping method, an adaptive controller is proposed to suppress chaos in the FOSMIB power system. Numerical simulation results demonstrate the validity of the proposed controller.

Cite this paper

Liang, Z. and Gao, J. (2016) Chaos in a Fractional-Order Single-Machine Infinite-Bus Power System and Its Adaptive Backstepping Control. International Journal of Modern Nonlinear Theory and Application, 5, 122-131. doi: 10.4236/ijmnta.2016.53013.

References

[1] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.
[2] Hilfer, R. (2001) Applications of Fractional Calculus in Physics. World Scientific, New Jersey.
[3] Grigorenko, I. and Grigorenko, E. (2003) Chaotic Dynamics of the Fractional Lorenz System. Physical Review Letters, 91, Article ID: 034101.
http://dx.doi.org/10.1103/PhysRevLett.91.034101
[4] Hartley, T.T., Lorenzo, C.F. and Qammer, H.K. (1995) Chaos in a Fractional Order Chua’s System. IEEE Transactions on Circuits and Systems CAS-I, 42, 485-490.
http://dx.doi.org/10.1109/81.404062
[5] Gao, X. and Yu, J.B. (2005) Chaos in the Fractional Order Periodically Forced Complex Duffing’s Oscillators. Chaos, Solitons and Fractals, 24, 1097-1104.
http://dx.doi.org/10.1016/j.chaos.2004.09.090
[6] Li, C.G. and Chen, G.R. (2004) Chaos and Hyperchaos in the Fractional-Order Rossler Equations. Physica A, 341, 55-61.
http://dx.doi.org/10.1016/j.physa.2004.04.113
[7] Li, C.P. and Peng, G.J. (2005) Chaos in Chen’s System with a Fractional Order. Chaos, Solitons and Fractals, 22, 443-450.
http://dx.doi.org/10.1016/j.chaos.2004.02.013
[8] Kopell, N. and Washburn, R.B. (1982) Chaotic Motions in the Two-Degree-Of-Freedom Swing Equations. IEEE Transactions on Circuits and Systems, 29, 738-746.
http://dx.doi.org/10.1109/TCS.1982.1085094
[9] Lee, B. and Ajjarapu, V. (1993) Period-Doubling Route to Chaos in an Electrical Power System. IEE Proceedings C-Generation, Transmission and Distribution, 140, 490-496.
http://dx.doi.org/10.1049/ip-c.1993.0071
[10] Chiang, H.D., Conneen, T.P. and Flueck, A.J. (1994) Bifurcations and Chaos in Electric Power Systems: Numerical Studies. Journal of the Franklin Institute, 331, 1001-1036.
http://dx.doi.org/10.1016/0016-0032(94)90095-7
[11] Ji, W. and Venkatasubramanian, V. (1996) Hard-Limit Induced Chaos in a Fundamental Power System Model. International Journal of Electrical Power and Energy Systems, 18, 279-295.
http://dx.doi.org/10.1016/0142-0615(95)00066-6
[12] Chen, H.K., Lin, T.N. and Chen, J.H. (2005) Dynamic Analysis, Controlling Chaos and Chaotification of a SMIB Power System. Chaos, Solitons and Fractals, 22, 1307-1315.
http://dx.doi.org/10.1016/j.chaos.2004.09.081
[13] Gholizadeh, H., Hassannia, A. and Azarfar, A. (2013) Chaos Detection and Control in a Typical Power System. Chinese Physics B, 22, 10503-10507.
http://dx.doi.org/10.1088/1674-1056/22/1/010503
[14] Yu, Y.X., Jia, H.J., Li, P. and Su, J.F. (2003) Power System Instability and Chaos. Electric Power Systems Research, 65, 187-195.
http://dx.doi.org/10.1016/S0378-7796(02)00229-8
[15] Tan, W., Zhang, M. and Li, Z.P. (2011) Chaotic Oscillation of Interconnected Power System and Its Synchronization. Journal of Hunan University of Science and Technology (Natural Science Edition), 26, 74-78. (In Chinese)
[16] Sun, F.Y. and Li, Q. (2014) Dynamic Analysis and Chaos of the 4D Fractional-Order Power System. Abstract and Applied Analysis, 2014, Article ID: 534896.
http://dx.doi.org/10.1155/2014/534896
[17] Ahmed, E., El-Sayed, A.M.A. and El-Saka, H.A.A. (2007) Equilibrium Points, Stability and Numerical Solutions of Fractional-Order Predator—Prey and Rabies Models. Journal of Mathematical Analysis and Applications, 325, 542-553.
http://dx.doi.org/10.1016/j.jmaa.2006.01.087
[18] Tavazoei, M.S. and Haeri, M. (2007) A Necessary Condition for Double Scroll Attractor Existence in Fractional-Order Systems. Physics Letters A, 367, 102-113.
http://dx.doi.org/10.1016/j.physleta.2007.05.081
[19] Tavazoei, M.S. and Haeri, M. (2008) Chaotic Attractors in Incommensurate Fractional Order Systems. Physica D: Nonlinear Phenomena, 237, 2628-2637.
http://dx.doi.org/10.1016/j.physd.2008.03.037
[20] Aguila-Camacho, N., Duarte-Mermoud, M.A. and Gallegos, J.A. (2014) Lyapunov Functions for Fractional Order Systems. Communications in Nonlinear Science and Numerical Simulation, 19, 2951-2957.
http://dx.doi.org/10.1016/j.cnsns.2014.01.022
[21] Li, Y., Chen, Y.Q. and Podlubny, I. (2009) Mittag-Leffler Stability of Fractional Order Nonlinear Dynamic Systems. Automatica, 45, 1965-1969.
http://dx.doi.org/10.1016/j.automatica.2009.04.003
[22] Diethelm, K., Ford, N.J. and Freed, A.D. (2002) A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics, 29, 3-22.
http://dx.doi.org/10.1023/A:1016592219341
[23] Diethelm, K. (1997) An Algorithm for the Numerical Solution of Differential Equations of Fractional Order. Electronic Transactions on Numerical Analysis, 5, 1-6.
[24] Diethelm, K. and Ford, N.J. (2002) Analysis of Fractional Differential Equations. Journal of Mathematical Analysis and Applications, 265, 229-248.
http://dx.doi.org/10.1006/jmaa.2000.7194
[25] Wolf, A., Swift, J.B., Swinneya, H.L. and Vastano, J.A. (1985) Determining Lyapunov Exponents from a Time Series. Physica D: Nonlinear Phenomena, 16, 285-317.
http://dx.doi.org/10.1016/0167-2789(85)90011-9

  
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