×

Effect of noise on erosion of safe basin in power system. (English) Zbl 1204.70028

Summary: We study the effect of Gaussian white noise on erosion of safe basin in a simple model of power system whose safe basin is integral in the absence of noise. The stochastic Melnikov method is first applied to predict the onset of basin erosion when the noise excitation is present in system. And then the eroded basins are simulated according to the necessary restrictions for the system’s parameters. It is found that for the noisy power system when the noise intensity \(\sigma \) is greater than a threshold, basin erosion occurs and as \(\sigma \) is further increased basin erosion is aggravated. These studies imply that random noise excitation can induce and enhance the basin erosion in the power system.

MSC:

70Q05 Control of mechanical systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Soliman, M.S., Thompson, J.M.T.: Integrity measures quantifying the erosion of smooth and fractal basins of attraction. J. Sound Vib. 135, 453–475 (1989) · Zbl 1235.70106 · doi:10.1016/0022-460X(89)90699-8
[2] Soliman, M.S., Thompson, J.M.T.: Global dynamics underlying sharp basin erosion in nonlinear driven oscillators. Phys. Rev. A 45, 3425–3431 (1992) · doi:10.1103/PhysRevA.45.3425
[3] Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos, Geometrical Methods for Engineers and Scientists, 2nd edn. Wiley, Chichester (2002)
[4] Santee, D.M., Gonçalves Paulo, B.: Oscillations of a beam on a non-linear elastic foundation under periodic loads. Shock Vib. 13, 273–284 (2006)
[5] Aguirre, J., Viana Ricardo, L., Sanjuán, M.A.F.: Fractal structures in nonlinear dynamics. Rev. Mod. Phys. 81, 333–387 (2009) · doi:10.1103/RevModPhys.81.333
[6] Nayfeh, A.H., Sanchez, N.E.: Bifurcations in a softening Duffing oscillator. Int. J. Non-Linear Mech. 24, 483–497 (1989) · Zbl 0712.70045 · doi:10.1016/0020-7462(89)90014-0
[7] Sanjuán, M.A.F.: The effect of nonlinear damping on the universal escape oscillator. Int. J. Bifurc. Chaos 9, 735–744 (1999) · Zbl 0977.34041 · doi:10.1142/S0218127499000523
[8] Rega, G., Lenci, S.: Identifying, evaluating, and controlling dynamical integrity measures in non-linear mechanical oscillators. Nonlinear Anal. 63, 902–914 (2005) · Zbl 1153.70307 · doi:10.1016/j.na.2005.01.084
[9] Tchawoua, C., Siewe Siewe, M., Tchatchueng, S., Moukam Kakmeni, F.M.: Nonlinear dynamics of parametrically driven particles in a {\(\Phi\)} 6 potential. Nonlinearity 21, 1041–1055 (2008) · Zbl 1153.34025 · doi:10.1088/0951-7715/21/5/008
[10] Gan, C.B.: Noise-Induced chaos and basin erosion in softening Duffing oscillator. Chaos Solutions Fractals 25, 1069–1081 (2005) · Zbl 1099.37065 · doi:10.1016/j.chaos.2004.11.070
[11] Gan, C.B.: Noise-induced chaos in Duffing oscillator with double wells. Nonlinear Dyn. 45, 305–317 (2006) · Zbl 1123.70021 · doi:10.1007/s11071-005-9008-6
[12] Rong, H., Wang, X., Xu, W., Fang, T.: Erosion of safe basins in a nonlinear oscillator under bounded noise excitation. J. Sound Vib. 313, 46–56 (2008) · doi:10.1016/j.jsv.2007.11.046
[13] Kopell, N., Washburn, R.B.: Chaotic motions in the two-degree-of-freedom swing equations. IEEE Trans. Circuits Syst. 29, 738–746 (1982) · Zbl 0506.93029 · doi:10.1109/TCS.1982.1085094
[14] Venkatasubramanian, V., Ji, W.: Coexistence of four different attractors in a fundamental powersystem model. IEEE Trans. Circuits Syst. I 46, 405–409 (1999) · doi:10.1109/81.751316
[15] Carreras, B.A., Lynch, V.E., Dobson, I.: Critical points and transitions in an electric power transmission model for cascading failure blackouts. Chaos 12, 985–994 (2002) · Zbl 1080.82579 · doi:10.1063/1.1505810
[16] Nayfeh, M.A., Hamdan, A.M.A., Nayfeh, A.H.: Chaos and instability in a power system: subharmonic-resonant case. Nonlinear Dyn. 2, 53–72 (1991) · doi:10.1007/BF00045055
[17] Dhamala, M., Lai, Y.-C.: Controlling transient chaos in deterministic flows with applications to electrical power systems and ecology. Phys. Rev. E 59, 1646–1655 (1999) · doi:10.1103/PhysRevE.59.1646
[18] Marcos, S.H.C., Lopes, S.R., Viana, R.L.: Boundary crises, fractal basin boundaries and electric power collapses. Chaos Solitons Fractals 15, 417–424 (2003) · doi:10.1016/S0960-0779(02)00108-X
[19] Zhang, Q., Wang, B.H., Yang, C.W.: Fractal erosion of safe basins in power system and its control. Power Syst. Technol. 29, 63–67 (2005) (in Chinese)
[20] Lu, Q., Sun, Y.Z.: Nonlinear Control of Power System. China Science Press, Beijing (1993)
[21] Chen, X., Zhang, W., Zhang, W.: Chaotic and subharmonic oscillations of a nonlinear power system. IEEE Trans. Circuits Syst. II 52, 811–815 (2005) · doi:10.1109/TCSII.2005.853512
[22] Wei, D.Q., Luo, X.S.: Passivity-based adaptive control of chaotic oscillations in power system. Chaos Solitons Fractals 31, 665–671 (2007) · Zbl 1143.93339 · doi:10.1016/j.chaos.2005.10.097
[23] Lin, H., Yim, S.C.S.: Analysis of a nonlinear system exhibiting chaotic, noisy chaotic, and random behaviors. ASME J. Appl. Mech. 63, 509–516 (1996) · Zbl 0875.70115 · doi:10.1115/1.2788897
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.