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Journal of Control Science and Engineering
Volume 2017, Article ID 2796090, 14 pages
Research Article

Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

1School of Information Engineering, Gansu Forestry Technological College, Tianshui, Gansu 741020, China
2School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China
3School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China
4School of Applied Mathematics, Xinjiang University of Finance & Economics, Wulumuqi, Xinjiang 830000, China

Correspondence should be addressed to Jiangang Zhang; moc.621@6775177gjgnahz

Received 11 February 2017; Revised 3 July 2017; Accepted 26 July 2017; Published 12 September 2017

Academic Editor: Yongji Wang

Copyright © 2017 Hongwei Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.