Nonlinear modeling and dynamic control of hydro-turbine governing system with upstream surge tank and sloping ceiling tailrace tunnel

Using the Hopf bifurcation theory, the nonlinear dynamic characteristics of hydro-turbine governing system of hydropower station with upstream surge tank and sloping ceiling tailrace tunnel are studied. Firstly, a novel and rational nonlinear mathematical model of the hydro-turbine governing system is proposed. This model contains the nonlinear dynamic equation of pipeline system, which can accurately describe the motion characteristics of the interface of free surface-pressurized flow in sloping ceiling tailrace tunnel. According to the nonlinear mathematical model, the existence and direction of Hopf bifurcation of the nonlinear dynamic system are analyzed. Furthermore, the algebraic criterion of the occurrence of Hopf bifurcation is derived. Then the stability domain and bifurcation diagram of hydro-turbine governing system are drawn by the algebraic criterion. Finally, the dynamic characteristics under different state parameters are investigated and the dynamic control method is proposed. The results indicate that: For the example in this paper, the Hopf bifurcation of hydro-turbine governing system is supercritical. The phase space trajectories of characteristic variables stabilize at the equilibrium points and stable limit cycles when the system state parameter point locates in the stable domain and unstable domain, respectively. The dynamic response processes of the characteristic variables of the hydro-turbine governing system under load disturbance show an obvious feature of wave superposition. To make the dynamic response process of the hydro-turbine governing system with upstream surge tank and sloping ceiling tailrace tunnel more stable and attenuate faster, the governor parameters $$(K_{p}, K_{i})$$(Kp,Ki) should locate in the stable domain and keep away from the bifurcation point as much as possible.

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