Fractional-order Dynamic Model Of A Hydropower Generation System And Its Stability

All systems need a fractional order differential equation to description,and the dynamic behaviors of systems for prediction should not only know the present operating states(t=t0)but also the past history(t<t0).For a hydropower station consisting of penstock,hydro-turbine generator units and servicing facilities,its stability partly depends on the performance of subsystems.From the viewpoint of time,the stability of the hydropower station can be divided into the electromagnetic oscillation with high speed,the governing oscillation process with middle speed and the hydraulic oscillation process with slow speed.In other words,it is can be considered that the three parts of the electromagnetic oscillation,the mechanical action and the hydraulic process cannot be uniquely determined on large time scale.Therefore,some novel fractional order models based on the shafting model,the nonlinear limiting model,the grid model,the elastic model,the one penstock model and the common penstock model are proposed.The dynamic stabilities of the above models are analyzed in detail.The contents and the conclusions of this paper include:(1)In order to study the stability of a hydro-turbine-generator unit in further depth,we establish a novel nonlinear fractional-order mathematical model considering a fractional-order damping force,a fractional-order oil-film force,an asymmetric magnetic pull and a hydraulic-asymmetric force.Furthermore,the nonlinear dynamics of the above fractional-order hydro-turbine-generator unit system with six typical fractional orders are studied in detail.Based on these,we analyze the effect of the fractional order on bifurcation points,the orbit of centroid of the rotor,the power and the frequency of the rotor.Fortunately,some variable laws are found from numerical simulation results.(2)From the viewpoint of energy flow,the fractional models of the hydro-turbine governing system are established respectively considering the hydraulic subsystem,the mechanical subsystem and the electromagnetic subsystem.Comparing the simulation results from the above models,we obtain the change laws of bifurcation points,the chaotic ranges with the increasing value of fractional orders the different moving speed of bifurcation points for different fractional order subsystems,which reflect the three parts of the electromagnetic oscillation,the mechanical action and the hydraulic process cannot be uniquely determined on large time scale.Then,we proposed a universal solution about the relationship of two parameters in higher-degree equations according to the stability theorem of a fractional order system.According to the above theorem,we presented a variable law of stable regions of the fractional-order hydro-turbine governing system and analyzed the effect of various degree of elastic water hammer on the stable regions of the PID parameters with the increase of fractional order.The nonlinear dynamic behaviors of the system are also studied in detail.(3)This paper focuses on the Hamiltonian mathematical modeling and dynamic characteristics of multi-hydro-turbine governing systems with sharing common penstock under the excitation of stochastic and shock load.Considering the hydraulic-coupling problem in the common penstock,we propose a universal dynamic mathematical model of the multi-hydro-turbine governing system.Then,the proposed model is fitted into the theoretical framework of the generalized Hamiltonian system,utilizing the method of orthogonal decomposition.The dissipation energy,the produced energy and the energy supplied from the external sources are derived from the Hamiltonian model and compared with the physical energy flow.Furthermore,numerical experiments based on a real hydropower station demonstrate that the Hamiltonian function can describe accurately the energy variation of the hydro-turbine system in the transient and stable processes.Moreover,in order to deal with the randomness and mutability of the electrical load,we introduce a Gaussian function and a jump function to the control signal of the PID controller to analyze the dynamic characteristics of system dynamic parameters.In addition,the level of shock load is discussed before the system loses its stability.