Research Of Time-delay Fractional Calculus And Its Application On The Hydro-turbine Governing System

Hydro-turbine governing system,which involves hydrodynamics,mechanical dynamics and electromagnetic mechanics,is a nonlinear control system.And it consists of the hydroelectric generating set,the penstock system,the hydraulic servo governor system and the power grid load.In practice,time delay exists in the hydraulic servo system,the transfer coefficients of the hydro-turbine always alter once the operating condition changes,and the power grid load has random disturbance.These phenomena not only threaten the stable operation of the governing system,but also influence the power quality of the electrical system.Hence,there is a tremendous need to introduce these unstable factors into the modelling of the hydro-turbine governing system and further investigate its stable theorem.In this paper,we first establish a more appropriate mathematic model for the hydro-turbine governing system by introducing the time delay into the hydraulic servo system and the fractional-order calculus into the penstock system.Then,we investigate the system stability.Subsequently,we study the existence and uniqueness theorem of the system solution,the finite-time stability and the asymptotic stability for a class of nonlinear fractional-order system with the discrete time delay.Finally,the main contents and conclusions of this paper are listed below.(1)Considering that studying the existence and uniqueness of the solution to the system is the basis of its stability investigation,we reduce the mathematical model of the hydro-turbine governing system to a class of nonlinear fractional-order system with time delay,and investigate the existence and uniqueness of the system solution.More specifically,using related properties of the fractional derivative and the generalized Gronwall inequality,we give the sufficient and necessary condition to ensure the existence of the solution,and present a sufficient condition to guarantee the uniqueness of the solution.Also,we get the estimate value of the solution.(2)Employing the Laplace transform,the Mittag-Leffler function and its properties,the sufficient conditions are given to guarantee the finite-time stability of the system.That is to say,no matter what the system initial state is,if the system meets the sufficient condition of the finite-time stability,the system always tends to be stable during the finite time interval.Subsequently,we carry out the numerical simulations to demonstrate the effectiveness of the proposed results.(3)We investigate the asymptotic stability of a class of nonlinear fractional-order system with the discrete time delay,and give the sufficient conditions.Also,comparing the proposed asymptotic stability theorem with the existing stability theory,we present the superiority of the proposed asymptotic stability theorem,and then two numerical examples displaying chaotic nonlinear behaviors are given to verify the effectiveness of the theoretical results.(4)We choose the Francis hydro-turbine governing system with complex penstocks as this section research object.Specifically,we introduce the fractional-order calculus into the modelling of the complex penstock system,and then establish a fractional-order nonlinear mathematic model for the governing system.Next,using the the stability theory of fractional-order nonlinear system,a basic law of the bifurcation points of the system is discovered along with the change of the fractional order,and the stable region of the system is also investigated in detail.Finally,the nonlinear dynamical behaviors of the system are identified and studied exhaustively via bifurcation diagrams,time waveforms,phase orbits,Poincare maps,power spectrums and spectrograms.(5)In view of the fact that the dead zone of the main control valve causes the servomotor to remain stationary and the displacement response of the servomotor piston has a lag in time,we introduce the time delay into the modelling of the hydraulic servo system and then establish a fractional-order time-delay nonlinear mathematical model for the hydro-turbine governing system.Further,using the modified Admas-Bashforth-Moulton algorithm and the principle of statistical physics,we carry out the numerical experiments based on the MATLAB,and investigate the effects of the fractional order and the time delay on the stable region of the system.Although this paper gives the finite-time stability theorem and the asymptotic stability theorem,the two theorems still cannot be applied on the hydro-turbine governing system,and need to be studied further.