Research On Bifurcation And Chaos Of Hydro Turbine Governing System With Fractional Order

The bifurcation and chaos theories of nonlinear dynamical systems are applied to different fractional order and integer order models of hydraulic turbine governing systems in this dissertation respectively. The nonlinear traits of the above systems with different orders are systematically studied and some novel and meaningful results are obtained.We first evaluate the feasibility of combining fractional calculus with turbine regulation system,then the continuous-time mathematical models of the hydraulic turbine governing systems are employed with all the essential parts included, i.e. inearized model, nonliear model, linearized model containing some typical nonlinear parts, nonlinear model with inelastic and elastic water columns. Based onthe abtained models above, the bifurcation and chaos theories of nonlinear systems are used to calculated and analyzed the nonlinear dynamical behaviors in hydraulic-mechanical-electro systems. Moreover, the bifurcation diagrams are employed to deeply investigate the nonlinear traits in supercritical Hopf bifurcation situation. The necessary conditions for the existence of double-Hopf bifurcation judging in four and six-dimensional nonlinear systems are investigated also. The method is simple and quite useful to deal with the double-Hopf bifurcation problems in practise. It has been approved for the four to six dimensional nonlinear models of hydraulic turbine governing systems that the double-Hopf bifurcation phenomenon are existed in both elastic and inelastie water columns models that considering nonlinear dynamics in hydraulic turbine and penstock. The critical values of double-Hopf bifurcation are obtained by numerical calculations and correctness of theoretical analysis results can be validated by simulations. Also, the nonlinear dynamical theories are applied for nonlinear models analysis of hydraulic turbine governing systems with second-order dynamical model in generator, chaos and its abundant nonlinear phenomena are verified and showed from bifurcation diagrams, Lyapunov exponents, chaotic attractors, sensitivety and dependence on the initial value and Poincare maps perspective. It is found that with the control parameters Kd increasing, the system will experience a series of complex motion including stable-Hopf bifurcation-unstable-doubling bifurcation-Chaos-Hyperchaotic series of complex motionThe mathematical control strategy of the system is deduced based on the proportional and integral sliding mode control chosen in the paper. By selecting appropriate system gain coefficient,the eliminate chattering problem occurred in control system is well solve.Simulations show that the chaotic vibration is eliminated and the system could be controlled to any fixed point and any periodic orbit strictly and smoothly with short transition time by means of sliding mode method. Simulation results verify the feasibility of the method.It is proved that the fractional order hydraulic turbine governing system has better control performance and greater stability than the traditional integer order model under the same parameters circumstance.The stability domain solution method for fractional order hydraulic turbine governing systems is proposed based on the proof of fractional order system stable condition. It is verified the stable conditions method herein is consistent with that of integer order system, proving that stable condition method of fractional order sytem can be applied equally to an integer order systems also. An example rigid and hydraulic turbine governing system is used to analyze the changes of system stability region with the changing order, finding that in certain circumstances, when the order is reduced, the stable region would gradually expand and the bifurcation point moves slowly to the right, system stability gets enhanced; By comparison, PID system stability region and the surface with the present method is completely consistent with the results from the third chapter integer order in calculation method when the fractional order is 1; Dynamic behaviors on both sides of bifurcation point of the dynamic system with fractional order are calculated; Taking parameter Twas an example to analyze and compare the effects of order system parameters on stability region of the systemchaos are also verified and showed from bifurcation diagrams, Lyapunov exponents, chaotic attractors, sensitivety and dependence on the initial value and Poincare maps perspective in fractional order system; What's more, system motion with the changing order are analyzed, finding that in certain PID parameters, the motion state of the system is in a substantially symmetrical shape in the order axis; Thus, the required operational status can be obtained by setting a reasonable system order. It is found that in certain scope of PID parameters and fractional order, the system will experience a series of complex motion including hyperchaotic-chaos-doubling bifurcation-unstable-Hopf bifurcation-stable-Hopf bifurcation-unstable-doubling bifurcation-chaos-hyperchaotic with the order changes.Therefore, to find a suitable model order of the turbine regulation system to describe the physical characteristics more accurately will be a key work in the future.An active sliding mode control method that combining the sliding mode control theory and active control theory to the fractional-order chaotic systems is proposed in this paper. An active sliding controller gain matrix can be obtained by setting reasonable parameters with the pole placement method. Simulation results verifythe effectiveness of the proposed method. A fractional order PID (FOPID) controller with improved firefly algorithm is employed for hydraulic turbine governing systems simultaneous optimization. Simulation results show that designed controller based on fractional principles has strong robustnessThis article combines fractional theory with hydraulic turbine governing systems, stability conditions of the system are studied under a new perspective, and analysis results verifythe feasibilities and advantages of fractional order hydraulic turbine governing systems.This work, therefore, presents a new theoretical analysis approach for the nonlinear dynamical phenomena and stability researches in the hydraulic-mechanical-electro systems.