Modeling and stability analysis of a fractional-order Francis hydro-turbine governing system

In this paper, a fractional order mathematical model of a hydro-turbine governing system is presented to analyze the dynamic stability of the hydro-turbine governing system in the process of operation. The fractional order hydro-turbine governing system is composed of a hydro-turbine and penstock system, a generator system and a hydraulic servo system. As a pioneering work, 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. Based on 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 parameters kd and kp with the increase of fractional order α. The nonlinear dynamic behaviors of the system are also studied in detail. Finally, all of these results supply some basic theories for the running of a hydropower plant.