There is a region of pump-turbine operation, often called the S-shaped region, in which one unit rotational speed corresponds to three unit flows or torques. In this paper, the dynamic model of the pump-turbine in S-shaped regions is established by introducing the nonlinear piecewise function of relative parameters. Then, the global bifurcation diagrams of the pump-turbine are presented to analyze its dynamic characteristics in the S-shaped regions. Meanwhile, a stability criterion of runaway point is given based on the established theoretical model. The numerical experiments are conducted on the model and the results are in good agreement with the theoretical analysis. Furthermore, the effects of the characteristic curve slopes on the stability of the pump-turbine are studied by an innovative use of the three-dimensional bifurcation diagrams. Finally, the factors influencing the runaway stability of pump-turbines are also discussed, based on the dynamic analysis.
Zhang, H., Chen, D., Wu, C., Wang, X., Lee, J. M., & Jung, K. H. (2017). Dynamic modeling and dynamical analysis of pump-turbines in S-shaped regions during runaway operation. Energy Conversion and Management, 138, 375–382. https://doi.org/10.1016/j.enconman.2017.01.053