This paper presents a mathematical model of hydro power units, especially the governor system model for different operating conditions, based on the basic version of the software TOPSYS. The mathematical model consists of eight turbine equations, one generator equation, and one governor equation, which are solved for ten unknown variables. The generator and governor equations, which are different under various operating conditions, are presented and discussed in detail. All the essential non-linear factors in the governor system (dead-zone, saturation, rate limiting, and backlash) are also considered. Case studies are conducted based on one Swedish hydro power plant (HPP) and three Chinese plants. The simulation and on-site measurements are compared for start-up, no-load operation, normal operation, and load rejection in different control modes (frequency, opening, and power feedback). The main error in each simulation is also discussed in detail. As a result, the model application is proved trustworthy for simulating different physical quantities of the unit (e.g., guide vane opening, active power, rotation speed, and pressures at volute and draft tube). The model has already been applied effectively in consultant analyses and scientific studies.

During the transitional processes of load rejection in a pumped-storage station, the S-shaped characteristics of the pump-turbines can result in relatively large water-hammer and pulsating pressures. These pressures and the high runaway speed during transient processes may directly damage the penstocks and shorten the life of the turbine. In this study, different guide-vane closing schemes for reducing the maximum transient pressures, including the water-hammer and pulsating pressures, and runaway speed were investigated, and the principles for improving the closing schemes were theoretically analyzed based on the transient characteristics in the S-shaped region. First, an analytical expression for the rate of change of relative water head during the transitional processes was deduced based on a simplified mathematical model. It reveals the relationship between the slopes of the trajectory at the pump-turbine operating points (defined as trajectory slopes) and the rigid water-column pressure, which approximates the water-hammer pressure considering compressibility. Then, based on the characteristics of the rigid water-column pressure during the transient process and the effects of guide-vane closure on the trajectory slopes, the selection method for a two-phase guide-vane closing scheme was proposed. The method included the technique for choosing the coordinates of the turning point and the closing speed of the guide vane. Furthermore, the pulsating pressures of pump-turbines were discussed under different working regions and guide-vane openings (GVOs). Considering the characteristics of the pulsating pressures and the runaway speed during the transient processes, the advantage of three-phase valve-closing schemes in controlling the pulsating pressures and the runaway speed was clarified. Finally, a series of model tests were conducted on a pumped-storage station model and the measured data fully validated the correctness of our analyses in this work.

Pump-turbine characteristics greatly affect the operational stability of pumped-storage plants. In particular, the S-shaped region of the characteristic curves leads to severe instability during runaway conditions with servomotor failure. Thus, this paper aims to investigate the runaway stability criterion by considering all of the important effects in the hydromechanical system. The criterion also helps to judge the S-characteristics of pump-turbines and can provide a guide for plant design and turbine optimization. First, the pump-turbine characteristic curves are locally linearized to obtain formulae for the relative changes of discharge and torque, which depend on the relative changes of rotational speed and water head. Control theory is then applied to analyze the high-order system, by importing the transfer function of the conduits in the elastic mode. Two different kinds of oscillation are found, associated with water inertia and elasticity, based on the established theoretical mathematical model. New stability criteria for the inertia wave in both rigid and elastic modes are developed and compared. The comparison reveals the effect of the water elasticity on runaway instability, which has often been neglected in the previous work. Other effects, such as friction loss and the timescales of water flow and machinery, are also discussed. Furthermore, the elastic wave, which often has a higher frequency than the inertia wave, is also studied. The stability criterion is deduced with analyses of its effects. Based on the stability criteria for the inertia wave and elastic wave, the unstable regions for two waves of the S-shaped curves are plotted. The results are applied to explain the development from inertia wave to elastic wave during transient behavior at runaway conditions. Model tests of runaway conditions were conducted on a model pumped storage station and the experimental data show good agreement with the theoretical analyses regarding the instability of the inertia wave. Further analyses and validations are made based on transient simulations. The simulation software topsys, which uses the method of characteristics (MOC) and a unit boundary represented by a spatial pump-turbine characteristic surface, was applied to analyze the elastic wave. This also supports the conclusions of the theoretical research.