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.

The pressure fluctuations and runner loads on a pump-turbine runner during runaway process are very violent and the corresponding flow evolution is complicated. To study these phenomena and their correlations in depth, the runaway processes of a model pump-turbine at four guide vane openings (GVOs) were simulated by three-dimensional computational fluid dynamics (3D-CFD). The results show that the flow structures around runner inlet have regular development and transition patterns—the reverse flow occurs when the trajectory moves to the turbine-brake region and the main reverse velocity shifts locations among the hub side, the shroud side and the midspan as the trajectory comes forward and backward in the S-shape region. The locally distributed reverse flow vortex structures (RFVS) enhance the local rotor–stator interaction (RSI) and make the pressure fluctuations in vaneless space at the corresponding section stronger than at the rest sections along the spanwise direction. The transitions of RFVS, turning from the hub side to midspan, facilitate the inception and development of rotating stall, which propagates at approximately 45–72% of the runner rotation frequency. The evolving rotating stall induces asymmetrical pressure distribution on the runner blade, resulting in intensive fluctuations of runner torque and radial force. During the runaway process, the changing characteristics of the reactive axial force are dominated by the change rate of flow discharge, and the amplitude of low frequency component of axial force is in proportion to the amplitude of discharge change rate.

Understanding the formation mechanism of the S-shaped characteristics (SSCs) and the relationship between flow structures and the runaway instability (RI) is the prerequisite for optimizing runner design to promote operational reliability and flexibility. In this study, a new turbine equation is derived to reveal the prime cause of the SSCs, and the influence of geometric parameters on the SSCs is analyzed. Moreover, the flow patterns in three model turbines of different specific-speeds are simulated by unsteady Computational fluid dynamics (CFD), and the correlation between inverse flow vortex structures (IFVSs) and the RI in the SSCs region is identified. Theoretical analysis shows that the turbine equation can theoretically predict the change trend of the first quadrant SSCs curves of the pump-turbines; the flow losses caused by small blade inlet angle, instead of the diameter ratio, are the primary cause of the SSCs. The numerical simulation results reveal that the IFVSs at the hub side of the runner inlet are the origin of the RI; when operating points are far away from the best efficiency point (BEP), the IFVS locations change regularly. For large guide vane openings (GVOs), the IFVSs first incept at the shroud side, and then translate to the hub side, and further back to the midspan, when the discharge decreases. The inception points (IPs) of the SSCs correspond to the onset of the IFVSs at the hub side, which are in advance of the zero-torque operating points (ZTOPs); therefore, the ZTOPs are located in the positive slope region, leading to RI. For small GVOs, however, the IFVSs only locate at the midspan; the IPs of the SSCs, having no definite correlation with the IFVSs, are coincided with or are below the ZTOPs, because the ZTOPs are in the negative slope region and RI disappears. It is also found that the IPs of SSCs are the turning points of the predominant states between the turbine effect and pump effect. These results are valuable for design and optimization of pump-turbine runners.

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.