**1. Introduction**

With the substantial increase of electricity consumption and the rapid development of green sustainable energies, pumped-storage power undertakes the functions of peak load regulation, valley filling, frequency modulation, phase modulation, and emergency standby in the power grids [1,2]. Its match-up with nuclear power and complement with wind and solar powers make it an indispensable tool to ensure safety, stability, and efficiency of clean energies [3–5]. To undertake these important functions, the stability and safety of pumped-storage power systems are essential. However, some stability problems in operating pumped-storage power stations, such as violent vibration of pump-turbine units [6], grid connection failure [7,8], runner lifting-up [9], and rotor-stator crashing [10], were frequently reported. These problems were generally attributed to the frequent conversions of operating conditions, especially when the working points pass through the so-called S- and hump-shaped characteristics regions, in which intense flow and pressure fluctuations occur. To know the mechanism, solve the stability problems, and predict working conditions, many studies on the transient processes of pump-turbine generator units were conducted in recent years [11].

Among many transient processes, the runaway process is the most dangerous one. Even if this scenario, it is rarely seen in practical operation, predicting the risk in the design phase is always required. The runaway process happens if the generator is cut from the power grid but the guide-vanes fail to close. Without retarding torque, the runner will be driven only by the unceasing water power, and the unit will be accelerated to the runaway speed. During this process, the working point slides rapidly through the S-shaped characteristic region that is comprised of the high-speed turbine, turbine braking and reverse pump modes, and violent vibrations in the unit happen due to quick flow pattern transitions and strong pressure pulsations [1]. Therefore, it is very important to ensure the safety and stability of the unit by analyzing the laws of flow pattern transitions and pressure pulsation changes, and revealing the interrelations of these key factors.

The existing studies about the runaway instability of pump-turbines mainly focused on unsteady flow patterns and pressure pulsations in the runner and vaneless space [12,13]. Two main situations [11,12], working at a runaway point and running away from a turbine working point, were both investigated. They concluded that strong backflows and vortices in the runner and the vaneless space lead to large pressure pulsations, channel blockage, discharge decrease, and pressure increase [12]. As for the simulations about static working at a runaway point, Gentner et al. [14] found toroid-like vortex structures around the vaneless space, and claimed that the secondary vortex in each runner channel can cause negative head gradient and pressure rise. Wang et al. [15] captured the obviously detached vortexes on the pressure sides of blades near the crown and pointed out that they may be the very reason for huge pressure fluctuations. Widmer et al. [16] showed the flow separation, recirculation, and vortex formation in every runner channels of a pump-turbine operating at the speed-no-load conition, and observed the obvious backflows and pressure fluctuations. Hasmatuchi et al. [17] investigated the flow distribution near the runaway point through experiments and found that the low-frequency pressure components can be captured in the spiral-casing and the guide-vanes channel. Jacquet et al. [18] pointed out that the position of backflows at the runner inlet depended on the operating point, and the accompanying pressure fluctuations can reach the maximum at the speed-no-load condition.

As for transient process studies, Trivedi et al. [19–22] concluded that the highest amplitudes of pressure fluctuations in pump-turbine were under the running away condition, according to the measurement of pressure fluctuations in the speed-no-load, running away, total load rejection, start-up, and shut-down conditions. Yin et al. [23] showed that the vortex formation at the runner inlet severely blocks the runner passages periodically, inducing torque and rotational speed fluctuations. Zhang et al. [24] also simulated the runaway process by computational fluid dynamics (CFD) and found that the successive features of transient flow patterns may induce pressure differences between the similar dynamic operating points in different moving directions. Xia et al. [1] conducted simulations of runaway processes of a model pump-turbine with different guide-vane openings (GVOs), and found that the backflows at the runner inlet can lead to quite different pressure fluctuations. Other research investigating the runaway instability by specifying discharge oscillating boundary condition at the turbine inlet or draft-tube outlet were also conducted. For example, Widmer et al. [16] decreased the discharge at the boundary starting from the runaway point, and found that the pressure pulsations can generate abnormal low-frequency signals with the number of stalled channels increased, which was similar to those in the runaway process.

The research discussed above shows that whether at the runaway point or during runaway process, flow blockages and severe pressure fluctuations are strong in the runner and vaneless space, which are the common features in pump-turbines. However, in much reported research, the problems encountered by different pump-turbines are mostly different. For example, a runner lifting-up happened in Tianhuangping power station during a load increase process [9], many grid connecting failures occurred in Baoquan power station under low head conditions [8], and a rotor-stator collision happened in Huizhou power station during a load rejection process [25]. Besides these accidents, there are still many other accidents that need to be paid attention to. Although these accidents are related to many factors, it is undeniable that the characteristics of the pump-turbine itself have a great influence on them. Most obviously, different pump-turbines have different S-characteristics because of their rated output, head, discharge, rotational speed, along with their runner shapes being different. Therefore, the flow patterns and pressure pulsations may not be similar in local and detailed perspectives, which may be related to the different problems mentioned above. For example, the conclusions in Hasmatuchi [17] and Jacquet [18] are different. In Hasmatuchi's paper, the low-frequency component will further increase in amplitude as the zero-discharge condition is approached, while those in Jacquet's paper reach at the maximum at the no-load conditions. In addition, Zhou et al. [26] optimized the blade inlet and showed the different developing trends of flow patterns and pressure fluctuations of two turbines during the runaway processes, though other geometry features of turbines were kept unchanged.

Therefore, we should not only focus on the common phenomena, but also the differences in different pump-turbines, in order to better understand the mechanism and solve the problems. As a common convention, the characteristics of pump-turbines are always labelled by their specific speeds. However, no research shows whether runaway process characteristics are related to the specific speed. These characteristics include the attenuation of runaway, the transition of flow patterns, the fluctuations of pressure pulsations, and runner forces. In order to answer these questions, we selected four prototype pump-turbines with different water heads, and simulated their runaway transient processes from the turbine mode. The evolutions of pressure pulsations and flow patterns were analyzed, their similarities and differences were discussed, and the mechanism was revealed. The paper will be arranged as follows: the Section 2 describes the basic simulation model and parameters; the Section 3 shows the resulting histories of macro parameters, and the evolutions of flow structures and pressure pulsations, along with their relations with specific speeds; the Section 4 explains the influences of runner shapes for the differences in the evolutions of flow structures and pressure pulsations; and conclusions are drawn in the Section 5.

#### **2. Three-Dimensional CFD Setups**

Software for simulation: Three-dimensional (3D) CFD simulations were carried out by using commercial software ANSYS FLUENT 17.0 (ANSYS, Canonsburg, PA, USA).

Computational domain: four pump-turbines with different specific speeds were selected. Because of their main parameters, such as head, discharge, output, and layout out of water conveyance systems are different, it is difficult to ensure that all the settings in the simulations are the same, which is also unrealistic. Therefore, in order to fully reflect the characteristics of the pump-turbines during the transient process, the actual water conveyance systems were removed in the simulations to eliminate the impact of flow inertia in water conveyance systems [1]. This removal will affect the variation period and maximum value of macro parameters due to the flow inertia in pipelines, but we mainly focused on the evolutions of flow patterns and pressure pulsations, which are more affected by the pump-turbine unit. In addition, two extended tubes were added to the inlets of spiral-casings and the outlets of draft-tubes for setting boundary conditions at the locations with smooth flow patterns. Also, a conventional hydraulic turbine was chosen to compare with the above four pump-turbines. The 3D computational domains and monitoring points of the five turbines are shown in Figure 1, and the main parameters are listed in Table 1. The specific speed is defined by *n*<sup>s</sup> = *n*<sup>r</sup> √ *N*r/*H*r1.25, in which *n*r, *N*r, and *H*r are the rated rotational speed, output, and head, respectively. The flow patterns have a certain regularity in *n*11-*Q*<sup>11</sup> plane under large guide vane opening conditions, especially at the runner inlets (in one pump-turbine) [1]. Therefore, the runaway processes of the four pump-turbines are all started near their corresponding rated turbine working conditions, while that of the conventional hydraulic turbine is started from a large guide vane opening condition, in which the runaway characteristics are similar to those in the rated one.

**Figure 1.** Computational domains of the pump-turbines (PT) and conventional turbine (CT), the schematic of monitoring points, and mesh information.


**Table 1.** Main parameters of the four pump-turbines and a turbine.

Mesh Generation: The upstream and downstream extended tubes, spiral-casings, runners, and draft-tubes were discretized by hexahedral structure grids, while the vane regions were discretized by wedge grids. Also, the areas near the blades and guide-vanes were locally refined. Grid refinement evaluations were performed for each pump-turbine and we found that when the grid number is more than 5.0 million, the relative differences in resulting macro parameters under steady conditions are negligible. Therefore, the cell numbers of the five turbines are 5.42 million, 5.58 million, 5.76 million, 5.97 million, and 5.54 million, respectively.

Numerical Scheme: After many comparisons, considering the calculation time and accuracy at the same time, we selected the timesteps for the five turbines as 0.00125, 0.001, 0.001, 0.001, and 0.00166667 s, corresponding to the times needed for the runner to rotate 1.5, 1.5, 3.0, 3.0, and 1.5 degrees, respectively. The SST-based scale-adaptive simulation model (SAS-SST) turbulence model [1] was adopted, and all the convergence criteria of residuals at each timestep were set to 1.0 <sup>×</sup> 10−4, including continuity, x-velocity, y-velocity, z-velocity, k, and omega. For both steady and unsteady simulations, the SIMPLEC algorithm was chosen to achieve the coupling solution of the velocity and pressure equations [1].

Boundary Conditions: The total pressure was defined at the inlet of the extended pipe of the spiral-casing, and the static pressure was defined at the outlet of the extended pipe of the draft-tube. The remaining solid walls were imposed with the no-slip wall condition.
