Hydraulic Characterization of Variable-Speed Pump Turbine under Typical Pumping Modes

: The pump turbine is a crucial component of pumped storage hydropower plants. When operated at a constant speed, it does not respond well to variations in the grid frequency. To improve the hydraulic efﬁciency of pumped storage units, variable-speed units have been introduced. However, the mechanism of variable-speed pump turbines has not been extensively studied numerically. In this study, the ﬂow characteristics of a variable-speed pump turbine were computed under two typical pumping modes, the maximum head and minimum ﬂow rate condition, as well as the minimum head and maximum ﬂow rate condition. The computed results aligned with experimental results, and the changing trends of hydraulic thrust under these two pumping modes were discussed. The error for the Hmax, Qmin condition was 1.3%, and the error for the Hmin, Qmax condition was − 1.9%. These error values fell within a reasonable range. The research ﬁndings indicate that in the H max , Q min condition, the ﬂow within the ﬂow passage exhibited higher velocity, which was 84.87 m/s, increased ﬂow turbulence, larger pressure ﬂuctuations, and poorer unit stability. On the other hand, in the H min , Q max condition, both the axial hydraulic thrust and radial forces were greater, and there were sudden changes in the extreme values of pressure ﬂuctuations over a certain period of time. It is recommended to avoid operating the variable-speed pump turbine under these two conditions during pumping operations.


Introduction
Pumped storage technology is an effective method to mitigate the impact of renewable energy generation on grid stability.It provides peak shaving, valley filling, frequency regulation, phase modulation, and emergency backup functions.It is currently one of the most mature, cost-effective, and widely installed energy storage technologies in the market [1][2][3].The hydraulic pump turbine is the core component of pumped storage power plants [4].However, most of the pumped storage units in the country are constant-speed units, which have a poor response to grid frequency [5].Variable-speed pump turbine units can use variable-speed devices to provide load frequency control [6].Furthermore, due to the variable-speed capability, the input frequency during the pumping operation becomes continuously adjustable; this enhances the safe and stable operation of the grid and provides a higher level of reliability [7,8].
Improving the efficiency of pumped storage units can be effectively achieved through variable-speed operation [9].However, there is limited research on variable-speed pump turbines.To gain a better understanding of this issue, it would be helpful to refer to existing studies that have focused on improving the overall efficiency of the units through variable-speed operation [10][11][12][13].Presas et al. [10] studied the hydraulic-saving potential of mixed-flow turbines during variable-speed operation.They found that compared to fixed-speed units, variable-speed units can achieve hydraulic savings of over 2% when the unit operates at partial load and experiences lower head at maximum power.Juan et al. [11] calculated the additional energy that can be extracted by using a variable speed in an irrigation reservoir.They estimated an energy increase of 20% when a variablespeed operation mode is adopted in reservoirs where irrigation takes precedence over hydropower.Delgado et al. [12] conducted experimental research on the variable-speed characteristics of pumps operating as turbines.The test results showed that the specific energy range discharged by the pump in variable-speed mode expanded with improved efficiency, and no unstable region (S-region) close to the runaway condition was observed.Damdoum et al. [13] used control algorithms to study the performance of hydroelectric units with reaction turbines during variable-speed operation.They found that variablespeed operation could improve the efficiency of the turbine by reducing the fluid dynamic impact on the generator and expanding the operating power range.From these studies, it is evident that variable-speed operation can indeed enhance the efficiency of the units during operation.This conclusion applies equally to pump turbines.However, the operating conditions of pump turbines are much more complex compared to single-mode turbines or pumps.When pump turbines operate in variable-speed mode, there are even more variations in operating conditions [14,15].Therefore, it is crucial to study whether variablespeed pump turbines operate under appropriate conditions to ensure the efficiency, safety, and stability of the units.
The stability of operation is indeed a key indicator in evaluating pump turbines [16][17][18], and pressure pulsation is a major factor associated with the stability of pump turbine operation.Various studies have been conducted to investigate the factors influencing pressure pulsation.Svarstadand and Nielsen [19] performed laboratory and field measurements of reversible pump turbines during the transition process and concluded that the pressure pulsation changes during the rapid switch from pump mode to turbine mode had a small amplitude but a significant numerical value.Liao et al. [20] conducted research on the low compressibility of hydraulic fluids and simulated both stable and unstable internal flow conditions in turbines.They found significant pressure fluctuations in both the runner region and the draft tube under conditions of small blade openings.Pankaj P. Gohil and R.P. Saini [21] explored the non-steady pressure pulsation in low-head mixed-flow turbines under severe cavitation effects.They discovered that the influence of the tail hydraulic submergence level on the dominant frequency can be neglected under rated and maximum partial load conditions.Based on these studies, it can be concluded that the maximum pressure pulsation occurs in both the runner region and the draft tube.Due to the limitations of conventional fixed-speed units in pump mode [22], in recent years, there has been growing interest among researchers in variable-speed pump turbines.I. Iliev et al. [23] tested the pressure pulsation of reversible pump turbines under variable-speed operation; the findings indicated that compared to the synchronous-speed pump turbine operating at its optimal condition, the hydraulic efficiency was improved by 1%.At the same time, the amplitude of pressure fluctuations was also reduced.Pavesi et al. [24] conducted numerical simulations to investigate the unstable behavior resulting from a decrease in pumping power.They decreased the speed from 100% to 88% and observed that as the speed decreased, the vortex became stronger and the pressure pulsation reduced at lower speeds.Many researchers have indeed found that variable-speed operation can effectively improve the hydraulic performance of pump turbines.Due to the difficulty of pump turbine calculations, many studies have proposed optimization strategies combining fuzzy logic (FL) and genetic algorithms (GA) [25][26][27].In summary, variable-speed operation has been shown to enhance the efficiency and stability of pump turbines [28].However, given the complex operating conditions of pump turbines, it is crucial to study and ensure appropriate operating conditions to maintain the efficiency, safety, and stability of these units.
In this paper, the incompressible Reynolds-averaged Navier-Stokes equations are solved based on the finite volume method in commercial computational fluid dynamics (CFD) software ANSYS CFX 18.0, and the shear stress transport k-ω model is applied.The hydraulic characteristics are analyzed for two operating points: the minimum head with maximum flow rate and the maximum head with minimum flow rate under pump conditions.In contrast to previous works, this study analyzes the hydraulic characteristics of two typical pumping conditions of variable-speed pump turbines, which have been less studied so far, including pressure pulsation and hydraulic thrust analysis.The research findings are of significant importance in explaining the flow mechanisms under extreme operating conditions of variable-speed pump turbines and ensuring the safe and stable operation of the units.
The rest of the paper is organized as follows.Section 2 presents the model used for calculations in this paper, the mesh partitioning, and the numerical simulation method.Section 3 demonstrates the outcomes of the numerical simulation analysis of the internal flow field velocity streamline diagram, runner pressure distribution, pressure pulsation, and hydraulic thrust.In Section 4, the conclusions drawn from the research are presented.

Geometric Model and Numerical Schemes 2.1. Variable-Speed Pump Turbine Model and Parameters
Figure 1 represents a three-dimensional solid model of the entire flow passage components of a variable-speed pump turbine.It includes elements such as the pressure balance pipe, spiral casing, stay vanes, guide vanes, runner, draft tube, crown, and band.The main parameters of the flow passage are provided in Table 1.
Processes 2023, 11, x FOR PEER REVIEW 3 of 17 In this paper, the incompressible Reynolds-averaged Navier-Stokes equations are solved based on the finite volume method in commercial computational fluid dynamics (CFD) software ANSYS CFX 18.0, and the shear stress transport k-ω model is applied.The hydraulic characteristics are analyzed for two operating points: the minimum head with maximum flow rate and the maximum head with minimum flow rate under pump conditions.In contrast to previous works, this study analyzes the hydraulic characteristics of two typical pumping conditions of variable-speed pump turbines, which have been less studied so far, including pressure pulsation and hydraulic thrust analysis.The research findings are of significant importance in explaining the flow mechanisms under extreme operating conditions of variable-speed pump turbines and ensuring the safe and stable operation of the units.
The rest of the paper is organized as follows.Section 2 presents the model used for calculations in this paper, the mesh partitioning, and the numerical simulation method.Section 3 demonstrates the outcomes of the numerical simulation analysis of the internal flow field velocity streamline diagram, runner pressure distribution, pressure pulsation, and hydraulic thrust.In Section 4, the conclusions drawn from the research are presented.

Variable-Speed Pump Turbine Model and Parameters
Figure 1 represents a three-dimensional solid model of the entire flow passage components of a variable-speed pump turbine.It includes elements such as the pressure balance pipe, spiral casing, stay vanes, guide vanes, runner, draft tube, crown, and band.The main parameters of the flow passage are provided in Table 1.

Calculated Operating Conditions
In this study, two operating points were selected for analysis: the maximum head with minimum flow rate (Case 1) and the minimum head with maximum flow rate (Case

Calculated Operating Conditions
In this study, two operating points were selected for analysis: the maximum head with minimum flow rate (Case 1) and the minimum head with maximum flow rate (Case 2) under pump conditions.These operating points are prone to cavitation and pressure fluctuations.The selection of these points on the pump operating Q-H curve can be seen in Figure 2. The guide vane openings for the two operating points are 27 • and 8.4 • , and the corresponding speeds are 447.49rpm and 421.09 rpm, as shown in Table 2.These parameters were used in the calculations and analysis conducted in the study.
Processes 2023, 11, x FOR PEER REVIEW 4 of 17 2) under pump conditions.These operating points are prone to cavitation and pressure fluctuations.The selection of these points on the pump operating Q-H curve can be seen in Figure 2. The guide vane openings for the two operating points are 27° and 8.4°, and the corresponding speeds are 447.49rpm and 421.09 rpm, as shown in Table 2.These parameters were used in the calculations and analysis conducted in the study.

Grid generation
As shown in Figure 3, the efficiency and head are selected as judgment criteria to perform mesh independence verification.A total of five groups of grids, N1, N2, N3, N4, N5, are compared with the known net head of 406.85 m as a benchmark for comparison.It is found that the calculated head of N4 is 411.7 m, with a difference of 4.85 m, and the difference of N3 is 3.35 m.However, the efficiency obtained by the calculation of N4 will be higher compared to that of N3.Although N5 is more efficient, the difference in calculated head is larger, and the number of grids is more, taking into account the cost savings of the calculation and improving the accuracy of the calculation of the comprehensive considerations under the choice of the N4 grid.

Grid Generation
As shown in Figure 3, the efficiency and head are selected as judgment criteria to perform mesh independence verification.A total of five groups of grids, N1, N2, N3, N4, N5, are compared with the known net head of 406.85 m as a benchmark for comparison.It is found that the calculated head of N4 is 411.7 m, with a difference of 4.85 m, and the difference of N3 is 3.35 m.However, the efficiency obtained by the calculation of N4 will be higher compared to that of N3.Although N5 is more efficient, the difference in calculated head is larger, and the number of grids is more, taking into account the cost savings of the calculation and improving the accuracy of the calculation of the comprehensive considerations under the choice of the N4 grid.
2) under pump conditions.These operating points are prone to cavitation and pressure fluctuations.The selection of these points on the pump operating Q-H curve can be seen in Figure 2. The guide vane openings for the two operating points are 27° and 8.4°, and the corresponding speeds are 447.49rpm and 421.09 rpm, as shown in Table 2.These parameters were used in the calculations and analysis conducted in the study.

Grid generation
As shown in Figure 3, the efficiency and head are selected as judgment criteria to perform mesh independence verification.A total of five groups of grids, N1, N2, N3, N4, N5, are compared with the known net head of 406.85 m as a benchmark for comparison.It is found that the calculated head of N4 is 411.7 m, with a difference of 4.85 m, and the difference of N3 is 3.35 m.However, the efficiency obtained by the calculation of N4 will be higher compared to that of N3.Although N5 is more efficient, the difference in calculated head is larger, and the number of grids is more, taking into account the cost savings of the calculation and improving the accuracy of the calculation of the comprehensive considerations under the choice of the N4 grid.The accuracy of numerical simulation results is directly influenced by the appropriateness of grid division.In this study, a grid division method combining hexahedral structured grids and tetrahedral unstructured grids was adopted for the entire fluid domain to ensure computational accuracy and save computing time to a certain extent.The stay/guide vanes, runner, gap, and draft tube use hexahedral grids, while the volute uses tetrahedral grids, and the pressure balance pipe uses a hybrid grid composed of tetrahedral unstructured grids and hexahedral structured grids.The final grid generation scheme is shown in Figure 4, which contains 4,389,241 elements after the grid independence check, and the number of N4 grids for each component is shown in Table 3.
The accuracy of numerical simulation results is directly influenced by the appropriateness of grid division.In this study, a grid division method combining hexahedral structured grids and tetrahedral unstructured grids was adopted for the entire fluid domain to ensure computational accuracy and save computing time to a certain extent.The stay/guide vanes, runner, gap, and draft tube use hexahedral grids, while the volute uses tetrahedral grids, and the pressure balance pipe uses a hybrid grid composed of tetrahedral unstructured grids and hexahedral structured grids.The final grid generation scheme is shown in Figure 4, which contains 4,389,241 elements after the grid independence check, and the number of N4 grids for each component is shown in Table 3.

Numerical Setup and Boundary Conditions
In the calculations, the model's inlet was set as the total pressure inlet, and the outlet was set as the mass flow outlet; the runner domain was set as the rotating domain, and the other domains were set as the stationary domain.All the walls were set as no-slip walls and the interfaces between the stationary domains were set as GGI.The time steps for Case 1 and Case 2 were 0.00074 s and 0.00079 s, respectively.The fluid domain was set as a rotating domain and all wall surfaces were designated as no-slip walls.Furthermore, the initial flow field for transient simulation was determined using the results obtained from the steady-state simulation, with a maximum iteration number of 1000 steps.Data transfer interfaces were established at the interfaces between each pair of components.In the unsteady calculations, Case 1 had a rotation period of the runner set as 0.134 s for a total of

Numerical Setup and Boundary Conditions
In the calculations, the model's inlet was set as the total pressure inlet, and the outlet was set as the mass flow outlet; the runner domain was set as the rotating domain, and the other domains were set as the stationary domain.All the walls were set as no-slip walls and the interfaces between the stationary domains were set as GGI.The time steps for Case 1 and Case 2 were 0.00074 s and 0.00079 s, respectively.The fluid domain was set as a rotating domain and all wall surfaces were designated as no-slip walls.Furthermore, the initial flow field for transient simulation was determined using the results obtained from the steady-state simulation, with a maximum iteration number of 1000 steps.Data transfer interfaces were established at the interfaces between each pair of components.In the unsteady calculations, Case 1 had a rotation period of the runner set as 0.134 s for a total of 7 periods, while Case 2 had a rotation period of 0.142 s for a total of 7 periods.The convergence criteria for both the continuity equation and momentum equation were set to 10 −5 .
In the numerical simulation process, the choice of turbulence model is crucial as it affects the accuracy and convergence of the calculations.The SST k-ω model takes into account the transfer of turbulence shear stress and can accurately predict the flow initiation and separation under negative pressure gradients.It also exhibits high accuracy in simulating free shear turbulence, attached boundary layer turbulence, and moderate separation turbulence [29].Compared to other turbulence models, it not only enhances the functionality of wall modeling but also avoids excessive mesh refinement near the walls, which could lead to the sensitivity of the solution to the wall grid.This model demonstrates higher inclusiveness and good convergence properties and provides reliable results in calculating the hydraulic characteristics of variable-speed hydraulic pump turbines [30][31][32][33].Therefore, choosing the SST k-ω model for the calculation is reasonable.
where u is velocity, p is pressure, x i is the turbulent flow, ρ is density, and ν is kinematic viscosity.In order to close the equations mentioned above, a turbulence model is required.
The Reynolds-averaged Navier-Stokes (RANS) equation for the Shear Stress Transport (SST) k-ω model can be expressed as follows (Equations ( 3) and ( 4)): where l k−ω = k 1 2 β k ω represents the turbulence scale, µ is viscosity, P represents the production term, C ω is the production term coefficient, and F 1 is the blending function.

Monitoring Points
The distribution of monitoring points is shown in Figure 5. PHCM1 is used to monitor the pressure fluctuations in the vaneless region, PHCM2 is used to monitor the pressure fluctuations at the guide vane, and RV1 and RV2 are used to monitor the pressure fluctuations at the runner location.

Hydraulic Performance
The comparison between the numerical simulation and the numerical values of the net head obtained from the design performance parameters is performed to validate the accuracy of the simulation results.The comparison results are shown in Figure 6.The error for Case 1 is 1.3%, and the error for Case 2 is −1.9%.These error values fall within a reasonable range, which demonstrates that the numerical simulation is viable.

Hydraulic Performance
The comparison between the numerical simulation and the numerical values of the net head obtained from the design performance parameters is performed to validate the accuracy of the simulation results.The comparison results are shown in Figure 6.The error for Case 1 is 1.3%, and the error for Case 2 is −1.9%.These error values fall within a reasonable range, which demonstrates that the numerical simulation is viable.
The comparison between the numerical simulation and the numerical values of the net head obtained from the design performance parameters is performed to validate the accuracy of the simulation results.The comparison results are shown in Figure 6.The error for Case 1 is 1.3%, and the error for Case 2 is −1.9%.These error values fall within a reasonable range, which demonstrates that the numerical simulation is viable.Figure 7 shows the pressure distribution on the runner surface under two typical pumping modes.The pressure is uniformly distributed along the circumference, and it gradually increases from the inner edge to the outer edge of the flow passage.In Case 1, the maximum pressure value reaches around 5 MPa, while, in Case 2, it reaches around 4.7 MPa.It is evident that the maximum pressure is concentrated at the outlet of the runner blade surface.Figure 7 shows the pressure distribution on the runner surface under two typical pumping modes.The pressure is uniformly distributed along the circumference, and it gradually increases from the inner edge to the outer edge of the flow passage.In Case 1, the maximum pressure value reaches around 5 MPa, while, in Case 2, it reaches around 4.7 MPa.It is evident that the maximum pressure is concentrated at the outlet of the runner blade surface.Figure 8 shows the streamline in the flow passage.In the pumping mode, the fluid flows into the draft tube, and its flow direction is opposite to the pump conditions.It is evident that the streamlines inside the draft tube exhibit a smooth flow, whereas vortices can be observed inside the spiral case.The maximum flow velocity in the entire flow passage is 84.87 m/s for Case 1, while it is 74.19 m/s for Case 2.
Based on Figure 9, it is evident that the streamlines inside the runner are smooth, indicating the absence of vortices or backflow.In both conditions, the maximum flow velocity in the entire flow passage is distributed between the runner and the guide vanes, specifically in the vaneless region.The vaneless region is also one of the most unstable areas during operation in the pumping mode.Furthermore, the streamline within the flow passage in Case 1 is clearly more organized compared to the chaotic streamline within the flow passage in Case 2. Figure 8 shows the streamline in the flow passage.In the pumping mode, the fluid flows into the draft tube, and its flow direction is opposite to the pump conditions.It is evident that the streamlines inside the draft tube exhibit a smooth flow, whereas vortices can be observed inside the spiral case.The maximum flow velocity in the entire flow passage is 84.87 m/s for Case 1, while it is 74.19 m/s for Case 2.
locity in the entire flow passage is distributed between the runner and the guide vanes, specifically in the vaneless region.The vaneless region is also one of the most unstable areas during operation in the pumping mode.Furthermore, the streamline within the flow passage in Case 1 is clearly more organized compared to the chaotic streamline within the flow passage in Case 2.  Based on Figure 9, it is evident that the streamlines inside the runner are smooth, indicating the absence of vortices or backflow.In both conditions, the maximum flow velocity in the entire flow passage is distributed between the runner and the guide vanes, specifically in the vaneless region.The vaneless region is also one of the most unstable areas during operation in the pumping mode.Furthermore, the streamline within the flow passage in Case 1 is clearly more organized compared to the chaotic streamline within the flow passage in Case 2.

Pressure Pulsation Characteristics under Two Typical Pumping Modes
Figures 10-13 show the time-domain and frequency spectrum diagrams of pressure pulsations for Case 1 and Case 2 within a duration of 0.7 s.The left side displays the timedomain plots of pressure pulsations, with the horizontal axis representing the time required for five rotations of the runner and the vertical axis representing the pressure pulsation values.To further understand the characteristics of pressure pulsations in different parts, the time-domain data obtained are subjected to fast Fourier transform (FFT) analysis.The resulting frequency spectrum plot on the right side show the pressure pulsation spectra for each location.The horizontal axis represents multiples of the runner rotation frequency (fn), and the vertical axis represents the amplitude of pressure pulsations.
Based on Figure 10, in Case 1, the peak values of pressure pulsations at the PHCM1 location in the vaneless region are 4575 Kpa and 4340 Kpa.The dominant frequency of pressure pulsations at PHCM1 in the vaneless region is 9fn, with a subharmonic frequency   From Figure 11, in Case 2, the peak values of pressure pulsations at the PHCM3 monitoring point are 5400 KPa and 4300 KPa.The dominant frequency of pressure pulsations at PHCM3 is also 9fn, and there is a subharmonic frequency at 27fn.In Case 1, the peak values of pressure pulsations at the PHCM2 monitoring point in the guide vane region are 4800 KPa and 4670 KPa.The dominant frequency of pressure pulsations at PHCM2 is 9fn, and there is a subharmonic frequency at 27fn.The pressure fluctuation at the guide vane location clearly shows that case 1 exhibits more irregularity compared to Case 2, with larger magnitudes.Although both cases have a dominant frequency of 9fn, Case 1 has a significant low-frequency component.In the time-domain plot of Case 2, it can be observed that the extreme values of pressure fluctuation decrease noticeably during the time intervals of 0-0.05 s and 0.35-0.45s, and this behavior persists for a certain duration.This indicates that the pressure fluctuation in Case 2 is unstable.Based on Figure 10, in Case 1, the peak values of pressure pulsations at the PHCM1 location in the vaneless region are 4575 Kpa and 4340 Kpa.The dominant frequency of pressure pulsations at PHCM1 in the vaneless region is 9f n , with a subharmonic frequency of 18f n .In Case 2, the peak values of pressure pulsations at the PHCM1 location in the vaneless region are 4295 KPa and 4100 KPa.The dominant frequency of pressure pulsations in the vaneless region at PHCM1 is 18f n , with a subharmonic frequency of 27f n .In Case 2, the pressure pulsations in the vaneless region exhibit clear periodicity.Within the time it takes for the runner to complete one rotation, there are 18 peaks and 18 valleys of pressure pulsations, indicating that it occurs within two rounds of the runner.This suggests that the dominant frequency is twice the blade passing frequency.Compared to other monitoring points, the PHCM1 location in the vaneless region is significantly affected by dynamic and static interference.It can be observed from the time-domain plot that the pressure variation in Case 1 is more pronounced.This could be attributed to the minimal guide vane opening, and there is also a certain proportion of low-frequency pulsations present.On the other hand, the variation in the pressure pulsations in the vaneless region in Case 2 is more regular, indicating that the operation of the unit under Case 2 conditions would be more stable.Based on Figure 12, in Case 1, the peak pressure pulsation at monitoring point RV1 is 4515 KPa and 4175 KPa, and the RV1 has a dominant pulsation frequency of 3fn and a sub-frequency of 22fn.In Case 2, the peak pressure pulsation at monitoring point RV1 is 4200 KPa and 4275 KPa, and the runner monitoring point RV1 has a dominant pulsation frequency of 22fn and a sub-frequency of 3fn.RV1 is a monitoring point located closer to the crown.In Case 1 and Case 2, the dominant frequency and sub-frequency are exactly the opposite.In Case 1, with a smaller guide vane opening, there might be secondary flow, which can be observed as the low-frequency becoming the dominant frequency.Case 1 exhibits poorer stability, but the pressure variation in Case 1 also follows a more regular pattern.From Figure 11, in Case 2, the peak values of pressure pulsations at the PHCM3 monitoring point are 5400 KPa and 4300 KPa.The dominant frequency of pressure pulsations at PHCM3 is also 9f n , and there is a subharmonic frequency at 27f n .In Case 1, the peak values of pressure pulsations at the PHCM2 monitoring point in the guide vane region are 4800 KPa and 4670 KPa.The dominant frequency of pressure pulsations at PHCM2 is 9f n , and there is a subharmonic frequency at 27f n .The pressure fluctuation at the guide vane location clearly shows that case 1 exhibits more irregularity compared to Case 2, with larger magnitudes.Although both cases have a dominant frequency of 9f n , Case 1 has a significant low-frequency component.In the time-domain plot of Case 2, it can be observed that the extreme values of pressure fluctuation decrease noticeably during the time intervals of 0-0.05 s and 0.35-0.45s, and this behavior persists for a certain duration.This indicates that the pressure fluctuation in Case 2 is unstable.Based on Figure 13, in Case 1, the peak pressure pulsation at monitoring point RV1 is 4565 KPa and 4135 KPa, and RV1 has a dominant pulsation frequency of 22fn and a subfrequency of 4.6fn.In Case 2, the peak pressure pulsation at monitoring point RV2 is 4200 KPa and 3980 KPa, and the runner monitoring point RV2 has a dominant pulsation frequency of 22fn and a sub-frequency of 1fn.The pressure fluctuation at RV2 is larger than at RV1, and this is due to the location of the monitoring points.RV2 is closer to the tailrace conduit.In Case 2, the magnitude of the pressure fluctuation is greater, indicating a more unstable pressure fluctuation.

The Axial Hydraulic Thrust and Radial Force of the Unit in the Two Typical Pumping Modes
For Case 1 and case 2, the axial hydraulic thrust can be seen in Figure 14, where itive values indicate an upward direction of the axial hydraulic thrust.In Case 2, the m imum axial hydraulic thrust reaches 375 t, fluctuating between 75 and 375 tons.In Ca the maximum axial hydraulic thrust reaches 215 t, fluctuating between 155 and 215 Case 2, the thrust bearing will experience fluctuating loads with an amplitude of 3 while, in Case 1, the thrust bearing will experience fluctuating loads with an amplitud 60 t.In both operating conditions, the thrust bearing is at risk of failure, with Case 2 b more severe.Unlike the radial force, the complexity of the axial hydraulic thrust freque distribution is higher.The axial hydraulic thrust dominant frequency in Case 1 is 0 while, in Case 2, it is 0.4fn.Low-frequency components of 0.4 fn and 0.2 fn, which are pre in the pressure pulsations, can also be observed in the axial hydraulic thrust, explai the source of the axial hydraulic thrust frequency.
Figure 15 illustrates the temporal and spectral characteristics of the radial force v ations.It is evident that the radial force is greatly influenced by the operating conditi Both selected conditions exhibit notable instability.In Case 2, the maximum radial f

The Axial Hydraulic Thrust and Radial Force of the Unit in the Two Typical Pumping Modes
For Case 1 and case 2, the axial hydraulic thrust can be seen in Figure 14, where positive values indicate an upward direction of the axial hydraulic thrust.In Case 2, the maximum axial hydraulic thrust reaches 375 t, fluctuating between 75 and 375 tons.In Case 1, the maximum axial hydraulic thrust reaches 215 t, fluctuating between 155 and 215 t.In Case 2, the thrust bearing will experience fluctuating loads with an amplitude of 300 t, while, in Case 1, the thrust bearing will experience fluctuating loads with an amplitude of 60 t.In both operating conditions, the thrust bearing is at risk of failure, with Case 2 being more severe.Unlike the radial force, the complexity of the axial hydraulic thrust frequency distribution is higher.The axial hydraulic thrust dominant frequency in Case 1 is 0.2 f n , while, in Case 2, it is 0.4f n .Low-frequency components of 0.4 f n and 0.2 f n , which are present in the pressure pulsations, can also be observed in the axial hydraulic thrust, explaining the source of the axial hydraulic thrust frequency.Figure 15 illustrates the temporal and spectral characteristics of the radial force variations.It is evident that the radial force is greatly influenced by the operating conditions.Both selected conditions exhibit notable instability.In Case 2, the maximum radial force reaches 70 t, while, in Case 1, it reaches 45 t.In other words, the increase in radial force during unstable operating conditions is not only detrimental to the flow passage but also poses a threat to the safety of the shaft system.The radial force dominant frequency in Case 1 is 1 f n , while, in Case 2, it is 0.2 f n .

Figure 1 .
Figure 1.A 3D model for entire fluid passages.

Figure 1 .
Figure 1.A 3D model for entire fluid passages.

Figure 2 .
Figure 2. Stable operating zone of the unit.

Figure 2 .
Figure 2. Stable operating zone of the unit.

Figure 2 .
Figure 2. Stable operating zone of the unit.

Figure 4 .
Figure 4. Variable-speed pump turbine CFD domain and mesh.

Figure 4 .
Figure 4. Variable-speed pump turbine CFD domain and mesh.

Figure 5 .
Figure 5.The positions of monitoring points in fluid domain.(a) PHCM1 and PHCM2 (b) RV1 and RV2

Figure 5 .
Figure 5.The positions of monitoring points in fluid domain.(a) PHCM1 and PHCM2 (b) RV1 and RV2.

Figure 6 .
Figure 6.The design parameters and simulation results comparison.

Figure 6 .
Figure 6.The design parameters and simulation results comparison.

Figure 7 .
Figure 7.The pressure distribution on the runner surface.(a) The pressure on the runner surface in Case 1 (b) The pressure on the runner surface in Case 2

Figure 7 .
Figure 7.The pressure distribution on the runner surface.(a) The pressure on the runner surface in Case 1 (b) The pressure on the runner surface in Case 2.

Figure 8 .
Figure 8.The streamline diagram of the entire flow passage.(a) Case 1 (b) Case 2.

Figure 8 .
Figure 8.The streamline diagram of the entire flow passage.(a) Case 1 (b) Case 2.

Figure 9 .
Figure 9.The streamline of the runner from blade to blade at different spans. .(a) Case 1 (b) Case 2.

Figure 9 .
Figure 9.The streamline of the runner from blade to blade at different spans.(a) Case 1 (b) Case 2.

Figures 10 -
Figures 10-13 show the time-domain and frequency spectrum diagrams of pressure pulsations for Case 1 and Case 2 within a duration of 0.7 s.The left side displays the timedomain plots of pressure pulsations, with the horizontal axis representing the time required for five rotations of the runner and the vertical axis representing the pressure pulsation values.To further understand the characteristics of pressure pulsations in different parts, the time-domain data obtained are subjected to fast Fourier transform (FFT) analysis.The resulting frequency spectrum plot on the right side show the pressure pulsation spectra for each location.The horizontal axis represents multiples of the runner rotation frequency (f n ), and the vertical axis represents the amplitude of pressure pulsations.

Figure 10 .
Figure 10.The time-domain and frequency spectra of pressure pulsations at the PHCM1 monitoring point for both Case 1 and Case 2. (a) Time-domain plot of pressure pulsations in Case 1 (b) Frequency spectrum plot of pressure pulsations in Case 1 (c) Time-domain plot of pressure pulsations in Case 2 (d) Frequency spectrum plot of pressure pulsations in Case 2.

Figure 10 .
Figure 10.The time-domain and frequency spectra of pressure pulsations at the PHCM1 monitoring point for both Case 1 and Case 2. (a) Time-domain plot of pressure pulsations in Case 1 (b) Frequency spectrum plot of pressure pulsations in Case 1 (c) Time-domain plot of pressure pulsations in Case 2 (d) Frequency spectrum plot of pressure pulsations in Case 2.

Figure 11 .
Figure 11.The time-domain and frequency spectra of pressure pulsations at the PHCM2 monitoring point for both Case 1 and Case 2. (a) Time-domain plot of pressure pulsations in Case 1 (b) Frequency spectrum plot of pressure pulsations in Case 1 (c) Time-domain plot of pressure pulsations in Case 2 (d) Frequency spectrum plot of pressure pulsations in Case 2.

Figure 11 .
Figure 11.The time-domain and frequency spectra of pressure pulsations at the PHCM2 monitoring point for both Case 1 and Case 2. (a) Time-domain plot of pressure pulsations in Case 1 (b) Frequency spectrum plot of pressure pulsations in Case 1 (c) Time-domain plot of pressure pulsations in Case 2 (d) Frequency spectrum plot of pressure pulsations in Case 2.

Figure 12 .
Figure 12.The time-domain and frequency spectra of pressure pulsations at the RV1 monitoring point for both Case 1 and Case 2. (a) Time-domain plot of pressure pulsations in Case 1 (b) Frequency spectrum plot of pressure pulsations in Case 1 (c) Time-domain plot of pressure pulsations in Case 2 (d) Frequency spectrum plot of pressure pulsations in Case 2.

Figure 12 .Figure 13 .
Figure 12.The time-domain and frequency spectra of pressure pulsations at the RV1 monitoring point for both Case 1 and Case 2. (a) Time-domain plot of pressure pulsations in Case 1 (b) Frequency spectrum plot of pressure pulsations in Case 1 (c) Time-domain plot of pressure pulsations in Case 2 (d) Frequency spectrum plot of pressure pulsations in Case 2.

Figure 13 .
Figure 13.The time-domain and frequency spectra of pressure pulsations at the RV2 monitoring point for both Case 1 and Case 2. (a) Time-domain plot of pressure pulsations in Case 1 (b) Frequency spectrum plot of pressure pulsations in Case 1 (c) Time-domain plot of pressure pulsations in Case 2 (d) Frequency spectrum plot of pressure pulsations in Case 2.

Figure 14 .Figure 14 .
Figure 14.Time-domain and frequency spectrum plots of axial hydraulic thrust in Case 1 and Case 2. (a) Time-domain plot in Case 1 (b) Frequency spectrum plot in Case 1 (c) Time-domain plot in Case 2 (d) Frequency spectrum plot in Case 2.

Figure 14 .Figure 15 .
Figure 14.Time-domain and frequency spectrum plots of axial hydraulic thrust in Case 1 and Case 2. (a) Time-domain plot in Case 1 (b) Frequency spectrum plot in Case 1 (c) Time-domain plot in Case 2 (d) Frequency spectrum plot in Case 2.

Table 1 .
The specification parameters for variable-speed pump turbines.

Table 1 .
The specification parameters for variable-speed pump turbines.

Table 3 .
Grid number of each component.

Table 3 .
Grid number of each component.