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Numerical analysis of the hydraulic force of a pump turbine under partial load conditions in turbine mode

The power grid demand keeps changing at any time, so the pump turbine often operates in partial load condition. The noise and vibration of the unit caused by hydraulic force have an adverse effect on the safe and stable operation of the power plant. In order to study the mechanism of hydraulic force, a model pump turbine of a pumped-storage power station was set as the research object in this paper, both the steady and unsteady numerical simulation were carried out to simulate the flow field and hydraulic force of pump turbine. Axial force will increase significantly when the unit overloaded, but it has almost no change at partial load conditions. And the axial force on the runner is mainly produced by the hub and shroud, while it can be ignored on the blade. The radial force increases sharply when the condition deviates from the rated operating point. The frequency of pressure fluctuation is 20fn which is caused by rotor stator. There is more low frequency pressure fluctuation at small discharge condition because of the flow separation. The amplitude and deviation coefficient increased. The heterogeneity of pressure pulsation leads to the increase of radial force at partial load condition.


Introduction
With the fast development of new energy such as solar power and wind power, the proportion of new energy in the power grid is increasing. It will adversely affect the safety and stability of the power grid, because of the obvious instability and intermittent which influenced by the environment [1] . The pumped storage power plant has flexible operation ability, fast starting, real-time tracking of load change, so it is an indispensable tool to respond rapid changes and to realize optimal dispatching of power grid [2] . The pump turbine often operates in partial load condition. And the hydraulic force will significantly increased compared to the optimal condition. It will cause the unit severe vibration and affect the stable operation of the unit.
Among the forces on the reversible pump turbine, there are three kinds of excitation forces according to their origins: the hydraulic forces, the mechanical forces and electromagnetic forces [3] . What the present paper focuses on is hydraulic force. J W Li [4] found that the axial force is prominent in the turbine mode while the radial force is the dominant near the runaway and turbine break modes. The component perpendicular to the shaft caused by the asymmetric flow inside runner is the dominant one in runaway and turbine break regions. The runner is subjected to an asymmetric radial force whose amplitude varies periodically，the number of cycles are equal to the number of blades， The double-rotation stall phenomenon appears in the runner，and the stalls can impact on blades intermittently，resulting in an additional alternating stress on the blades [5] [6] . The abnormal hydraulic force affects the safe operations of the pumped srotage power station seriously. Such as, Tianhuangping pumped storage power station, the frequent lifts of the rotational components of the reversible pump turbines were observed during the load increment from 200 MW to 300 MW in the generating mode, leading to the abrasive damage of the machine and hence unplanned closing down of the machine [7] . The literature review shows that the hydraulic force of the pump turbine runner under partial load conditions in turbine mode remains an issue to be explored.

Turbulence model
In this paper, the SST turbulence model was used to close the Reynolds averaged Navier-Stokes (RANS) solver for the solution of the turbulence flows. The continuity equation and momentum equation are adopted in simulation. Continuity equation

Compressible model
In engineering application, given the constant temperature and the compressibility of water is always represented by the bulk modulus of elasticity. The water density is defined as: Where p is the static pressure and ρ is the density of water and K is the modulus of elasticity. Under the operating conditions for hydraulic machinery, the modulus of elasticity can be assumed to be constant. In the present study, K=2×10 9 Pa, p 0 =101325Pa and ρ 0 =998kg/m 3 . By means of user defined function, the variable sound speed is linked to the flow solver CFX.

Pump Turbine Specification
As shown in Figure 1, the whole pump turbine configuration including spiral case, stay vane, guide vane, runner and draft tube is considered in the physical model. Specifications of the pump turbine are listed in Table.1.

Grid generation
Mesh generation is an important component of numerical discretization in flow governing equations, mesh quality has a crucial impact on calculation accuracy. Structured hexahedral grid was used in this study to discretize all components in the ANSYS ICEM software. Grid quality values in ANSYS ICEM usually range from 0 to 1, with a higher grid value representing higher grid quality. Grids of all components details are shown in Figure 2.  Figure 3 shows the efficiency and head of the pump turbine from the simulations for the five different grid nodes. The total number of grid elements used was about 5.46 million, while the mesh for each component is shown in Table 2. (2) Outlet condition. Pressure outlet is used at the draft tube of pump turbine, static pressure is specified for all cases.
(3) Other boundary conditions. No-slip condition has applied on the all solid walls, and standard wall function is used to calculate the turbulence kinetic energy and turbulence dissipation frequency near the wall. (4) Time step. During the numerical simulations, the results of steady RANS simulations were used as the initial flow field for the transient simulations. Vlad Hasmatuchi [8] compared unsteady numerical results of the three time steps that runner rotated 2, 1, 0.5 degree spent and argued that the time step which runner rotates 1 degree is reasonable in terms of computation time and accuracy. So the time that runner rotated 1 degree spent is adopted as the time step.

Calculation conditions
The numerical simulation of inflow pattern in pump turbine at different conditions was calculated in this paper is shown in Table.3. For convinence the operating conditions are plotted using unit parameters, which for hydro-turbines are defined as follows. The unit speed and unit flow rate is Where D is runner diameter at low pressure side, H is the net head across the pump turbine, n is the speed of runner in rpm and Q is the volume flow rate in m 3 /s.

Hydraulic force
In order to analyse the radial force and axial force characteristic of pump turbine, a dimensionless parameter K is defined as follow [9] : where F is the radial force Fxy or axial force Fz, ρ is the density of water, u 1 , D 1 and b 1 are the circumferential velocity, diameter and width of runner inlet respectively. The variation of axial force Kz and radial force Kxy under different working conditions from steady calculation results are as shown in Figure 4, It can be found that the axial force will increase significantly when overloaded, which is 6.7% higher than OP5. At the large flow condition, the differential pressure between the pressure and suction surface increases because of the rise of the blade load. This is the fundamental reason which leads to the increase of axial force. As the flow rate decrease, and the axial force almost has no change at partial load condition.
For the radial force, it is very small at the rated operating point and radial force coefficient Kxy is equal to 0.165. When the condition deviates from the rated operating point, Kxy increases sharply. The radial force coefficient Kxy is equal to 17.1 at OP1 and it is 103 times than rated operating point. At small flow condition, there is a positive angle of the runner inlet, which causes the flow separation on the impeller pressure surface easily, and the smaller the flow rate is, the more serious the flow separation phenomenon is. At the same time, the inlet flow separation will affect the circumferential pressure distribution of the wheel inlet pressure, which makes the radial force increase sharply.
(a) Axial force (b) Radial Force Figure 4. The variation of radial force Kxy and axial force Kz of the runner As shown in Figure 5, it can be clearly seen that the distribution of pressure at OP5 is uniform. There are 9 peaks along the circumferential direction which is equal to blade number. The range of pressure fluctuation increased obviously when the working condition deviates from rated condition. For example at OP6, the maximum pressure is largest and the minimal pressure is smallest than other condition. Especially there is a low pressure area at OP1. In addition, the law of pressure distribution along the Span direction is also different. Those phenomena are the main reason that resulting in the increasing of radial force.

Hydraulic force pulsation characteristics
In order to ensure the accuracy of the calculation results, the data after 10 revolutions is analysed. The distribution of radial force with the rotation of the runner is shown in Figure 6. It can conclude that, the runner rotated almost four revolutions but one revolution of radial force at OP1, and the radial force shows the characteristics of high frequency fluctuation at OP4. Figure 7 shows the vector distribution of radial force. The coordinates of each point indicate the size and direction of the radial force. The radial force acting on the runner at different conditions is always changing and regularly distributed. The radial force is very small in the rated condition, but the value is not zero, which is deviated from the theoretical value. The flow inside the impeller is not completely symmetrical due to the asymmetry structure of the pump turbine. It will result in uneven distribution of velocity and pressure. The angle of runner inlet increases with the decrease of flow rate, and the radial force increases significantly.

Pressure fluctuation
The dimensionless number C p (pressure fluctuation coefficient) is introduced the in the treatment of pressure fluctuation data of various monitoring points, the definition of C p as shown in equation (6) Figure 8. In the pressure surface, the dominant frequency is 20f n (rotating frequency), and it was caused by rotor stator. The amplitude on the pressure surface is bigger than suction surface. The flow separation is more obvious and there is more low frequency pressure fluctuation at small discharge condition. The dominant frequency on suction surface is 0.25f n at OP1. With the decrease of discharge, the amplitude of low frequency pressure fluctuation on suction surface increases.  Figure 9. the deviation coefficient of amplitude on pressure surface The deviation coefficient of monitor points on the suction surface is shown in and Figure 10. For dominant frequency, the S of all monitor points is very small. But for secondary frequency, the fluctuation of S is extremely volatile. Such as the minimum value is almost zero, but the maximum reached 0.2 at OP2.
To summarize, the low frequency and heterogeneity of pressure pulsation leads to the increase of radial force at partial load condition.
(a) Dominant frequency (b) Secondary frequency Figure 10. the deviation coefficient of amplitude on suction surface

Conclusion
The comprehensive calculation of pump turbine at partial load condition is carried out in this paper by using the numerical simulation method. And the analysis of hydraulic force and pressure fluctuation was also carried out. Comparison and analysis of calculation results of different calculation conditions were conducted, and the following conclusions can be drawn: (1) Axial force will increase significantly when the unit overloaded, it has almost no change at part load conditions. For the radial force, it is very small at the rated operating point. However the radial force increases sharply when the condition deviates from the rated operating point. With the increase of the flow rate, the axial force of the hub and shroud tends to decrease first and then increase, and the axial force on the runner is mainly produced by the hub and shroud, the axial force on the blade can be ignored.
(2) The runner rotated almost four revolutions but one revolution of radial force at OP1, and the radial force shows the characteristics of high frequency fluctuation at OP4.
(2) The frequency of pressure fluctuation is 20f n which is caused by rotor stator. There is more low frequency pressure fluctuation at small discharge condition because of the flow separation. And the amplitude increased.
(3) The deviation coefficient of monitor point bp1 which location near the tongue is bigger than other points. The S of all monitor points dominant frequency on suction surface is very small. But for secondary frequency, the fluctuation of S is extremely volatile at partial load condition.
(4) The low frequency and heterogeneity of pressure pulsation leads to the increase of radial force at partial load condition. It will cause more vibration and threaten the safe operation of the unit.