## 4 Output Power Modelling for the Switching Process of the DFVSPS Unit
### Pump Mode
#### 4.2.2 On-Grid Operation Stage  
In the previous section, the control methods of generating mode and pump mode are researched. The pump turbine model are different under the two working modes, and the control strategies are the same. Therefore, when the model is simplified under pump mode, the analysis method is consistent with that of the generating mode. Only a few differences are analyzed below. During the load ramping stage and load rejection stage, the pump turbine operates at the small guide vane opening, so the control process is exactly the same as the generating mode, except that the same guide vane opening corresponds to different mechanical power. Since the converter flows through the slip power, the working mode does not affect the power of the converter, and the power flow direction is opposite to that in the generating mode. The conclusion is that the more power is regulated and the shorter the response time, the larger the capacity the converter required. The equivalent method of the load ramping stage and load rejection stage are consistent with generating mode. In addition, it can be expressed by the following formula:  
\[p_{sp}=p_{P0}\frac{t-\tau_{1}}{N_{SP}},\ \tau_{1}\leq t<\tau_{1}+N_{SP}, \tag{27}\]  
Figure 22: Control diagram of start up in pump mode. \[p_{PS}=p_{P1}\left(1-\frac{t-\tau_{2}}{N_{PS}}\right),\ \tau_{2}\leq t<\tau_{2}+N_{PS}, \tag{28}\]  
where \(P_{p0}\) is the power increased during the load ramping stage, \(P_{p1}\) is the power reduced during the load rejection stage, \(\tau_{2}\) is the start time of the load rejection stage, \(N_{SP}\) and \(N_{PS}\) are the time used in the load ramping stage and load rejection stage, and Equation (27) is the load ramping stage. Equation (28) indicates the load rejection stage. The parameter selected in the simulation is: \(k_{p1}\) = 0.000005, \(k_{i1}\) = 0.00005, so the response time of each segment is 5 s, and each segment is superimposed 30 MW, superimposed seven times to reach 240 MW. This choice requires small capacity, easy implementation, and high economy. According to the selected parameters, \(P_{p0}\) = 210 MW, \(P_{p1}\) = 210 MW, \(N_{SP}\) = 35 s, \(N_{PS}\) = 35 s, the comparison of the waveforms before and after the equivalent of the load ramping stage and load rejection stage is shown in Figure 23. The red line in the figure is unit active power of simplified model obtained by Equations (27) and (28). In addition, the black line is the unit active power simulation waveform obtained by the electromagnetic transient simulation model. Stable operation stage: When the DFVSPS runs at stable operation stage under pump mode, the pump turbine is located in the part of large guide vane opening, and the input power of the pump turbine is proportional to the cube of the unit speed. The control block diagram of this stage is shown in Figure 24.  
Compared with the generating mode, the power control closed loop is the same. The difference mainly exists with the speed control closed loop. Since this part only models the output power of DFVSPS unit, the equivalent formula similar with Formula (22), as shown in Equation (29). According to the parameters of the pump mode, \(k_{p2}\) = 0.00005, \(k_{i2}\) = 0.0005. Bring the parameter to Formula (29), the equivalent effect is shown in Figure 25. The red line in the figure is unit active power of a simplified  
Figure 23: Comparison of active power waveform before and after equivalent. (**a**) load ramping stage; (**b**) load rejection stage. Figure 24: Control block diagram of stable operation stage under pump mode. model obtained by Equation (29) and the black line is the unit active power simulation waveform obtained by the electromagnetic transient simulation model:  
\[p_{P}=G_{1}(s)P_{ref},\ \tau_{1}+N_{SP}\leq t<\tau_{2}. \tag{29}\]  
\(\tau_{2}\) is the end time of the stable operation stage. In the simulation of this paper, after the power increases by 30 MW in this stage, it needs to be reduced by 30 MW and returned to 240 MW, in order to facilitate the load rejection stage.