# A Reversible Hydropump-Turbine System
## 4 Results of Simulations  
The system presented in this paper has been simulated in order to obtain validation data. Table 1 shows the main parameters for pump-turbine components. The initial conditions are as follows. \[\text{col}\Big{(}\lambda_{s_{d}},\lambda_{s_{q}},\lambda_{s_{d}},\lambda_{r_{q }},p_{M-P}\Big{)}=\text{col}(0,0,0,0,0). \tag{48}\]  
We have chosen three different scenarios where the initial conditions for the pressure \(P_{r}(0)\) and flow \(Q(0)\) are defined accordingly. Furthermore, we assume that \(Q\geq 0\) and \(\omega_{m}\geq 0\) for all scenarios even though the direction (sign) changes in a real system for the pump or the turbine mode. First, the system is operated as a hydropump system, i.e., \(s=0\) in (45). The initial conditions for the flow and the water height are \(Q(0)=0.0\) (\(\text{m}^{3}/s\)), \(P_{r}(0)=0.0\) (\(\text{Pa}\)), and \(h_{w}=0.0\,(\mathrm{m})\). Figure 3 shows the resulting dynamics of rotor speed \(\omega_{m}\), hydraulic flow \(Q\), hydraulic pressure \(P_{r}\), and water height \(h_{w}\) in the penstock and the reservoir. The water increases its pressure and height in the pipeline at \(t\leq 70\,[\mathrm{s}]\). Then, the reservoir begins to be filled at \(t>70\,[\mathrm{s}]\) and due to the cross-section area; i.e., \(A_{r}\gg A_{p}\) (see Table 1) both \(P_{r}\) and \(h_{w}\) have a lower increasing rate. We finally see in the first scenario how the hydraulic flow grows faster during the first seconds, and later on, such an acceleration rate decreases \((12<t<75\,)(\mathrm{s})\) due to the impulse pressure and the accumulated pressure in the system. The second scenario simulates the hydroturbine system, i.e., \(s=1\) in (45). The simulation results are shown in Figure 4. The initial conditions for the simulation are \(Q(0)=0.0\,(\mathrm{m}^{3}/s)\), \(P_{r}(0)=3.4275\times 10^{5}\,(\mathrm{Pa})\), and \(h_{w}=35\,(\mathrm{m})\), which means that the upper reservoir is full of water. The asynchronous machine starts its operation as a motor in \(0\leq t<10\,(\mathrm{s})\). Then, valve \(v_{h}\) is opened such that the hydro turbine system is engaged at \(t\geq 10\,[\mathrm{s}]\). The rotor speed increases above the synchronous speed, which means that the asynchronous machine delivers electricity to the power grid (Figure 1). In Figure 5, we see how the hydraulic flow reaches a stable value while the pressure in the upper reservoir decreases together with the water head of the system. Finally, the operation of the reversible hydropump-turbine system corresponds to the third simulated scenario. The simulation results are shown in Figure 5. The initial conditions are \(Q(0)=0.0\,(\mathrm{m}^{3}/s)\), \(P_{r}(0)=3.1825\times 10^{5}\,(\mathrm{Pa})\), and \(h_{w}=32.5\,(\mathrm{m})\). The system operates first as a hydropump system at \(t\leq 10\,(\mathrm{s})\) when the asynchronous machine configured as a motor _is powered_ by the electrical network. The water level rises with the pressure in the upper reservoir, which is initially half-loaded. Then, the pumping action is switched off, and the hydraulic valve \(v_{h}\) in the penstock is closed at \(t=85\,(\mathrm{s})\). Both actions co-occur. Similarly to the second scenario, the asynchronous machine starts its operation as a motor in \(100\leq t<115\,(\mathrm{s})\). The hydraulic valve \(v_{h}\) opens at \(t=115\,(\mathrm{s})\). It follows that the hydroturbine system configuration begins it operation at \(t=115\,(\mathrm{s})\). The action over the hydro pump as a turbine system increases the velocity of the rotor \(\omega_{m}\) above the synchronous speed, creating a negative slip in the motor that induces a reverse current in the asynchronous machine's stator. Consequently, the machine ultimately _powers_ the electrical network since it is configured as a generator. It is commonly known that the electrical network regulates the voltage and frequency of the induction generators, which means that a bank of capacitors is unnecessary. Figure 3: First scenario: operation of the hydro pump system, i.e., \(s=0\).