# Simulation and Optimisation of Utility-Scale PV-Wind Systems with Pumped Hydro Storage
## 2 Modelling Approach
### Simulation
#### 2.1.3 Pump Model  
Variable-speed reversible pump turbines are considered in this study. For each time step, the control strategy determines the value of the pump input power \(P_{P}(t)\) (W), which cannot be higher than the nominal rated power (\(P_{P}(t)\leq P_{P_{\_}N}\)) depending on the type of system and on the different variables (Section 2, Table 1). The pump flow rate (water taken from the lower reservoir and stored in the upper reservoir) depends on the pump efficiency and head. The pump model [40] calculates the flow rate of the pump \(Q_{P}(t)\) (m\({}^{3}\)/s) as a function of the input power (first term in Equation (8)). The second term of Equation (8) shows that the pump flow rate cannot exceed the nominal flow rate \(Q_{P\_}N\) (m\({}^{3}\)/s). The third and fourth term of that equation show that, at the end of the time step (after \(\Delta t\) = 15 \(\times\) 60 = 900 s, which corresponds to the time step of 15 min), the lower reservoir volume \(V_{LR}(t)\) (m\({}^{3}\)) cannot be lower than its minimum allowed \(V_{LR\_min}\) (m\({}^{3}\)), and the upper reservoir volume \(V_{UR}(t)\)(m\({}^{3}\)) cannot be higher than its maximum allowed \(V_{UR\_max}\) (m\({}^{3}\)). \[Q_{P}(t)=\min\,\bigg{(}\frac{P_{P}(t)\eta_{P}(t)}{\rho gH_{P}(t)},\ Q_{P\_}N,\frac{V_{LR}(t)-V_{LR\_min}}{\Delta t},\frac{V_{UR\_max}-V_{UR}(t)}{\Delta t }\bigg{)}, \tag{8}\]  
where \(\eta_{P}(t)\) represents the pump efficiency (which includes the motor efficiency) and is a function of \(Q_{P}(t)\), which can be found in the pump technical datasheet; \(\rho\) represents the water density (\(\rho\) = 997 Kg/m\({}^{3}\) at 25 \({}^{\circ}\)C), \(g\) represents the gravity acceleration (\(g\) = 9.81 m/s\({}^{2}\)), and \(H_{P}(t)\) (m) represents the pump head, which is the sum of the static head \(H_{PS}(t)\) and head loss of the pump mode \(H_{Pl}(t)\). \[H_{P}(t)=H_{PS}(t)+H_{Pl}(t). \tag{9}\]  
The static head is the difference in the water level of the upper and lower reservoirs, which considers the difference in heights between the minimum level of the upper reservoir and maximum level of the lower reservoir (\(H_{diff}\)), and the variations over the minimum level of the upper reservoir \(H_{UR}(t)\) and below the maximum level of the lower reservoir \(H_{LR}(t)\) (Figure 1). \[H_{PS}(t)=H_{diff}+H_{UR}(t)+H_{LR}(t). \tag{10}\]  
Figure 2: Wind turbine output curve of a 500 kW wind turbine with pitch control. Standard (green curve) and other conditions (dotted red curve). The water level in the upper reservoir \(H_{\text{{U}R}}(t)\) can be calculated by assuming that the reservoir has a cuboid shape. \[H_{\text{{U}R}}(t)=\frac{V_{\text{{U}R}}(t)}{V_{\text{{U}R}\_max}}H_{\text{{U}R} \_max}, \tag{11}\]  
where \(H_{\text{{U}R}\_max}\) represents the maximum height of the water in the upper reservoir (m). The variation below the maximum level of the lower reservoir \(H_{\text{{L}R}}(t)\) can be calculated by assuming that the lower reservoir has a cuboid shape given by  
\[H_{\text{{L}R}}(t)=\frac{V_{\text{{L}R}\_max}-V_{\text{{L}R}}(t)}{V_{\text{{L}R }\_max}}H_{\text{{L}R}\_max}, \tag{12}\]  
where \(V_{\text{{L}R}\_max}\) represents the maximum volume of the stored water in the lower reservoir (m\({}^{3}\)) and \(H_{\text{{L}R}\_max}\) represents the maximum height of the water in the upper reservoir (m). The head loss of the pump mode \(H_{\text{{P}I}}(t)\) represents the hydraulic losses of the pump mode because of the friction between the water and inner surface of the pipes and fittings. \(H_{\text{{P}I}}(t)\) is calculated as a function of \(Q_{\text{{P}}}(t)\) using the Darcy-Weisbach equation [40]  
\[H_{\text{{P}I}}(t)=K\frac{v(t)^{2}}{2g}. \tag{13}\]  
\[v(t)=\frac{Q_{\text{{P}}}(t)}{0.25\pi\text{{L}{D_{{P}}}^{2}}}. \tag{14}\]  
\[K=K_{pipe}+K_{fittings}. \tag{15}\]  
\[K_{pipe}=f\frac{L_{\text{{P}}}}{D_{\text{{P}}}}. \tag{16}\]  
For laminar flow, the friction factor (\(f\)) is calculated as x  
\[H_{P}(t)=H_{PS}(t)+H_{Pl}(t). \tag{9}\]0  
where \(Re(t)\) represents the Reynolds number.