# Simulation and Optimisation of Utility-Scale PV-Wind Systems with Pumped Hydro Storage
## 2 Modelling Approach
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Type A: Allowed to Buy Energy for Arbitrage**} & \multicolumn{2}{c}{**Type B: Not Allowed to Buy Energy for Arbitrage**} \\ \hline Allowed to purchase electricity from the grid for arbitrage (for & ✓ & ✗ \\ pumping) & & & ✗ \\ \hline Arbitrage sell electricity price & & & \\ set points & & ✗ & ✓ \\ \((\Pr_{E\textit{sell\_High\_stp}},\Pr_{E\textit{sell\_Low\_stp}})\) & & & \\ \hline Arbitrage purchase electricity & & & \\ price set points & ✓ & & ✗ \\ \((\Pr_{E\textit{purch\_High\_stp}},\Pr_{E\textit{purch\_low\_stp}})\) & & & \\ \hline \multicolumn{2}{c}{**Operation principle:** During each time step \(t\):} & & \\ \hline \hline \end{tabular} \
\begin{tabular}{c c} - & If buy electricity price is higher than the set point \\  & & \(\left(\Pr_{E\textit{purch}}(t)\geq\Pr_{E\textit{purch\_High\_stp}}\right)\to\) \\  & & _Energy arbitrage (discharge):_ priority is \\  & & injecting (sell) electricity to the grid (net \\  & & renewable generation + hydro t. max. \\  & & power *): \\  & & \(\left.\begin{array}{l}P_{\textit{sell}}(t)=P_{\textit{Nlet\_ren}}(t)+P_{ \textit{HT\_max}}(t)\\ \end{array}\right.\) \\ - & If there is net renewable generation & & \(\left.\begin{array}{l}P_{\textit{sell}}(t)=P_{\textit{Nlet\_ren}}(t)+P_{ \textit{HT\_max}}(t)\\ \end{array}\right.\) \\ - & If there is net renewable generation & & \(\left.\begin{array}{l}P_{\textit{sell\_ren}}(t)\geq 0\end{array}\right.\) or no \\  & & & \\ surplus power; if it is lower than pump \\  & & rated power, buy electricity to run the \\  & & pump at the rated power: \\  & & \(\left.\begin{array}{l}P_{\textit{p}}(t)=P_{\textit{P\_N}}\\ \end{array}\right.\) \\  & & \(\left.\begin{array}{l}P_{\textit{purch}}(t)=\max\left(0,P_{\textit{P\_N}}-P_{ \textit{Nlet\_ren}}(t)\right)\\ \end{array}\right.\) \\ - & If \(\Pr_{E\textit{purch\_High\_stp}}>\Pr_{E\textit{purch\_(t)}}(t)>\) & & \\  & & \(\left.\begin{array}{l}P_{\textit{F\_Eurch\_Low\_stp}}\to\textit{No\ arbitrage:}\text{priority}\\ \end{array}\right.\) \\  & & & \(\left.\begin{array}{l}P_{\textit{F\_Eurch\_Low\_stp}}\to\textit{No\ arbitrage:}\text{priority}\\ \end{array}\right.\) \\  & & & \(\left.\begin{array}{l}P_{\textit{sell}}(t)=P_{\textit{Nlet\_ren}}(t)\\ \end{array}\right.\) \\ - & If there is net load & & \\  & \(\left.\begin{array}{l}P_{\textit{Nlet\_load\_stp}}(t)>0\end{array}\right.\) \\ - & Otherwise, use the grid to supply the net load: & \(\left.\begin{array}{l}P_{\textit{purch}}(t)=P_{\textit{Nlet\_load}}(t).\\ \end{array}\right.\) \\ \hline \hline \end{tabular}
\begin{tabular}{c} - & If there is net load \\ \((\Pr_{\textit{Nlet\_load\_(t)}}(t)>0)\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance of the two different types of projects (A or B) of PV–wind–PHS systems. during the time steps when arbitrage (charge or discharge) was selected, this setpoint had no meaning. Minimum hydro turbine load: minimum percentage of the hydro turbine rated power to supply the net load (\(P_{\%}\)). For each time step, the net load \(P_{Net\_load}(t)\) is defined as the load minus the renewable power. \[P_{Net\_load}(t)=\max\left(0,P_{L}(t)-P_{PV}(t)-P_{WT}(t)\right) \tag{1}\]  
The net renewable power is expressed as  
\[P_{Net\_ren}(t)=\max\left(0,P_{PV}(t)+P_{WT}(t)-P_{L}(t)\right) \tag{2}\]  
The energy arbitrage performances of the two types of systems are explained as follows (see Table 1). **Type A:** Energy arbitrage is considered during each time step. If the purchase electricity price is higher than the high set point (\(Pr_{E\_purch}(t)>Pr_{E\_purch\_High\_stp}\)) \(\rightarrow\)**'arbitrage: discharge'**, the priority is injecting the (sell) electricity into the grid (net renewable generation + hydro turbine at maximum power). If the purchase electricity price is lower than the low set point (\(Pr_{E\_purch}(t)\leq Pr_{E\_purch\_Low\_stp}\)) \(\rightarrow\)**'arbitrage: charge'**, the priority is storing the energy (to pump water with renewable surplus power).