# Game Approach to HDR-TS-PV Hybrid Power System Dispatching
## 4 Case Study
### Dispatching Results  
Each payoff function and characteristic function is linear programming or mixed-integer linear programming, which was solved by software MATLAB2016b and CPLEX12.8. The results are shown in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**System Parameters** & **Values** \\ \hline Installed capacity of \(HDR\) power station \(P_{H}^{install}\) (MW) & 6.6 \\ Installed capacity of \(PV\) plant \(P_{PV}^{install}\) (MW) & 200 \\ Installed capacity of \(TS\) power station \(P_{H}^{install}\) (MW) & 6.6 \\ Brine temperature at the producing well \(T_{HDR}\) (\({}^{\circ}\)C) & 200 \\ Brine mass flow \(m_{t}^{t}\) (kg/s) & 75 \\ Brine temperature at the reinjection well \(T_{Well}\) (\({}^{\circ}\)C) & 75 \\ Initial temperature of \(HTO\)\(T_{I}\) (\({}^{\circ}\)C) & 25 \\ Specific heat capacity of \(HTO\)\(c_{po}\) (k/(kg.\({}^{\circ}\)C) & 1.938 \\ Specific heat capacity of brine \(c_{pr}\) (k/(kg.\({}^{\circ}\)C) & 4.2 \\ \(ORC\) generator efficiency \(\eta_{P}\) (\%) & 13.2 [12] \\ Heat preservation coefficient \(\eta_{h}\)(\%) & 99 \\ Efficiency of \(TESE\)\(\eta_{ex}\) (\%) & 90 \\ Power predicted error of the \(PV\) plant \(\varphi\) (\%) & 20 \\ Allowed fluctuation rate \(\sigma\) (\%) & 0–10 \\ Penalty coefficient \(p\) & 2–5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. Figure 4: Solar irradiance and \(TOU\) price curve. It can be observed from Table 2 that the payoff of the _TS_ power station is zero because it cannot profit in the non-cooperative game. The payoff of the _HDR-PV_ coalition is lower than the sum of the two non-cooperative game payoffs, i.e.,  
\[V_{\{\text{H,P}\}}<I_{\text{H}}+I_{\text{P}}. \tag{36}\]  
These results show that the _HDR_ power station and the _PV_ plant cannot form a stable coalition. Figure 5 shows the results for the coordinated operation of the _HDR-PV_ coalition. The _HDR_ power station provides reserves to increase the credible capacity of the _PV_ plant by replacing geothermal energy with the curtailed power and the abandoned loads. However, the cost of abandoning geothermal energy is greater than the sum of the curtailed power and penalty for abandoning loads. Therefore, the coordinated operation of the _HDR_ power station and _PV_ plant cannot increase the overall payoff, making the _HDR-PV_ coalition unstable. When the _TS_ power station joins the _HDR-PV_ coalition to form a grand coalition, the abandoned geothermal energy used to stabilize the fluctuation of the _PV_ plant is reused by the _TS_ power station to generate new profits. The results of the grand coalition cooperative game dispatching are shown in Figure 6. During the low TOU price period (0:00-7:00), both the _HDR_ and _TS_ power stations maintain minimum output power. The continuous geothermal energy is stored in the thermal energy storage tank of the _TS_ power station while ensuring the safety of the geothermal mining cycle. During the _PV_ generation period (8:00-18:00), the _HDR_ and _TS_ power stations provide reserves corresponding to the credible output power of the _PV_ plant. The _HDR_ power station is then operated at full power to maximize profits. The _TS_ power station converts all the stored thermal energy into electricity during the high TOU price period (19:00-22:00). In the end, the three players tend to form an _HDR-TS-PV_ grand coalition. The overall payoff of the hybrid power system in each game form is shown in Table 3.  
The coordinated dispatching results of the grand coalition obtained by the cooperative game dispatching model proposed in this paper are shown in Figure 7.  
Figure 5: The coordinated operation of hot dry rock-photovoltaic (_HDR-PV_) coalition. where \(core(G)\) is the core set of the coalition and \(\varepsilon\) is the relaxation variable [44]. If the lower bound of \(\varepsilon\) is equal to zero, then the \(core(G)\) is non-empty before relaxation. Thus, the coalition is stable.