## 5 Case Studies  
The proposed method is tested based on real data collected from an industrial park near the city of Xining, Qinghai Province, China. Electricity and heat demands in four typical days in spring, summer, autumn and winter are used. The maximum electricity/heat demand is 240 MW/200 MW, both happening in winter. The solar radiation data in 2019 is available at [53]. Parameters of system components are provided in Table 1. Using the model in Section 4.1, the CHP unit must have a rated electrical capacity of 273 MW and thermal capacity of 450 MW. As the electrical and thermal outputs of CHP unit are not independent, the capacity is larger than the respective maximum demands. The cost function is \(\mathrm{C}(p,h)=0.09p^{2}+280p+0.05h^{2}+70h+0.03 ph+12000\) CNY. Next, we investigate the value of EES and TES. If the demand is served by the CHP, the storage is barely dispatched because of the inevitable energy losses. Only when solar energy output exceeds the demand could the value of energy storage be reflected. In this bunch of tests, we first examine the accuracy of the proposed method in Algorithm 1. The capacity of PV panel and STC is fixed at 300 MW. We use different number of initial samples. The surfaces and critical regions of the obtained optimal value function (OVF) are portrayed in Figure 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**ESS Unit**} & \multicolumn{2}{c}{**TES Unit**} & \multicolumn{2}{c}{**CHP Unit**} \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(\eta_{E}^{c}/\eta_{d}^{c}\) & 0.95/0.95 & \(\eta_{L}^{h}/\eta_{d}^{h}\) & 0.98/0.98 & \(\eta_{P}^{g}\) & 0.30 \\ \(\beta_{t}^{s}/\beta_{m}^{c}\) & 0.20/0.95 & \(\beta_{t}^{h}/\beta_{m}^{h}\) & 0.10/1.00 & \(\eta_{R}^{g}\) & 0.90 \\ \(\beta_{Em}^{cd}\) & 0.20 & \(\beta_{hm}^{cd}\) & 0.20 & \(\alpha_{h}^{h}\) & 0.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of System components. With \(16\times 21\) initial samples, the OVF retrieved from Algorithm 1 is exact. For verification, we solve problem (2) with \(101\times 101\) sampled values of \(\theta\), and compare the optimum with the optimal value offered by the OVF in Figure 2c, the maximum relative error is below \(10^{-4}\). Then, we reduce the number of initial samples to \(9\times 11\); the same result is obtained, demonstrating that the proposed method does not rely on a set of very dense sampling points, and is robust once sufficient samples are available. Figure 2: OVFs and critical regions (CRs) with different initial samples. provided. However, if the number of samples is insufficient, say \(3\times 3\), the result is shown in Figure 2a: the approximate OVF underestimates the true one, and fewer critical regions are found. From Figure 2, the value of energy storage is clear. Without energy storage, the operation cost is as high as 902,698 CNY. With the increase of the sizes of the EES unit and TES unit, the cost gradually decreases. When \(E_{S}=621.3\) MWh and \(H_{S}=857.7\) MWh, the cost reduces to the minimum of 785,671 CNY. Further enlarging storage capacity brings no additional benefit, as there is no more energy to be stored. It is also observed that the marginal cost reduction diminishes with the increase of storage capacity, which follows from the convexity of the OVF. The impact of solar generation is investigated. The capacities of PV panel and STC are changed from 100 MW to 500 MW; the results are shown in Figure 3 and Table 2. When \(C_{PV}=C_{STC}=100\) MW, the electric load is always larger than solar power output, so EES is not dispatched. The cost decreases with a larger TES capacity, and reaches the minimum when \(H_{S}=105\) MWh; further increasing \(H_{S}\) has no impact on the cost, because there is no surplus energy to be stored. The maximum cost reduction is only 0.2%. When \(C_{PV}=C_{STC}=200\) MW, the minimum cost is achieved at \(E_{S}=40\) MWh and \(H_{S}=570\) MWh. The cost reduction is mainly attributed to the TES unit.