## 4 Case Study
### Analysis of the Thermal Energy Storage System in Power Units  
In this section, a 600 MW thermal power unit in CSs 3 and 4 is equipped with TPUM-STSR under RLVs, with parameters consistent with Scheme 2 from the previous section. To investigate the effectiveness of multi-stage economic dispatch in an MCS, the Gurobi solver is employed to solve the decoupled and nonlinear relaxed model. To verify the robustness of the method proposed in this study, typical summer and winter days with different load curves and renewable energy output are selected for a case study; specifically, for the summer non-heating period (May to October), the PR revenue correction coefficient \(k_{\rm m}^{\rm TPU}=0.5\), the wind power output correction coefficient \(z_{t}^{\rm WTS}=0.8\), and the PV output correction coefficient \(z_{t}^{\rm PV,S}=1\); for the winter heating period (November to April of the following year), these coefficients are set to \(k_{\rm m}^{\rm TPU}=1\), \(z_{t}^{\rm WTS}=1.6\), and \(z_{t}^{\rm PVS}=2\). The simulation was carried out on a desktop workstation with an AMD Ryzen 7 5700G CPU @ 3.8 GHz and 16 GB RAM. The overall computation time was 588 s (9.8 min) and the total number of iterations was 400,000. The computation time for each MCS was less than 10 min, meeting the real-time dispatch requirements. The detailed power outputs of various generators over two typical days are illustrated in Figure 9 (thermal power unit numbering follows the sequence in Table 2). As shown in Figure 9, the upper-level subgraphs represent the output of the internal power sources of coupling systems 1 and 2 from left to right, respectively, and the lower-level subgraphs represent the output of the internal power sources of coupling systems 3 and 4 from left to right, respectively. The power output of each CS aligns with dispatch demands at each time period on two typical days, and the wind/PV generation, follow a similar trend to their maximum output curves. This demonstrates that the proposed three-stage optimization approach effectively maintains dispatch feasibility while preserving privacy among multiple entities, achieving efficient coordination of coal-fired and renewable units. In the typical summer scenario, wind power output within each CS remains relatively low throughout the day, generally staying within 100 MW. PV output starts increasing at 5:00 a.m., reaching its peak (100-300 MW) between 11:00 a.m. and 1:00 p.m. Dispatch demand remains low from 1:00 a.m. to 7:00 a.m., increases from 7:00 a.m. to 10:00 a.m., and stabilizes between 11:00 a.m. and 11:00 p.m. The midday renewable energy output can offset part of the dispatch demand, but thermal power units primarily balance the load throughout the day. Only from 4:00 a.m. to 8:00 a.m. do CSs with flexibility-retrofitted units participate in DPR, leading to lower CA. Simultaneously, the corrected output imposes constraints on DPR, thereby preventing the CS from universally mandating that its internally flexible retrofitted units continuously engage in DPR solely as a profit-seeking mechanism and fostering a healthy competitive market environment. Owing to the relatively low value of \(k_{\rm m}^{\rm TPU}\) during summer and the modest DPR revenue from thermal power units, DPR is infrequently employed to reduce operational losses. Meanwhile, in CSs 3 and 4, the higher charge/discharge frequency of the MSTS system expands the electrical output range of the thermal power units, thereby extending the duration of their participation in DPR. Consequently, the DPR revenue for each CS is enhanced, and the corresponding CA is also increased. In the typical winter scenario, wind power output in each CS remains high during most periods, with a slight decrease at midday. During this time, PV generation, which experiences a relative increase at midday, compensates for the dip in wind output, resulting in overall renewable energy output that is comparable to that observed in summer.