## 2 The TPUMSTSR Under RLV Model
### Internal Operation Model of MCSs
#### 3.3.1 Objective Function
\begin{array}{ll}f_{i,u,t}^{\rm UD}=\left\{\begin{array}{ll}0&,&\mu_{1}< \mu_{i,u,t}^{\rm TPU}\leq 1\\ C_{1}k_{m}^{\rm TPU}(\mu_{1}P_{i,u}^{\rm TPU,max}-P_{i,u,t}^{\rm TPU})&,&\mu_ {2}<\mu_{i,u,t}^{\rm TPU}\leq\mu_{1}\\ C_{2}k_{m}^{\rm TPU}(\mu_{1}P_{i,u}^{\rm TPU,max}-P_{i,u,t}^{\rm TPU})&,&0<\mu _{i,u,t}^{\rm TPU}\leq\mu_{2}\end{array}\right.\\ \\ \mu_{i,u,t}^{\rm TPU}=\frac{P_{i,u,t}^{\rm TPU}}{P_{i,u}^{\rm TPU,max}}\end{array}  \tag{16}\]  
where \(P_{i,u,t}^{\rm TPU,max}\) is the maximum output of the thermal power unit; \(C_{1}\) and \(C_{2}\) are the unit electricity sales prices corresponding to different compensation thresholds, which can only be obtained if the unit's output is below the corresponding threshold; \(\mu_{i,u,t}^{\rm TPU}\) is the load rate of the thermal power unit within the CS; \(\mu_{1}\) and \(\mu_{2}\) are the load rates corresponding to compensation thresholds; and \(k_{m}^{\rm TPU}\) is the seasonal revenue adjustment coefficient. (2) Operating Costs of Thermal Power Units \(f_{i,u,t}^{\text{UO}}\)  
This cost accounts for two operational states: regular peak regulation (RPR) and DPR. In RPR mode, operating costs only consider the fuel consumption of the unit, whereas in DPR mode, additional losses due to output reduction are also included. The operating cost \(f_{i,u,t}^{\text{UO}}\) is calculated as follows:  
\[f_{i,u,t}^{\text{UO}}= \begin{cases}&\text{C}^{\text{coal}}[a_{i,u}^{\text{RPR}} (P_{i,u,t}^{\text{BURN}})^{2}+b_{i,u}^{\text{RPR}}P_{i,u,t}^{\text{BURN}}+c_{ i,u}^{\text{RPR}}]\quad\mu_{1}<\mu_{i,u,t}^{\text{TPU}}\leq 1\\ &\text{C}^{\text{coal}}[a_{i,u}^{\text{RPR}}(P_{i,u,t}^{\text{BURN}})^{2}+b_{i,u }^{\text{RPR}}P_{i,u,t}^{\text{BURN}}+c_{i,u}^{\text{RPR}}]\quad\quad 0<\mu_{i,u,t}^{ \text{TPU}}\leq\mu_{1}\\ &+a_{i,u}^{\text{DPR}}P_{i,u,t}^{\text{BURN}}+b_{i,u}^{\text{DPR}}\end{cases}  \tag{17}\]  
where \(\text{C}^{\text{coal}}\) is the unit price of coal per ton; \(a_{i,u}^{\text{RPR}}\), \(b_{i,u}^{\text{RPR}}\), and \(c_{i,u}^{\text{RPR}}\) are the fitting coefficients for RPR loss; and \(a_{i,u}^{\text{DPR}}\) and \(b_{i,u}^{\text{DPR}}\) are the fitting coefficients for DPR loss. The operating costs under RPR mode mainly consider coal consumption and purchase costs. Under DPR mode, additional losses incurred due to DPR are also considered [39]. (3) Pollutant Emission Costs of Thermal Power Units \(f_{i,u,t}^{\text{UE}}\)  
Coal-fired thermal power generation results in the emission of various pollutants, necessitating an assessment of pollutant emission costs:  
\[f_{i,u,t}^{\text{UE}}=\sum_{h=1}^{H}c_{h}^{\text{TE}}\frac{\rho_{h}^{\text{TE} }}{\tau_{h}^{\text{TE}}}(a_{h}^{\text{TE}}P_{i,u,t}^{\text{BURN}}+b_{h}^{\text {TE}}) \tag{18}\]  
where \(c_{h}^{\text{TE}}\), \(\rho_{h}^{\text{TE}}\), and \(\tau_{h}^{\text{TE}}\) denote the unit treatment cost, unit generation emission factor (kg/MW), and pollutant conversion coefficient, respectively; \(a_{h}^{\text{TE}}\) and \(b_{h}^{\text{TE}}\) are pollutant fitting coefficients; \(h\) represents pollutant types; and \(H\) is the total number of pollutants. The considered pollutants include dust, SO\({}_{2}\), and NO\({}_{\text{x}}\). (4) CA of Thermal Power Units \(f_{i,u,t}^{\text{US}}\)  
The compensation revenue for thermal power units participating in DPR must be shared among external IPPs, non-participating thermal power units within the CS, and renewable energy units. Given the capacity differences among internal power sources in the CS, a fair and reasonable CA method requires determining the corrected output of each internal power source involved in the allocation. Ultimately, the PR compensation revenue is distributed based on the proportion of the corrected output of each power source. Based on the existing ancillary service market rules applicable to generation entities in northeast China [22], the mathematical expression for computing the CA of thermal power units that do not participate in DPR is formulated as follows:  
\[f_{i,u,t}^{\text{US}}=