## 2 The TPUMSTSR Under RLV Model
### Internal Operation Model of MCSs
#### 3.3.1 Objective Function
\begin{cases}\text{C}_{t}^{\text{TN,D}}\frac{P_{i,u,t}^ {\text{TPU,C}}}{\sum\limits_{g\in G}c_{g,g}^{\text{TE}}},\mu_{1}<\mu_{i,u,t}^ {\text{TPU}}\leq 1\\ 0,0<\mu_{i,u,t}^{\text{TPU}}\leq\mu_{1}\end{cases}  \tag{19}\]  
\(C_{t}^{\text{TN,D}}\) is computed as follows:  
\[C_{t}^{\text{TN,D}}=\sum_{i=1}^{I}\sum_{u=1}^{U}f_{i,u,t}^{\text{UD}} \tag{20}\]  
\(P_{i,u,t}^{\text{TPU,C}}\) is given by the following equation:\[P_{i,u,t}^{\text{TPU,C}}=\left\{ \begin{array}{ll}0,&0<\mu_{i,u,t}^{\text{TPU}} \leq\mu_{1}\\ z_{1}^{\text{TPU}}P_{i,u,t}^{\text{TPU},\text{max}}+z_{2}^{\text{TPU}}(P_{i,u,t }^{\text{TPU}}-A_{1}P_{i,u,t}^{\text{TPU,max}}),&P_{i,u,t}^{\text{TPU}}\in[P_{ i,u,t}^{\text{TPU,min}},A_{1}P_{i,u,t}^{\text{TPU,max}}],\mu_{i,u,t}^{\text{TPU}} \geq\mu_{1}\\ z_{1}^{\text{TPU}}A_{1}P_{i,u,t}^{\text{TPU,max}}+z_{2}^{\text{TPU}}(A_{2}-A_{ 1})P_{i,u,t}^{\text{TPU,max}}&P_{i,u,t}^{\text{TPU}}\in[A_{1}P_{i,u,t}^{\text{ TPU,max}},A_{2}P_{i,u,t}^{\text{TPU,max}}],\mu_{i,u,t}^{\text{TPU}} \geq\mu_{1}\\ +z_{3}^{\text{TPU}}(P_{i,u,t}^{\text{TPU}}-A_{1}P_{i,u,t}^{\text{TPU,max}}),&P _{i,u,t}^{\text{TPU,max}}\in[A_{2}P_{i,u,t}^{\text{TPU,max}},P_{i,u,t}^{\text {TPU,max}}],\mu_{i,u,t}^{\text{TPU}}\geq\mu\end{array} \right. \tag{21}\]  
where \(A_{1}\) and \(A_{2}\) represent different allocation tiers; \(z_{1}^{\text{TPU}}\), \(z_{2}^{\text{TPU}}\), and \(z_{3}^{\text{TPU}}\) are correction coefficients for thermal power; \(P_{i,u,t}^{\text{TPU}}\) is the actual power output; and \(P_{i,u,t}^{\text{TPU,max}}\) and \(P_{i,u,t}^{\text{TPU,min}}\) are the maximum and minimum power outputs, respectively. \(P_{g,t}^{\text{C}}\) is expressed as follows:  
\[\left\{ \begin{array}{ll}P_{i,t}^{\text{C}}=\sum\limits_{u=1}^{U}P_{i,u,t}^{ \text{TPU,C}}+P_{i,t}^{\text{WT,C}}+P_{i,t}^{\text{PV,C}}\\ P_{g,t}^{\text{C}}=\sum\limits_{i=1}^{U}P_{i,t}^{\text{C}}+\sum\limits_{d=1}^{D }P_{d,t}^{\text{C}}\end{array} \right. \tag{22}\]  
where \(P_{i,t}^{\text{C}}\) is the corrected output of the CS; \(d\) indexes the IPPs connected to the main grid; \(D\) is the total number of such sources; and \(P_{d,t}^{\text{C}}\) is the corrected output of the \(d\) IPP. (5) Comprehensive Operational Costs of TPUMSTSR Systems \(f_{i,e,t}^{\text{HS}}\)  
The integration of TPUMSTSR systems introduces comprehensive operational costs, including maintenance costs \(f_{i,e,t}^{\text{OB}}\) and lifespan degradation costs \(f_{i,e,t}^{\text{LB}}\)  
\[f_{i,e,t}^{\text{HS}}=\sum\limits_{e=1}^{E}\left(f_{i,e,t}^{\text{OB}}+f_{i,e, t}^{\text{LB}}\right) \tag{23}\]  
where \(e\) represents the index of TPUMSTSR and \(E\) denotes the total number of TPUMSTSR. Maintenance Costs \(f_{i,e,t}^{\text{OB}}\)  
Maintenance costs are incurred during the charging and discharging processes of TPUMSTSR:  
\[f_{i,e,t}^{\text{OB}}=\text{sum}\left(\left|P_{i,e,t}^{\text{HD}}+P_{i,e,t}^{ \text{HC}}+P_{i,e,t}^{\text{LD}}+P_{i,e,t}^{\text{LC}}\right|\right)C_{i,e}^{ \text{OB}} \tag{24}\]  
where \(C_{i,e}^{\text{OB}}\) is the maintenance price per cycle for the TPUMSTSR system. \(\text{\textcircled{2}}\) Lifespan Degradation Costs \(f_{i,e,t}^{\text{LB}}\)  
Lifespan degradation costs arise from the wear and tear that occur during the charging and discharging cycles:  
\[f_{i,e,t}^{\text{LB}}=\frac{C_{i,e}^{\text{LB}}}{V_{i,e}^{\text{LC}}}\frac{s_ {e,t}^{\text{HB}}+s_{e,t}^{\text{LB}}}{2} \tag{25}\]  
where \(C_{i,e}^{\text{LB}}\) is the initial investment cost, \(V_{i,e}^{\text{LC}}\) is the total number of life cycles, and \(s_{e,t}^{\text{HB}}/s_{e,t}^{\text{LB}}\) is a binary variable indicating the state change (1 for a change, 0 for no change) of the TPUMSTSR system at time \(t\). (6) CA for Renewable Energy Systems \(f_{i,t}^{\text{WTS}}+f_{i,t}^{\text{PVS}}\)  
The CA for renewable energy systems, including wind and PV, follows a similar methodology as for TPUs:\[\left\{