## 3 Model Development
### Decision Variables and Model Linearization  
The operation model of the CGS established in this paper is mixed integer nonlinear programming (MINLP). The principal decision variables are the output of the power sources: \(P_{u,t}^{\text{TPU}}\) for TPUs, \(P_{t}^{\text{WT}}\) for WT, and \(P_{t}^{\text{PV}}\) for PV. To address the MINLP, the big-M method [33] and the piecewise linearization method [34] are employed to reformulate the model into a mixed integer linear optimization programming (MILP) for an easier solution. Specifically, for the generation revenue of TPU in different compensation standards, as outlined in Equations (3) and (4), binary variables \(\beta_{u,t}^{h_{0}}\), \(\beta_{u,t}^{h_{1}}\), and \(\beta_{u,t}^{h_{2}}\) are introduced. These variables serve to indicate the TPU's current operation status within the non-DPR compensation state as well as the first and second-tier DPR compensation states, respectively. By also introducing a sufficiently large positive parameter, \(M\), the original constraints can be transformed into  
\[\left\{ \begin{array}{l}\beta_{u,t}^{h_{0}}+\beta_{u,t}^{h_{1}}+ \beta_{u,t}^{h_{2}}=1\\ \mu_{1}P_{u,t}^{\text{TPU}}P_{u}^{\text{TPU,max}}-M\Big{(}1-\beta_{u,t}^{h_{0} }\Big{)}\leq P_{u,t}^{\text{TPU}}\leq\alpha_{u,t}^{\text{TPU}}P_{u}^{\text{ TPU,max}}+M\Big{(}1-\beta_{u,t}^{h_{0}}\Big{)}\\ \mu_{2}P_{u,t}^{\text{TPU}}P_{u}^{\text{TPU,max}}-M\Big{(}1-\beta_{u,t}^{h_{1} }\Big{)}\leq P_{u,t}^{\text{TPU}}\leq\mu_{1}\alpha_{u,t}^{\text{TPU}}P_{u}^{ \text{TPU,max}}+M\Big{(}1-\beta_{u,t}^{h_{1}}\Big{)}\\ -M\Big{(}1-\beta_{u,t}^{h_{2}}\Big{)}\leq P_{u,t}^{\text{TPU}}\leq\mu_{2}\alpha _{u,t}^{\text{TPU}}P_{u}^{\text{TPU,max}}+M\Big{(}1-\beta_{u,t}^{h_{2}}\Big{)} \\ F_{u,t}^{\text{TPU,GI}}\leq C_{0}P_{u,t}^{\text{TPU}}+M\Big{(}1-\beta_{u,t}^{h_{ 0}}\Big{)}\\ F_{u,t}^{\text{TPU,GI}}\leq C_{0}P_{u,t}^{\text{TPU}}+C_{1k}\Big{(}\mu_{1} \alpha_{u,t}^{\text{TPU}}P_{u}^{\text{TPU,max}}-P_{u,t}^{\text{TPU}}\Big{)}+ M\Big{(}1-\beta_{u,t}^{h_{1}}\Big{)}\\ F_{u,t}^{\text{TPU,GI}}\leq C_{0}P_{u,t}^{\text{TPU}}+C_{2k}\Big{(}\mu_{1} \alpha_{u,t}^{\text{TPU}}P_{u}^{\text{TPU,max}}-P_{u,t}^{\text{TPU}}\Big{)}+ M\Big{(}1-\beta_{u,t}^{h_{2}}\Big{)}\\ F_{u,t}^{\text{TPU,GI}}\leq C_{0}P_{u,t}^{\text{TPU}}+C_{2k}\Big{(}\mu_{1} \alpha_{u,t}^{\text{TPU,max}}-P_{u,t}^{\text{TPU}}\Big{)}-M\Big{(}1-\beta_{u,t}^{h_ {2}}\Big{)}\end{array} \right.. \tag{18}\]For the operation cost of the TPU in Equation (6), piecewise linearization is used with the help of the big-M method. Specifically, the aforementioned cost function is segmented into \(H_{1}\) and \(H_{2}\) parts based on the output range of TPU in RPR state \(P_{u,t}^{\text{TPU}}\in\left[P_{u}^{\text{TPUR,min}},P_{u}^{\text{TPU,max}}\right]\) and DPR state \(P_{u,t}^{\text{TPU}}\in\left[P_{u}^{\text{TPUD,min}},P_{u}^{\text{TPUR,min}}\right]\) respectively, with corresponding indices \(h_{1}\) and \(h_{2}\). For the start of each segment, the output power at those points are \(P_{u,t}^{\text{TPU}}(h_{1}=1)\), \(P_{u,t}^{\text{TPU}}(h_{1}=2)\),..., \(P_{u,t}^{\text{TPU}}(h_{1}=H_{1})\), and \(P_{u,t}^{\text{TPU}}(h_{2}=1)\), \(P_{u,t}^{\text{TPU}}(h_{2}=2)\),..., \(P_{u,t}^{\text{TPU}}(h_{2}=H_{2})\), the corresponding cost at those points can be calculated as \(f_{u}^{\text{OP,RPR}}\left[P_{u,t}^{\text{TPU}}(h_{1}=1)\right],\ldots,f_{u}^{ \text{OP,RPR}}\left[P_{u,t}^{\text{TPU}}(h_{1}=H_{1})\right]\) and \(f_{u}^{\text{OP,DPR}}\left[P_{u,t}^{\text{TPU}}(h_{2}=1)\right],\ldots,f_{u}^{ \text{OP,DPR}}\left[P_{u,t}^{\text{TPU}}(h_{2}=H_{2})\right]\).