## 3 Model Development
### Objective Function  
From the perspective of CGS operators, their operation goal is to achieve the economic operation of the system. Therefore, the objective function is formulated to maximize total annual revenues, taking into account the generation of revenue from each power source, revenue from DPR auxiliary service revenue, operational costs, and pollution emission costs. The objective function \(F^{\text{A}}\) is presented as follows:  
\[\text{max}F^{\text{A}}=F^{\text{TPU,GI}}-F^{\text{TPU,OP}}-F^{\text{TPU,ON}}-F ^{\text{TPU,E}}+F^{\text{WT,GI}}+F^{\text{PV,GI}}-F^{\text{GU}}, \tag{1}\]  
In (1), \(F^{\text{TPU,GI}}\) is the total generation revenue of TPUs; it is related to the real-time DPR service they can provide at a time step [25]  
\[F^{\text{TPU,GI}}=\sum_{d=1}^{T}\sum_{u=1}^{N_{\text{TPU}}}F^{\text{TPU,GI}}_{ u,t}\Delta t, \tag{2}\]  
\[\mu^{\text{TPU}}_{u,t}=\frac{P^{\text{TPU}}_{u,t}}{\sigma^{\text{TPU}}_{u,t}P^ {\text{TPU,max}}_{u}}, \tag{3}\]  
\[F^{\text{TPU,GI}}_{u,t}= \begin{cases}C_{0}P^{\text{TPU}}_{u,t}&\mu_{1}<\mu^{ \text{TPU}}_{u,t}\leq 1\\ C_{0}P^{\text{TPU}}_{u,t}+C_{1}k_{\text{t}}\left(\mu_{1}\sigma^{\text{TPU}}_{u,t }P^{\text{TPU,max}}_{u}-P^{\text{TPU}}_{u,t}\right)&\mu_{2}<\mu^{\text{TPU}}_{ u,t}\leq\mu_{1}\\ C_{0}P^{\text{TPU}}_{u,t}+C_{2}k_{\text{t}}\left(\mu_{1}\sigma^{\text{TPU}}_{u,t }P^{\text{TPU,max}}_{u}-P^{\text{TPU}}_{u,t}\right)&0\leq\mu^{\text{TPU}}_{ u,t}\leq\mu_{2}\end{cases} . \tag{4}\]  
The function of Equation (4) serves to calculate the revenue that a startup TPU can obtain. \(F^{\text{TPU,GI}}_{t}\) comprises basic revenue and DPR service revenue. The basic revenue corresponds to \(\mu^{\text{TPU}}_{u,t}\), and the unit price is \(C_{0}\). The first line in the equation representsDPR service revenue only, which can be obtained when the output rate of a TPU \(\mu_{u,t}^{\text{TPU}}\) is lower than the compensation standard \(\mu_{1}\). The second line of the equation indicates that the current active output of TPU can obtain additional first-tier DPR service revenue with a unit price of \(C_{1}\). This revenue is calculated based on the difference between the active output corresponding to the compensation standard \(\mu_{1}P_{u}^{\text{TPU,max}}\) and the current active output \(P_{u,t}^{\text{TPU}}\). The third line of Equation (4) represents the current active output of TPU, which is lower than the compensation standard \(\mu_{2}\), and can obtain second-tier DPR service revenue. The calculation method remains consistent with the first tier, except that the unit price changes to \(C_{2}\). \(F^{\text{TPU,OP}}\) is the operation cost of the startup TPUs, encompassing coal consumption, maintenance expenses, and extra costs associated with the DPR state [26]. \[F^{\text{TPU,OP}}=\sum_{t=1}^{T}\sum_{u=1}^{N_{\text{TPU}}}\left[\beta_{u,t}^ {\text{RPR}}f_{u}^{\text{OP,RPR}}\Big{(}P_{u,t}^{\text{TPU}}\Big{)}+\beta_{u,t }^{\text{DPR}}f_{u}^{\text{OP,DPR}}\Big{(}P_{u,t}^{\text{TPU}}\Big{)}\right] \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
To accurately describe the scheduling characteristics of power sources before and after the coupling integration of retrofitted TPUs with RESs, this paper establishes the following two power balance constraints:  
(a) Before implementing coupling integration (independent operation), the main power grid dispatches separate instructions to different power sources. For TPUs, the dispatch instructions are based on the forecasted net load, which is calculated as the forecast load demand minus the forecast outputs of WT and PV. The dispatch instructions for WT and PV are derived from their predicted outputs. The flexibility of TPUs is solely employed to mitigate minor fluctuations based on the dispatch instructions without addressing the variability from WT and PV outputs. Fluctuations from WT and PV are partially absorbed by the main power grid. Any excess generation from WT and PV that cannot be accommodated results in curtailment.