## 3 Model Development
### Objective Function
Given these settings, the power balance constraints disregarding the integration are as follows:  
\[\left\{ \begin{array}{l}\sum_{u=1}^{N_{\text{TPU}}}P_{u,t}^{\text{TPU}}=P_{t} ^{\text{L*}}-\left(P_{t}^{\text{PV,P}}+P_{t}^{\text{WT,P}}\right)\\ P_{t}^{\text{c,min}}\leq P_{t}^{\text{WT}}-P_{t}^{\text{WT,P}}+P_{t}^{\text{ PV}}-P_{t}^{\text{PV,P}}\leq P_{t}^{\text{c,max}}\end{array} \right., \tag{11}\]  
where \(P_{t}^{\text{c,max}}\) and \(P_{t}^{\text{c,min}}\) represents the upper and lower limits of the main power grid's capability to accommodate fluctuations from RESs. Due to limitations imposed by power-flow constraints and insufficient information, power sources within the main grid often struggle to coordinate efficiently without coupling coordination, resulting in these capability limits for accommodating fluctuations being typically low. (b) After coupling integration, the main power grid provides a consolidated dispatch instruction for the CGS. The outputs from power sources within the CGS are optimally coordinated to fulfill the requirements of this unified dispatch instruction. In this setup, the flexible capabilities of TPUs can be directly leveraged to mitigate fluctuations. The power balance constraint considering the coupling integration is as follows:  
\[\sum_{u=1}^{N_{\text{TPU}}}P_{u,t}^{\text{TPU}}+P_{t}^{\text{WT}}+P_{t}^{\text {PV}}+P_{t}^{\text{GS}}=P_{t}^{\text{L*}}+P_{t}^{\text{GO}}, \tag{12}\]  
where \(P_{t}^{\text{GO}}\geq 0\). The constraints of TPUs comprise output constraints, ramping constraints, flexible constraints, and on-off state constraints, which are formulated as  
\[\left\{ \begin{array}{l}P_{u,t+1}^{\text{TPU}}-P_{u,t}^{\text{TPU}}\geq- \sigma_{u,t+1}^{\text{TPU}}R_{u}^{\text{I}}\Delta t+\left(\sigma_{u,t+1}^{ \text{TPU}}-\sigma_{u,t}^{\text{TPU}}\right)P_{u}^{\text{TPUR,min}}\\ P_{u,t+1}^{\text{TPU}}-P_{u,t}^{\text{TPU}}\leq\sigma_{u,t}^{\text{TPU}}R_{u}^ {\text{I}}\Delta t+\left(\sigma_{u,t+1}^{\text{TPU}}-\sigma_{u,t}^{\text{TPU }}\right)P_{u}^{\text{TPUR,min}}\\ \frac{R_{u}^{\text{II}}}{R_{u}^{\text{II}}}P_{u,t}^{\text{TPU}}-P_{u,t+1}^{ \text{TPU}}\leq\sigma_{u,t}^{\text{TPU}}\Big{[}R_{u}^{\text{II}}\Delta t- \left(1-\frac{R_{u}^{\text{II}}}{R_{u}^{\text{II}}}\right)P_{u}^{\text{TPUR, min}}\Big{]}-\left(\sigma_{u,t+1}^{\text{TPU}}-\sigma_{u,t}^{\text{TPU}}\right) \frac{R_{u}^{\text{II}}}{R_{u}^{\text{II}}}P_{u}^{\text{TPUR,min}}\\ \end{array} \right., \tag{13}\]  
\[\left\{ \begin{array}{l}P_{u,t}^{\text{TPU}}=P_{u,t}^{\text{TPU},B}+P_{u,t} ^{\text{TPU,F}}\\ -\tau_{u}^{\text{TPU}}\Delta P_{u}^{\text{TPU,max}}\leq P_{u,t}^{\text{TPU,F}} \leq\tau_{u}^{\text{TPU}}\Delta tP_{u}^{\text{TPU,max}}\\ 0_{u,t}^{\text{TPU}}P_{u}^{\text{UD,min}}\leq P_{u,t}^{\text{TPU}}\leq\sigma_ {u,t}^{\text{TPU}}P_{u}^{\text{TPU,max}}\\ \end{array} \right., \tag{14}\]  
\[\left\{ \begin{array}{l}\left(\sigma_{u,t}^{\text{TPU}}-\sigma_{u,t+1}^{ \text{TPU}}\right)\left(D_{u,t}^{\text{TPU,on}}-D_{u}^{\text{TPU,on,min}} \right)\geq 0\\ \left(\sigma_{u,t+1}^{\text{TPU}}-\sigma_{u,t}^{\text{TPU}}\right)\left(D_{u,t }^{\text{TPU,off}}-D_{u}^{\text{TPU,off,min}}\right)\geq 0\end{array} \right., \tag{15}\]  
where Equation (13) outlines the ladder-type ramping constraints for the TPU that satisfies \(\Delta t\geq\left(P_{u}^{\text{TPUR,min}}-P_{u}^{\text{TPU,min}}\right)/R_{u}^ {\text{II}}\)[30]. It represents the maximum ramping rate variation in the TPU under different operation states. Specifically, the maximum ramping rate is \(R_{u}^{\text{I}}\) when \(P_{u}^{\text{TPUR,min}}\leq P_{u,t}^{\text{TPU}}\leq P_{u}^{\text{TPU,max}}\), and \(R_{u}^{\text{II}}\) when \(P_{u}^{\text{TPU,min}}\leq P_{u,t}^{\text{TPU}}\leq P_{u}^{\text{TPUR,min}}\). The variation in maximum ramping rates is attributable to the fact that a reduction in TPU output during the DPR state can lead to increased losses and diminished performance [31].