## 4 Source-Grid Collaborative Planning Model  
### Objective Function  
Considering frequency security and the retirement of coal-fired power units, the source-grid collaborative expansion planning model's objective function is to minimize the total investment and operational costs \(C^{overall}\). As shown in Equations (23)-(29). The investment costs include the total processing cost of aging units \(C^{re\_total}\), the construction cost of coal-fired power units \(C^{f}\), the construction cost of wind power units \(C^{wind}\), the construction cost of photovoltaic units \(C^{pv}\), and the construction cost of transmission lines \(C^{net}\). The operational costs include fuel costs \(C^{fuel}\), wind curtailment costs \(C^{w\_cut}\), solar curtailment costs \(C^{pv\_cut}\), load shedding costs \(C^{load\_cut}\), and unit start-stop costs \(C^{start}\). \[\min C^{overall}=A(C^{re\_totle}+C^{f}+C^{wind}+C^{pv}+C^{net})+C^{pp}; \tag{23}\]  
\[C^{f}=\sum_{g\in\mathcal{G}_{Gfire}^{cntal}}C_{f}S^{f}_{g}\lambda^{f}_{g}; \tag{24}\]  
\[C^{wind}=\sum_{g\in\mathcal{G}_{Grail}^{cntal}}C_{g}S^{wind}_{g}\lambda^{wind}_ {g}; \tag{25}\]  
\[C^{pv}=\sum_{g\in\mathcal{G}_{Gfire}^{cntal}}C_{pv}S^{pv}_{g}\lambda^{pv}_{g}; \tag{26}\]  
\[C^{net}=\sum_{l\in\mathcal{C}^{cntal}}C_{l}\lambda^{\mathcal{L}}_{l}; \tag{27}\]\[C^{\text{\emph{p}}pe}=C^{\text{\emph{fuel}}}+C^{\text{\emph{w}},\text{\emph{c}ut}}+ C^{\text{\emph{p}}pe\_{\text{c}ut}}+C^{\text{\emph{load}}\_\text{\emph{c}ut}}+C^{ \text{\emph{carbon}}}+C^{\text{\emph{start}}}; \tag{28}\] \[C^{\text{\emph{fuel}}}=\sum_{\rho\in\mathcal{P}}\sum_{t\in T}\sum_{S \in G_{\text{\emph{G}}firn}}C_{g}^{t}P_{g,\rho,t};\] (29) \[C^{\text{\emph{w}}\_\text{\emph{c}ut}}=\sum_{\rho\in\mathcal{P}} \sum_{t\in T}\sum_{S\in G_{\text{\emph{G}}firn}}C_{g}^{\text{\emph{w}}\_\text{ \emph{g}}}P_{g,\rho,t}^{c};\] (30) \[C^{\text{\emph{p}}\_\text{\emph{v}}\_\text{\emph{c}ut}}=\sum_{ \rho\in\mathcal{P}}\sum_{t\in T}\sum_{S\in G_{\text{\emph{G}}firn}}C_{g}^{\text {\emph{p}}\_\text{\emph{v}}\_\text{\emph{g}}}P_{g,\rho,t}^{c};\] (31) \[C^{\text{\emph{load}}\_\text{\emph{c}ut}}=\sum_{\rho\in\mathcal{P}} \sum_{t\in T}\sum_{b\in B}C_{b}^{\text{\emph{h}}}P_{b,\rho,t}^{c};\] (32) \[C^{f}=\sum_{g\in\mathcal{G}_{Gfire}^{cntal}}C_{f}S^{f}_{g}\lambda^{f}_{g}; \tag{24}\]0 (33) \[C^{f}=\sum_{g\in\mathcal{G}_{Gfire}^{cntal}}C_{f}S^{f}_{g}\lambda^{f}_{g}; \tag{24}\]1  
In Equations (23)-(34), \(C_{f}\), \(C_{\text{\emph{w}}}\), and \(C_{\text{\emph{p}}pe}\) represent the construction costs per unit capacity for coal-fired, wind, and photovoltaic power units, respectively. \(S_{g}^{f}\), \(S_{g}^{wind}\), and \(S_{g}^{pv}\) represent the capacities of the constructed coal-fired, wind, and photovoltaic power units, respectively. \(\lambda_{g}^{f}\), \(\lambda_{g}^{wind}\), and \(\lambda_{g}^{pv}\) are the decision variables for the construction of coal-fired, wind, and photovoltaic power units, respectively. \(\mathcal{G}_{\text{\emph{G}}firn}^{\text{\emph{c}}mid}\), \(\mathcal{G}_{\text{\emph{G}}firn}^{\text{\emph{anti}}}\), and \(\mathcal{G}_{\text{\emph{G}}firn}^{\text{\emph{anti}}}\) represent the candidate sets of coal-fired, wind, and photovoltaic power units, respectively. \(C_{l}\) represents the construction cost of line \(l\). \(\lambda_{l}^{\mathcal{L}}\) is the decision variable for the construction of the line. \(\mathcal{L}^{\text{\emph{candid}}}\) denotes the candidate set of lines; \(C_{g}^{f}\) represents the unit fuel cost of coal-fired power unit g. \(C_{g}^{\text{\emph{v}}\_\text{\emph{c}}}\) denotes the unit wind curtailment cost of wind power unit g. \(C_{g}^{\text{\emph{v}}\_\text{\emph{c}}}\) represents the unit solar curtailment cost of photovoltaic unit \(g\). \(C_{b}^{\text{\emph{h}}}\) denotes the unit load shedding cost at node \(b\). \(C_{g}^{\text{\emph{c}}}\) represents the unit carbon emission cost of coal-fired power unit g. \(C_{g}^{\text{\emph{st}}}\) denotes the startup cost of coal-fired power unit g. \(p_{g,\rho,t}\) represents the power output of unit \(g\) at time \(t\) in scenario \(\rho\).