## 4 Source-Grid Collaborative Planning Model
### Constraints  
The constraints related to unit retirement, overall system frequency security, and system node inertia requirements in the planning model are shown in Equations (8), (23) and (27)-(29). In addition, investment decision constraints and operational simulation constraints for the planning model must also be considered, as detailed below. 1. Node power balance constraints: \[\sum_{g\in\mathcal{G}_{full}^{b}}p_{g,\rho,t}-\sum_{g\in\mathcal{G}_{\text{ \emph{G}}firn}^{\text{\emph{b}}}\cup\mathcal{G}_{\text{\emph{G}}firn}}p_{g, \rho,t}^{c}+\sum_{l\in\mathcal{L}}a_{b,l}^{L}f_{l,\rho,t}=d_{b,\rho,t}-p_{b,\rho,t}^{c}.\] (35)  
In Equation (35), \(\mathcal{G}_{full}^{b}\) represents the set of generators at node \(b\). \(\mathcal{G}_{\text{\emph{G}}firn}^{b}\) denotes the set of wind power units at node \(b\). \(\mathcal{G}_{\text{\emph{G}}firn}^{b}\) represents the set of photovoltaic units at node \(b\). \(\mathcal{L}\) denotes the set of lines. \(a_{b,l}^{L}\) represents the elements of the node-line association matrix, \(a_{b,l}^{L}=1\) indicates that flow from line \(l\) is entering node \(b\), while \(a_{b,l}^{L}=-1\) indicates that flow from line \(l\) is exiting node \(b\). \(f_{l,\rho,t}\) represents the flow on the line \(l\). \(d_{b,\rho,t}\) represents the load at node \(b\). Constraints on the output of coal-fired power units: \[(1-\tau_{g}^{rt})h_{g,\rho,t}p_{g,\min}\leq p_{g,\rho,t}\leq(1-\tau_{g}^{rt})h_{ g,\rho,t}p_{g,\max},g\in\mathcal{G}_{Gfire}.\] (36)  
In Equation (36), \(h_{g,\rho,t}\) represents the operating status of the unit, \(h_{g,\rho,t}=1\) indicating that the unit is in operation; otherwise, it is in a shut-down state. \(p_{g,\min}\) represents the minimum output of generator g. \(p_{g,\max}\) represents the maximum output of generator. Constraints on the start-up and shut-down logic and duration of coal-fired power units:  
\[\kappa_{g,\rho,t}^{on}+\kappa_{g,\rho,t}^{off}\leq 1; \tag{37}\]  
\[\sum_{k=t-UT_{g}+1}^{t}\kappa_{g,\rho,k}^{off}\leq h_{g,\rho,t},g\in\mathcal{ G}_{Gfire},t\in[T_{g,\mu},T_{N}]; \tag{38}\]  
\[\sum_{k=t-DT_{g}+1}^{t}\kappa_{g,\rho,k}^{off}\leq 1-h_{g,\rho,t},g\in\mathcal{ G}_{Gfire},t\in[T_{g,d},T_{N}]. \tag{39}\]  
\[\kappa_{g,\rho,t}^{on}-\kappa_{g,\rho,t}^{off}=h_{g,\rho,t}-h_{g,\rho,t-1}; \tag{40}\]  
In Equations (37)-(40), \(\kappa_{g,\rho,t}^{off}\) represents the decision variable for the shut-down action of coal-fired power unit g, where \(\kappa_{g,\rho,t}^{off}=1\) indicates the shut-down action. \(UT_{g}\) represents the minimum up time of coal-fired power unit g. \(DT_{g}\) represents the minimum down time of coal-fired power unit g. \(T_{g,\mu}\) represents the duration of operation for coal-fired power unit \(g\) at the initial time. \(T_{g,d}\) represents the duration of shut-down for coal-fired power unit \(g\) at the initial time. \(T_{N}\) represents the total number of time instances within the scenario. Constraints on the ramp-up/ramp-down of coal-fired power units:  
\[p_{g,\rho,t}-p_{g,\rho,t-1}\leq\xi_{g,\mu p},g\in\mathcal{G}_{Gfire}; \tag{41}\]  
\[p_{g,\rho,t-1}-p_{g,\rho,t}\leq\xi_{g,down},g\in\mathcal{G}_{Gfire} \tag{42}\]  
In Equations (41) and (42), \(\xi_{g,\mu p}\) represents the maximum ramp-up power of coal-fired power unit g. \(\xi_{g,down}\) represents the maximum ramp-down power of coal-fired power unit g.  
Load shedding constraints:  
\[0\leq p_{g,\rho,t}^{c}\leq d_{b,\rho,t},b\in\mathcal{B} \tag{43}\]  
Wind and solar curtailment constraint:  
\[0\leq p_{g,\rho,t}^{c}\leq\beta_{w}h_{g,\rho,t}p_{g,\rho,t},g\in\mathcal{G}_{ Gwind}; \tag{44}\]  
\[(1-\tau_{g}^{rt})h_{g,\rho,t}p_{g,\min}\leq p_{g,\rho,t}\leq(1-\tau_{g}^{rt})h_{ g,\rho,t}p_{g,\max},g\in\mathcal{G}_{Gfire}.\]0  
In Equations (44) and (45), \(\beta_{w}\) represents the maximum curtailment coefficient of wind power; \(\beta_{pv}\) represents the maximum curtailment coefficient of photovoltaic power.