## 3 Robust Emergency Dispatch Strategy Considering Spatiotemporal Grid Failures During Typhoons
### Operational Constraints of Emergency Dispatch  
The output power \(P_{j,t}^{\text{g}}\) of thermal unit \(j\) at time period \(t\) should satisfy the upper and lower output limit constraints:  
\[P_{j}^{\text{g,min}}\leq P_{j,t}^{\text{g}}\leq P_{j}^{\text{g,max}} \tag{17}\]  
where \(P_{j}^{\text{g,min}}\) and \(P_{j}^{\text{g,max}}\) represent the lower and upper power limits of thermal unit \(j\). The ramping limit of thermal units is expressed by:  
\[P_{j}^{\text{rd}}\leq P_{j,t}^{\text{g}}-P_{j,t}^{\text{g}}\leq P_{j}^{\text{ ru}} \tag{18}\]  
where \(P_{j}^{\text{rd}}\) and \(P_{j}^{\text{ru}}\) represent the ramp-down and ramp-up limits of unit \(j\). The discharge and charge power limits of ESS \(j\) satisfy:  
\[0\leq P_{j,t}^{\rm dis}\leq\beta_{j,t}^{\rm dis}\,P_{j}^{\rm dis,max} \tag{19}\]  
\[0\leq P_{j,t}^{\rm cha}\leq\beta_{j,t}^{\rm cha}\,P_{j}^{\rm cha,max} \tag{20}\]  
where \(P_{j,t}^{\rm dis}\) and \(P_{j,t}^{\rm cha}\) represent the discharge and charge power of ESS \(j\) at time period \(t\); \(\beta_{j,t}^{\rm dis}\) and \(\beta_{j,t}^{\rm cha}\) (\(\beta_{j,t}^{\rm dis}\), \(\beta_{j,t}^{\rm cha}\in\{0,1\}\)) represent the discharge and charge status of ESS \(j\) at time period \(t\); \(P_{j}^{\rm dis,max}\) and \(P_{j}^{\rm cha,max}\) represent the maximum discharge and charge power of ESS \(j\). An ESS cannot be in both discharging and charging states simultaneously:  
\[\beta_{j,t}^{\rm dis}+\beta_{j,t}^{\rm cha}=1 \tag{21}\]  
The relation between the stored energy and the discharge or charge power of an ESS is:  
\[E_{j,t}=E_{j,t-1}-P_{j,t}^{\rm dis}\,\Big{/}\eta_{j}^{\rm dis}+P_{j,t}^{\rm cha }\eta_{j}^{\rm cha} \tag{22}\]  
where \(E_{j,t}\) represent the stored energy of ESS \(j\) at period \(t\); \(\eta_{j}^{\rm cha}\) and \(\eta_{j}^{\rm dis}\) represent the charge and discharge efficiency coefficient of ESS \(j\). The stored energy \(E_{j,t}\) of ESS \(j\) at period \(t\) should satisfy:  
\[E_{j,\rm min}\leq E_{j,t}\leq E_{j,\rm max} \tag{23}\]  
where \(E_{j,\rm min}\) and \(E_{j,\rm max}\) represent the minimum and maximum energy of ESS \(j\). The nodal power balance of node \(j\) is written by:  
\[ \begin{array}{l}\alpha_{j}^{\rm g}P_{j,t}^{\rm g}+\alpha_{j}^{\rm e}\left(P _{j,t}^{\rm dis}-P_{j,t}^{\rm cha}\right)+\alpha_{j}^{\rm w}P_{j,t}^{\rm w}\\ -\alpha_{j}^{\rm d}P_{j,t}^{\rm d}=\sum\limits_{k\in\Omega_{t}}P_{jk,t}-\sum \limits_{i\in\Omega_{j}}P_{ij,t}\end{array}  \tag{24}\]  
where \(\alpha_{j}^{\rm g}\), \(\alpha_{j}^{\rm e}\), \(\alpha_{j}^{\rm w}\), \(\alpha_{j}^{\rm d}\) represent the identifier, indicating whether node \(j\) is installed with a thermal unit, ESS, wind farm, or load. \(\alpha_{j}^{\rm g,e,w,d}=1\) indicate whether node \(j\) is installed with a thermal unit, ESS, wind farm, or load; \(\alpha_{j}^{\rm g,e,w,d}=0\) indicates that node \(j\) is not installed with a thermal unit, ESS, wind farm, or load. \(\Omega_{t}\) represents the set of to-nodes interconnected with node \(j\); \(\Omega_{f}\) represents the set of from-nodes interconnected with node \(j\); \(P_{jk,t}\) represent the power flow from node \(j\) to node \(k\) at time period \(t\); \(P_{ij,t}\) represent the power flow from node \(i\) to node \(j\) at time period \(t\). The power flow equation describing the relationship between the transmitted power and the phase angles at both ends of the line can be expressed by:  
\[P_{ij,t}=B_{ij}\big{(}\theta_{i,t}-\theta_{j,t}\big{)} \tag{25}\]  
where \(P_{ij,t}\) represents the transmitted power from node \(i\) to node \(j\) at time period \(t\); \(B_{ij}\) denotes the susceptance of the line between node \(i\) and \(j\); \(\theta_{i,t}\) and \(\theta_{j,t}\) represent the phase angle of node \(i\) and \(j\) at time period \(t\). If the line between node \(i\) and \(j\) is disconnected due to the fatigue damage caused by typhoons, Equation (25) does not hold and \(P_{ij,t}\) can be any value; while Equation (25) holds if the line between node \(i\) and \(j\) is connected.