## 3 Robust Emergency Dispatch Strategy Considering Spatiotemporal Grid Failures During Typhoons
### Robust Emergency Dispatch Strategy During Typloons  
Based on Section 3.1, the two-stage robust optimization (TSRO) model under the attacker-defender framework is recast as:  
\[ \begin{split}&\min_{y\in\Omega_{y}}\max_{\mathbf{r}\in\Omega_{y}}\min_{ \mathbf{x}}obj\\ &\text{s.t.}\quad\mathbf{G_{1}x}+\mathbf{G_{2}y}+\mathbf{G_{3}\mu}+\mathbf{G_{4}h }\leq 0\end{split}  \tag{28}\]  
where the binary ESS status \(\mathbf{y}=\left[\rho_{j,t}^{\text{dis}},\rho_{j,t}^{\text{cha}}\right]\) are the first-stage decision variables; while \(\mathbf{x}=\left[P_{j,t}^{\text{dis}},P_{j,t}^{\text{dis}},P_{j,t}^{\text{S}}\right]\) are the second-stage decision variables. \(\mathbf{\mu}\in\Omega_{y}\) represents the set of \(R\) emergency scenarios constructed from Section 2.2. The constraint represents (17)-(27). \(\mathbf{G_{1}}\), \(\mathbf{G_{2}}\), \(\mathbf{G_{3}}\), and \(\mathbf{G_{4}}\) represent the corresponding matrices; \(\mathbf{h}=\left[E_{j,t},P_{ij,t},\theta_{ij,t}\right]\) are the intermediate variables. As can be seen, the max-min operator in (28) makes the second stage a defender-attacker model. The attacker (typhoon) attempts to maximize the emergency dispatch cost by applying the severest emergency scenario; while the defender (dispatcher) attempts to minimize the dispatch cost. The max operator means that the dispatch cost under the severest emergency scenario is larger than that under other emergency scenarios. Therefore, Equation (28) can be reformulated as the following:  
\[ \begin{split}&\min_{y\in\Omega_{y}}\eta\\ & s.t.\quad\eta\geq obj_{r},\quad r=1,\cdots,R\\ &\mathbf{G_{1}x}_{r}+\mathbf{G_{2}y}+\mathbf{G_{3}\mu}_{r}+\mathbf{G_{4}h}_{r} \leq 0,\quad r=1,\cdots,R\end{split}  \tag{29}\]  
where \(\mathbf{x}_{r}\) represent the optimal solutions of the second-stage decision variables corresponding to emergency scenario \(r\). \(\mathbf{\mu}_{r}\) represent the on-off line statuses under emergency scenario \(r\). \(\mathbf{h}_{r}\) represent the optimal solutions of the second-stage intermediate variables corresponding to emergency scenario \(r\). As a result, solving (28) reduces to solve an equivalent (29). Rather than enumerating all the emergency scenarios and solving (29), a column-and-constraint generation (C&CG) procedure is implemented in a master-subproblem framework. The procedure is as follows:  
1. Set the lower boundary to the negative infinity \(LB=-\infty\) and the upper boundary to the positive infinity \(UB=\infty\); set the counter to zero \(k=0\); initialize the set of counter \(\mathbf{O}=\emptyset\); initialize the error threshold \(\epsilon\). 2. Solve the master problem \[ \begin{split}&\min_{y,t}\eta\\ & s.t.\quad\eta\geq obj_{r},\qquad\forall r\in\mathbf{O}\\ &\mathbf{G_{1}x}_{r}+\mathbf{G_{2}y}+\mathbf{G_{3}\mu}_{r}^{*}+\mathbf{G_{4}h}_{r} \leq 0,\forall r\leq k\\ &\mathbf{y}\in\Omega_{y}\end{split} \] (30) Derive the optimal solutions (\(\mathbf{y}_{k+1}^{*}\), \(\eta_{k+1}^{*}\), \(\mathbf{x}_{1}^{*}\), \(\cdots\), \(\mathbf{x}_{r}^{*}\)) and update the lower boundary \(LB=\eta_{k+1}^{*}\)
3. Solve the sub-problem \(\mathcal{Q}(\mathbf{y}_{k+1})=\left\{\max_{\mathbf{\mu}\in\Omega_{y}}\min_{\mathbf{x}}obj: \mathbf{G_{1}x}+\mathbf{G_{2}y}+\mathbf{G_{3}\mu}+\mathbf{G_{4}h}\leq 0\right\}\) and update the upper boundary \(UB=\mathcal{Q}(\mathbf{y}_{k+1}^{*})\). 4. If \(UB-LB\leq\epsilon\), return \(\mathbf{y}_{k+1}^{*}\), \(\mathbf{x}_{k}^{*}\), and terminate. Otherwise,* (a) if \(\mathcal{Q}(\mathbf{y}_{k+1}^{*})<\infty\), create variables \(\mathbf{x}_{k+1}\) and \(\mathbf{h}_{k+1}\) in the \(k+1\)-th iteration and add the optimality cuts \[\eta\geq obj_{k+1}\] (31) \[\mathbf{G}_{1}\mathbf{x}_{k+1}+\mathbf{G}_{2}\mathbf{y}+\mathbf{G}_{3}\mathbf{y}_{k+1}^{*}+\mathbf{G}_{4} \mathbf{h}_{k+1}\leq 0\] (32) to the master problem, where \(\mathbf{\mu}_{k+1}^{*}\) is the optimal emergency scenario by solving \(\mathcal{Q}(\mathbf{y}_{k+1})\). Update \(k=k+1\), \(\mathbf{Q}=\mathbf{Q}\cup k+1\), go to step 2.