## 4 Case Studies
### Coordinated Emergency Dispatch of Thermal Units and ESSs  
Based on the generated emergency scenarios in Section 4.2, the emergency dispatch model in (28) is developed and solved by the procedures in Section 3.3. Figure 5 shows the time-series output power of thermal units (i.e., G1, G2, and G3), ESS during Typhoon Maria. Since G1 has the lowest cost among thermal units and the ESS, G1 bears the main responsibility of power generation. Due to the ramping limits, the three thermal units cannot immediately satisfy the demand from 1-2 h, and the ESS bears the remainder that cannot be covered by thermal units. After 2 h, the expensive ESS withdraws, since the power of thermal units has increased high enough to cover the load. To evaluate the robustness of the proposed strategy, the non-robust strategy is generated singly for the emergency scenario 1 in Figure 4a. Then the non-robust strategy is tested on the emergency scenario 2-4 in Figure 4b-d. The unexpected load and wind power curtailment cost is defined to evaluate the robustness. The calculation of the unexpected load and wind power curtailment cost is defined as an optimization problem. The objective is to minimize the unexpected load and wind power curtailment cost. \[\min\sum_{t}\sum_{j}\lambda_{j}^{d}\Delta P_{j,t}^{d}+\sum_{t}\sum_{j}\lambda_ {j}^{w}\Delta P_{j,t}^{w} \tag{34}\]  
Since the component-level regulation characteristic constraints (17)-(23) of respective emergency dispatch resources always hold for the given optimal \(\{P_{j,t}^{s},P_{j,t}^{\text{diss}},P_{j,t}^{\text{chas}},E_{j,t}^{s}\}\) using either the non-robust or the robust strategy, only the system-level power flow-related constraints are considered. The revised nodal power balance of node \(j\) is written by:  
\[ \begin{array}{l}\alpha_{j}^{\text{g}}P_{j,t}^{s*}+\alpha_{j}^{\text{r}}\left( P_{j,t}^{\text{dis}*}-P_{j,t}^{\text{cha*}}\right)+\alpha_{j}^{w}\left(P_{j,t}^{ w}-\Delta P_{j,t}^{w}\right)\\ -\alpha_{j}^{\text{d}}(P_{j,t}^{\text{d}}-\Delta P_{j,t}^{\text{d}})=\sum \limits_{k\in\Omega_{k}}P_{j,t}-\sum\limits_{i\in\Omega_{j}}P_{ij,t}\end{array}  \tag{35}\]  
The revised power flow equations are:  
\[B_{ij}\big{(}\theta_{i,t}-\theta_{j,t}\big{)}+P_{ij,t}\leq\left(1-\mu_{ij,t}^ {\epsilon}\right)M \tag{36}\]  
Figure 5: Sequential output power of emergency dispatch resources. \[B_{ij}\big{(}\theta_{i,t}-\theta_{j,t}\big{)}+P_{ij,t}\geq\left(\mu_{ij,t}^{c}-1 \right)M \tag{37}\]  
where \(\mu_{ij,t}^{c}\) represent the on-off status of the line between node \(i\) and \(j\) at time period \(t\) under emergency scenario \(\epsilon\). The definition of other variables or parameters are the same as those in (36) and (37). By running the non-robust and robust strategy using (35)-(37) under four emergency scenarios in Figure 4, the unexpected load and wind power curtailment costs are given in Table 6. Due to the similarity between emergency scenario 1 and 2, the non-robust strategy for emergency scenario 1 can be applied to emergency scenario 2 without incurring load and wind power curtailment costs. However, when applying non-robust strategy to emergency scenario 3 and 4, there will be unexpected load and wind power curtailment costs (2.31e55). In contrast, the proposed robust strategy can guarantee no more curtailment costs under all four emergency scenarios. Previous findings assume that Typhoon Maria exhibits fixed features, including its trajectory and meteorological factors. The proposed robust emergency dispatch strategy is designed to address emergency scenarios arising from these rigid typhon characteristics. However, inherent uncertainties may introduce flexible features, leading to multiple typhoon scenarios. Each scenario could yield a distinct set of emergency scenarios, similar to those generated by Typhoon Maria with fixed parameters, as described in Section 4.1.