## 6 Case Studies
### Case I: Modified IEEE 33-Node Distribution System  
The modified IEEE 33-node distribution system undergoes modifications through the reservation of certain lines and the alteration of node characteristics and distribution lines. Consequently, this system comprises five independent microgrid subsystems when the switches on boundary lines are in the open state. The capacities of the five generation nodes are listed in the first two columns of Table 2. Given the unavailability of actual damage data, this paper employs a normal distribution to generate ten hypothetical damage scenarios, as depicted in Table 3. The load priorities and transportation speeds of MES are randomly assigned from the sets \(\{5,3,1\}\) and \(\{60\) km/h, \(90\) Kkm/h, \(100\) km/h\(\}\), respectively, with equal probabilities. The maximum purchasing capacity of the MES is set to \(1000\) kW. Figure 3: A modified IEEE 69-node test system. Figure 2: A modified IEEE 33-node test system. Subsequently, the transportation time for each MES device along the designed routes is computed based on the fixed speed parameters and their Euclidean distances. Taking Scenario 1 of Table 3 as an example, we illustrate the application of the proposed two-stage optimization method. Prior to the occurrence of natural disasters, the optimal pre-positioning and capacity purchase schemes for MES devices are determined by solving the first-stage problem using the ten generated scenarios, as shown in the second columns of Tables 4 and 5. The first-stage solutions aim to optimize the effectiveness of the MES allocation scheme in the second stage. Notably, Table 5 reveals that the purchased capacities of the three MES devices are smaller than the maximum design capacity (1000 kW), demonstrating noticeable economic benefits. In Figure 4, the nodes connected to quadrangular-star nodes denote the presence of MES devices. From Figure 4a, it can be observed that the MES devices are pre-positioned on nodes 7, 18, and 31 in the first stage. Upon the occurrence of a natural disaster, the damaged lines in the actual scenario are detected and marked with a cross in Figure 4b. The optimal re-allocation scheme for MES is derived by solving the second-stage problem using the selected reality scenario. Subsequently, the MES devices are dispatched from the pre-positioning nodes to the re-allocation nodes, with the corresponding decisions listed in the forth column of Table 4. Once the MES devices arrive at the re-allocation nodes, they are integrated into the system \begin{table}
\begin{tabular}{c c c} \hline \hline
**MES** & **Pre-Positioning Nodes** & **Re-Allocation Nodes** \\ \hline I & node 7 & node 7 \\ II & node 18 & node 22 \\ III & node 31 & node 33 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The pre-positioning and re-allocation scheme of MES devices in Case I. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**IEEE 33-Node** & \multicolumn{3}{c}{**IEEE 69-Node**} \\ \hline Generation node & Capacity (kW) & Generation node & Capacity (kW) \\
1 & 10,000 & 1 & 10,000 \\
16 & 800 & 22 & 1000 \\
21 & 600 & 31 & 1000 \\
24 & 1300 & 38 & 1000 \\
29 & 1300 & 47 & 3000 \\ \(\backslash\) & \(\backslash\) & 58 & 5000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The capacity of generation nodes of the modified IEEE 33/69-node distribution system.