## 6 Case Studies
### Impact of Microgrid Boundary Lines  
As described in the above two cases, the boundary lines play an essential role in microgrid reconfiguration, where the switches of boundary lines can form microgrid subsystems dynamically. To better illustrate the superiority of dynamic boundary lines, comparisons are conducted between the proposed model and a model without boundary lines, where the compared model is represented by several stand-alone microgrids. Based on the proposed two-stage optimization framework, the objectives of this compared model are presented in Equations (32) and (33), followed by the constraints that need to be satisfied. We use the same damage scenarios in Case II to examine the model without boundary lines; the optimization results are shown in Figure 6. It can be seen from Figure 6 that more loads need to be shed compared to the results in Case I. Apart from node 25 remaining unpowered, the nodes in set \(\{10,11,12,22\}\) are also without electricity supply.  
\[\min\sum\limits_{n\in N}W_{n}\left\{\eta_{1}\frac{\sum\limits_{p\in I_{1}}\sum \limits_{k\in I_{2}}\sum\limits_{m\in M}\beta_{mpkn}T_{skn}}{BM\,\textbf{.}\,T_ {\max}}+\eta_{2}\frac{\sum\limits_{i\in I}Z_{i}sl_{in}}{\sum\limits_{i\in I}Z_ {i}P_{i}}+\eta_{3}\frac{\sum\limits_{m\in M}G_{m}}{BM\,\textbf{.}\,G_{M}}\right\} \tag{32}\]  
\[\min\left\{\eta_{1}\frac{\sum\limits_{p\in I_{1}}\sum\limits_{k\in I_{2}}\sum \limits_{m\in M}\beta_{mpkn}T_{skn}}{BM\,\textbf{.}\,T_{\max}}+\eta_{2}\frac{ \sum\limits_{i\in I}Z_{i}sl_{in}}{\sum\limits_{i\in I}Z_{i}P_{i}}+\eta_{3} \frac{\sum\limits_{m\in M}G_{m}}{BM\,\textbf{.}\,G_{M}}\right\} \tag{33}\]  
\[\theta_{ijn}+\theta_{jin}=tl_{ijn},\forall i,j\in I,\forall n\in N \tag{34}\]  
\[p_{l_{ijn}}r_{l_{ijn}}=tl_{ijn}(1-al_{ijn})(\delta_{o(l_{ij})}-\delta_{d(l_{ ij})}),\forall i,j\in I,\forall n\in N \tag{35}\]  
\[\text{s.t. (\ref{eq:1})-(\ref{eq:2}), (\ref{eq:3}), (\ref{eq:4})-(\ref{eq:5}), (\ref{eq:6})-(\ref{eq:7}), (\ref{eq:8}), (\ref{eq:9})-(\ref{eq:10}) \tag{21}\]  
Furthermore, as summarized in Tables 10 and 11, the optimization process based on both of the models is simulated for 20 rounds under different damage scenario settings. To assess the effects, load shedding is employed and measured by using Equation (36). \(|N|\) denotes the number of scenarios. The results of the two tables clearly indicate that the model without boundary lines produces higher load shedding compared to the proposed model with boundary lines. Moreover, when the experimental parameters shift towards conditions unfavorable for optimization, such as an increase in damaged line number and a decrease in MES budget, it makes sense that the proposed model with boundary lines exhibits a smaller increase in load shedding compared to the model without boundary lines. This observation further confirms the superior robustness of the model with boundary lines. Overall, these results demonstrate that by operating the switches on boundary lines, each microgrid subsystem is capable of expanding its boundaries to interconnect with other microgrid subsystems and form NMGs, thus achieving energy mutual assistance and reducing the shed loads.  
\[\text{Load Shedding}=\frac{\sum\limits_{n\in N}\left(\sum\limits_{i\in I}Z_ {i}sl_{in}/\sum\limits_{i\in I}Z_{i}P_{in}\right)}{|N|}\times 100\%,\forall i\in I \tag{36}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Boundary Line** & **First Stage** & & **Second Stage** \\ \hline Line (3, 28) & open & \(\rightarrow\) & open \\ Line (3, 36) & open & \(\rightarrow\) & open \\ Line (4, 47) & open & \(\rightarrow\) & open \\ Line (9, 53) & open & \(\rightarrow\) & closed \\ Line (16, 17) & open & \(\rightarrow\) & closed \\ \hline \end{tabular}
\end{table}
Table 9: The switch operation scheme of boundary lines in Case II.