## 7 Conclusions  
This paper proposes a two-stage stochastic mixed-integer model for the restoration of NMGs,which simultaneously optimizes the sizing, pre-positioning, and reallocation of MES devices, as well as the dynamic reconfiguration of NMG boundaries. Our numerical studies on the modified IEEE 33-node and 69-node distribution systems demonstrate the effectiveness, robustness and economic benefits of the proposed model and algorithm. The main research results are summarized as follows:  
(1) Simultaneous Pre-Disaster Optimization: The proposed approach enables the joint optimization of MES capacity and pre-positioning before a disaster occurs. This not only enhances system resilience, but also significantly reduces economic costs. By providing a comprehensive strategy for pre-disaster planning, it offers decision makers a valuable tool for developing proactive prevention measures. (2) Post-Disaster Reallocation and Dynamic Boundary Reconfiguration: After a disaster, the re-allocation of MES devices and the operation of boundary switches can dynamically adjust the operational boundaries of microgrids. This flexibility allows previously isolated microgrids to reconnect to power sources, forming new NMGs. As a result, the risk of prolonged isolation is mitigated, and overall system resilience is greatly enhanced. (3) Effectiveness of the Customized PHA: The customized PHA employed in this study proves highly effective in addressing multi-stage stochastic programming problems. It demonstrates strong performance in optimizing complex, uncertain scenarios, making it a suitable method for solving large-scale problems in power systems. Several promising areas for future research are worth investigating to further enhance the proposed framework. First, incorporating more detailed and realistic disaster damage scenarios, tailored to different types of natural disasters, would be beneficial. By generating representative disaster patterns, more accurate modeling of post-disaster recovery and resource allocation strategies can be achieved. Additionally, deeper integration with transportation networks, coupled with the coordination of repair crews, should be explored. This would involve optimizing the scheduling of MES devices in conjunction with the sequence of repairing damaged components, ensuring a coordinated approach that improves the overall efficiency of the recovery process. Furthermore, the potential to leverage private electric vehicles (EVs) as supplementary mobile energy storage units warrants investigation. By integrating privately owned EVs into the framework, enhanced system flexibility can be achieved, particularly in scenarios where additional energy resources are limited during disaster recovery operations. Conceptualization, B.J. and H.L. ; methodology, B.J. and H.L. ; software, B.J. and C.Z. ; validation, B.J. and Y.L. ; formal analysis, B.J. ; data curation, B.J. ; writing--original draft preparation, B.J. ; writing--review and editing, B.J., H.L., and Y.L. ; visualization, B.J. and Y.L. ; supervision, H.L. and C.Z. ; project administration, T.Z. All authors have read and agreed to the published version of the manuscript. This work was partly supported by the Hunan Provincial Natural Science Foundation of China under grant no. 2022 JJ10069, and by the Hunan Provincial Science and Technology Innovation Program of China under grant no. 2023 RC1002. This support is greatly appreciated. Not applicable. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Power System**} & \multirow{2}{*}{**Solution Method**} & \multicolumn{4}{c}{**Scenario Number**} \\ \cline{3-5}  & & **10** & **20** & **30** \\ \hline \multirow{2}{*}{33-node} & Gurobi 11.0.1 & 0.22 & 1.34 & 4.57 \\ \cline{2-5}  & PHA & 0.19 & 0.96 & 3.06 \\ \hline \multirow{2}{*}{69-node} & Gurobi 11.0.1 & 15.63 & 140.13 & 379.22 \\ \cline{2-5}  & PHA & 7.88 & 79.91 & 252.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Comparison of computation time (minutes) for problem solving. **Informed Consent Statement:** Not applicable.