## 2 Problem Description  
The problem is an optimization problem of MES sizing, pre-positioning, and re-allocation for resilient NMGs with dynamic boundaries. After a natural-disaster-induced attack occurs, the distribution lines connecting to the microgrid may be damaged, which leads to power outages in the microgrid subsystems of the networked system. To mitigate the power outages resulting from these attacks, the resilient operations of NMG systems are proposed. On the one hand, the boundary of the microgrid subsystem can be adjusted, so that a subsystem with abundant power supply can support the damaged subsystem after line damage occurs; on the other hand, the MES devices are pre-positioned on the nodes with key load for rapid restoration, and the MES can be re-allocated to effectively balance the power supply across all microgrid subsystem regions. By optimizing the power flow and adjusting the boundaries within the NMG system, the overall system resilience can be returned to the highest level. Notably, the size capacity of MES is also a focus of this paper, and it is optimized to maximize cost-effectiveness.  
As depicted in Figure 1, the problem includes two stages. In the first stage, before the natural disaster occurs, three MES devices are pre-positioned at suitable nodes based on the optimal solutions. Three boundary lines connect different microgrid subsystems with the switch status set to 'open'. In the second stage, after the damage event occurs, the  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Objectives**} & & & & \\
**Ref.** & \begin{tabular}{c} **Increase** \\ **Supply** \\ **Reliability** \\ \end{tabular} & \begin{tabular}{c} **Minimize** \\ **Economic** \\ **Cost** \\ \end{tabular} & \begin{tabular}{c} **Minimize** \\ **Outage** \\ **Duration** \\ \end{tabular} & \begin{tabular}{c} **Minimize** \\ **Power** \\ **Exchange** \\ \end{tabular} & \begin{tabular}{c} **Considering** \\ **Recon-** \\ **figuration** \\ \end{tabular} & \begin{tabular}{c} **Considering** \\ **MES** \\ **Usage** \\ \end{tabular} & \
\begin{tabular}{c} **Considering** \\ **MES** \\ **Sizing** \\ \end{tabular} &
\begin{tabular}{c} **Considering** \\ **Test** \\ **System** \\ \end{tabular} \\ \hline
[9] & ✓ & \(\times\) & \(\times\) & \(\times\) & ✓ & \(\times\) & \(\times\) & 123-node \\
[10] & ✓ & \(\times\) & \(\times\) & \(\times\) & ✓ & \(\times\) & \(\times\) & 16-node \\
[11] & ✓ & ✓ & \(\times\) & \(\times\) & \(\times\) & \(\times\  
damaged lines will be detected, which are labeled with red crosses in the figure. The status of each boundary line switch is then determined accordingly. Meanwhile, the three MES devices are re-allocated to the optimal locations while coordinating with the adjustment of boundary line switch status. In the figure, eight new microgrid subsystems are formed to enable each node to receive power after the boundary adjustment of microgrid subsystems in the networked system.  
This problem is based on five reasonable assumptions: (1) damage scenarios can be derived from expert knowledge or natural disaster damage models; (2) the system can rapidly detect damaged components and promptly update its status; (3) the amount of MES that can be pre-positioned at each node is limited; (4) all nodes are assumed to be reachable to one another; (5) hardened boundary lines are assumed to remain undamaged.  
Figure 1: A schematic diagram of the proposed two-stage networked microgrid reconfiguration optimization method.