## 2 Vulnerable Node Identification  
A vulnerable node refers to the node whose transient stability has a relatively strong coupling with the operational stability of the distribution system. That is to say, when a vulnerable node experiences a transient fluctuation due to a fault, it will propagate to other nodes along the route through the distribution line and, thus, cause larger fluctuation amplitude and scope than a fault at non-vulnerable nodes. In a normal complex network theory, the distance between two nodes reflects the difficulty of flow transmission from one node to another node. According to [14; 15], the distance in a power grid should be considered as the electrical distance in terms of equivalent impedance \(Z_{ij}\):  
\[Z_{ij}=\frac{U_{ij}}{I_{i}} \tag{1}\]  
where \(U_{ij}\) is the voltage between nodes; \(I_{i}\) is the unit current of injection node \(i\). The value of electrical equivalent impedance is equal to the voltage between nodes \(i\) and \(j\) after injecting unit current into the node \(i\). For \(I_{i}=1\), Equation (1) can be converted into:  
\[Z_{ij}=U_{ij}=U_{i}-U_{j} \tag{2}\]  
According to electrical network theory, the node impedance matrix containing \(J\) nodes can be expressed as:  
\[\mathbf{U}=\mathbf{Z}\mathbf{I}=\left[ \begin{array}{cccc}z_{11}&z_{12}&\cdots&z_{1J}\\ z_{21}&z_{22}&\cdots&z_{2J}\\ \vdots&\vdots&\ddots&\vdots\\ z_{j1}&z_{j2}&\cdots&z_{JJ}\end{array} \right]\left[ \begin{array}{c}I_{1}\\ I_{2}\\ \vdots\\ I_{J}\end{array} \right] \tag{3}\]  
Assume that the unit current only flows through nodes \(i\) and \(j\) in measuring electrical equivalent impedance, then Equation (3) can be rewritten into:  
\[\left[ \begin{array}{c}\vdots\\ U_{i}\\ \vdots\\ U_{j}\\ \vdots\end{array} \right]=\left[ \begin{array}{cccc}&\vdots&&\vdots&\\ \cdots&z_{ii}&\cdots&z_{ij}&\cdots\\ &\vdots&\ddots&\vdots&\\ \cdots&z_{ji}&\cdots&z_{jj}&\cdots\\ &\vdots&&\vdots&\end{array} \right]\left[ \begin{array}{c}\mathbf{0}\\ \mathbf{1}\\ \mathbf{0}\\ -\mathbf{1}\\ \mathbf{0}\end{array} \right] \tag{4}\]  
where \(U_{i}\) and \(U_{j}\) are, respectively, node voltages to ground; \(z_{ij}\) is the mutual impedance; \(z_{ii}\) and \(z_{jj}\) are the self-impedances of the nodes. \(z\) is the measurement value. According to Equation (4), Equation (2) can be further rewritten into:  
\[Z_{ij}=\ U_{i}-U_{j}=\ (z_{ii}-z_{ij})-(z_{ij}-z_{jj})\ =\ z_{ii}+z_{jj}-2z_{ij} \tag{5}\]  
where \(z_{ij}=z_{ii}\), \(Z_{ij}\) reflects the difficulty of the fault at node \(i\) being transmitted to \(j\) when current flows from node \(i\) to \(j\). Due to that transient coupling of non-neighboring nodes needs to consider the reaction of the nodes on transmission route to transient fluctuation, nodes \(i\) and \(j\) herein specifically refer to neighboring ones. The equivalent electrical impedance reflects the difficulty of node \(v_{i}\) affecting the operation state of node \(v_{j}\). Inspired by [PERSON] et al. [14], we defined \(c_{ij}\), the electrical coupling coefficient, to express the state correlation between node \(v_{i}\) and \(v_{j}\). \[c_{ij}= \begin{cases}\ln\ \underline{1}_{ij}&\textit{i}\textit{and}\_\textit{j} \textit{are}\ \textit{neighboring}\ \textit{nodes}\\ 0&\textit{others}\end{cases}  \tag{6}\]  
where \(c_{ij}\) reflects how much node \(v_{i}\) affects a single neighbor node \(v_{j}\) when it fails. Therefore, we defined \(C_{i}\), the regional electric coupling coefficient (RECE), as the electrical coupling efficiency of node \(v_{i}\) that reflects the ability of \(v_{i}\) to affect all its neighboring nodes. \[C_{i}=\sum_{j\in\Omega_{i}}c_{ij}=\sum_{j\in\Omega_{i}}\ln(\frac{1}{Z_{ij}})=\ln( \frac{1}{\prod_{j\in\Omega_{i}}Z_{ij}}) \tag{7}\]  
where \(\Omega_{i}\) is the neighboring node set of node \(i\). When a transient fluctuation or failure occurs in one node, the operational states of all nodes in the power system are influenced through the power transmission line, not only the neighboring nodes.