## 2 Critical Line Set and Its Search Method
### The Establishment and Dimension Reduction of the Graph Model  
According to the characteristics of the CLS mentioned above, before searching for the CLS of the resected branch, it is necessary to transform the power system into a graph model and then perform a dimensionality reduction process on it so as to realize CLS searching in a smaller area, which greatly reduces the burden of further analyses and calculations and narrows down the target range, thus improving the searching speed. According to the graph theory, the power system can be abstracted into a graph model, represented by \(T\), and \(T=(V,E)\), where  
* \(V=\{v_{1},\,v_{2},\,\ldots,\,v_{m}\}\), and \(V\) represents the set of grid nodes; \(v_{i}\) represents the node abstracted from the generator or bus of the power grid;
* \(E=\{e_{1},\,e_{2},\,\ldots,\,e_{m}\}\), and \(E\) represents the edge set of the grid; \(e_{i}\) represents the edge abstracted from the transmission line or power transformer. If \(e_{i}\) has a direction, then \(T\) is a directed graph; otherwise, \(T\) is an undirected graph. If \(e_{i}\) has weight, then \(T\) is a weighted graph; otherwise, it is an unweighted graph. In the weighted graph, the weight of \(e_{i}\) is represented by \(w_{ij}\). In this paper, the branch reactance is used to represent the branch weight, which is also called branch length. Suppose \(T\) has \(n\) nodes. If \(v_{i}\) and \(v_{j}\) are directly connected by \(e_{i}\), then \(v_{i}\) and \(v_{j}\) are called adjacent nodes. If \(v_{i}\) and \(v_{j}\) are connected by a finite number of edges, then all branches from \(v_{i}\) to \(v_{j}\) form a path, denoted by \(P_{i}\), and the length of the path is the sum of the weights of all branches on the path, denoted by:  
\[Z(P_{i})=\sum_{e_{i}\in P_{i}}w_{ij} \tag{2}\]  
The topological relations of a graph are commonly represented by an adjacency matrix, denoted as \(A\), and \(A\) is an Nth-order square matrix. The search method for the CLS proposed in this paper is based on undirected graphs. The elements in the adjacency matrix of the undirected graph are denoted by \(A_{ij}\), as shown in Equation (3). \[A_{ij}= \begin{cases}0&i=j\\ w_{ij}&i\
eq j\big{(}v_{i},v_{j}\big{)}\in E\end{cases}  \tag{3}\]  
Reducing the dimension of the graph model is based on the hanging nodes and breakpoints. The definition of hanging nodes and breakpoints is as follows:  
1. Hanging nodes According to whether it can form a loop with other lines in the graph, the edges are distinguished. The edges that cannot form a loop with other edges are defined as hanging edges, and the nodes only connected to the hanging edges are defined as hanging nodes. The number of branches associated with the node is defined as the node's branch correlation degree (BCD). The BCD of the node in an undirected graph is equal to the number of non-zero elements in the row/column where the node is located in \(A\). According to the above definition of hanging nodes and BCD, hanging nodes include:  
1. Nodes whose BCD is one. 2. After deleting all nodes in the graph whose BCD is one, the nodes whose BCD becomes one in all remaining nodes. The BCD of each node in the graph and its neighboring nodes can be quickly obtained by searching \(A\), thus quickly obtaining the hanging nodes. 2. Breakpoints Suppose \(T\) is a connected graph and \(v_{a}\) is a node in \(T\). If \(T\) becomes a non-connected graph after \(v_{a}\) is removed (removing a node requires removing all the edges associated with it together), then \(v_{a}\) is called a breakpoint. If \(T\) contains breakpoints, then \(T\) is called a divisible graph. The breakpoints of the graph can be searched using the Tarjan algorithm based on the depth-first algorithm. It follows from the graph theory characteristics of the CLS that if there is a breakpoint in a graph, the branch in the CLS and the removal branch should be located in the same subgraph.