## 2 Critical Line Set and Its Search Method
### The Definition and Graph Theory Characteristics of the CLS  
Rapid judgment of the impact of branch removal on the whole network is of great significance in suppressing the development and spread of grid accidents. Before searching the CLS of the removed branch, it is necessary to find out its characteristics.  
Under unfavorable operating conditions, a wide range of LE in the power system is likely to happen after removing the faulty branch, which may lead to cascading tripping accidents. The possible operating states of cascading trips caused by LE are as follows:* The overall load of the system is large, and some branches in the system are in a heavy load state. Once the faulty line is removed, LE will likely cause branch overload.
* One line in the system is out of power for maintenance, and the other line is removed by the main protection due to a short-circuit fault, resulting in LE, and the remaining branch is prone to overload.  
The set of dangerous lines greatly affected by the LE is called the CLS. The definition of the CLS is given according to the load encroachment-related proportional factor (LERF): the set of all branches satisfying Equation (1) after the faulty line is removed is called the CLS of the removed branch.  
\[d>d_{0} \tag{1}\]  
where \(d\) is the LERPF of the remaining branches to the faulty branch after it is removed; its value is related to the power grid's topological structure and branch parameters. The definition of its concept and the derivation of its mathematical expression will be discussed in Section 3 of this paper. The value of \(d\) can reflect the active power change value of the remaining branches in the network caused by the LE after the faulty branch is removed. The larger the value of \(d\), the larger the branch is affected by the LE, and vice versa, the smaller the value of \(d\), the smaller the branch is affected by the LE. \(d_{0}\) is a fixed value; the larger the value of \(d_{0}\), the fewer the number of branches included in the CLS of the removed branch and the higher the search speed of the CLS can be improved, but there is a possibility that some of the branches affected by the LE are missed; the smaller the value of \(d_{0}\), the more branches affected by the LE can be included, but it will increase the collection range of the wide-area information and reduce the speed. Therefore, the value of \(d_{0}\) should not be too small or too large; many scholars have studied this area and finally concluded that, according to the actual operational requirements, the value of \(d_{0}\) in the range of 0.2-0.3 is the best [23].  
In fact, the effect of the removed branch on the power flow of the whole network decreases gradually outwards, centered on the removed branch. Thus, in the early stages of incident development, the overloaded branch tends to be electrically close to the resected branch. According to the circuit principle, the graph theory characteristics of the CLS are summarized as follows:  
1. The branches in the CLS can form a loop with the removed branches.
2. The smaller the loop reactance value is, the greater the influence of LE is. When the loop reactance value is larger, the influence of LE can be ignored.  
The fast search method for the CLS proposed in this paper consists of two modules: (1) grid abstraction and dimensionality reduction; (2) path search to determine the CLS. The overall steps for searching for the CLS are shown in Figure 1.  
All the branches within the CLS make up the area highly affected by LE, and if this area can be quickly identified, the analysis can be narrowed down and time saved. Therefore, if backup protection activation exists after a line is removed, the CLS of that removed line is quickly searched to determine the scope of the analysis, and the wide-area information within that scope is used for the identification of the cause of branch overloading.  
Figure 1: The overall steps for searching for the CLS.