## 2 The Comprehensive Vulnerability Rate Model
### Multisource Situational Awareness Analysis
#### 2.3.5 Cascading Faults Factor  
The cascading faults of the distribution line in the ice weather is related to the influence of the power flow. The power flow disturbance will make some distribution lines in the self-organized critical state under certain conditions [24], according to the theory of complex systems. It is easy to cause cascading faults when the power system is in a self-organized critical state. The criteria for judging a critical line is as follows:  
\[H\;=\;\frac{|L_{i}|/L_{i,max}}{\sum_{l}|P_{li}|/\sum_{l}P_{li,max}}\;\geq\;\alpha \tag{19}\]\[\sum_{l}\lvert P_{li}\rvert e(0.56\ -\ 1.00)\sum_{l}P_{li,max}\]  
where \(H\) is the critical coefficient, \(L_{i,max}\) is the rated power, \(L_{i}\) is the actual power, \(P_{li}\) is the active power flow of the branch, \(P_{li,max}\) is the rated active transmission capacity of the branch, \(I\) is composed of all lines and transformers set, \(\alpha\) is the critical reference coefficient which the empirical value is generally 1.05.  
Let the total number of lines in the area be \(N\), and the chain failure propagation degree of the long-distance connection line \(k\) is defined as:  
\[g_{k}\ =\ \frac{N_{count}\sum_{y=1,\cdots,s}f(k,y)}{N_{count}\sum_{i=1,\cdots,N}k} \tag{20}\]  
where \(N_{count}\) is the counting function. \(N_{count}\sum\limits_{y=1,\cdots,s}f(k,y)\) is the correlation between distribution line \(k\) and distribution line \(y\). If \(k\) fails, \(y\) also fails, meaning they are related [25], \(f(k,y)\ =\ 1\), otherwise \(f(k,y)\ =\ 0\). \(N_{count}\sum\limits_{i=1,\cdots,N}k\) is the number of all distribution lines associated with distribution line \(k\).  
Define the average failure rate of the ice disaster is \(R_{ax}\ =\ n_{2}/n_{a2}\), then cascading faults situation index of the distribution line \(k\) that considering the power flow disturbance is as follows:  
\[\eta_{k,5}^{t}\ =\ R_{ax}\cdot g_{k} \tag{21}\]  
Any extreme situation index value may cause distribution fault. In order to meet the actual situation, standard deviation is proposed in this paper to jointly describe the impact of the internal operating situation on the failure rate of the distribution network. Thus, use Equation (22) to determine the number of useful situation indicators \(N\) from \(\eta_{k,i}^{t}\in\left\{\eta_{k,1}^{t},\eta_{k,2}^{t},\eta_{k,3}^{t},\eta_{k,4 }^{t},\eta_{k,5}^{t}\right\}\). The initial value of \(N\) is 5. \(\sigma_{0}\) is critical value of the standard deviation. If \(\sigma>\sigma_{0}\), the smallest indicator is ignored and the value of \(N\) decreases by 1. Repeat the verification by Equation (22) until \(\sigma\ \leq\ \sigma_{0}\) and determine the value of \(N\). Finally, the internal operation situation factors \(\eta_{k,sit}^{t}\) can be calculated by using Equation (23).  
\[\sigma\ =\ \sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\eta_{k,i}^{t}-\overline{ \eta_{k}^{t}}\right)^{2}} \tag{22}\]  
\[\eta_{k,sit}^{t}\ =\ exp\left[\sum_{i=1}^{N}\theta_{i}\ln\ \eta_{k,i}^{t}\right] \tag{23}\]  
where \(\theta_{i}\) indicates the index weight, \(\sum\ \theta_{i}\ =\ 1\).