## 3 State Evaluation and Fault Prediction of Protection System Based on Digital Twin Technology
### State Evaluation of Protection System Based on Matter-Element Extension
#### 3.2.3 Characteristic Quantities Weight of State Evaluation  
The occurrence of different characteristic quantities indicates that the equipment may have different fault types, and the probability of each fault type is different, and the influence on the equipment is also different. In this paper, the improved analytic hierarchy process (AHP) is used to give the initial weight to the characteristic quantity [26]. The posterior probability in Bayesian theory can characterize the influence of the characteristic quantity on the equipment state. In Bayesian theory, the prior probability is the probability of a fault type; the conditional probability is the probability of obtaining the corresponding fault symptom information, namely the probability of characteristic quantity, under the condition that a certain fault occurs in the equipment. According to the existing characteristic quantities, the probability of the corresponding fault type is the posterior probability. Bayesian formula is  
\[p(f_{a}\mid c_{b})=\frac{p(c_{b}\mid f_{a})*p(f_{a})}{\sum_{k=1}^{m}p(c_{b} \mid f_{k})*p(f_{k})} \tag{6}\]  
where \(m\) is the number of faults in the fault set \(F\), and the fault set is detailed in Section 3.3.1. \(p(f)\) is the prior probability of occurrence of fault type \(f_{a}\), \(p(c_{b}\mid f_{a})\) is the probability of occurrence of feature \(c_{b}\), i.e., conditional probability, when fault type \(f_{a}\) occurs. \(p(f_{a}\mid c_{b})\) is the probability of fault type \(f_{a}\) when characteristic \(c_{b}\) appears, namely the posterior probability. When \(c_{b}\) appears, the comprehensive posterior probability (CPP) of equipment failure is \(p(f\mid c_{b})=\sum_{k=1}^{m}\ p(f_{a}\mid c_{b})\). The higher the probability is, the higher the possibility of equipment failure occurring when the characteristic \(c_{b}\) occurs, and it can be considered that the influence of the characteristic \(c_{b}\) on the equipment state is greater. By referring to the relevant literature, the prior probability and conditional probability of the characteristic quantity of common faults of the protection system can be obtained [27; 28]. Considering the prior probability of all faults and the conditional probability of the occurrence of  
Figure 5: Channel operation scoring criteria. characteristic quantities when faults occur, the comprehensive posterior probabilities of nine characteristic quantities \(c_{1-9}\) of the protection system faults are calculated, as shown in Table 3.  
According to the comprehensive posterior probability of characteristic quantities \(c_{1-9}\) of equipment failure, combined with the experience of equipment manufacturers and high voltage station operation and maintenance experts, the fuzzy complementary judgment matrix \(M\) can be obtained by comparing the influence of each feature on the equipment state. \[M=\left[ \begin{array}{ccccccccc}0.5&0.6&0.7&0.9&0.7&0.7&0.9&0.7&0.9\\ 0.4&0.5&0.6&0.8&0.6&0.7&0.2&0.7&0.9\\ 0.3&0.4&0.5&0.7&0.5&0.6&0.7&0.6&0.8\\ 0.1&0.2&0.3&0.5&0.3&0.4&0.6&0.3&0.4\\ 0.3&0.4&0.5&0.7&0.5&0.7&0.8&0.5&0.8\\ 0.3&0.3&0.4&0.6&0.3&0.5&0.7&0.4&0.7\\ 0.1&0.8&0.3&0.4&0.2&0.3&0.5&0.3&0.6\\ 0.3&0.3&0.4&0.7&0.5&0.6&0.7&0.5&0.8\\ 0.1&0.1&0.2&0.6&0.2&0.3&0.4&0.2&0.5\end{array} \right] \tag{7}\]  
The element \(m_{ij}\) in column \(j\) of row \(i\) of the matrix means the influence of the first characteristic quantities \(c_{i}\) on the device state compared with the \(j\)-th characteristic quantities \(c_{j}\). The meanings of the 0 to 1 scales in the matrix are shown in Table 4.  
The formula for calculating the weight of characteristic quantity by the improved AHP method is:  
\[w_{i}=\frac{1}{n}-\frac{1}{2a}+\frac{1}{na}\sum_{k=1}^{n}m_{ik},\ (i=1,2\cdots,n) \tag{8}\]  
where \(a\) is an adjustment parameter and \(n\) is the number of characteristic quantity. In order to avoid the large gap between the weight of each characteristic quantity, generally take \(a=(n-1)/2\). The initial weight \(w_{i}\) of each characteristic quantity is obtained as shown in Table 5.