## 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
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scale** & **Implication** & **Explanation** \\ \hline
0.5 & Equally important & \(c_{i}\) and \(c_{j}\) are equally important for equipment status \\
0.6 & Slightly important & \(c_{i}\) is slightly more important than \(c_{j}\) \\
0.7 & Generally important & \(c_{i}\) is a little more important than \(c_{j}\) \\
0.8 & Clearly important & \(c_{i}\) is obviously more important than \(c_{j}\) \\
0.9 & Much more important & \(c_{i}\) is much more important than \(c_{j}\) \\ \(<\)0.5 & Comparing \(c_{i}\) with \(c_{j}\) to judge \(m_{ij}\), then comparing \(c_{j}\) with \(c_{i}\) to judge \\ \(m_{ji}\ =\ (1-m_{ij})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Scale meaning and explanation in matrix \(M\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \(c_{1}\) & \(c_{2}\) & \(c_{3}\) & \(c_{4}\) & \(c_{5}\) & \(c_{6}\) & \(c_{7}\) & \(c_{8}\) & \(c_{9}\) \\ \hline CPP & 0.5975 & 0.5285 & 0.4760 & 0.3075 & 0.5065 & 0.4245 & 0.0972 & 0.4655 & 0.2675 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comprehensive posterior probability of characteristic quantities \(c_{1-9}\). The AHP method is a multi-objective decision-making method combining qualitative analysis and quantitative analysis, which inevitably has subjective factors of expert experience. The improved AHP method proposed in this paper introduces complete and detailed mathematical statistics, and uses the posterior probability in Bayesian theory to characterize the comprehensive influence degree to balance the influence of subjective factors, which has more application value and rationality. In the actual operation process, when certain piece of equipment of the protection system issues a certain type of forewarning alarm signal, it often indicates a trend of deterioration of the relevant equipment condition. If the alarm signal continues to appear afterwards, it is necessary to increase the weight of these characteristics indicators at this time, so that the status rating of the protection system continues to decline and the operation and maintenance personnel can carry out preventive maintenance as early as possible. In this paper, a variable weight synthesis model based on Factor Space Theory [29] is used to modify the characteristic quantity weight on the basis of the initial weight, combined with the information of the equipment operation data monitored in real time. Assuming that the factor state vector is \(\mathbf{X}=(x_{1},x_{2},\cdots,x_{n})\), the factor constant weight vector is \(\mathbf{W}=(w_{1},w_{2},\cdots,w_{n})\), and the state variable weight vector is \(S(\mathbf{X})=(S_{1}(\mathbf{X}),S_{2}(\mathbf{X}),\cdots,S_{n}(\mathbf{X}))\), then the normalized Hadamard product of \(\mathbf{W}\) and \(S(\mathbf{X})\) can obtain the variable weight vector \(\mathbf{W}\)(\(\mathbf{X}\)):  
\[w_{i}(\mathbf{X})=\frac{w_{i}S_{i}(\mathbf{X})}{\sum_{k=1}^{n}w_{k}S_{k}(\mathbf{X})},(i=1,2,\cdots,n) \tag{9}\]  
In the formula \(S_{i}(\mathbf{X})=e^{\mathbf{i}(x_{i}-\mathbf{\gamma})}\), \(\alpha\) is a variable weight factor, and the degree of heavy change of control rights. In this paper, the state vector is \(\mathbf{V}=(v_{1},v_{2},\cdots,v_{9})\), \(v_{i}\) is the value of the \(i\)-th characteristic quantity \(c_{i}\) at the current time. Factor constant weight vector is the initial weight of nine characteristic quantities. Formula (9) can be changed into  
\[w_{i}(\mathbf{X})=\frac{w_{i}\exp\Bigl{[}\alpha\Bigl{(}d_{iap}-d_{ibp}\Bigr{)} \Bigr{]}}{\sum_{k=1}^{n}w_{k}\exp\Bigl{[}\alpha\Bigl{(}d_{kap}-d_{kbp}\Bigr{)} \Bigr{]}},(i=1,2,\cdots,9) \tag{10}\]  
In the above formula, the general variable weight factor \(\alpha\)= \(-1\). \(w_{i}\) is the initial weight of the \(i\)-th characteristic quantity. \(d_{iap}\) and \(d_{ibp}\) are the distance between the characteristic quantity \(c_{i}\) and the left and right boundary of the node domain, \(d_{iap}=\bigl{|}v_{i}-a_{pi}\bigr{|}\), \(d_{ibp}=\bigl{|}v_{i}-b_{pi}\bigr{|}\).