## 3 Lifespan Prediction Method
### Prediction Method for Aging Time of Composite Insulators  
A BP neural network is a kind of multi-layer feed-forward neural network. Because of its simple structure, many adjustable parameters, training algorithms and good maneuverability, BP neural networks have been extensively applied [26]. In this case, a BP neural network was employed to propose a prediction method for determining the aging time of composite insulators by means of the four parameters which are significantly correlated with aging time. The structure of the BP neural network used in this paper is shown in Figure 7.  
In Figure 7, there are \(n\) neurons in the input layer of the BP neural network, \(m\) neurons in the hidden layer and \(l\) neurons in the output layer. Based on Reference [27], the optimal number of neurons in the hidden layer are given by Equations (6)-(8). \[m<n-1 \tag{6}\]  
\[m<\sqrt{n+l}+a \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Correlation Coefficient** & **Significance** & **Correlation** \\ \hline \(\theta_{av}\) & \(-0.997\) & \(\alpha=0.0006\) & Significant negative correlation \\ \(A\) & \(0.915\) & \(\alpha=0.0106\) & Positive correlation \\ \(S_{em}\) & \(0.906\) & \(\alpha=0.0136\) & Positive correlation \\ \(H\) & \(-0.861\) & \(\alpha=0.0286\) & Negative correlation \\ \(X_{\text{Si}}\) & \(-0.992\) & \(\alpha=0.0006\) & Significant negative correlation \\ \(X_{\text{C}}\) & \(-0.904\) & \(\alpha=0.0136\) & Positive correlation \\ \(X_{\text{O}}\) & \(0.996\) & \(\alpha=0.0006\) & Significant positive correlation \\ \(U_{f}\) & \(-0.963\) & \(\alpha=0.0026\) & Significant negative correlation \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Correlation between lifespan prediction parameters and aging time. Figure 7: Structure of back propagation (BP) neural network. \[m=\log_{2}n \tag{8}\]  
where \(n\) is the number of neurons in the input layer; \(l\) is the number of neurons in the output layer; \(m\) is the number of neurons in the hidden layer; \(a\) is a constant between 0 and 10. This paper employs four parameters, namely, the static contact angle \(\theta_{\text{av}}\), the relative content of Si element \(X_{Si}\), the relative content of O element \(X_{O}\) and the salt-fog flashover voltage \(U_{f}\), which are significantly correlated with aging time, as the input neurons, and the aging time as the output neuron. Thus, the number of neurons in the hidden layer is 2. Moreover, the goal of the error is set as \(10^{-8}\). In BP neural network testing, parameters of composite insulators aged for 1-10 years were employed as training samples, and parameters of composite insulators aged for 12 years were employed as test samples. The error changes during training are shown in Figure 8.  
In the process of BP neural network training for aging time prediction, the mean squared error of the predicted aging time of composite insulators scarcely changed at first. During the process from 30 to 35 iterations, the mean squared error of predicted aging time decreased rapidly and achieved convergence in 35 iterations. In this case, the mean squared error of predicted aging time is \(10^{-8}\), which indicates that the equivalent relationship between significantly correlated parameters and aging time can be established accurately based on the BP neural network, and that the aging time of composite insulators can be predicted accurately based on the method proposed in this paper. In this paper, \(\sigma_{1}\) is defined as the relative error between the predicted aging time (denoted \(a_{1}\)) and actual aging time (denoted \(a\)). Results for \(a_{1}\), \(a\) and \(\sigma_{1}\) of samples aged for 12 years are shown in Table 10.  
\[\sigma_{1}=\frac{|a_{1}-a|}{a}\times 100\% \tag{9}\]  
Table 10 shows that the relative error \(\sigma_{1}\) between the predicted and actual values is less than 5%, which is within the range of high accuracy in engineering practice.