# Research on Lifespan Prediction of Cross-Linked Polyethylene Material for XLPE Cables
## 3 Lifespan Prediction Method
### Prediction Method for Aging Time of XLPE Cables  
Due to the multi-dimensional and non-linear relation between the lifespan prediction parameters and the operating time of XLPE cables, there are some deviations in using a traditional simplified statistical method to characterize the aging degree and predict the lifespan of XLPE cables compared with the actual results. A BP neural network, a system with self-learning, self-organization and self-adaptive ability, can be used to obtain the implied relationship between input and output data through learning and training the input and output data in its network model [18]. In this case, a BP neural network was employed in this paper to establish the multi-dimensional nonlinear relationship between four lifespan prediction parameters obtained in Section 3.1 and the operating time. The structure of the BP neural network in this paper is shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Correlation Coefficient** & **Significance** & **Correlation** \\ \hline \(C_{ge}\) & \(-0.914\) & \(\alpha=0.0019\) & Significant negative correlation \\ \(X_{C}\) & \(-0.972\) & \(\alpha=0.0009\) & Significant negative correlation \\ \(\sigma\) & \(-0.977\) & \(\alpha=0.0009\) & Significant negative correlation \\ \(\delta\) & \(-0.995\) & \(\alpha=0.0009\) & Significant negative correlation \\  
In Figure 6, 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 [19], the best number for the neurons in the hidden layer should maintain the following Equation (8):  
\[m<\sqrt{n+l}+a \tag{8}\]  
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 employed four parameters including the gel content \(C_{ge}\), the crystallinity \(X_{C}\), tensile strength \(\sigma\), ultimate elongation \(\delta\), the dielectric permittivity \(\varepsilon\) and the dielectric loss \(I_{tan}\), which are significantly correlated with the operating time, as the input neurons, while the operating time is the output neuron. Thus, the number of neurons in the hidden layer is five. Moreover, the goal of error is set as \(10^{-8}\).  
In this paper, XLPE cables aged for 1-7 years were employed as training samples in a BP neural network, along with samples aged for 8 years as testing samples. The error changes during training are shown in Figure 7.  
In Figure 7, the iterative process can be divided into two ranges (0-26 iterations and 26-28 iterations). In the 0-26 iterations range, there are tiny changes occurring in the mean squared error of the predicted operating time of XLPE cables in the iterative calculation. When the iteration step reaches 26 iterations,  
Figure 6: Structure of Backpropagation (BP) neural network.  
Figure 7: Drop in deviation by training.  
the decreasing rate of the mean squared error gradually accelerates. At 28 iterations, the mean squared error reaches the stop trigger condition and the computation of the BP neural network is completed, which indicates that an equivalent relationship between the significantly correlated parameters and the operating time has been established.  
In this paper, \(\sigma_{1}\) is defined as the relative error between the predicted aging time (denoted as \(a_{1}\)) and actual aging time (denoted as \(a\)). The results of \(a_{1}\), \(a\) and \(\sigma_{1}\) in samples aged for 8 years are shown in Table 8.  
According to the results obtained in Table 8, the error between the actual operating time and the predicted operating time is less than 5%, which is within the allowable error range of engineering practice. In future research, the method proposed in this paper can be used for predicting the operating time of XLPE cables from Factory A.