# A Data-Driven Model of Cable Insulation Defect Based on Convolutional Neural Networks
## 2 Finite Element Modeling Method of Cable  
A cross-linked polyethylene (XLPE) three-core cable of 10 kV commonly used in the model Y]V22-8.7/10-3 \(\times\) 240 mm\({}^{2}\) was chosen as the object of finite element calculations. The cable model containing impurities was constructed by the finite element calculation software COMSOL Multiphysics and the cable model was constructed as shown in Figure 1.  
The core, semiconductor layer, insulation layer, shielding layer, filler, armor layer and outer sheath of the three-phase axisymmetric cable are set. The geometric parameters of each structure of the cable are shown in Table 1.  
After constructing the model using COMSOL Multiphysics finite element calculation software, the control equations involved are analyzed. This paper is a study of the cable electric field under steady state conditions, so the set of equations involved is as follows.  
\[\mathbf{E}=-\
abla\varphi \tag{1}\] \[\
abla\cdot\mathbf{D}=\rho\] (2) \[\
abla\cdot\mathbf{J}=-j\omega\rho \tag{3}\]  
In the above equation, \(\mathbf{E}\), \(\mathbf{D}\) and \(\mathbf{J}\) are electric field strength, potential shift vector and current density. \(\varphi\) is electric potential, \(\rho\) is charge density.  
The constitutive relationship of the material properties is  
\[\mathbf{D}=\varepsilon\mathbf{E} \tag{4}\]  
\[\mathbf{J}=\sigma\mathbf{E} \tag{5}\]  
where, \(\varepsilon\) is relative electric permittivity, \(\sigma\) is electrical conductivity.  
Based on Equations (1)-(5), the differential equation for \(\varphi\) is obtained as follows.  
\[\
abla\cdot((\sigma+j\omega\varepsilon)\
abla\varphi)=0 \tag{6}\]  
After choosing the control equations for the model, the boundary conditions for the model are then determined.  
In a three-phase AC cable, the phase difference between the voltages of the three phases A, B and C is 120 degrees. The A-phase voltage \(\varphi_{a}\), B-phase voltage \(\varphi_{b}\), and C-phase voltage \(\varphi_{c}\) are set as follows.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structure** & **Materials** & **Inner Diameter/mm** & **Outer Diameter/mm** \\ \hline \hline \multirow{2}{*}{Cable cores} & Copper & 11.9 & 11.9 \\ \cline{2-4}  & Semiconductor & 11.9 & 12.7 \\ \hline \hline \multirow{2}{*}{Insulation layer} & XLPE & 12.7 & 17.2 \\ \hline \hline \multirow{2}{*}{Shielding layer} & Copper & 17.2 & 17.3 \\ \cline{2-4}  & \multirow{2}{*}{Polyppropylene} & 41.1 & 41.1 \\ \hline \hline \multicolumn{1}{c}{Armor layer} & Aluminum & 41.1 & 41.11 \\ \hline \hline \multicolumn{1}{c}{Outer sheath} & Polyethylene & 41.11 & 43.11 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric parameters of cable structure.  
Figure 1: Finite element calculation modeling of three-phase cable containing impurities.  
\[\begin{split}\varphi_{a}=\varphi_{0}e^{j0}\\ \varphi_{b}=\varphi_{0}e^{-j\frac{2\pi}{3}}\\ \varphi_{c}=\varphi_{0}e^{+j\frac{2\pi}{3}}\end{split} \tag{7}\]  
After determining the boundary conditions, the finite element model is meshed and then the finite element calculations are performed on the model.