# Fault Diagnosis of HV Cable Metal Sheath Grounding System Based on LSTM
## 2 Model of High Voltage Cable Grounding System
### Sheath Induced Current  
According to the principle of electromagnetic induction, the cable core will generate an induced voltage on the metal sheath due to the flow of AC, and the induced voltage generates an induced current in the metal sheath loops. Figure 4 represents the equivalent induction circuit diagram of the metal sheath of the three-phase cable cross-bonded. \(U_{\rm mA1}\), \(U_{\rm mBn}\), and \(U_{\rm mCn}\) (\(n=1,\,2,\,3\)) represent the equivalent induction voltage of the metal sheath of each minor section of the cable, including both the core-to-sheath induction voltage and the sheath-to-sheath induction voltage; \(I_{\rm i1}\), \(I_{\rm i2}\), and \(I_{\rm i3}\) represent the induction currents of the three sheath loops.  
Figure 3: The A1 minor section cable equivalent two-port network.  
Taking the sheath loop L1 as an example, the induced current of the sheath loop can be calculated by Equations (7) and (8).  
\[I_{\rm{I1}}=\frac{U_{\rm{mA1}}+U_{\rm{mB2}}+U_{\rm{mC3}}}{Z_{\rm{mA1}}+Z_{\rm{ mB2}}+Z_{\rm{mC3}}+R_{e}+R_{g}} \tag{7}\]  
\[\left\{\begin{array}{l}U_{\rm{mA1}}=-\mathrm{i}\omega(I_{\rm{A}}L_{\rm{AA}}+ (I_{\rm{B}}+I_{\rm{I2}})M_{\rm{AB}}+(I_{\rm{C}}+I_{\rm{i3}})M_{\rm{AC}})I_{1} \\ U_{\rm{mB2}}=-\mathrm{i}\omega((I_{\rm{A}}+I_{\rm{I1}})M_{\rm{AB}}+I_{\rm{B}} L_{\rm{BB}}+(I_{\rm{C}}+I_{\rm{i3}})M_{\rm{BC}})I_{2}\\ U_{\rm{mC3}}=-\mathrm{i}\omega((I_{\rm{A}}+I_{\rm{I1}})M_{\rm{AC}}+(I_{\rm{B}} +I_{\rm{I2}})M_{\rm{BC}}+I_{\rm{C}}L_{\rm{CC}})I_{3}\end{array}\right. \tag{8}\]  
where \(I_{\rm{A}}\), \(I_{\rm{B}}\), and \(I_{\rm{C}}\) denote the load currents of the three-phase cable; \(I_{1}\), \(I_{2}\), and \(I_{3}\) denote the lengths of the three minor sections of the sheath loops.  
The core (or sheath) to sheath mutual inductance coefficients are shown in Equations (9) and (10) [14].  
\[L_{xx}=\frac{\mu_{0}}{2\pi}\ln\left(\frac{D_{\rm{s}}}{D_{\rm{c}}}\right) \tag{9}\]  
\[M_{xy}=\frac{\mu_{0}}{2\pi}\ln\left(\frac{2S_{xy}}{D_{\rm{c}}}\right) \tag{10}\]  
where, \(L_{\rm{xx}}\) said unit length of the \(x\)-phase cable core on the phase metal sheath mutual inductance, \(M_{\rm{xy}}\) said unit length of the \(x\)-phase core (or sheath) on the \(y\)-phase sheath mutual inductance, \(\mu_{0}\) for the vacuum permeability, \(D_{\rm{s}}\) for the metal sheath outer diameter, \(S_{xy}\) for the \(x\)-phase cable core and the y-phase cable core distance.  
According to Equations (11) and (12), the equivalent impedance of the \(n\)th minor section of the metal sheath of the \(x\)th phase cable can be obtained.  
\[Z_{\rm{mx}n}=R_{\rm{m}}I_{n} \tag{11}\]  
\[R_{\rm{m}}=\frac{4\rho_{\rm{m}}(1+k_{\rm{s}}(T-20))}{\pi(D_{\rm{se}}^{2}-D_{ \rm{si}}^{2})} \tag{12}\]  
where, \(l_{n}\) (\(n=1,\,2,\,3\)) is the length of a minor section of cable, \(R_{\rm{m}}\) is the equivalent impedance of the metal sheath per unit length, \(\rho_{\rm{m}}\) is the resistivity coefficient of the metal sheath, \(k_{\rm{s}}\) is the temperature coefficient of the metal sheath, \(T\) is the temperature of the metal sheath, \(D_{\rm{se}}\) is the outer diameter of the metal sheath, and \(D_{\rm{si}}\) is the inner diameter of the metal sheath.  
Similarly, the sheath induced currents of sheath circuits L2 and L3 are shown in Equation (13).  
\[\left\{\begin{array}{l}I_{\rm{I2}}=\frac{U_{\rm{mB1}}+U_{\rm{mC2}}+U_{\rm{ mA3}}}{Z_{\rm{mB1}}+Z_{\rm{mC2}}+Z_{\rm{mA3}}+R_{e}+R_{g}}\\ I_{\rm{i3}}=\frac{U_{\rm{mC1}}+U_{\rm{mA2}}+U_{\rm{mB}}}{Z_{\rm{mC1}}+Z_{\rm{ mA2}}+Z_{\rm{mB}}+R_{e}+R_{g}}\end{array}\right. \tag{13}\]