## 2 Fault Analysis and Feature Extraction
### Single-Phase Ground Fault Simulation and Analysis
#### 2.1.1 High Resistance Ground Fault  
When a high resistance ground fault occurs, the fault equivalent circuit is shown in Figure 1. The resistance \(R\) can be approximated as 3 times the transition resistance \(R_{f}\). Due to the large value of the transition resistance, the zero-sequence capacitance flowing through the system is small and the transient resonant frequency is low, so the line inductance \(L\) can be ignored. The fault transient process is mainly a parallel resonance process between the equivalent inductance \(L_{p}\) of the arc suppression coil and the ground capacitance \(C_{0\Sigma}\) of the system. Figure 1 shows that \(C_{0k}(k=1,2,3)\) is the zero-sequence distributed capacitance of line k to ground, \(C_{0\Sigma}=\sum\limits_{k=1}^{3}C_{0k};i_{0k}\) is zero sequence current of the k feeder, and \(i_{C0j}\) is zero sequence capacitance current to ground. \(i_{0f}\) and \(i_{0L_{p}}\) are the zero-sequence current at the fault point respectively and \(u_{f}\) is the zero-sequence voltage of the bus. The second-order nonlinear homogeneous differential equation is given in Equation (1):  
\[3R_{f}C_{0\Sigma}L_{p}\frac{d^{2}i_{0L_{p}}}{dt^{2}}+L_{p}\frac{di_{0L_{p}}}{ dt}+3R_{f}i_{0L_{p}}=u_{f}(t) \tag{1}\]  
The characteristic roots of this differential equation are given in Equation (2):  
\[\left\{ \begin{array}{c}p_{1}=-\frac{1}{6R_{f}C_{0\Sigma}}+\sqrt{\left(\left( \frac{1}{6R_{f}C_{0\Sigma}}\right)\right)^{2}-\frac{1}{L_{p}C_{0\Sigma}}}\\ p_{2}=-\frac{1}{6R_{f}C_{0\Sigma}}-\sqrt{\left(\frac{1}{6R_{f}C_{0\Sigma}} \right)^{2}-\frac{1}{L_{p}C_{0\Sigma}}}\end{array} \right. \tag{2}\]  
The zero-sequence current flowing through the arc suppression coil is given in Equation (3):  
\[i_{0L_{p}}(t)=Bsin(\omega_{0}t+\varphi_{2})+A_{1}e^{p_{1}t}+A_{2}e^{p_{2}t} \tag{3}\]  
Here, \(\omega_{0}\) is the angular frequency of zero-sequence current oscillation, and \(B\), \(A_{1}\), \(A_{2}\), and \(\varphi_{2}\) has the following meanings (Equation (4)):  
\[\left\{ \begin{array}{c}B=\frac{U_{m}}{\sqrt{\left(3R_{f}\right)^{2}\left(1- \omega_{0}^{2}L_{p}C_{0}\right)^{2}+\left(\omega_{0}^{2}L_{p}\right)^{2}}}\\ A_{1}=\frac{\omega_{0}Bcos\varphi_{2}-p_{2}Bsin\varphi_{2}}{p_{2}-p_{2}Bsin \varphi_{2}}\\ A_{2}=\frac{\omega_{0}Bcos\varphi_{2}-p_{1}Bsin\varphi_{2}}{p_{2}-p_{1}}\frac{ \omega_{0}L_{p}}{3R_{f}\left(1-\omega_{0}^{2}L_{p}C_{0}\right)}\\ \end{array} \right. \tag{4}\]  
Figure 1: High-resistance ground fault equivalent circuit. According to Equation (4), busbar zero sequence voltage is given in Equation (5):  
\[u_{0}(t)=L_{p}\frac{di_{0L_{p}}}{dt}=\omega_{0}L_{p}Bcos(\omega_{0}t+\varphi_{2} )+L_{p}p_{1}A_{1}e^{p_{1}t}+L_{p}p_{2}A_{2}e^{p_{2}t} \tag{5}\]  
The zero-sequence capacitance to the earth is given in Equation (6):  
\[i_{C0k}=C_{0k}\frac{du_{0}}{dt}=\omega_{0}^{2}L_{p}C_{0k}Bsin(\omega_{0}t+ \varphi_{2})+\ L_{p}C_{0k}\left(p_{1}^{2}A_{1}e^{p_{1}t}+p_{2}^{2}A_{2}e^{p_{2 }t}\right) \tag{6}\]  
The zero-sequence current at the fault point is given in Equation (7):  
\[i_{0f}=i_{0L_{p}}+i_{0}=\left(1-\omega_{0}^{2}L_{p}C_{0}\right)Bsin(\omega_{0} t+\varphi_{2})+\left[L_{p}C_{0}p_{2}^{2}+1\right]A_{1}e^{p_{1}t}+\left[L_{p}C_{0}p_{2 }^{2}+1\right]A_{2}e^{p_{2}t} \tag{7}\]  
According to Equation (7), each outgoing zero-sequence current consists of a steady-state component and a transient component. Considering that the initial value of the two-phase attenuated DC component is not 0 at the same time and the attenuation factors are not equal, the transient-free component always exists. The grounding resistance value is proportional to the transient duration, which is helpful for fault diagnosis. For high-resistance ground faults, the greater the ground resistance, the lower the zero-sequence current flowing through the system, and the charge and discharge speed of the zero-sequence capacitor on the line is slow, resulting in a low transient resonant frequency of the system. In this case, the inductive reactance of the circuit is low and can be essentially ignored.