## 6 Application of the Incipient Fault Model  
This section shows a cable incipient fault model application case. A low-voltage distribution system, as shown in Figure 22, is established in Simulink. The system parameters are as follows: The 10 kV system has an equivalent resistance of 0.05 \(\Omega\), an equivalent inductance of 1.58 mH, and a short-circuit capacity of 200 MVA. The transformer has a turn ratio of 10/0.4 kV and a capacity of 60 kVA. The system consists of three branch lines, each with a length of 0.5 km. The per-unit-length parameters are: resistance of 0.3 \(\Omega\)/km, inductance of 0.08 mH/km, and capacitance of 0.2 \(\mu\)F/km. Each line carries a load of 10 kW, resulting in a total system equivalent load of 30 kW with a power factor of 0.88. A power factor compensation capacitor bank with a capacity of 5 kvar is employed, targeting a power factor of 0.93. In the simulation, the grounding resistance is set to 10 \(\Omega\), and the fault point is positioned at the midpoint of the cable. The parameters used in the model are shown in Table 3 to simulate the typical incipient fault waveforms described in Section 3.1. Since incipient faults typically occur near the voltage peak, the fault resistance is inserted at the fault point during the peak. At the measurement point located at the sending end of the cable, phase-to-ground voltages and phase currents are recorded. Figure 21: Comparison of experimental and simulated waveforms: (**a**) sub-cycle arc waveform in liquid penetration or immersion condition; (**b**) sub-cycle arc waveform in humid air exposure condition; (**c**) periodic current waveform in liquid penetration or immersion condition. \begin{table}
\begin{tabular}{c c} \hline \hline
**Type** & **Value** \\ \hline Incipient fault model for sub-cycle disturbance & \(g_{\max}=100\) mS, \(f=101\) Hz \\  & \(\theta=32^{\circ}\), \(t_{\text{st}}=4.4\), \(t_{0}=4.8\) \\ \hline Incipient fault model for periodic disturbance & \(g_{\max}=1.2\) mS, \(g_{\min}=0.2\) mS \\  & \(t_{\text{st}}=3.2\) ms, \(t_{0}=4.2\) ms \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model parameters used in the simulation. Figure 22: Low-voltage distribution system simulation diagram. For sub-cycle arcs, when the incipient fault occurs, the fault-phase current measured at the sending end of the line increases suddenly, while the impact on the normal-phase current is negligible. However, due to the low internal impedance and high short-circuit capacity of the power supply side, the voltage disturbances measured at the sending end of the line are relatively small. As a result, the fault-phase voltage exhibits only minor disturbances, as shown in Figure 23. For periodic fault currents, since their amplitude is generally within 2 A, the voltage and current waveforms at the sending end do not exhibit significant changes. To analyze the characteristics of disturbances measured at the sending end under different system parameters, the incipient fault current is extracted by computing the phasor summation of the measured three-phase currents and is compared with the fault point current. Table 4 presents three sets of system parameters and calculates the normalized mean square deviation (MSD) between the fault point and sending-end waveforms under sub-cycle arc faults (\(\sigma_{1}\)) and periodic current faults (\(\sigma_{2}\)). The results indicate that as the system capacity increases, the MSD between the fault point and sending-end waveforms gradually increases, suggesting that system parameters significantly influence fault waveform propagation. Figure 24 compares the fault point and sending-end waveforms under three different system parameters. It can be observed that as system capacity increases, the waveform shape at the sending end remains largely unchanged, but its amplitude gradually decreases.