## 5 Simulation Results
### Effect of Applied Voltage on PD  
The electric field is extremely inhomogeneous at the electrode tip, as shown in Figure 10. The PD pattern of the electrode of the foil layer with time at different voltages obtained from the simulation, where the charge distribution and the maximum and minimum values of the charge density at different moments, is shown in Figure 11. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Electron mobility \(\mu_{e}\) [m\({}^{2}\)/(V\(\cdot\)s)] & \(1\times 10^{-9}\) \\ Positive ion mobility \(\mu_{p}\) [m\({}^{2}\)/(V\(\cdot\)s)] & \(1\times 10^{-9}\) \\ Negative ion mobility \(\mu_{n}\) [m\({}^{2}\)/(V\(\cdot\)s)] & \(1\times 10^{-4}\) \\ Positive ion–negative ion complex rate \(R_{pn}\) [m\({}^{3}\)/s] & \(1.64\times 10^{-17}\) \\ Positive ion–electron complex rate \(R_{pe}\) [m\({}^{3}\)/s] & \(1.64\times 10^{-17}\) \\ Electron attachment time constant \(t_{e}\) [s] & \(1\times 10^{-6}\) \\ Boltzmann constant \(k\) [J/K] & \(1.3806\times 10^{-23}\) \\ [PERSON]’s constant \(H\) [J\(\cdot\)s] & \(6.626\times 10^{-34}\) \\ [PERSON]’s constant A [MA\(/\)(m\({}^{2}\)\(\cdot\)K\({}^{2}\))] & \(1.2\) \\ Oil density \(r\) [kg\(/\)m]\({}^{3}\) & \(880\) \\ Specific heat capacity of oil \(c\) [kg\(/\)m]\({}^{3}\) & \(1700\) \\ Molecular separation distance \(a\) [m] & \(3\times 10^{-10}\) \\ Free electron mobility \(\mu_{e\mu}\) [m\({}^{2}\)/(V\(\cdot\)s)] & \(1\times 10^{-14}\) \\ Free hole mobility \(\mu_{h_{H}}\) [m\({}^{2}\)/(V\(\cdot\)s)] & \(1\times 10^{-14}\) \\ Maximum electron entrapment density \(N_{e0}\) [C\(/\)m]\({}^{2}\) & \(100\) \\ Maximum cavity entry density \(N_{hop}\) [C\(/\)m]\({}^{2}\) & \(100\) \\ Electron entry rate \(B_{e}\) [s\({}^{-1}\)] & \(5\times 10^{-3}\) \\ Cavity-in-trap rate \(B_{h}\) [s\({}^{-1}\)] & \(5\times 10^{-3}\) \\ Electron descent rate \(D_{e}\) [s\({}^{-1}\)] & \(3\times 10^{-4}\) \\ Cavity descent rate \(D_{h}\) [s\({}^{-1}\)] & \(3\times 10^{-4}\) \\ Trapped hole–trap electron complex rate \(S_{0}\) [m\({}^{3}\)/(C\(\cdot\)s)] & \(0\) \\ Trap hole–free electron complex rate \(S_{1}\) [m\({}^{3}\)/(C\(\cdot\)s)] & \(5\times 10^{-3}\) \\ Free hole–trap electron complex rate \(S_{2}\) [m\({}^{3}\)/(C\(\cdot\)s)] & \(5\times 10^{-3}\) \\ Free hole–free electron complex rate \(S_{3}\) [m\({}^{3}\)/(C\(\cdot\)s)] & \(5\times 10^{-3}\) \\ Temperature \(T\) [K] & \(293\) \\ Electron injection barrier \(\omega_{ei}\) [eV] & \(1.18\) \\ Vacancy injection potential \(\omega_{ni}\) [eV] & \(1.19\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the hydrodynamic model and bipolar charge transport model. Electrostatic attraction is the main mechanism that causes streamers to grow towards the dielectric, and some experiments demonstrated this phenomenon [28; 30]. In Figure 11, the discharge starts as a corona discharge at the tip of the broken foil layer, with the charge concentrated at the high voltage end; as the discharge time increases, the discharge progresses outwards, but does not reach the insulating paper surface and ends. This situation is significantly improved at 40 kV (As shown in Figure 12), where the discharge time is reduced from 400 ns to 325 ns. From Figures 13 and 14, the discharge develops to the oil-paper interface at about 150 ns at 50 kV, and, at 60 kV, the time for the discharge to reach the paper surface of the junction is reduced to about 100 ns, followed by flashover discharge along the surface, as shown in Figure 15, where the total discharge time is reduced to 175 ns. Figure 11: Discharge process at 30 kV. Figure 12: Discharge process at 40 kV. Figure 10: Voltage contour with unit in V/m distribution at the tip. As shown in Figures 11-14, with increasing voltage from 30 kV to 60 kV, the PD time decreased from 400 ns to 150 ns.