## 4 Simulation Models
### Three-Carrier Current Continuum Equation Model  
This paper uses a three-carrier current continuum equation model to simulate the charge transport processes in transformer oil by considering the generation, recombination, and dissipation of electrons (\(e\)), positive ions (\(p\)), and negative ions (\(n\)). \[\frac{\partial\rho_{p}}{\partial t}+\
abla\cdot\left(\rho_{p}\mu_{p}\vec{E} \right)=G_{FD}+G_{FP}+G_{T}+\frac{\rho_{p}\rho_{e}R_{pe}}{q}+\frac{\rho_{p}\rho _{n}R_{pn}}{q} \tag{2}\]  
\[\frac{\partial\rho_{n}}{\partial t}-\
abla\cdot\left(\rho_{n}\mu_{n}\vec{E} \right)=\frac{\rho_{e}}{\tau_{a}}+\frac{\rho_{p}\rho_{n}R_{pn}}{q} \tag{3}\]  
\[\frac{\partial\rho_{e}}{\partial t}-\
abla\cdot\left(\rho_{e}\mu_{e}\vec{E} \right)=-G_{FP}-G_{T}-\frac{\rho_{p}\rho_{e}R_{pe}}{q}-\frac{\rho_{e}}{\tau_{ a}} \tag{4}\]  
\[\
abla\cdot\left(\vec{E}\vec{E}\right)=\rho_{p}+\rho_{e}+\rho_{n} \tag{5}\]  
\[\vec{J}=\left(\mu_{p}\rho_{p}-\mu_{e}\rho_{e}-\mu_{n}\rho_{n}\right)\vec{E} \tag{6}\]  
where \(\varepsilon\), \(k_{T}\), \(c_{v}\) and \(\rho\) are the permittivity (\(\varepsilon_{0}\)), thermal conductivity, specific heat, and density of the oil, respectively. Since in the oil gap surface discharge occurs generally between 2 and 3 us, the speed of oil can be negligible; \(q\) is the electronic charge quantity and \(E\) is the field strength; \(\rho_{p}\), \(\rho_{n}\), \(\rho_{e}\) are the charge density of positive ions, negative ions, and electrons, respectively; \(\mu_{p}\), \(\mu_{n}\), \(\mu_{e}\) are the mobilities of positive ions, negative ions, and electrons, respectively; \(R_{\rm pn}\), \(R_{\rm pe}\) are the coefficients of positive ions-negative ions and positive ions-electrons, respectively; \(\tau_{a}\) is the electron attachment time constant. Equation (5) is [PERSON]'s equation, which represents the effect of space charge on the electric field. Equations (2)-(4) are the three-carrier current continuum equations, which satisfy [PERSON]'s law [24], where a is the quotient of the electron decay length and the electron velocity, which is the attachment time constant, and where the electron decay length \(\lambda_{A}\) is 1 mm in Equation (7). \[\tau_{a}=\lambda_{A}/\mu_{e}\left|\vec{E}\right| \tag{7}\]  
On the right side of the Equations (2)-(4) for the theory of ionization in liquids, there are three different mechanisms [24]: field ionization, impact ionization, and photoionization. One of the common ionization mechanisms is field ionization. The molecules are in a liquid under a high electric field, whose ionization charge density production rate is denoted by \(G_{F}\). Field ionization has two main components: [PERSON]'s theory of the formation of free positive and negative ions by neutral molecule pairs in a dielectric liquid with an applied high electric field [25], denoted as \(G_{FD}\); and the formation of positive ions and free electrons by neutral molecules in an electrolytic ion, denoted as \(G_{FP}\). The field ionization density production rate is summarized using the Zener model [25]. \[G_{FD}\left(\left|\vec{E}\right|\right)=\frac{\sigma^{2}I_{1}\left(4\sqrt{q^{3 }\left|\vec{E}\right|/16\pi\varepsilon_{ro}\varepsilon_{0}k_{B}^{2}T^{2}} \right)}{2b\varepsilon_{ro}\varepsilon_{0}\left(\mu_{p}+\mu_{n}\right)}\cdot \exp\left(-\frac{\pi^{2}ma\mathrm{I}\mathrm{P}\left(\vec{E}\right)}{qh\left| \vec{E}\right|}\right) \tag{8}\]  
\[G_\[\mathrm{IP}\left(\stackrel{{\rightarrow}}{{E}}\right)=\Delta-\gamma \sqrt{\left|\stackrel{{\rightarrow}}{{E}}\right|} \tag{10}\]  
According to the Townsend theory of impact ionization [26], the collision of free electrons with neutral molecules at high field strengths is accelerated and results in a surge of electron energy in the neutral molecule, producing another group of positive ions and free electrons, which is also an ionization mechanism.