## 2 Multi-Physics Coupling Mathematical Model
### Mathematical Modelling of Surface Charge Accumulation on the Insulator  
In DC GIL, the current density and bulk charge density satisfy the continuity equation:  
\[\
abla\cdot\mathbf{J}+\frac{\partial\rho}{\partial t}=0 \tag{8}\]  
where \(\mathbf{J}\) is the current density (unit: A/m\({}^{2}\)) and \(\rho\) is the bulk charge density (unit: C/m\({}^{3}\)). Considering the microscopic physical processes involved in the generation, recombination, and diffusion of positive and negative charge carriers in SF\({}_{6}\), the gas-side ion control equation can be articulated in accordance with the fluid dynamics equation:  
\[\frac{\partial n^{+}}{\partial t}=\frac{\partial n_{\text{IP}}}{\partial t}-n ^{+}k_{r}n^{-}-b^{+}\
abla\cdot(n^{+}\mathbf{E})+D^{+}\
abla^{2}n^{+} \tag{9}\]  
\[\frac{\partial n^{-}}{\partial t}=\frac{\partial n_{\text{IP}}}{\partial t}-n ^{-}k_{r}n^{+}+b^{-}\
abla\cdot(n^{-}\mathbf{E})+D^{-}\
abla^{2}n^{-} \tag{10}\]  
Here, \(\mathbf{E}\) is the electric field strength (unit: V/m); \(n^{+}\) and \(n^{-}\) are positive and negative ion concentrations, respectively (unit: 1/m\({}^{3}\)); \(\partial n_{\text{IP}}/\partial t\) is the ion-pair-generation rate in SF\({}_{6}\)[29]; \(k_{r}\) is the recombination rate, which varies with the gas pressure [30]; and \(b^{+}\) and \(b^{-}\) are positive and negative ion migration rates, respectively (unit: m\({}^{2}\)/(V\(\cdot\)s)). When the electric field is below 3 kV/mm, the mobility exhibits minimal variation, allowing the effect of the electric field on mobility to be disregarded [30]. \(D^{+}\) and \(D^{-}\) are positive and negative ion diffusion coefficients (unit: m\({}^{2}\)/s), which can be calculated from the [PERSON] equation. At a temperature of 20 \({}^{\circ}\)C, the mobility of positive ions is measured at 1.31 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/(V\(\cdot\)s), while the mobility of negative ions is recorded at 1.15 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/(V\(\cdot\)s). The ion mobility at various temperatures can be derived from the following formula:  
\[b(T)=b_{0}\frac{T}{T_{0}} \tag{11}\]where \(b_{0}\) is the mobility of positive and negative ions at the temperature \(T_{0}\) = 293 K and \(T\) represents the temperature. The diffusion coefficient is defined by (12). \[D^{\pm}=b^{\pm}\frac{k\cdot T}{e} \tag{12}\]  
where \(k\) is the Boltzmann constant and \(e\) is the elementary charge. The current density \(\mathbf{J}_{\mathrm{g}}\) through \(\mathrm{SF}_{6}\) gas can be defined as  
\[\mathbf{J}_{\mathrm{g}}=e\mathbf{E}(n^{+}b^{+}+n^{-}b^{-})-e\cdot\
abla(D^{+}n ^{+}-D^{-}n^{-}) \tag{13}\]  
We assumed a constant relative dielectric constant for \(\mathrm{SF}_{6}\) gas, as its variation within the 20-80 \({}^{\circ}\)C range at 0.5 MPa is less than 1%, calculated from density data (Using AP1700 chemistry webbook density data). The relationship between positive and negative ions on the gas side and space charge and potential can be expressed as  
\[\mathbf{E}=-\
abla\varphi \tag{14}\]  
\[\
abla^{2}\varphi=-\frac{\rho}{\varepsilon}=-\frac{e(n^{+}-n^{-})}{\varepsilon} \tag{15}\]  
In the temperature and electric field ranges examined in this paper, the insulator dielectric constant exhibits minimal variation; therefore, the effect of temperature on it can be neglected. The transient equation for the insulator bulk charge density can be formulated as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\sigma_{v}}{\mathbf{E}}\rho+\mathbf{E} \cdot\
abla\sigma_{v}=0 \tag{16}\]  
where \(\sigma_{v}\) is the bulk conductivity of the insulator (unit: \(\mathrm{S/m}\)). At the interface between different media, the normal component of the current density maintains continuity. Considering the surface conduction of the insulator, the equilibrium condition at the insulator surface is expressed as follows:  
\[