## 2 Hybrid Numerical Model
### Boundary Conditions and Initial Values  
The governing equations in the hybrid numerical model are solved based on Comsol Multiphysics 6.0. Actually, when there is a burr defect on the surface of a DC wire, corona discharges are concentrated at the tip of the burr. In order to simulate this phenomenon and reduce the solution difficulty, a two-dimensional geometric structure comprising a corona cage, wire, and burr tip defect is established, as shown in Figure 1.  
In Figure 1, the center of the model represents the wire and the tip defect structure, and their enlarged diagram is pointed out by the red arrow, where the wire radius is \(R\), the tip height is \(h\), and the radius of curvature of the tip is \(r\). The outer edge of the model simulates the corona cage wall, with a maximum radius of 200 mm from the center. The corona cage wall is set to be effectively grounded, and a positive voltage \(U\) is applied on the surface of the central conductor. The expression of the voltage is shown in Equation (6).  
Figure 1: Computation: geometric model.  
\[U=U_{\rm{app}}{\rm{tanh}}\left(1\times 10^{4}t\right), \tag{6}\]  
where the voltage unit is kV, the time unit is ns, and the model background temperature is 293.15 K.  
The initial density of charged particles is set to a Gaussian distribution with the tip as the origin. As shown in Figure 2, compared with the uniform distribution, the Gaussian distribution converges faster and has little effect on the corona discharge characteristics.  
The boundary's flux conditions of electrons, electron energy, and heavy particles at the metal electrode are set according to Equation (7):  
\[\left\{\begin{array}{l}n\cdot\Gamma_{\rm{e}}=\frac{1-\gamma_{\rm{e}}}{1+ \gamma_{\rm{e}}}\left(\frac{1}{2}v_{\rm{eth}}n_{\rm{e}}\right)-\sum\limits_{ \rm{P}}\gamma_{\rm{P}}(\Gamma_{\rm{P}}\cdot n)\\ n\cdot\Gamma_{\rm{e}}=\frac{1-\gamma_{\rm{e}}}{1+\gamma_{\rm{e}}}\left(\frac {5}{8}v_{\rm{eth}}n_{\rm{e}}\right)-\sum\limits_{\rm{P}}\gamma_{\rm{P}}v_{\rm {P}}(\Gamma_{\rm{P}}\cdot n)\\ n\cdot\Gamma_{\rm{k}}=\frac{1}{4}v_{\rm{kth}}n_{\rm{k}},\end{array}\right. \tag{7}\]  
where \(n\) is the boundary's normal vector, \(\gamma_{\rm{e}}\) is the reflection coefficient of electrons on the electrode surface, \(v_{\rm{eth}}\) is the thermodynamic velocity of electrons, \(\gamma_{\rm{P}}\) is the secondary emission coefficient of electrons on the electrode surface, \(\epsilon_{\rm{p}}\) is the average energy of secondary electrons emitted from the cathode, and \(v_{\rm{kth}}\) is the thermodynamic velocity of heavy particles.  
When the boundary condition is applied to the cathode, \(\gamma_{\rm{P}}\) is 0.004; and when it is applied to the anode, \(\gamma_{\rm{P}}\) is 0. The average energy of secondary electrons emitted from the electrode surface is always 5 eV. The thermodynamic velocities of electrons and heavy particles are shown in Equation (8):  
\[\begin{array}{l}v_{\rm{eth}}=\sqrt{\frac{8k_{\rm{B}}T_{\rm{e}}}{\pi m_{\rm{ e}}}},\\ v_{\rm{eth}}=\sqrt{\frac{8 kT}{\pi M_{\rm{e}}}},\end{array} \tag{8}\]  
where \(k_{\rm{B}}\) is the Boltzmann constant, \(m_{\rm{e}}\) is the electronic mass, \(R\) is the universal gas constant, and \(M_{\rm{n}}\) is the average molar mass of the mixture.  
The quality of mesh generation largely determines the accuracy and efficiency of model solutions. So, the triangular elements are used for mesh generation, and the mesh  
Figure 2: Gaussian distribution of initial electron density.  
inside the discharge channel and on the electrode's surface is divided more densely. In addition, depending on the defect geometry and wire radius, the size of mesh division varies. The number of basic triangular cells used in this study is at least 105,330 and at most 214,970, and the typical mesh is shown in Figure 3.