## 5 Insulator Surface Charge and Electric Field Distribution Under DC Voltage  
The electric field distribution and the flow of electric field lines within the DC GIL are depicted in Figure 10, which presents the results of finite element simulation (FEM). Under positive-polarity DC voltage, the electric field lines flow out of the high-voltage side and into the low-voltage side, resulting in the aggregation of positive ions on the grounded side and negative ions on the high-voltage side. We observed that the electric field strength on the insulator's convex surface significantly exceeds that on the concave surface. The electric field lines on the convex surface were dominated by outflow from the insulator, whereas those on the concave surface were dominated by inflow into the insulator. Consequently, negative ions in the SF\({}_{6}\) gas region gather near the convex side, whereas positive ions gather near the concave side. In DC GIL, the primary sources of surface charges on insulators can be categorized into three types: conduction from the SF\({}_{6}\) gas side, conduction through the insulator bulk, and conduction along the insulator surface [16]. The distribution characteristics of surface charges are governed by the electric-field-driven carrier migration process. The charge behavior on the SF6 gas side is described by the particle dynamics Equations (9) and (10),  
Figure 10: Cloud map of electric field line distribution. (The yellow arrows indicate a local magnification, and the red arrows indicate the direction of the electric field lines.) Figure 9: Temperature of insulator surface in different directions. where the gas-side current arises from the drift motion of positive and negative ions under Coulomb forces. Conduction through the insulator bulk and along its surface is characterized in this study by bulk conductivity and surface conductivity, as expressed in Equation (17). Under DC voltage, the electric field within the GIL transitions from a capacitive to a resistive state, a process that essentially reflects the gradual saturation of charge accumulation within the field domain. The accumulation of surface charges on the insulator results from the interplay and eventual equilibrium among the three conduction mechanisms. This interplay is governed by Equation (17), which also elucidates the mechanism of surface charge accumulation. The polarity of surface charges primarily depends on the direction of the normal component of the electric field lines at the insulator surface, as illustrated in Figure 10. Under DC voltage, charges from both the solid (insulator) and gas sides migrate along the electric field direction, ultimately accumulating at the gas-solid interface. These charges may also transfer along the insulator surface under the influence of the tangential electric field, affecting the electric field distribution along the insulator surface until surface charge accumulation reaches equilibrium. When this equilibrium is achieved (at \(\partial\rho/\partial t=0\)), the capacitive electric field fully transitions to a stable resistive electric field distribution. Under the application of DC voltage, the GIL electric field distribution gradually transitions from a capacitive field in the initial stage to a resistive field after stabilization [20], during which the evolution of the insulator surface charge density adheres to (17). Figure 11 illustrates the transient characteristics of the surface charge density distribution at the 3 o'clock direction of the insulator. We found that the convex surface predominantly accumulates positive charge, which increases gradually over time, indicating that \(\partial\rho/\partial t\) in Equation (17) exceeds zero, with solid-side conduction dominating. Conversely, on the concave side of the insulator, the negative charge around the conductor and the enclosure gradually increases, while the negative charge in the middle region progressively decreases while the positive charge gradually increases.