## 2 Characteristics of Surface Potential Distribution for DC Grounding Electrode
### Formation and Calculation Method of Surface Potential  
When a DC transmission line operates in monopolar ground return mode, the DC current from the grounding electrode of the sending-end power grid flows into the earth, penetrating to an infinite depth, and then exits through the grounding electrode of the receiving-end power grid. During this process, the DC current flows from the upper layers of the soil towards the deeper layers, exhibiting continuity in its flow through the soil, which subsequently generates a DC potential on Earth's surface. The structural diagram  
Figure 4: The current distribution under the DC grounding electrode breaking one feeder rod. (**a**) Straight-line region; (**b**) Ring region. Figure 5: The current distribution under the DC grounding electrode breaking three feeder rods. (**a**) Straight-line region; (**b**) Ring region. of the monopolar ground return operation mode of the DC transmission line is shown in Figure 6.  
The current passes through the grounding electrode and then enters the soil. The current in the soil and the surface potential satisfy the constant current field equation [22; 23]:  
\[\
abla\times\overset{\rightarrow}{E}=0 \tag{1}\]  
\[\
abla\cdot\overset{\rightarrow}{J}=-\frac{\partial\rho_{v}}{\partial t} \tag{2}\]  
\[\overset{\rightarrow}{J}=\gamma\overset{\rightarrow}{E} \tag{3}\]  
Based on Equations (1)-(3) above, it can be inferred that:  
\[E=-\
abla V \tag{4}\]  
\[\
abla^{2}=0 \tag{5}\]  
In the above equation, \(E\) represents the induced electric field strength at the ground surface excited by the current penetrating into the earth, V/m; \(J\) denotes the current density, A/m\({}^{2}\); \(\rho_{v}\) is the charge density of the conductor, C/m\({}^{3}\); \(\gamma\) represents the electrical conductivity, S/m; and \(V\) is the electric potential, V.  
In the constant flow field, the scalar potential \(V\) is selected as the degree of freedom, and the above equation is constant in the field of the grounding electrode. The interface between different conductivity substances satisfies the boundary conditions [24]:  
\[V_{i}=V_{j} \tag{6}\]  
\[\frac{1}{\rho_{i}}\frac{\mathrm{d}V_{i}}{dn}=\frac{1}{\rho_{j}}\frac{\mathrm{d }V_{j}}{dn} \tag{7}\]  
A layer of coke is first applied to the surface of the grounding electrode, followed by a layer of soil. The coke layer acts as a good conductor, allowing the entire surface of the grounding electrode to be considered as equipotential. The surface of the grounding electrode is treated as another boundary of the solution domain, and the boundary condition on this boundary satisfies that the potential is constant. The reference coordinate system is taken as a three-dimensional Cartesian coordinate system, and the partial differential equation satisfied by the potential is obtained as follows:  
Figure 5:0  
Figure 5:1 Schematic diagram of the monopolar ground circuit operation. The constant current field can be analogized to the electrostatic field; therefore, by applying the uniqueness theorem, the following first-type boundary condition is defined:  
Figure 5:2  
Figure 5:3  
And the second-type boundary condition:  
Figure 5:4  
According to the above equations, the unique solution for the constant current field can be determined. After establishing the field equation and boundary conditions, combined with the actual problem, finite element simulation software can be employed to solve the aforementioned partial differential equation, thereby obtaining the \(V\) values at various field points [25; 26]. In COMSOL Multiphysics 6.1, a simulation model of a single-ring DC grounding electrode, as shown in Figure 7, was established according to the parameters of Figure 5:5 The straight part of the single-ring DC grounding electrode is 250 m long, and the ring area is composed of two semicircles with a diameter of 120 m. Each feeder rod is a cylindrical iron rod with a length of 1.5 m and a diameter of 50 mm.