## 2 Interference Mechanism of the Grounding Current  
The high-voltage DC grounding electrode is a grounding facility that provides a low resistance path during the operation of the HVDC transmission system [17]. Figure 1 shows the interference process of the DC grounding electrode current on the buried metallic pipelines. The earth currents flow into the pipeline in the area close to the grounding electrode when the electrode is under anode operation. The currents flow out of the pipeline at the two ends which are far away from the electrode. As we can see, electrochemical corrosion occurs in the area currents flow out. The pipe-soil potential also has an obvious positive shift.  
Different electrochemical reactions occur for different interference areas [18]. The electrochemical reaction for the anode interference area is usually described by:  
\[2\mathrm{F_{e}}\to 2\mathrm{F_{e}}^{2+}+4\mathrm{e}\]  
Figure 1: Interference process of HVDC electrode currents on pipelines.  
For the cathode interference area, hydrogen evolution reaction or oxygen absorption reactions dominate under different soil medium environments. In general, it can be shown as follows:  
\[\begin{array}{ll}2\mathrm{H}^{+}+2\mathrm{e}^{-}\rightarrow\mathrm{H}_{2}& \mathrm{acid\ oxygen\text{-free\ condition}}\\ \mathrm{H}_{2}\mathrm{O}+2\mathrm{e}^{-}\rightarrow 2\mathrm{OH}^{-}+\mathrm{H}_{2}& \mathrm{neutral\ or\ alkaline\ oxygen\text{-free\ condition}}\\ \mathrm{O}_{2}+4\mathrm{H}^{+}+4\mathrm{e}^{-}\rightarrow 2\mathrm{H}_{2} \mathrm{O}&\mathrm{acid\ aerobic\ condition}\\ \mathrm{O}_{2}+2\mathrm{H}_{2}\mathrm{O}+4\mathrm{e}^{-}\rightarrow 4\mathrm{OH}^{-}& \mathrm{alkaline\ aerobic\ condition}\end{array}\]  
The above electrochemical reactions can finally lead to a specific relationship between the PSP and the leakage current which can be defined as follows:  
\[J=f_{\mathrm{P}}(\varphi_{\mathrm{P}}-\varphi_{\mathrm{s}}) \tag{1}\]  
where \(J\), \(\varphi_{\mathrm{P}}\) and \(\varphi_{\mathrm{s}}\) are the leakage current density, the pipe-side potential and the soil-side potential interface between the soil and pipe, respectively. The nonlinear function\(f_{\mathrm{P}}\), which describes the relationship between the pipe-to-soil potential and the leakage current density, can be measured by experiment. The three-electrode system is often used to measure the polarization curves with the potential sweep method [19]. Figure 2 shows the typical three-electrode system for measuring polarization curves.  
The discrete measurement data can be fitted by the Butler-Volmer equation which is one of the most fundamental equations in electrochemical kinetics. Equation (2) shows the typical Butler-Volmer equation [20]:  
\[J=f_{\mathrm{P}}(E)=J_{corr}(\exp(\frac{E-E_{corr}}{\beta_{a}})-\exp(-\frac{E- E_{corr}}{\beta_{c}})) \tag{2}\]  
where \(J\) and \(E\) are current density and PSP, respectively, and are obtained by measurement. \(E_{corr}\), \(J_{corr}\), \(\beta_{a}\), \(\beta_{c}\) are the metal self-corrosion potential, self-corrosion current density, Tafel slope of the anode reaction and Tafel slope of the cathode reaction, respectively. The four parameters to be fitted can be obtained using the least square method. Then, the fitted functions can be used in the proposed methodology. The methodology must consider the nonlinear polarization to calculate the current and potential distribution accurately.