# Non-Uniform Distribution of Contamination on Composite Insulators in HVDC Transmission Lines
## 5 Discussion
### Numerical Calculation and Analysis of Electric Field Force  
The uneven distribution of contamination along insulator string is due to the influence of electric field force on contamination particles. Before calculating the electric field force, we need to get the electric field distribution around the insulator. The electric field distribution of insulator strings is solved by COMSOL, the control equations of computational domain under DC electric field are as follows. \[\
abla\cdot\mathbf{D}=\rho,\ \mathbf{E}=-\
abla V,\ \mathbf{D}=\varepsilon_{0} \varepsilon_{1}\mathbf{E} \tag{4}\]  
where \(\mathbf{E}\) is the electric field strength, in V/m; \(\mathbf{D}\) is the electric flux density, in C/m\({}^{2}\); U is the potential value, in V; and the \(\varepsilon_{0}\) is the absolute dielectric constant of the vacuum, taking a value of 8.85 \(\times\) 10\({}^{-12}\) F/m; \(\varepsilon_{1}\) is relative dielectric constant of the medium, its value of silicon rubber insulator is 3.3; \(\rho\) is charge density, in C/m\({}^{3}\). The contour line of the calculated electric field around Type-A is shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Relatively Humidity** & **60\%** & **70\%** & **80\%** & **90\%** \\ \hline NSDD (mg/cm\({}^{2}\)) & 0.21 & 0.23 & 0.25 & 0.26 \\ \(\Sigma\) & 4.1\% & 5.6\% & 7.2\% & 7.7\% \\ K\({}_{\rm T/B}\) & 1/0.51 & 1/0.51 & 1/0.53 & 1/0.49 \\ K\({}_{\rm W/L}\) & 1/1.45 & 1/1.44 & 1/1.45 & 1/1.43 \\ K\({}_{\rm H/M}\) & 1/The electric field around insulators is distributed unevenly along the insulator surface, and the electric field at the high and low ends is larger. The maximum value occurs at the surface of the steel foot close to the high voltage end, reaching 13 kV/cm. The increase of contamination on the surface of the insulator is mainly caused by the force of the electric field in the vertical direction, because the vertical force will deflect the polluted particles into the insulator surface so as to increase the contamination amount. Therefore, we calculate \(Y\) axis component of electric field of the path A-C, as shown in Figure 9.  
It can be seen from the graph that the \(Y\) axis components of electric field at high voltage group and middle group are 0.305 kv/cm, 0.24 kv/cm, respectively. Based on the calculated field strength, we calculate the forces acting on the vertical direction of contamination particles. The movement of air ions will cause neutral atmosphere dust to be charged. Research shows that [19] 31% of the fly ash particles in nature have positive charges, 26% of fly ash particles have negative charges, and 43% of particles are not charged. The saturated charge of contaminated particles can be approximately expressed as:  
\[Q_{p}=3\pi\epsilon_{0}{E}{d_{p}}^{2}\bigg{(}\frac{\epsilon_{p}}{\epsilon_{p}+ 2}\bigg{)} \tag{5}\]  
where \(Q_{p}\) is quantity of electric charge of particle, in C; \(E\) is the electric field intensity in V/m; \(e\) is elementary charge; \(d_{p}\) is particle diameter, in m; \(\epsilon_{p}\) is the relative dielectric constant of particles,  
Figure 8: Contour line of the electric field around Type-A. Figure 9: \(Y\) axis component of electric field. its value of SiO\({}_{2}\) is 4.5. In order to simplify the analysis, only the extreme case of saturated electricity is considered, so Formula (5) is used to characterize the quantity of electric charge of polluted particles. The electric field force on polluted particles is expressed as Formula (6). \[f_{e}=Q_{p}E \tag{6}\]  
where \(f_{e}\) is electric field force. The calculated electric field force in the vertical direction on particle (take the diameter of 200 meshes as example) at the high voltage group and middle group are 1.67 \(\times\) 10\({}^{-10}\) N and 7.18 \(\times\) 10\({}^{-11}\) N respectively. We can see that the calculated electric field force in the vertical direction on particle at the high voltage group is almost 1.6 times larger than that at the middle group.