# Modeling of Positive Switching Impulse Discharge of UHV Transmission Line Air Gaps
## 2 Breakdown Voltage Calculation Method
### Method  
According to the above discharge voltage calculation model, the disparity in the continuous leader inception voltage results in the difference in breakdown voltage in various air gaps of transmission lines. The difference in the continuous leader inception voltage is caused by the difference in gap structure. Therefore, the calculation of the gap characteristic factor, which represents the structural difference, becomes the key to calculating the breakdown voltage of air gaps. In fact, the gap characteristic factor represents the ability of the induced charge on the surrounding grounding structure to generate the induced potential on the surface of the high voltage electrode. [PERSON] calculated the gap characteristic factors of some typical gaps by a mathematical method [9; 10]. A gap characteristic factor calculation method is proposed in [15]. This method sets an equivalent point charge in a finite element simulation software to determine the induced potential on the surface of the high voltage electrode so as to calculate the factor. It is difficult to calculate the gap characteristic factor of complex gaps by the mathematical method, whereas the finite element simulation method can calculate the gap characteristic factor of any gap. In order to calculate the breakdown voltage of different gaps, the finite element simulation method was adopted in this paper.  
The flow chart of the finite element simulation method is shown in Figure 2. First, a solid model of the gap was made in Solidworks. The model was then imported into Comsol Multiphyscis. Subsequently, the shortest distance between the conductor and the tower was determined to be the breakdown path, and the gap length was the discharge path length \(d\). Thereafter, on the breakdown path, a point charge \(Q_{0}\) was set at a distance of \(s\) to the surface of the conductor. In this study, the initial leader length was about 0.7 m [22; 23], and the value of \(s\) was in the range of 0.3-0.7 m and thus had little influence on the gap characteristic factor [15]. Therefore, the charge was considered away from the surface of the conductor 0.5 m in this paper. According to equations (8), (9), and (10), it can be considered that the value of the \(Q_{0}\) has no effect on the gap characteristic factor [15], so the value of \(Q_{0}\) was considered as 1C in this paper. That is to say, on the breakdown path, a point charge of 1C was set at the position of 0.5 m from the surface of the conductor. Then, the boundary conditions were set so that the potential at infinity was zero, consequently grounding the tower. The induced voltage calculated by this method was actually the induced voltage at the tip of the streamer stem. Because the length of the streamer stem was very small, it was considered that the tip of the streamer stem was on the surface of the conductor. Therefore, although the conductor was an equipotential body, the method actually calculated the potential of a small distance outside of the conductor, indicating that the relevant boundary conditions were not set. After the boundary conditions were set, the potential \(U_{in}\) of the point on the surface of the conductor was obtained by analysis and calculation. The gap characteristic factor and the breakdown voltage of the gap were calculated by the long air gap discharge model.