# Modeling of Positive Switching Impulse Discharge of UHV Transmission Line Air Gaps
## 4 Discharge Model  
The breakdown voltage calculation method can effectively calculate the breakdown voltage of UHV transmission lines with different tower types. However, this method still needs to build a solid model, a finite element simulation calculation, and other processes, which has certain complexity, and the discharge characteristics are not intuitive enough. The discharge model of UHV transmission lines is obtained by studying different gap characteristics factor of gaps. According to the general model of the continuous leader inception voltage, Equation (11), it can be concluded that the difference in continuous leader inception voltage in the same high voltage electrode gap is caused by the difference in gap characteristic factors. Therefore, according to the continuous leader inception voltage model of conductor-plane gaps, Equation (13), it can be considered that in gaps of the conductor electrode, the model of the continuous leader inception voltage is:  
\[U_{lc}=\frac{U_{c\infty}}{1+\frac{A}{R_{ct}}}=\frac{U_{c\infty}}{1+\frac{R_{ cp}}{R_{ct}}\cdot\frac{A}{R_{cp}}}=\frac{2247}{1+J\cdot\frac{5.15-5.49\ln a}{d\ln \frac{2d}{a}}} \tag{18}\]  
where \(J\) is the difference factor between the conductor-plane gap characteristic factor \(R_{cp}\) with a gap length \(d\) and the conductor-tower gap characteristic factor \(R_{ct}\) with a tower width of 1 m and a gap length \(d\). Its expression is as follows:  
\[J=\frac{R_{cp}}{R_{ct}} \tag{19}\]  
As shown in Figure 7, in order to determine the difference factor of different tower types, the gap characteristic factors of conductor-cathead-tower window gaps, conductor-cathead-tower side gaps, double-circuit-tower gaps, and conductor-plane gaps were calculated. The gap lengths of the above gaps were 5 m, 6 m, 7 m, and 8 m. The tower widths of the cathead-tower window, the cathead-tower side, and the double circuit-tower were 1 m. The radius of the conductor was 0.05 m, and the length was 20 m. The tower model was made according to the actual tower 1:1 size. Gap characteristic factors of different gaps after calculation are shown in Figure 8. According to Equation (19), the difference factors of the conductor-cathead-tower window gaps with different lengths were within the range of 1.21 \(\pm\) 3%. The difference factors of the conductor-cathead-tower side gaps with different lengths were within the range of 1.01 \(\pm\) 4%. The difference factors of the conductor-double-circuit-tower gaps with different lengths were within the range of 1.17 \(\pm\) 5%. As a result, the difference factor \(J\) of different tower types was 1.21 for the cathead-tower window gaps, 1.01 for the cathead-tower side gaps, and 1.17 for the double-circuit-tower gaps. Figure 6: Comparison results between the tower-width discharge model, previous formulas, and tests. Therefore, according to the principle of capacitance equivalence, the continuous leader inception voltage model of UHV transmission lines is:  
\[U_{lc}=\frac{2247}{1+J\cdot\frac{5.15-5.49\ln{a_{e}}}{d\ln{\frac{2d+r_{1}+s_{e}}{ \kappa}}}} \tag{20}\]  
\[\left[ \begin{array}{c}J_{ch-t-w}=1.21\\ J_{ch-t-s}=1.01\\ J_{d-c-r}=1.17\end{array} \right. \tag{21}\]  
where \(J_{ch-t-w}\) is the difference factor of the cathead-tower window gaps, \(J_{ch+s}\) is the difference factor of the cathead-tower side gaps, and \(J_{d-c-r}\) is the difference factor of the double-circuit-tower gaps. Combined with the long air gap discharge model, the breakdown voltage of the gaps with a tower width of 1 m can be calculated. After that, the breakdown voltage of the gaps with an actual tower width can be calculated by the tower-width discharge model. Figure 8: Gap characteristic factors of different gaps. Figure 7: Different conductor gaps. The correctness of the models is verified by actual test results.