# Modeling of Positive Switching Impulse Discharge of UHV Transmission Line Air Gaps
## 2 Breakdown Voltage Calculation Method  
### Long Air Gap Discharge Model  
The development process of air gap discharge in UHV transmission lines is mainly composed of some physical processes, such as the first corona initiation, the streamer-leader transformation, the continuous leader development, and the final jump under positive switching impulse [20; 21]. The voltage rise of the conductor causes the surface electric field distortion and the first corona initiates. After obtaining the free charge from the air, the first corona is transformed into the streamer and develops in the gap. In long air gaps, the streamer is unable to complete the breakdown of the whole gap. As the conductor voltage increases further, the streamer stem is transformed into a leader and continues to extend. When the streamer ahead of the leader reaches the ground potential electrode, the discharge goes into the final jump, as shown in Figure 1.  
In the course of the discharge development, gap breakdown must meet two basic conditions: the continuous leader inception, and the streamer in front of the leader developing to the ground potential electrode. When the free charges generated by the streamer are sufficient to maintain the continuous development of the leader, the applied voltage of the conductor is the continuous leader inception  
Figure 1: The development process of air gap discharge. voltage. After the inception of the continuous leader, the leader develops at an almost constant speed in the gap, and the potential of the leader head nearly remains the continuous leader inception voltage. When the streamer in front of the leader reaches the ground potential tower, the leader develops rapidly and gap breakdown becomes inevitable. At this time, the applied voltage of the conductor is the minimum breakdown voltage. It can be concluded that the minimum breakdown voltage of long air gaps consists of two parts [10]:  
\[U_{B}=U_{lc}+\Delta U_{l} \tag{1}\]  
where \(U_{B}\) is the minimum breakdown voltage, \(U_{lc}\) is the continuous leader inception voltage, and \(\Delta U_{l}\) is the leader voltage drop. The leader can be regarded as a channel with arc characteristics, and the expression of the leader voltage drop can be obtained:  
\[\Delta U_{l}=50l_{z}+37.5\ln\left[8-7e^{-1.33l_{z}}\right] \tag{2}\]  
where \(l_{z}\) is the length of the leader. When the streamer in front of the leader reaches the ground potential electrode, the length of the leader channel is:  
\[l_{z}=d-h_{f} \tag{3}\]  
where \(d\) is the gap length, and \(h_{f}\) is the length of the final jump. According to the characteristic that the potential gradient \(E_{s}\) of the streamer is almost constant, the relationship between the continuous leader inception voltage \(U_{lc}\) and the final jump length \(h_{f}\) can be obtained:  
\[h_{f}=\frac{U_{lc}}{E_{s}} \tag{4}\]  
The expression of the minimum breakdown voltage of long air gaps can be obtained from the above equations:  
\[U_{B}=U_{lc}+50(d-\frac{U_{lc}}{E_{s}})+37.5\ln\left[8-7e^{-1.33(d-\frac{U_{ lc}}{E_{s}})}\right] \tag{5}\]  
According to the dispersibility of the discharge process and the randomness of the breakdown path, the 50% breakdown voltage \(U_{50}\) of the air gaps under the critical positive switching impulse is related to the minimum breakdown voltage \(U_{B}\)[9]:  
\[U_{50}=\frac{U_{B}}{1-3\sigma} \tag{6}\]  
where \(\sigma\) is the standard deviation of the probability function of the breakdown, which is 3% in conductor-tower gaps. It can be seen that the continuous leader inception voltage is the key to calculating the long air gap breakdown voltage. According to the two basic conditions, [PERSON].