# Fault Location Method of Multi-Terminal Transmission Line Based on Fault Branch Judgment Matrix
## 2 Improved Double-Ended Traveling Wave Localization Method Based on Line Mode Component
### Main Errors of Double-Ended Traveling Wave Ranging  
The traditional traveling wave positioning method calculates the fault distance by using the time a which the fault traveling wave reaches both ends of the line. When a short circuit fault occurs at line F, the traveling wave propagation path is shown in Figure 1. In this figure, \(d_{M}\) and \(d_{N}\) represent the distance between the fault point and the endpoint M and N, respectively.  
According to the double-ended traveling wave positioning formula, the distance from the fault point F to the end of the line M, N is shown in Equation (6).  
\[\left\{\begin{array}{l}d_{M}=\frac{L_{MN}+v(t_{M}-t_{N})}{2}\\ d_{N}=\frac{L_{MN}+v(t_{N}-t_{M})}{2}\end{array}\right. \tag{6}\]  
where, \(L_{MN}\) is the total length of the line; \(t_{M}\), \(t_{N}\) are the times at which the wave head reaches the terminals M and N, respectively; \(v\) is the velocity of the traveling wave.  
From Equation (6), it can be seen that the wave speed, the arrival time of the wave head, and the line length are the main factors affecting the accuracy of fault location. In practice, the arc sag of the conductor leads to a difference between the horizontal distance and the actual length. The variation range of the conductor length is 0.9-1.1 times the horizontal distance of the tower.  
When the fault occurs at point F at the time \(t_{0}\), the distance between fault point F and the end of line M is given in Equation (7):  
\[d^{{}^{\prime}}_{M}=v(t_{M}-t_{0}) \tag{7}\]  
The distance of the wire between the endpoints MN is given in Equation (8):  
\[L^{{}^{\prime}}_{MN}=d^{{}^{\prime}}_{M}+d^{{}^{\prime}}_{N}=v(t_{M}-t_{0})+v( t_{N}-t_{0}) \tag{8}\]  
The horizontal distance from F to the M end is given in Equation (9):  
\[d_{M}=\frac{d_{M}}{L_{MN}}L_{MN}=\frac{t_{M}-t_{0}}{t_{M}+t_{N}-2t_{0}}L_{MN} \tag{9}\]  
According to Equations (6)-(9), it can be seen that the errors of double-ended traveling wave ranging mainly originate from:  
* Traveling wave velocity: In actual working conditions, the transmission line environment affects line parameters, making it difficult to determine the wave speed. Therefore, wave speed cannot be accurately calculated in practical applications, leading to large-ranging errors.
* Wave head capture accuracy: First arrival time of fault traveling wave at both ends of the line is a primary factor affecting the positioning accuracy. The error mainly consists of the synchronization error of the sampling clock and the extraction error of the faulty traveling wave head.
* Arc sag, ambient temperature, load current, and other factors on the line length.  
Due to the high synchronization accuracy of current GPS synchronization technology, the synchronization error of the sampling clock can be effectively solved. This paper does not delve into this issue in-depth. Further, as for factors affecting the actual length of the line, the main factor causing line length error is arc sag. To solve the problem of the traditional double-ended method's difficulty in determining wave speed and the impact of line arc sag on fault location, this paper improves upon the conventional double-ended traveling wave method.  
Figure 1: Diagram of traditional fault location.