# 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
### Decoupling of Three-Phase Transmission Lines  
In real conditions, the transmission lines are uneven. For the sake of analysis, all factors causing unevenness are usually ignored. The existing transmission line is treated as a homogeneous transmission line.  
The uniform transmission line equation is given by Equation (1)  
\[-\frac{dU}{dx}=ZI \tag{1}\]  
In Equation (1), \(\mathrm{U}=\left[\mathrm{U_{a}},\mathrm{U_{b}},\mathrm{U_{c}}\right]^{\mathrm{T}}\), \(\mathrm{I}=\left[\mathrm{I_{a}},\mathrm{I_{b}},\mathrm{I_{c}}\right]^{\mathrm{T}}\) are the three-phase voltage and current vectors, \(\left[Z\right]=\begin{bmatrix}Z_{\mathrm{nat}}&Z_{ab}&Z_{ac}\\ Z_{\mathrm{lu}}&Z_{bb}&Z_{bc}\\ Z_{ca}&Z_{cb}&Z_{cc}\end{bmatrix}\) is the line impedance matrix.  
There is a coupling relationship between the phases in the three-phase transmission line, which are not independent of each other. In order to eliminate coupling effects, phasors are usually converted to moduli, and the phase-to-mode conversion relationship is established in Equation (2).  
\[U_{s}=S^{-1}U,I_{s}=Q^{-1}I \tag{2}\]  
Bringing this equation into Equation (1), it can be organized by Equation (3)  
\[-\frac{dU_{s}}{dx}=Z_{s}I_{s} \tag{3}\]  
In Equation (3), \(\mathrm{U_{s}}=\left[\mathrm{U_{\alpha}},\mathrm{U_{\beta}},\mathrm{U_{0}} \right]^{\mathrm{T}}\), \(\mathrm{I_{s}}=\left[\mathrm{I_{\alpha}},\mathrm{I_{\beta}},\mathrm{I_{0}} \right]^{\mathrm{T}}\) denote the \(\alpha\), \(\beta\), and \(0\) moduli of voltage and current, respectively.  
\(Z_{s}=S^{-1}ZQ\) is the mode resistance matrix. Phase-mode transformation can be realized using various methods, such as Clark phase-mode transformation, Kellenberger phase-mode transformation, and symmetric component transformation matrix. However, the Clark phase-mode transform and Kellenberger phase-mode transform may fail to detect certain types of faults, so the symmetric component transform matrix is chosen to calculate the modulus in this paper.  
The symmetric component transformation for current is given in Equation (4).  
\[\begin{bmatrix}I_{a}\\ I_{\beta}\\ I_{0}\end{bmatrix}=\frac{1}{3}\begin{bmatrix}1&a&a^{2}\\ 1&a^{2}&a\\ 1&1&1\end{bmatrix}\begin{bmatrix}I_{a}\\ I_{b}\\ I_{c}\end{bmatrix} \tag{4}\]  
In Equation (4), \(I_{a},I_{\beta},I_{0}\) is the line mode and zero mode components under symmetric component transformation and \(I_{a},I_{b},I_{c}\) is the phase current, and \(a=e^{i\text{120}}\).  
Among them, the \(0\)-mode component becomes more severely attenuated as the frequency increases, so the line mode component is chosen as the detection object.  
Taking the alpha mode as an example, the Ia is given in Equation (5).  
\[I_{\alpha}=I_{a}+aI_{b}+a^{2}I_{c} \tag{5}\]  
From Equation (5), it can be seen that \(I_{a}\) is not equal to \(0\) regardless of the fault type, so the alpha mode component can detect all fault types. Therefore, this paper chooses the \(\alpha\)-mode component as the fault signal to detect the traveling wave head.