# An Improved Arc Fault Location Method of DC Distribution System Based on EMD-SVD Decomposition
## 2 Traveling Wave Characteristics in a DC Distribution System  
### Line Fault Characteristics  
The post-fault network can be equated to a superposition of the pre-fault normal network and the fault component network, per the circuit theory principle of superposition [24]. Thus, the analysis procedure is streamlined. In Figure 1, \(F\) represents the location of the fault, \(X_{\text{MF}}\) represents the distance from the M end to the fault point, and \(X_{\text{FN}}\) represents the distance from the fault point to the N end. The length of the line is \(L\). The values of \(U_{f}\) and \(U_{f}^{\prime}\) are equal in magnitude and are the voltages to the ground at the pre-fault point \(F\), but with opposite polarity. \(U_{f}^{\prime}\) is the source of an additional excitation voltage in the fault-component network. When a line fault occurs, the excitation voltage source is connected to the network in the fault component network. Affected by the distribution parameters of the transmission line, the fault traveling wave will start from the fault point and propagate along the line to both sides, and abrupt traveling wave signals can be detected at the measurement points [25]. As shown in Figure 2, the single-ended method of localization can be divided into two categories: one considers only the behavior of two reflections of the initial traveling wave between the M end and the faulty point; the other considers the first arrival of the initial traveling wave at the M end and the incident wave launched to the M end via the N end. The first single-end approach is employed in this research. \(t_{1}\) is the moment when the initial traveling wave leaves the fault point and propagates to the M end, \(t_{2}\) is the instant the faulty traveling wave is reflected by the M end and returns to the fault point, and \(t_{3}\) is the moment when the traveling wave reflected back from M has reflected back to M again through the fault point. The second captured traveling wave head is reflected twice. Consider that the speed of the traveling wave is a constant value \(v\). Then, the following equation can be derived:  
\[\left\{ \begin{array}{l}X_{MF}=v(t_{1}-t)\\ 3X_{MF}=v(t_{3}-t)\end{array} \right. \tag{1}\]  
Simplifying Equation (1) gives:  
\[X_{MF}=\frac{1}{2}v(t_{3}-t_{1}) \tag{2}\]  
The single-ended method has high real-time performance in calculating measurement results, but it relies on precise hardware equipment. If the wave head cannot be accurately captured, it will lead to significant calculation errors and positioning failures. Secondly, it is necessary to model and analyze the corresponding lines. This article adopts the Bergeron model shown in Figure 3 to simplify the analysis process. \(U_{M}(s)\), \(U_{N}(s)\), \(I_{M}(s)\), and \(I_{N}(s)\) represent the voltage and current at the M and N terminals of  
Figure 1: Post-fault network decomposition. (**a**) Post-fault network. (**b**) Normal network. (**c**) Fault component network. the line, respectively; \(Z_{c}\) represents the line's wave impedance. \(B_{M}(s)\) and \(B_{N}(s)\) are the Bergeron line's equivalent voltage sources, which may be written as:  
\[ \begin{cases}B_{M}(s)=e^{-\gamma(s)l}F_{N}(s)\\ B_{N}(s)=e^{-\gamma(s)l}F_{M}(s)\end{cases}  \tag{3}\]  
In the formula, \(F_{M}(s)\) and \(F_{N}(s)\) are the voltage forward waves at the M and N terminals of the line, respectively, and \(e^{-\gamma(s)l}\) is the transfer function of the line, where \(\gamma(s)\) is the propagation coefficient and \(l\) is the length of the line. Due to capacitive and inductive coupling between the poles in bipolar DC transmission systems, if there is a ground fault at one pole, it will cause overvoltage at the other healthy pole, which will cause overcurrent. Phase mode transformation converts the line to two independent module domain lines to reduce the complexity of the fault analysis process [24]. The DC line model for each module domain is the same as the unipolar DC line model. The formula for phase mode change is:  
\[\left\{