# An Improved Arc Fault Location Method of DC Distribution System Based on EMD-SVD Decomposition
## 2 Traveling Wave Characteristics in a DC Distribution System
### Analysis of Faults Characteristics  
In a two-terminal DC distribution system, if a fault occurs, an additional fault port will be added at the fault point. Using the above simplification method, a simplified 1-mode network model of the faulted line can be obtained, as shown in Figure 4. Where a single-pole ground fault occurs at point \(F\), \(R_{f}\) is the fault transition resistance, \(U^{\prime}_{f1}\) is the fault additional excitation power, and \(U_{F1}\) is the fault point voltage.  
Figure 3: Bergeron model for DC transmission lines.  
Figure 2: Principle of single-end fault location method. 1 The first traveling wave locating method. 2 The second traveling wave locating method.  
The model contains three Bergeron equivalent voltage sources whose values are determined by the forward-traveling wave generated by the faulty port circuit and the transfer function of the line. Before the fault traveling wave propagates to both ends of the line and reflects, \(B_{fM1}\) and \(B_{fN1}\) is both zero. Since this chapter only considers the voltage and current changes caused by the first traveling wave of the fault arriving at the protection and does not consider the reflexive behavior after that, only the initial excitation state of the fault needs to be taken into account in the calculation. Further, the model of Figure 4 can be simplified, as shown in Figure 5.  
\(U_{R1}\) and \(I_{R1}\) represent the 1-mode component of the fault voltage and current measured at the protection installation site.  
The boundary conditions of the phase domain are [26]:  
\[\left\{\begin{array}{l}U_{f+}=-U_{f}+I_{f+}R_{f}\\ I_{f-}=0\end{array}\right. \tag{5}\]  
\(U_{f}\), \(U_{f+}\), \(I_{f+}\) and \(I_{f-}\) respectively the DC voltage value at the fault point before the fault, the voltage at the positive fault point, the current at the positive fault point, and the current at the negative fault point.  
Furthermore, according to the domain network structure, it can be seen that:  
\[\left\{\begin{array}{l}I_{f1}=-\frac{2U_{F1}}{Z_{c1}}\\ I_{f0}=-\frac{2U_{R1}}{Z_{c0}}\end{array}\right. \tag{6}\]  
By combining Equations (5) and (6), the complex frequency domain expression of the fault point voltage can be obtained:  
\[U_{F1}(s)=\frac{-\sqrt{2}Z_{c1}}{Z_{c1}+Z_{c0}+4R_{f}}\frac{U_{f1}^{\prime}}{s} \tag{7}\]  
Furthermore, the equivalent voltage source of the 1-mode line Bergeron is:  
\[B_{Mf1}=2e^{-\gamma(s)}I\cdot U_{F1}(s) \tag{8}\]  
Figure 4: A simplified model of a double terminal DC system line under fault conditions.  
Figure 5: Minimalist computational model.  
Based on the network structure of the port where the protection is located, calculate the expression of the mode domain voltage and current fault components obtained from the protection measurement using the following formula:  
\[U_{R1}(s)=\frac{Z_{M}+sL_{MN}}{Z_{M}+sL_{MN}+Z_{c1}}\cdot B_{Mf1}(s) \tag{9}\]  
\[I_{R1}(s)=\frac{-B_{Mf1}(s)}{Z_{M}+sL_{MN}+Z_{c1}} \tag{10}\]  
By solving the Laplace inverse transformation for (9) and (10), the time-domain analytical expressions corresponding to modulus voltage and current can be obtained.  
Using the same analysis method, obtain the complex frequency domain and time domain analytical expressions of 0 mode voltage and current. Finally, the time-domain solutions of phase-domain voltage and current fault components are calculated using phase-mode inverse transformation.