# An Improved Arc Fault Location Method of DC Distribution System Based on EMD-SVD Decomposition
## 3 Fault Location Method Based on EMD-SVD
### Traveling Wave Head Calibration with SVD  
Singular value decomposition (SVD) is a mathematical method of orthogonal transformation, as shown in Equation (21). Where A is an arbitrary real matrix, \(U\) and \(V\) are orthogonal matrices of order m and n, respectively, and matrix \(S\) is a diagonal matrix whose diagonal elements are composed of singular values, Through SVD, matrix \(A\) can be decomposed into a superposition of a principal component and several detail components, with each singular value corresponding to a component. This decomposition ensures that the phase of each component is the same. \[A=USV^{T}=\sum_{i=1}^{k}A_{i} \tag{21}\]  
Given a discrete signal \(X=[x(1),x(2),\ldots,x(N)]\), if it is constructed into a Hankel matrix, SVD can be used to achieve singularity detection of the signal, that is, to detect sudden changes in the signal itself or its derivative at a certain moment. The Hankel matrix form is:  
\[A= \begin{bmatrix}x(1)&x(2)&\ldots&x(m)\\ x(2)&x(3)&\ldots&x(m+1)\\ \ldots&\ldots&\ldots&\ldots\\ x(n)&x(n+1)&\ldots&x(N)\end{bmatrix}  \tag{22}\]  
Construct signal \(X\) as a Hankel matrix and perform SVD, where the \(A\) matrix can be represented as:  
\[A=\sum_{i=1}^{k} \begin{bmatrix}x_{i}(1)&x_{i}(2)&\ldots&x_{i}(m)\\ x_{i}(2)&x_{i}(3)&\ldots&x_{i}(m+1)\\ \ldots&\ldots&\ldots&\ldots\\ x_{i}(n)&x_{i}(n+1)&\ldots&x_{i}(N)\end{bmatrix}  \tag{23}\]  
Through Equation (23), signal reconstruction can be performed on each SVD component: \(X_{i}=\{x_{i}(1),x_{i}(2),\ldots,x_{i}(n),x_{i}(n+1),\ldots,x_{i}(N)\},i\in(1,k)\). The elements in \(X_{i}\) are the combination of the elements of the first column and the last row in the matrix \(A_{i}\). The \(X_{1}\) component is the main component of the original signal \(X\), which approximates the original signal; the \(X_{i}\) component is the detail component of the signal \(X\), which can detect the signal itself and the singularity of a certain order derivative of the signal. If we take the mode value of the first detail component, the moment corresponding to the mode maximum point is the moment of signal mutation. The traveling wave method for distance measurement relies on accurate measurement of the time when the fault traveling wave head reaches the monitoring point. When a fault occurs in a certain part of a DC transmission line, the initial fault traveling wave propagates from the fault point to the monitoring points at both ends of the line. The voltage and current signals detected by the monitoring points show high-frequency mutations in the time-frequency map, and the mutation point marks the time when the fault traveling wave reaches the detection point. Due to the high-frequency mutation of the traveling wave signal head in the time-frequency map, the time corresponding to the first frequency mutation point in the instantaneous frequency graph can be considered the time when the fault traveling wave head reaches the monitoring point. From the analysis in Section 3.1, it is known that the first eigenmode function of the signal EMD is the highest frequency component with high temporal resolution. Therefore,the time at which the faulty traveling wave head arrives at the monitoring point can be determined by the mutation point in the time-frequency diagram of the \(imf_{1}(t)\) component. The locating steps in this paper are as follows: EMD is performed on the fault current signal; the highest frequency component \(imf_{1}(t)\) is taken and its Hankel matrix is constructed; the constructed Hankel matrix is decomposed into two layers of singular values, the first one being the main component and the second one being the detail component; the detail component detects the singularity of the \(imf_{1}(t)\) component itself.