# 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 Decomposition with EMD  
The EMD process uses mathematical methods to construct an envelope, which envelopes all of the data in the signal \(X(t)\). The envelope is then computed by subtracting the average from the original signal and subtracting the residual component. The mean of the envelope is then calculated, and the mean is subtracted from the original signal to obtain the residual component \(c_{1}(t)\).  
\[c_{1}(t)=X(t)-s_{1}(t) \tag{14}\]If \(c_{1}(t)\) meets the conditions defined by the IMF, then it is taken as the first component. If it does not meet the conditions defined by the IMF, then use it as the original signal and repeat steps 1 and 2 until the first component is obtained.  
\[c_{1k}(t)=c_{1(k-1)}(t)-s_{1(k-1)}(t) \tag{15}\]  
After separating \(X(t)\), the remaining signal is obtained. Repeat the above steps to obtain the 2, 3, 4,... nth IMF components in sequence. If the function obtained from the last decomposition meets the IMF definition, the loop ends. The above decomposition process can be simplified, as shown in Figure 6.  
In the EMD process, the IMF components are extracted in order from high frequency to low frequency. \(imf_{1}(t)\) is the signal with the highest frequency; \(r_{n}(t)\) is the signal with the lowest frequency and represents the average trend of the signal. This set of IMF components is all single-component signals, i.e., each moment contains only one frequency component.  
For any time-series signal \(X(t)\), define its Hilbert transform \(Y(t)\) as [28]:  
\[Y(t)=\frac{1}{\pi}P\int_{-\infty}^{+\infty}\frac{X(\delta)}{t-\delta}d\delta \tag{16}\]  
Its inverse transformation is:  
\[X(t)=\frac{1}{\pi}P\int_{-\infty}^{+\infty}\frac{Y(\delta)}{t-\delta}d\delta \tag{17}\]  
In the formula, \(P\) represents the principal value, and \(t\) represents time. \(X(t)\) and \(Y(t)\) are mutually complex conjugates. The expression of the analytic signal \(Z(t)\) is:  
\[Z(t)=X(t)+jY(t)=A(t)e^{\mathrm{i}\theta(t)} \tag{18}\]  
Among them:  
\[\left\{\begin{array}{l}A(t)=\sqrt{X^{2}(t)+Y^{2}(t)}\\ \theta(t)=\arctan\frac{Y(t)}{X(t)}\end{array}\right. \tag{19}\]  
In the formula, \(A(t)\) represents the instantaneous amplitude and \(\theta(t)\) represents the instantaneous phase. Given the phase, the instantaneous frequency can be calculated utilizing the following formula:  
\[f(t)=\frac{1}{2\pi}\frac{d\theta(t)}{dt} \tag{20}\]  
Often, the instantaneous phase and instantaneous frequency obtained by directly performing the Hilbert transform on the signal will have meaningless negative values because most signals do not only contain one vibration mode but are composed of multiple complex waves superimposed (the signal is locally asymmetric to the zero means). The premise of performing the Hilbert transform is to perform EMD on the signal to obtain a series of stable IMF with only one vibration mode. The accuracy of the time synchronization when the traveling wave head reaches the measurement point seriously affects the accuracy of the traveling wave method for distance measurement. When the fault traveling wave propagates to the measurement point, the signal detected by the measuring point shows a high-frequency mutation, and the mutation point marks the arrival time of the fault traveling wave. The characteristic of high-frequency sudden changes in the time-frequency map of the traveling wave signal head allows the time corresponding to the first frequency  
Figure 6: EMD decomposition diagram.  
sudden change point in the instantaneous frequency map to be regarded as the time when the fault traveling wave head reaches the measurement point.  
The first intrinsic modal function (\(imf_{1}(t)\)) of the signal EMD is the component with the highest frequency and has a high temporal resolution. Therefore, the moment when the faulty wave head arrives at the measurement point can be determined by the mutation point in the time-domain plot of this component.