## 5 Fault Location in Direct Current (DC) Grid Having Cable and Overhead Line Segments
### Fault Arrival Time Identification  
Precision of traveling wave-based dc fault location highly depends upon the accuracy of detection of faulty surge arrival time. Traveling wave front may present in the terminal voltages or it may flow through the surge capacitors, deployed at the converter stations for VSC protection [61]. For a permanent fault on dc-link, steepness of voltage profile decreases when a fault propagates along the length of the line as increase in TW wavelength results in attenuation. Here, if a level detection method is employed for determining arrival time of surge, then an error may occur due to change in shape of wave front for a fault location closer to the VSC station. This effect is not considerable in the event of occurrence of fault in middle of the TL.  
In case of high impedance fault, magnitude and steepness of the TW would be small, making it difficult for classical methods to proceed for analysis. Therefore, time frequency analysis, i.e., wavelet transform is employed because of its high efficiency and performance for high impedance  
Figure 5: Faulty segment identification algorithm.  
faults [61; 64; 66]. Normally continuous wavelet transform is used for transient identification within dc faults. Continuous wavelet transform of a signal \(f(t)\) is given by:  
\[CWT(f(t)):a,b)=\int_{-\infty}^{\infty}f(t)\varphi_{a,b}^{*}(t)dt \tag{24}\]  
\[\varphi_{a,b}^{*}(t)=\frac{1}{\sqrt{a}}\varphi\bigg{(}\frac{t-b}{a}\bigg{)} \tag{25}\]  
where \(\psi_{a,b}^{*}(t)\) is complex continuous function in time and frequency domains and is known as mother wavelet. \(a\) is a scaling (dilation) factor and \(b\) is the shifting (translation) factor. For discrete wavelet transform, mother wavelet may be dilated and translated in a dyadic block.  
Wavelet Based Surge Detection  
Transients in the fault can be identified by analyzing the coefficients of CWT of the measured terminal voltages or coefficients of current signals from surge capacitor [61]. Magnitude of CWT coefficients are evaluated on different scales: \(a~{}=~{}16,~{}32,~{}64\), and \(128\) at each terminal (station). These magnitudes are then compared against pre-determined thresholds to detect the arrival of initial surge. The instant at which the magnitude of CWT is lying above the threshold is known as arrival time of the traveling wave.  
Natural Frequency Based Fault Location Finding in Multi-Terminal High-Voltage Direct Current (MT-HVdc) Systems  
Traveling wave methods for fault locations are based on the time utilized by the generated traveling wave to reach the converter stations. These methods are highly efficient and are independent of the effects of parameters of transmission lines [34; 62; 75; 76]. But the velocity of the propagation of TW and the wave front of the TW are key parameters for traveling wave-based algorithms. However, it is difficult to detect these parameters for high resistance faults and for gradual change in resistance. Fault location calculation is dependent on the velocity and traveling time of TW from fault point to VSC station. However, the velocity of traveling wave is highly influenced by the electrical parameters of the transmission lines and thus raise concerns on the precision of the result of fault location calculated by TW based algorithms [77; 78; 79]. Therefore, in order to overcome aforementioned shortcomings, natural frequency of traveling wave is measured to predict and estimate fault location and distance from VSC station [35]. Traveling wave's spectrum of faulty voltage signals can be employed for calculation of fault distance from VSC stations. When system's impedance is zero or infinite, dc fault distance and natural frequencies of TW are defined by \(L~{}=~{}c/4f\)_and_\(f=1/4\tau\), respectively [80; 81; 82].