## 7 Fault Location Method Based on Similarity of Voltage Signals
### Obsevance of the First Traveling Wave  
First zero and positive sequence components of traveling waves are observed at converter station and are:  
\[\left\{ \begin{array}{l}b_{0}(t)=\Delta u_{0}-Z_{0c}\Delta i_{0}=2\Delta u_{ f0}e^{-\alpha_{0}t}e(t-\tau_{0})\\ b_{1}(t)=\Delta u_{1}-Z_{1c}\Delta i_{1}=2\Delta u_{f1}e^{-\alpha_{1}t}e(t- \tau_{1})\end{array} \right. \tag{78}\]  
and  
\[\frac{b_{0}}{b_{1}}=\frac{\Delta u_{f0}}{\Delta u_{f1}}=\frac{e^{-\alpha_{0}t} e(t-\tau_{0})}{e^{-\alpha_{1}t}e(t-\tau_{1})} \tag{79}\]  
where \(\Delta u_{0}\), \(\Delta u_{1}\), \(\Delta i_{0}\), and \(\Delta i_{1}\) are the superimposed zero and positive sequence voltage and currents at relay location. \(Z_{0c}\) and \(Z_{1c}\) are zero and positive sequence surge impedances. \(\alpha_{0}\) and \(\alpha_{1}\) are attenuated coefficients of zero and positive sequence and \(\tau_{0}\) and \(\tau_{1}\) are traveling times of TWs from fault to the VSCs. \(\varepsilon(t)\) is a step input. When time period equals twice the traveling time of TW, \(b_{0}\) and \(b_{1}\) will remain constant at \(2\Delta u_{f0}\) and \(2\Delta u_{f1}\), respectively. It is clear from Equation (79) that ratio of positive and negative sequence waves not only depends on the initial values of the traveling waves but also depends on the attenuation and the velocity. The polarities of zero and positive sequence TWs for pole to ground fault are the same at faulty pole and are different at healthy pole, as expressed in Equations (70) and (78). It is obvious from the Equations (71), (75), (77), and (79) that the ratio of positive and zero sequence traveling waves is greater for single pole fault than double pole fault after the fault initiation. This is an important result in designing the identification of fault. Protection Scheme for High-Voltage Direct Current (HVdc) Transmission Line Using Symmetrical Components Method  
Ideally, relaying signal \(x_{1}\) as shown in Figure 14, is zero under steady state conditions. When a dc fault occurs, fault current starts building up immediately. A data window is then achieved which must obey the criterion given in Equation (80) to select the first sample for integration:  
\[|x_{1}|>M_{1} \tag{80}\]  
where \(M_{1}\) is preset data window of fault current signal. Smoothing reactors or in some cases harmonic filters or surge capacitors, are present in HVdc systems and are connected pole to ground. Current cannot change abruptly because of smoothing reactors and dc filters. This results in the decaying of traveling waves. Gradient of traveling wave is lower for external fault than for internal faults [91]. Thus, criterion for internal fault detection is described as:  
\[|\Delta x_{1}|>M_{2} \tag{81}\]  
where \(M_{2}\) is minimum value of gradient required for internal fault detection. The ratio of positive to zero sequence components is greater for single pole fault than double pole fault for a small moment of time after fault initiation. Because of short duration, it is difficult to detect the type of fault in a classical way. Therefore, integration of sub-durations of the short duration of time is done to set up a criterion for detection of fault type and to classify faulty pole [92]. \[k=\frac{\int_{x_{0}}^{t+\tau_{0}}|x_{0}|}{\int_{x_{1}}^{t+\tau_{1}}|x_{1}|} \tag{82}\]where \(\tau_{1}\) is the time of protection startup and \(\tau\) is the data window of fault which is chosen to be nearly equal to 1 ms. \(\tau_{0}\) is the arrival time of the first zero sequence traveling wave and the criterion for detection of zero sequence traveling wave is given by:  
\[|x_{0}|>M_{0} \tag{83}\]  
and from Figure 15, if \(k\,>k_{0}\), fault is single pole to ground, otherwise it would be double pole fault. Faulty pole can be detected by viewing the polarity of positive and zero sequence backward traveling waves. If polarity is same in case of single pole to ground fault then it is faulty pole and if the polarity is different, then it is healthy pole.