## 7 Fault Location Method Based on Similarity of Voltage Signals
### Multi-Terminal High-Voltage Direct Current (MT-HVdc) Test System  
When a dc fault \(F\) occurs on transmission line \(L_{1}\) connected to converter station, traveling wave of voltage at fault instant starts propagating towards the series inductor deployed at both ends of transmission line. At the converter station, traveling waves are reflected back and transmitted to other transmission lines connected to this converter station. Therefore, ratio of transmitted wave at line side of series inductor of a normal line to transmitted wave at line side of series inductor of faulted line helps to determine the faulted line in MT-HVdc transmission system. For this, equivalent impedance \(z_{t}\) of lines connected to converter station with the exclusion of line \(L_{1}\) is given by:  
\[z_{t}=(sL_{2}+z_{2})\big{|}\ldots\big{|}\big{(}sL_{n}+z_{n})\big{|}\big{|}z_{ conv} \tag{89}\]  
where \(z_{conv}=\ R_{conv}+sL_{conv}+\frac{1}{sC_{conv}}\) and \(R_{conv}\), \(L_{conv}\) and \(C_{conv}\) are the equivalent resistance, inductance and capacitance of converter, respectively. \(s\) is the Laplace operator. Capacitive discharge stage of MT-HVdc system during dc faults is modelled by RLC circuit. Converter topology helps to determine the parameters of this circuit.  
Figure 16 shows the network of \(n\) HVdc transmission lines connected to MT-HVdc converter station. \(z_{k}\) and \(L_{k}\) represent characteristics impedance and series inductor of \(k\) transmission lines. \(F\) is dc fault on line \(L_{1}\). \(u_{ik}\), \(u_{rk}\), and \(u_{tk}\) denote incident, reflected, and transmitted traveling waves of voltage, respectively, at the converter station connected to \(k\) lines. The ratios of transmitted traveling  
waves of voltage and transmitted traveling wave of current to incident traveling waves of voltage at \(L_{1}\) are given by:  
\[\frac{u_{t1}}{u_{i1}}=\ \frac{2(sL_{1}+z_{t})}{sL_{1}+z_{t}+z_{1}} \tag{90}\]  
\[\frac{i_{t1}}{u_{i1}}=\ \frac{2}{sL_{1}+z_{t}+z_{1}} \tag{91}\]  
The ratio of transmitted traveling waves of voltage to \(k\) lines and incident traveling waves of voltage at \(L_{1}\) is:  
\[\frac{u_{kk}}{u_{i1}}=\frac{2}{sL_{1}+z_{t}+z_{1}}\ \times z_{t}\times\frac{z_{k}}{ sL_{k}+z_{k}} \tag{92}\]  
The ratio of transmitted traveling waves of voltage to \(k\) lines to transmitted traveling waves of voltage at line \(L_{1}\) is represented by \(k_{t1t1}\) and given by:  
\[k_{ttt1}=\ \frac{u_{tk}}{u_{t1}}=\ \frac{z_{t}z_{k}}{(sL_{1}+z_{t})(sL_{k}+z_{ k})} \tag{93}\]  
By considering the basic values of characteristics impedance and series inductor, the ratio \(k_{ttt1}\) works as a low pass filter. This low pass filter is used to attenuate high frequencies. Figure 17 shows the behavior of \(k_{ttt1}\). It is observed that traveling waves originated due to fault contain higher frequency components as compared to traveling waves transmitted on normal lines. Therefore, faulted line is identified with the help of this characteristic behavior and MG. Proposed flow chart of MG algorithm is shown in Figure 18.  
Figure 16: Multi-terminal high-voltage direct current (MT-HVdc) converter station under fault condition.  
Figure 17: Characteristics curve of ratio of transmitted traveling wave to k non faulted high-voltage direct current (HVdc) lines and transmitted traveling wave to faulted HVdc line.