## 5 Fault Location in Direct Current (DC) Grid Having Cable and Overhead Line Segments
### Relationship between Distance and Dominant Component of Natural Frequency  
Thevenin's equivalent of two conductor transmission line circuit is shown in Figure 7.  
The terminal voltages are expressed in Laplace Transform as:  
\[V_{1}(s) =\frac{Z_{1}(s)}{Z_{1}(s)+Z_{c}(s)}W_{1}(s)+\frac{Z_{c}(s)}{Z_{1}(s)+Z_{c} (s)}E_{1}(s) \tag{26}\] \[V_{2}(s) =\frac{Z_{2}(s)}{Z_{2}(s)+Z_{c}(s)}W_{2}(s)+\frac{Z_{c}(s)}{Z_{2}( s)+Z_{c}(s)}E_{2}(s) \tag{27}\]  
where \(V_{1}(s)\) and \(V_{2}(s)\) are the terminal voltages, respectively. \(E_{1}(s)\) and \(E_{2}(s)\) are Thevenin's voltage sources of systems 1 and 2, respectively. \(Z_{1}(s)\) and \(Z_{2}(s)\) are the equivalent impedance of system 1 and  
Figure 6: Surge reflection routes in a faulted line. Figure 7: Equivalent circuit of bipolar transmission line. 2, respectively. \(Z_{c}(s)\) is the characteristic impedance of TL. The state variables \(W_{1}(s)\) and \(W_{2}(s)\) are given by following expressions:  
\[W_{1}(s)=\frac{P(s)}{1-\tau_{1}(s)\tau_{2}(s)P^{2}(s)}\Bigg{[}\frac{2Z_{c}(s)}{Z _{2}(s)+Z_{c}(s)}E_{2}(s)+\tau_{2}(s)P(s)\frac{2Z_{c}(s)}{Z_{1}(s)+Z_{2}(s)}E_{ 1}(s)\Bigg{]} \tag{28}\]  
\[W_{1}(s)=\frac{P(s)}{1-\tau_{1}(s)\tau_{2}(s)P^{2}(s)}\Bigg{[}\frac{2Z_{c}(s)}{ Z_{1}(s)+Z_{2}(s)}E_{1}(s)+\tau_{1}(s)P(s)\frac{2Z_{c}(s)}{Z_{2}(s)+Z_{c}(s)}E_{ 2}(s)\Bigg{]} \tag{29}\]  
where \(\tau_{1}(s)\) and \(\tau_{2}(s)\) are reflection coefficients of TL ends:  
\[\tau_{1}(s)=|\tau_{1}|e^{i\theta_{1}}=\frac{Z_{1}(s)-Z_{2}(s)}{Z_{1}(s)+Z_{c}( s)} \tag{30}\]  
\[\tau_{2}(s)=|\tau_{2}|e^{i\theta_{2}}=\frac{Z_{2}(s)-Z_{2}(s)}{Z_{2}(s)+Z_{c}( s)} \tag{31}\]  
and \(P(s)\) is the delay operator of the line:  
\[P(s)=\ e^{-sT} \tag{32}\]  
where \(T\) is the time required by the TW to reach at second end from first end of the line. The roots of the natural frequencies of the line is given by:  
\[V_{1}(s) =\frac{Z_{1}(s)}{Z_{1}(s)+Z_{c}(s)}W_{1}(s)+\frac{Z_{c}(s)}{Z_{1}(s)+Z_{c} (s)}E_{1}(s) \tag{26}\]0  
Equation (33) can produce infinite roots, which is an indication that traveling wave is composed of infinite number of oscillatory components. Thus, frequencies of these oscillatory components are determined by Equation (30). Equating denominator of Equation (33) to zero provides the information about all natural frequencies induced by the dc fault:  
\[V_{1}(s) =\frac{Z_{1}(s)}{Z_{1}(s)+Z_{c}(s)}W_{1}(s)+\frac{Z_{c}(s)}{Z_{1}(s)+Z_{c} (s)}E_{1}(s) \tag{26}\]1  
Using \(s=\ \sigma\ +\ j\omega\) in (32) for \(n^{th}\) component of natural frequency:  
\[V_{1}(s) =\frac{Z_{1}(s)}{Z_{1}(s)+Z_{c}(s)}W_{1}(s)+\frac{Z_{c}(s)}{Z_{1}(s)+Z_{c} (s)}E_{1}(s) \tag{26}\]2  
where \(\sigma_{n}\) and \(\omega_{n}\) are the damping constant and angular frequency of the \(n^{th}\) component of the natural frequency, respectively. Substituting the delay operator in Equation (34) results in:  
\[V_{1}(s) =\frac{Z_{1}(s)}{Z_{1}(s)+Z_{c}(s)}W_{1}(s)+\frac{Z_{c}(s)}{Z_{1}(s)+Z_{c} (s)}E_{1}(s) \tag{26}\]3  
The dominant component of natural frequency possesses largest amplitude among all the components and therefore, it is used for the evaluation of fault distance. When a fault occurs in an HVdc transmission line, the reflection coefficient at the point of fault is given by:  
\[V_{1}(s) =\frac{Z_{1}(s)}{Z_{1}(s)+Z_{c}(s)}W_{1}(s)+\frac{Z_{c}(s)}{Z_{1}(s)+Z_{c} (s)}E_{1}(s) \tag{26}\]4  
or  
\[V_{1}(s) =\frac{Z_{1}(s)}{Z_{1}(s)+Z_{c}(s)}W_{1}(s)+\frac{Z_{c}(s)}{Z_{1}(s)+Z_{c} (s)}E_{1}(s) \tag{26}\]5  
where \(Y_{f}(s)\) is an admittance of the fault. Thus, Equation (36) implies:  
\[V_{1}(s) =\frac{Z_{1}(s)}{Z_{1}(s)+Z_{c}(s)}W_{1}(s)+\frac{Z_{c}(s)}{Z_{1}(s)+Z_{c} (s)}E_{1}(s) \tag{26}\]6Since \(T=d\;v\), using this substitution in Equation (39):  
\[V_{1}(s) =\frac{Z_{1}(s)}{Z_{1}(s)+Z_{c}(s)}W_{1}(s)+\frac{Z_{c}(s)}{Z_{1}(s)+Z_{c} (s)}E_{1}(s) \tag{26}\]7  
where \(f_{n}\) is component of natural frequency and \(n\;=1,2,3,\ldots\).