## 2 EMTR and Correlation Estimator Methods
### Correlation-Estimator-Based Fault Location Method
The problem depicted in Figure 3 is represented mathematically as follows:  
\[[V]=\tfrac{1}{2}([S]+[T])([1]-[S][\Gamma])^{-1}\Big{[}W^{(S)}\Big{]} \tag{11}\]  
\[[I]=\tfrac{1}{2}[Y_{\text{C}}]([S]-[T])([1]-[S][\Gamma])^{-1}\Big{[}W^{(S)} \Big{]} \tag{12}\]  
\[\Big{[}W^{(S)}\Big{]}=[W_{S1}\;W_{S2}\;W_{S3}\;W_{S4}] \tag{13}\]  
\[[\Gamma]{=}\,diag\Big{[}e^{-\gamma x_{f}}\,e^{-\gamma x_{f}}\,e^{-\gamma\left( L-x_{f}\right)}\,e^{-\gamma\left(L-x_{f}\right)}\Big{]} \tag{14}\]  
\[[S]=\left[ \begin{array}{cccc}0&S_{1}&0&0\\ S_{F11}&0&0&S_{F12}\\ S_{F21}&0&0&S_{F22}\\ 0&0&S_{2}&0\end{array} \right];[T]=\left[ \begin{array}{cccc}0&1&0&0\\ 1&0&0&1\\ 1&0&0&1\\ 0&0&1&0\end{array} \right] \tag{15}\]  
\[[Y_{\text{C}}]=\Big{[}Z_{\text{C}}{}^{-1}\Big{]};\;[1]=diag[1\;\;1\;1\;1] \tag{16}\]  
where \(W^{(S)}\) and \(\Gamma\) are the source super-vector and propagation super-matrix, respectively, while \(S\), \(T\), and \(Y_{\text{C}}\) are the scattering super-, identity super-, and admittance super-matrices, respectively. Finally, the transfer functions can be easily obtained from the voltage/current responses, denoted by (11) and (12). Now, with Figure 1 as a reference, a test fault location of 8 km on a line length of 10 km is considered by employing EMTP-RV [26,27]. Figure 4 shows the transients recorded at x = 0 due to fault voltage source \(V_{f}(t)\) (described by Equation (7), with \(T_{r}=10\) us, \(V_{o}=2\) kV). The time step and the time window used to record the transients are 1 us and 7 ms, respectively. The related information of the studied analysis is given in Table 1.  
Figure 3: Fault emulation by the BLT formulation. Following the steps of the EMTR conventional metric (energy metric), presented in Section 2.1, the fault location is estimated and shown in Figure 5, whereas, for the correlation estimator metric, in order to obtain the transfer function from the recorded transients, first of all, the surge compression technique is applied to these transients. It is necessary to mention that, for the accurate estimation of the transfer function, surge compression is essential. Because the discrete sampling will inevitably produce a sampling bias, this will produce additionallocation error. The surge compression technique is adopted to erase this sampling bias in the first step. Then, the inverse filter is used to filter out the fault transient signal in order to increase the location's spatial resolution; more details can be found in [20]. The corresponding result is presented in Figure 6. Then, the FFT is applied to the fault source voltage and to the surge-compressed transients (\(V_{m-comp}\)) to finally obtain the fault occurrence stage transfer function (\(H_{fos}\)), defined as  
\[H_{fos}=\frac{V_{m-comp}(f)}{V_{f}(f)} \tag{17}\]  
The fault location is then estimated using the correlation of \(H_{fos}\) and \(H_{fes}\), which is also shown in Figure 5. The frequency spectrum from DC to 1 MHz with a step of 10 Hz is adopted during the back-propagation phase of the EMTR energy metric and the computation of \(H_{fos}\). The correlation estimator metric is computationally more efficient than the EMTR energy metric, as shown in Table 2, where it can be seen that, for the correlation estimator metric, the only time spent is during the pre-fault processing, whereas, during post-fault processing, it only takes a fraction of ascond to estimate the fault location. All the simulations presented in this paper were performed on an HP laptop with an Intel Core i7-4600U CPU running @ 2.10 GHz with 16 GB of RAM.