## 7 Fault Location Method Based on Similarity of Voltage Signals  
Several researches are conducted for enhancing the traveling wave methods performance by considering different topologies and configurations [7; 33; 34; 36; 37; 84; 85; 86; 87]. Some intrinsic problems like wave front identification and precision are associated with TW method. Machine learning and pattern recognition approaches played a key role in reducing the inherent problems associated with TW methods [38]. In this section, learning based method for identification of fault location in HVdc transmission line is discussed. In this method, samples from a time window of a post fault voltage is referred as input feature. Pearson correlation coefficient is calculated to find similarity between samples of the input feature and existing feature. Then this similarity is used to estimate the fault location [88]. The sampling frequency is 80 kHz. Post fault voltage signal depends on the oscillations of the energy stored in the capacitors and the transients generated by the fault because of the reflection of traveling waves [80; 89; 90]. Post fault voltage signal would have considerable correlation with the distance from fault point. Therefore, Pearson correlation coefficient is employed to measure the similarity of the voltage signals and expressed as:  
\[r(x,y)=\frac{\sum_{i=1}^{n}(x_{i}-\frac{1}{n}\sum_{j=1}^{n}x_{j})}{\sqrt{\sum _{i=1}^{n}\left(x_{i}-\frac{1}{n}\sum_{j=1}^{n}x_{j}\right)^{2}}}\times\frac{ \left(y_{i}-\frac{1}{n}\sum_{j=1}^{n}y_{j}\right)}{\sqrt{\sum_{i=1}^{n}\left(y_ {i}-\frac{1}{n}\sum_{j=1}^{n}y_{j}\right)^{2}}} \tag{50}\]  
where \(r(x,y)\) is the Pearson correlation coefficient of signals \(x\) and \(y\) and \(r(x,y)\ \in\ [-1,+1]\). Correlation coefficient \(+1\) illustrates positive linear correlation. Correlation coefficient \(-1\) represents negative linear correlation. No linear correlation present at all correlation coefficient \(0\). The voltage signals for fault location are measured at one terminal of the HVdc system. In this method, whence a dc fault is occurred, one pole (either positive or negative) is selected for voltage (signal) measurements for determination of fault location. Appropriate voltage samples are collected, when an absolute value of voltage is less than the threshold value of the voltage. These samples are employed as feature of the pattern. Similarity of the new pattern extracted from voltage samples is measured by comparing with the existing patterns available in the database. Fault location is estimated by the weighted average of the pointed values obtained from the \(k\) most similar patterns, which is called \(k\) nearest neighbor principle. Therefore, correlation is converted to positive value distance matrix and expressed as [75]:  
\[d_{r}(x,y)=1-r(x,y) \tag{51}\]  
where \(d_{r}(x,y)\) is Pearson distance of signal \(x\) and \(y\) and, \(d_{r}(x,y)\ \in\ [0,2]\). Equation (52) is employed to estimate the fault location obtained via new pattern [75]. \[FF_{L}(x^{\prime})=\ \frac{\sum\ x\ \in\ S\ (x^{\prime})\ F_{L}(x)\ \times\ e^{\ (-\ d_{r}\ (x,\ x^{\prime}))\ ^{2}}}{\sum\ x\ \in\ S\ (x^{\prime})\ e^{\ (-\ d_{r}\ (x,\ x^{\prime}))^{2}}} \tag{52}\]  
where \(F_{L}(x)\) is corresponding fault location of the existing pattern \(x\) and \(b\). \(F_{L}(x_{0})\) is the estimated fault location corresponding to new pattern \(x_{0}\). \(S(x_{0})\) is set of \(k\) patterns similar to new pattern \(x_{0}\). \(d_{r}(x,x_{0})\) is Pearson distance of existing pattern x and the new pattern \(x_{0}\). Voltage samples are collected for a time window of length of 10 ms if voltage values are less than the threshold values of voltage in the case of fault. Features are extracted from these voltage samples and are compared with the samples available in data base for the fault. \(k\) most similar patterns are determined and weighted average of patterns are employed to evaluate location of fault.