## 5 Results and Discussion
### Fault Location  
As presented in Section 3.2, the DC fault events were simulated in the considered HVDC systems with varying fault resistances and locations. After fault events were classified as DC internal faults, the accurate detection of fault location was considered as an important task of the protection system. In this work, a fault location algorithm was also designed for the protection system, as shown in Figure 2. Similar to the fault classification task, the location of the fault was accomplished through the training and testing of different data regression techniques based on data-mining approaches (such as ANN, SVR, ELM, LR, RF, and DBN). To test the performance of each DMT, several performance indices or error indices (mathematically represented in Equations (9)-(13) were calculated and analyzed. The mean square error (MSE) and root-mean-square error  
Figure 8: Confusion matrix corresponding to the highest accuracy shown in Table 8: (**a**) confusion matrix showing numerical values (**b**) TPR and FNR (in percentages). Figure 9: Confusion matrix corresponding to the highest accuracy shown in Table 9: (**a**) confusion matrix showing numerical values (**b**) TPR and FNR (in percentages). (RMSE) were the two most familiar performance indicators which generally certified how close the predicted values were to real values. Here, a smaller error signified that the predicted value was closer to the actual one. The normalized RMSE (NRMSE) provided a comparative assessment between models with different scales. Similar to MSE and RMSE, a smaller value of calculated MAPE showed how close the forecast values were to the actual value. The scale-independencies and interpretability were two major advantages of MAPE. As compared with other performance indicators, a higher rate (maximum '1') of R-value indicated how well the prediction was achieved by the model. Mean Square Error \(\mathrm{(MSE)}=\)\(\dfrac{1}{k}\sum\limits_{i=1}^{k}\left(AV_{i}-PV_{i}\right)^{2}\) (9) \[\mathrm{R-Value}=\] (10) \[\mathrm{Root\ Mean\ Square\ Error\ (\mathrm{RMSE})=}\sqrt{\dfrac{\sum\limits_{i=1}^{k}\left\| \left(AV_{i}-PV_{i}\right)^{2}\right\|}{k}}\] (11) \[\mathrm{Normalized\ Root\ Mean\ Square\ Error(\mathrm{NRMSE})=\dfrac{\sqrt{\sum\limits_{i=1}^{k}\left\| \left(AV_{i}-PV_{i}\right)^{2}\right\|}/k}{AV_{\mathrm{max}}-AV_{\mathrm{min}}}\] (12) \[\mathrm{Mean\ absolute\ percentage\ error\ (\mathrm{MAPE})=}\dfrac{1}{k}\sum \limits_{i=1}^{k}\left\|\dfrac{\left(AV_{i}-PV_{i}\right)}{AV_{i}}\right\|\] (13)  
where '\(k\)' signifies the number of samples. \(AV\) and \(PV\) signify the actual value and predicted value, respectively. All the simulated internal DC faults as signals (3744 numbers, as presented in Section 2) were trained and tested by the DMTs with all 14 features as input data and fault location (or distance) from the respective relay points as target data. The error indices for each DMT are  
Figure 10: Confusion matrix corresponding to DBN: (**a**) confusion matrix showing numerical values (**b**) TPR and FNR (in percentages). presented in Figure 8:0 As presented in Table 11, the SVR-, ELM-, and LR-based regression techniques were found to be worst compared with other DMTs. The studied performance indicators of SVR, ELM, and LR were found, respectively, as: MSE: 654.7105, 591.977, and 595.2575; RMSE: 25.5872, 22.3306, and 24.3979; NRMSE: 0.3198, 0.3041, and 0.3050; MAPE: 75.2735, 76.5244, and 75.9142; R-Value: 0.7932, 0.8131, and 0.8120. However, the ANN and KNN techniques showed an improved performance compared with SVR and ELM. For example, the MSE of ANN and KNN were found to be 75.6485 and 65.0271 compared with SVR (654.71) and ELM (591.97), which was an 8-10 times lesser error of prediction. The RF showed a significantly better result with the lowest error rate (such as MSE = 2.2737, RMSE = 1.5079, NRMSE = 0.0188, and MAPE = 2.6403) compared with other ML-based DMTs. The R-value value for RF was noted as 0.9993 (very close to 'one'). However, DBN (a DL-based DMT) achieved a similar kind of result as RF achieved.