## 3 Results and Discussion
### Power-Loss Soft Sensor  
Table 12 shows the final input weights, output weights, and biases obtained for the two-layer feed-forward FSEV PL soft sensor design. Table 13 shows the details of the R-squared (0.9999) and MSE values for the training (MSE 5.74 \(\times\) 10\({}^{-3}\)), validation (MSE 5.76 \(\times\) 10\({}^{-3}\)), and testing (5.55 \(\times\) 10\({}^{-3}\)) sets of the FSEV PL data. The FSEV PL soft sensor attained higher R-squared metrics over the SOC as well as the SOE soft sensors. Its MSE values fared better than those of the SOE neural network but poorer in comparison to the SOC soft sensor. Figure 21 shows the PL soft-sensor regression plots under the various dataset partitions. The y label of the test regression plot indicates the following relation between output (predicted) and target (derived) PL values. \[\text{Output (predicted PL)}=\text{Target (derived PL)}+4\times 10^{-4}\] (  
Figure 22 shows the extremely nonlinear behaviour of the PL (watts) spikes experienced by the FSEV during one lap of the designated track. This figure depicts the dynamic effects of charging and discharging cycles of the FSEV battery on PL characteristics. In spite of the depicted power fluctuations, the proposed neural-network-based PL soft sensor can accurately follow the actual output throughout the lap. Figure 23 shows the error histogram of the data predicted by the PL soft sensor. It depicts the maximal number of testing data instances (around 10,000) at the error bin of \(-0.01007\) followed by comparatively lower instances at 0.03951 (around 4000) and \(-0.05965\) (approximately 2500) bins. The remaining error bins (0.08909, 0.1387, \(-0.1092\) and others) contained lesser instances. The error histogram shows the concentration of a majority of testing error instances at low magnitudes contributing towards the almost perfect predictability of the proposed FSEV PL soft-sensor design. Figure 24 presents the training state indicators for the PL soft-sensor generation. In this case, the training algorithm is terminated at epoch 211 because of reaching the maximal number (6) of validation fails permissible (Table 6). The gradient and mu at epoch 211 were 0.0034063 and 0.0001, respectively. Figure 25 shows that the best validation performance of the proposed PL soft sensor was recorded at 0.00057628, attained at epoch 205. This value was comparatively higher than the best validation performance metrics of the SOC and SOE soft sensors. Figure 21: Regression plot of FSEV PL soft sensor. Figure 23: Error histogram of FSEV PL soft sensor. Figure 24: Training states of FSEV PL soft sensor. Figure 22: Derived versus soft-sensor outputs of FSEV PL. Below are the regression and parametric model structures explored for the FSEV PL considered in the current work: Regression--Linear  
\[\text{PL}=2.074-0.02694*t+0.3897*I \tag{24}\]  
Regression--Nonlinear  
\[\text{PL}=-0.04033-0.006566*t+0.05491*I-8.419\times 10^{-5}*t^{2}-0.0009587*t*I+0.0 06543*I^{2} \tag{25}\]  
\[\text{ARX 441 }A(z)y(t)=B(z)u(t)+e(t)\]  
\[A(z) =1-1.345z^{-1}+0.2108z^{-2}+0.07419z^{-3}+0.06137z^{-4}\] \[B(z) =-0.03105z^{-1}+0.02019z^{-2}+0.009579z^{-3}+0.0006876z^{-4}\]  
Figure 21:0  
Figure 21:1 Figure 21:2 Figure 21:3  
Figure 21:4  
Figure 21:5 Figure 21:6 Figure 21:7 Figure 21:8  
Figure 21:9  
Figure 23:0 Figure 23:1  
Figures 26 and 27 show the R-squared/FIT% and mean squared errors of the FSEV PL models. The ANN and nonlinear regression PL models attained near perfect prediction accuracy. Linear regression and OE models achieved comparatively worse estimation performance. The mean squared error of the linear regression model was 47.7908 and is  
Figure 23:2 Validation performance of FSEV PL soft sensor. designs.