## 3 Results and Discussion
### State of Charge Soft Sensor  
Table 8 shows the final hidden or output layer weights and biases for the FSEV SOC neural network, and Table 9 depicts the mean squared error (MSE) and R squared (R-sq) values of the proposed SOC soft sensor as per various dataset partitions. The training, validation, and testing datasets attained very impressive MSEs of 8.28 \(\times\) 10\({}^{8}\), 8.30 \(\times\) 10\({}^{8}\) and 8.34 \(\times\) 10\({}^{8}\), respectively. All datasets achieved R squared values of 0.9996 each, showing excellent prediction accuracy for the proposed FSEC SOC soft sensor. Figure 7 shows the regression plot of the FSEV SOC sensor performance relative to various dataset partitions. Figure 8 shows a graphical presentation of the derived versus soft-sensor-predicted SOC values over one lap time of 118 s. The graph shows that the SOC decreased from 97% at the beginning of the lap to 93.5% at the end of the lap. The soft-sensor model output closely matched that of the derived SOC. Figure 9 shows the error histogram for the FSEV SOC soft-sensor predictions. It depicts a maximal number of testing instances (around 4000) at an error bin of 1.99 \(\times\) 10\({}^{-5}\), followed by comparatively lower instances (around 2500) at 1.34 \(\times\) 10\({}^{-4}\) and \(-\)9.4 \(\times\) 10\({}^{-5}\). There are around a 1000 instances at the bin corresponding to 2.47 \(\times\) 10\({}^{-4}\) error. Remaining error bins (\(-\)4.3 \(\times\) 10\({}^{-4}\), \(-\)3.2 \(\times\) 10\({}^{-4}\), \(-\)2.1 \(\times\) 10\({}^{-4}\), 3.61 \(\times\) 10\({}^{-4}\), 4.75 \(\times\) 10\({}^{-4}\) and others) contain lesser instances. This distribution of error instances at very low magnitudes supports the excellent predictability of the proposed FSEV SOC soft-sensor design. Figure 10 depicts the gradient, mu, and validation checks as indicators of the FSEV SOC soft-sensor training states. The gradient continuously decreased till 5.76 \(\times\) 10\({}^{-8}\) at epoch 25, whereas mu settled at 1 \(\times\) 10\({}^{-10}\) at the same epoch. The gradient is an indicator of the learning rate and step size of the training algorithm. The sloping reduction in gradient over successive epochs indicated the systematic progression of the training algorithm towards the global minimum of the proposed soft-sensor prediction error. In fact, the training algorithm terminated further epochs because the stopping criterion of minimal performance gradient of 1 \(\times\) 10\({}^{-7}\) (Table 6) had been attained. A low mu indicated that the Levenberg-Marqaurdt algorithm moved towards [PERSON]'s method and away from the gradient descent methodology for faster and more accurate convergence [55]. The Figure 10 validation check depicts that there was no validation fail till epoch 25. Figure 11 shows that the best validation performance of the proposed soft sensor was recorded to be a very low MSE value of 8.30 \(\times\) 10\({}^{-8}\) attained at epoch 25. The y label of the test regression plot exhibits the following relation between derived and soft-sensor-predicted SOC. \[\text{Output (predicted SOC)}=\text{Target (derived SOC)}+6.8\times 10^{-8} \tag{17}\]  
Figure 8: Derived versus soft sensor outputs of FSEV SOC. Figure 7: Regression plots of FSEV SOC soft sensor. Figure 11: Validation performance of FSEV SOC soft sensor. Figure 10: Training states of FSEV SOC soft sensor. Figure 9: Error histogram of FSEV SOC soft sensor. The SOC of the FSEV considered in the current study was also modelled using linear and nonlinear regression as follows: Regression-Linear  
\[\text{SOC}=0.9724-0.0003157*t+1.911\times 10^{-6}*I \tag{18}\]  
Regression--Nonlinear  
\[\text{SOC}=0.9721-0.0002997*t+2.342\times 10^{-5}*I-1.191\times 10^{-7}*t^{2} \tag{19}\] \[-1.225\times 10^{-7}*t*I-2.432\times 10^{-7}*I^{2}\]  
The FSEV SOC was also modelled using parametric structures generally considered for the system identification of control systems. ARX 441 \(A(z)y(t)=B(z)u(t)+e(t)\)  
\[