## 3 Results and Discussion
### State of Charge Soft Sensor
\begin{split} A(z)&=1-0.3499z^{-1}-0.1942z^{-2}-0. 2977z^{-3}-0.1582z^{-4}\\ B(z)&=-1.964\times 10^{-7}z^{-1}-6.007\times 10^{-8}z^{-2} -6.836\times 10^{-8}z^{-3}+4.473\times 10^{-9}z^{-4}\end{split} \]  
ARMAX 2221 \(A(z)y(t)=B(z)u(t)+C(z)e(t)\)  
\[ \begin{split} A(z)&=1-1.647z^{-1}+0.6465z^{-2}\\ B(z)&=-1.663\times 10^{-7}z^{-1}+1.162\times 10^{-7}z^{-2 }\\ C(z)&=1-1.42z^{-1}+0.4327z^{-2}\end{split} \]  
BJ 22221 \(y(t)=[B(z)/F(z)]u(t)+[C(z)/D(z)]e(t)\)  
\[ \begin{split} B(z)&=-2.316\times 10^{-7}z^{-1}-9.932 \times 10^{-9}z^{-2}\\ C(z)&=1-0.3207z^{-1}-0.4296z^{-2}\\ D(z)&=1-0.5763z^{-1}-0.402z^{-2}\\ F(z)&=1-0.2971z^{-1}-0.7029z^{-2}\end{split} \]  
OE 221 \(y(t)=[B(z)/F(z)]u(t)+e(t)\)  
\[ \begin{split} B(z)&=-1.886\times 10^{-7}z^{-1}+8.593 \times 10^{-8}z^{-2}\\ F(z)&=1-1.276z^{-1}+0.2759z^{-2}\end{split} \]  
Figure 12 shows the composite plot of R-squared (regression) and FIT% (system identification) models explored in the current study. The ANN-based soft sensor attained the highest accuracy among all models. However, the parametric models scored lower mean squared errors as compared to ANN and regression models (Figure 13). All models attained comparable performance with regard to SOC estimation. However, ANN is capable of handling unseen data better than other structures are [56]. Therefore, the ANN-based soft sensor is expected to exhibit higher robustness in real-time applications involving system variations and dynamic operating conditions. The ANN modelling procedure incorporates the partitioning of data into training, validation, and testing sets, ensuring the reliable predictability of the final model. Other modelling techniques such as regression and system identification do not provide this advantage, making them inferior to ANN with regard to soft-sensor applications. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Weight** & **Value** & **Weight** & **Value** & **Weight** & **Value** & **Bias** & **Value** \\ \hline \(W_{1,1}\) & 6.8392 & \(W_{2,1}\) & \(-0.0811\) & \(W_{0,1}\) & \(-0.1686\) & \(b_{1}\) & \(-5.2122\) \\ \(W_{1,2}\) & 1.5481 & \(W_{2,2}\) & 1.9788 & \(W_{0,2}\) & \(-0.0767\) & \(b_{2}\) & \(-1.6627\) \\ \(W_{1,3}\) & \(-2.7665\) & \(W_{2,3}\) & 2.5017 & \(W_{0,3}\) & \(0.0277\) & \(b_{3}\) & \(2.2216\) \\ \(W_{1,4}\) & 2.6111 & \(W_{2,4}\) & \(-0.0231\) & \(W_{0,4}\) & \(-0.3693\) & \(b_{4}\) & \(-0.6678\) \\ \(W_{15}\) & 1.1701 & \(W_{2,5}\) & 9.4484 & \(W_{0,5}\) & \(0.0191\) & \(b_{5}\) & \(2.1594\) \\ \(W_{1,6}\) & \(-2.6643\) & \(W_{2,6}\) & \(-2.7645\) & \(W_{0,6}\) & \(0.0396\) & \(b_{6}\) & \(-0.0218\) \\ \(W_{1,7}\) & \(-2.36749\) & \(W_{2,7}\) & 0.7517 & \(W_{0,7}\) & \(0.1327\) & \(b_{7}\) & \(-8.4033\) \\ \(W_{1,8}\) & \(-1.0208\) & \(W_{2,8}\) & 7.0473 & \(W_{0,8}\) & \(0.0214\) & \(b_{8}\) & \(7.0193\) \\ \(W_{1,9}\) & 8.2776 & \(W_{2,9}\) & 8.9437 & \(W_{0,9}\) & \(-0.0232\) & \(b_{9}\) & \(6.5266\) \\ \(W_{1,10}\) & 3.856 & \(W_{2,10}\) & \(-0.076\) & \(W_{0,10}\) & \(-0.2172\) & \(b_{10}\) & \(2.0931\) \\  & & & & & & \(b_{e1}\) & \(-0.0677\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Input and output weights and biases for the two-layer feed-forward FSEV SOE soft sensor. Figure 14: Regression plots of FSEV SOE soft sensor. Figure 15 shows the plots of derived and predicted SOE outputs for one lap time (118 s) as per the FSEV drive cycle. It is evident that the predicted SOE plot closely matches the derived SOE curve. The figure shows that the SOE dropped from 97% to approximately 93.8% during one lap time period. The final SOE was slightly better than the SOC (93.5%) at 118 s. Figure 16 shows the error histogram for the SOE soft sensor. The maximal number of testing error instances (around 2500) were generated at the error value of 3.88 \(\times\) 10\({}^{-5}\). More than 2000 testing error instances were detected at the \(-6.5\times\) 10\({}^{-6}\) error bin.