## 4 Transient Fuel Consumption Model and Its Validation  
To improve the accuracy of the fuel consumption calculation from the backward model, TCF was used to calibrate the steady-state fuel consumption rate of the model (\(\dot{m}_{ss\_m}\)) with the experimental one (\(\dot{m}_{exp}\)). As mentioned above in this paper, the transient model was developed based on the approach described in [14]. In that work, three different \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Driving Cycle**} & \multirow{2}{*}{**Experimental \(fc_{exp}\)**} & **Calculated:** & **Calculated:** \\  & & **Steady-State Map, Experimental** & **Steady State Map, Estimated** \\  & & **Torque and Speed \(fc_{ss\_exp}\)** & **Torque and Speed \(fc_{ss\_m}\)** \\ \hline UDDS & 741.6 g & 712 g & 695 g \\ \hline HW & 787 g & 758 g & 750.6 g \\ \hline US06 & 1005 g & 952 g & 940 g \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of overall fuel consumption from the model and experiment for the conventional vehicle. Figure 2: Fuel consumption map of Mazda CX9 2016 internal combustion engine obtained in steady-state conditions [18]. representations for TCF were proposed. The most accurate fuel usage estimation was obtained when TCF was represented by the following expression [14]:  
\[ TCF=A\cdot\omega_{ice}+B\cdot\frac{d\omega_{ice}}{dt}+C\cdot T_{ice}+D\cdot\frac{ dT_{ice}}{dt}+E \tag{2}\]  
Regression coefficients \(A,\ B,\ C,\ D,\ E\) can be estimated separately for the following use cases:  
\[\text{Case }(a):\ \frac{d\omega_{ice}}{dt}>0\text{ }and\ \frac{dT_{ice}}{dt}>0;\]  
\[\text{Case }(b):\ \frac{d\omega_{ice}}{dt}>0\text{ }and\ \frac{dT_{ice}}{dt}<0;\]  
\[\text{Case }(c):\ \frac{d\omega_{ice}}{dt}\leq 0\text{ }and\ \frac{dT_{ice}}{dt}>0;\]  
\[\text{Case }(d):\ \frac{d\omega_{ice}}{dt}\leq 0\text{ }and\ \frac{dT_{ice}}{dt}\leq 0;\]  
In [14], it was established that coefficients \(A,\ B,C\) and \(D\) have an order of exponential magnitude \(-5,\ -5,\ -4\) and \(-3\) for all the cases, respectively. This means that coefficient \(D\) is the dominant one. That indicates that the torque change rate mostly affects the transient fuel consumption increase. Therefore, TCF can be approximated with good accuracy using the following simplified form of Equation (2):  
\[ TCF=D\cdot\frac{dT_{ice}}{dt}+E \tag{3}\]  
In the current work, coefficient \(E\) was set to one (\(E=1\)), so that that TCF was applied only if there was a torque change. Otherwise, TCF was 1, which means that the predicted fuel consumption was equal to the one computed using the steady-state map. Regression analysis was carried out using experimental data on UDDS, US06, and HW driving cycles available on the ANL database [20]. The calculated values of \(T_{ice\_m}\) and \(\omega_{ice\_m}\) from the backward model was used to determine the torque change rate and fuel consumption \(\dot{m}_{ss\_m}\). Then, TCF was calculated as a ratio of measured \(\dot{m}_{exp}\) and steady-state fuel consumption \(\dot{m}_{ss\_m}\). Depending on the sign of the \(T_{ice\_m},\omega_{ice\_m}\) changing rate, the data were split into 4 cases. The data sampling time during the experiments was 0.1 s and a moving average filter of order 10 was used for smoothing. Moreover, operating points with low distribution density were excluded from the analysis. In the case shown in Figure 2(a), the few operating points with a torque change rate higher than 200 Nm/s were omitted as outliers. The values of coefficient \(D\) derived from this linear regression analysis are shown in Figure 3 and summarized in Table 3. These values of regression coefficient \(D\) are aligned with those obtained in [14]. The fuel consumption obtained from the steady-state map can then be corrected with TCF as:  
\[\dot{m}_{TCF}=TCF\cdot\dot{m}_{ss\_m} \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cases** & \(D\) & **E** \\ \hline (a) & 0.0026 & 1 \\ \hline (b) & \(-0.0003\) & 1 \\ \hline (c) & 0.0033 & 1 \\ \hline (d) & 0.0011 & 1 \\ \hline \hline \end{tabular}
\end{table}