# Optimal Fuel Consumption Modelling, Simulation, and Analysis for Hybrid Electric Vehicles
## 4 Fuel Economy Modelling and Regressions  
The comprehensive HEV model in Simulink in Figure 11 and other Simulink schemes are tested on different HEV models, shown in Table 1, and at the link references [28; 29; 30; 31; 32; 33; 34; 35]. Different HEV models are simulated with different tires running on FTP75 (standard urban  
Figure 11: Simulink model for the HEV. Figure 10: Modelling control system. driving cycles). These 8 HEV models with different weights and different tires, and with 90% fuel in the tank, are run for 100 s on FTP75. Ten different compatible tire sizes are selected and simulated with the above eight HEV models. These ten different compatible tire parameters are indicated in Table 2.  
The \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **HEV Models** & **HEV Weights in Kg** \\ \hline
1. & Hybrid Toyota Prius 2021 & 1335 [28] \\ \hline
2. & Hybrid Honda Insight 2021 & 1247 [29] \\ \hline
3. & Hybrid Kia Niro 2021 & 1419 [30] \\ \hline
4. & Hyundai Sonata 2021 & 1578 [31] \\ \hline
5. & Hybrid Kia Optima 2021 & 1595 [32] \\ \hline
6. & Hybrid Hyundai Avante LPi 2021 & 1287 [33] \\ \hline
7. & Hybrid Toyota Camry 2021 & 1639 [34] \\ \hline
8. & Hybrid Honda Civic 2021 & 1270 [35] \\ \hline \end{tabular}
\end{table} 
Table 1: HEV weights. \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Parameters** & **Rolling Radius in Meters** & **Rim Width in Inch** \\ \hline
1. & 154/65/R13 & 0.242 & 4.5 \\ \hline
2. & 154/80/R13 & 0.262 & 4.5 \\ \hline
3. & 164/60/R14 & 0.252 & 5.0 \\ \hline
4. & 164/65/R13 & 0.246 & 5.0 \\ \hline
5. & 164/65/R14 & 0.257 & 5.0 \\ \hline
6. & 164/70/R14 & 0.267 & 5.0 \\ \hline
7. & 164/80/R13 & 0.268 & 4.5 \\ \hline
8. & 174/65/R14 & 0.265 & 5.0 \\ \hline
9. & 174/65/R15 & 0.275 & 5.0 \\ \hline
10. & 204/50/R15 & 0.276 & 6.5 \\ \hline \end{tabular}
\end{table} 
Table 2: Ten different tires for HEVs. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **1335** & **1247** & **1419** & **1578** & **1595** & **1287** & **1639** & **1270** \\ \hline
0.242 & 0.01983 & 0.01983 & 0.01984 & 0.01992 & 0.01994 & 0.01983 & 0.02008 & 0.01983 \\ \hline
0.262 & 0.01962 & 0.01965 & 0.02033 & 0. Data for the fuel consumption in Table 3 are processed in MATLAB 2021b to attain the different fuel consumption regression models for HEVs based on their weights and tires. The fuel consumption is a function of the above data and depending on the function order:  
\[\text{\it Consumption of Fuel }F=\text{\it function }(\text{\it HEVs weight }W,\text{\it and radius Tires }T) \tag{41}\]  
Now, we consider the first-order fuel regression model in the following form:  
\[F_{1}=f_{1}(W,T)=P_{00}+P_{10}W+P_{01}T \tag{42}\]  
where \(F_{1}\) is the first-order regression model for the fuel consumption, \(W\) is the HEV's _Weight_, \(T\) is the _Tire_ radius, and the coefficients of \(P_{00}\), \(P_{10}\), and \(P_{01}\) are generated as \(P_{00}\) = 0.02015; \(P_{10}\) = 0.0006672; and \(P_{01}\) = 0.0002534, correspondingly. The first-order fuel regression model from (42) has the determination coefficient of R-square = 0.5364, or this regression prediction fits the data of 53.64%. Next, we consider the second-order fuel regression model in the following form:  
\[F_{2}=f_{2}(W,T)=P_{00}+P_{10}W+P_{01}T+P_{20}W^{2}+P_{11}WT+P_{02}T^{2} \tag{43}\]  
where \(F_{2}\) is the consumption of fuel in the second-order regression model; \(W\) is the HEV's _Weight_, \(T\) is the _Tire_ radius, and the coefficient of \(P_{00}\), \(P_{10}\), \(P_{01}\), \(P_{20}\), \(P_{11}\), and \(P_{02}\) are generated as \(P_{00}\) = 0.01946; \(P_{10}\) = 0.0004894; \(P_{01}\) = 0.0002958; \(P_{20}\)= 0.0005422; \(P_{11}\) = 0.0004614; and \(P_{02}\) = 0.0001364, correspondingly. The second-order regression equation model in (43) has the determination coefficient of R-square = 0.8875, or this regression prediction fits the data of 88.75%.