# Optimal Fuel Consumption Modelling, Simulation, and Analysis for Hybrid Electric Vehicles
## 4 Fuel Economy Modelling and Regressions
Finally, we consider the third-order fuel regression model in the following form:  
\[F_{3}=f_{3}\left(W,T\right)=P_{00}+P_{10}W+P_{01}T+P_{20}W^{2}+P_{11}WT+P_{02} T^{2}+P_{30}W^{3}+P_{21}W^{2}T+P_{12}WT^{2}+P_{03}T^{3} \tag{44}\]  
where \(F_{3}\) is the consumption of fuel in the third regression model; \(W\) is the HEV's _Weight_, \(T\) is the _Tire_ radius, and the coefficients of \(P_{00}\), \(P_{10}\), \(P_{01}\), \(P_{20}\), \(P_{11}\), \(P_{02}\), \(P_{30}\), \(P_{21}\), \(P_{12}\), and \(P_{03}\) are generated as \(P_{00}\) = 0.01948; \(P_{10}\) = \(-8.057\times 10^{-5}\); \(P_{01}\) = \(-8.597\times 10^{-5}\); \(P_{20}\) = 0.0004235; \(P_{11}\) = 0.0004004; \(P_{02}\) = 0.0001391; \(P_{30}\) = 0.0003122; \(P_{21}\) = 0.0003401; \(P_{12}\) = 0.000162; and \(P_{03}\) = 2.06 \(\times\) 10\({}^{-5}\). The third-order regression equation model in (44) has the determination coefficient of R-square = 0.9914, or this regression prediction fits the data of 99.14%. The 3D graphic map of this HEV's fuel consumption is drawn out in \(x\), \(y\), \(z\) coordinates, where the \(z\)-axis is the fuel consumption in litres, the \(x\)-axis is the HEV weights in kg, and the \(y\)-axis is the tire radius in meters. The range of fuel consumption is from 0.02 to 0.025 litres for the HEVs running for 100 s surrounding the standard driving urban cycle of FEP75. Details of this equation in a 3D graphic map for the fuel consumption vs. HEV weights vs. tires are shown in Figure 12.  
The minimum values of the HEV performances on the 3D graphic map and from the third-order regression model in (44) can be attained. When we obtain the first derivative of (44), we can find out the local maximum and the local minimum values of this function. When we continue to obtain the second derivative of this function and if the second derivative function results in \(f^{\prime\prime}(W,T)>0\), then the original function, \(f(W,T)\), has the local minimum values\(f_{min}\) at \(W_{min}\) and \(T_{min}\). Taking the derivative Function (44) for \(F_{3}=f_{3}(W,T)\), leads to the optimal values of \(T\) and \(W\):  
\[3.0541\times 10^{-6}\cdot W+0.0211\cdot T-0.0095=0 \tag{45}\]  
Equation (45) permits the achievement of the optimal values of HEVs for fuel consumption from specified weights and tires. The optimal HEV weights by the tires are indicated in Table 4.  
Finally, the optimal fuel consumption in (44) and (45) is tested and compared to similar HEVs to observe the efficiency of the optimal HEV's fuel consumption performances during the different standard driving cycles of EUDC, HWFET, NYCC, and FTP75. In this test, we selected the optimal HEV weight of 1222 kg vs. the rolling tire radius of 0.278 m from the right-hand side column in Table 4. We compared the optimal HEVs to the standard lighter HEVs with a weight of 1200 kg and with a smaller tire radius of 0.262 m. The results for the fuel consumption for optimal HEVs vs. normal HEVs during different driving cycles are shown in Table 5.  
Table 5 proves that fuel consumption for optimal HEV models is always less than the similar normal HEV performances with lighter weights and smaller tire radiuses. The results of the simulation re-affirmed that the optimal HEV data in Equation (44) and in the 3D graphic map in Figure 12 are the correct regressions. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Rolling Tire Radius** & 0.242 & 0.248 & 0.252 & 0.259 & 0.262 & 0.265 & 0.267 & 0.269 & 0.277 & 0.278 \\
**in Meters** & & & & & & & & & \\ \hline
**Optimal HEV** & 1472 & 1429 & 1402 & 1354 & 1333 & 1312 & 1298 & 1277 & 1229 & 1222 \\
**Weights in Kg** & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimal HEV weights vs. tire radiuses. Figure 12: The best-fit fuel consumption model, 99.14%.