## 3 Identification of the Optimal Operating Maps
### Calculation of Optimal Operating Maps in Power-Split Operations  
For a given vehicle speed \(V_{veh}\) and subsequent output power \(P_{out}\), the powertrain operation can be analyzed by considering each couple of ICE speed and torque (\(\omega_{in}\), \(T_{in}\)) ranging from their minimum and maximum values within the ICE operation range of Figure 4. This leads to the calculation of an overall speed ratio matrix, having the same dimension of the ICE efficiency map, where each element is \(\tau=V_{veh}/(R_{w}\cdot\omega_{in})\). A corresponding overall power ratio matrix containing \(\eta=-P_{out}/(\omega_{in}\cdot T_{in})\) can be obtained, too. Hence, these matrices can be used to interpolate the dimensionless results of Figure 2, in \begin{table}
\begin{tabular}{c c c c c} \hline
**Unladen Mass** & **Rolling Resistance** & **Drag Coefficient** & **Frontal Area** & **Wheel Radius** \\ \(m_{0}\) [kg] & \(f_{r}\) & \(C_{d}\) & \(A_{f}\) [m\({}^{2}\)] & \(R_{w}\) [m] \\ \hline
1603 & 0.0113 & 0.28 & 2.20 & 0.323 \\ \hline \end{tabular}
\end{table} 
Table 4: Chevrolet Volt parameters. order to identify the speed and power ratios of the electric machines for each combination of \(\tau\) and \(\eta\). Then, these ratios can be multiplied by the corresponding \(\omega_{in}\) and \(P_{in}\) to assess the dimensional rotational speed of the electric machines (\(\omega_{i}\), \(\omega_{o}\)) and their mechanical power (\(\overline{P}_{i}\), \(\overline{P}_{o}\)). In this way, the operating point of both electric machines is univocally determined, as is their efficiency which enables the calculation of the net electric power flowing to or from the battery as follows:  
\[P_{batt}(V_{vch},\omega_{in},T_{in})=\overline{P}_{i}\eta_{I}^{-sign( \overline{P}_{i})}+\overline{P}_{o}\eta_{O}^{-sign(\overline{P}_{o})} \tag{8}\]  
The positive sign of the power flows is shown in Figure 1.  
Figure 4: Efficiency map of the Chevrolet Volt ICE derived from [37]. Figure 5: Efficiency maps of Chevrolet Volt electric machines derived from [38]; (**a**) Motor Generator I; (**b**) Motor Generator O. Exploited for both motoring and generating operation. Hence, by the simple procedure herein addressed, it is possible to obtain a set of matrices containing all data describing the powertrain steady-state operation, which can be exploited as a basis of the desired energy management strategy. As an example, in this paper, a procedure for pursuing maximum global efficiency is proposed. According to the direction of the battery power, \(P_{batt}\) can be an output or input power in the powertrain. Therefore, if \(P_{batt}>0\) the battery supports the ICE in the vehicle propulsion, and the global efficiency is:  
\[\eta_{gl}(V_{vch},\omega_{in},T_{in})=-\frac{P_{out}}{P_{fuel}+P_{ batt}} \tag{9}\]  
where \(P_{fuel}=P_{in}/\eta_{ICE}\) is the power provided by fuel combustion. \(\eta_{ICE}\) can be derived for each combination of \(\omega_{in}\) and \(T_{in}\) from Figure 4. If \(P_{batt}<0\) the ICE delivers power in surplus which can be used to recharge the battery, and the global efficiency is:  
\[\eta_{gl}(V_{vch},\omega_{in},T_{in})=-\frac{P_{out}+P_{ batt}}{P_{fuel}} \tag{10}\]  
Eventually, it is possible to extract the maximum value from the matrix \(\eta_{gl}\) for a given vehicle speed and find the related ICE and electric machines operation resulting in the most efficient driving. Nevertheless, the working points which violate a constructive constraint of propulsors, power converters, or batteries should not be included among the potential optimal ones. Therefore, the final operating maps do not include the functioning points whereby the ICE or electric machines operation is not included within the maps of Figures 4 and 5, or the electric powers overcome their respective maximum limits of power converters or batteries (Table 5).