## 2 Multimode Hybrid Powertrain Model
### Power-Weighted Efficiency  
The main task for the design of a hybrid powertrain is to search for the optimal torque distribution that satisfies a desired target. The approach adopted within this paper is to solve an energy loss minimization problem through the criterion of the power-weighted efficiency proposed by [PERSON] et al. [15]. The power-weighted efficiency \(\eta_{PW}\) is evaluated as (see also [36]):  
\[\eta_{PW}=\left\{\begin{array}{cc}\frac{P_{out,tr}P_{out,tr}}{W_{c,max}V_{c, max}}+\frac{P_{on}C_{MM}}{W_{c,max}V_{c,max}V_{c,max}V_{c,max}V_{c,max}}+\frac{P_{d}}{W_{ c,max}}+\frac{P_{d}}{W_{c,max}}+\frac{P_{d}T_{init}T_{MM}}{W_{c,max}}\\ \frac{P_{tot}+T_{init}}{W_{c,max}}\\ \frac{P_{tot}+T_{init}}{W_{c,max}V_{c,max}}\\ \frac{P_{tot}+P_{d}T_{init}}{W_{c,max}V_{c,max}}\\ \frac{P_{tot}+P_{d}T_{init}}{W_{c,max}V_{c,max}}\\ \frac{P_{tot}+P_{d}T_{init}-P_{d}T_{d}-P_{d}S_{init}T_{MM}}\end{array}\right. \quad for\ P_{out,tr}<0 \tag{5}\]  
where \(P_{out,tr}=T_{out,tr}\omega_{out}\) represents the output power of the transmission with a transmission output torque \(T_{out,tr}=T_{load}+(I_{out}+I_{c1}+I_{c2})\dot{\omega}_{out}\). In the definition of \(\eta_{PW}\), two energy sources are considered for satisfying the demanded power at the transmission output shaft: the battery and the fuel tank. The transmission efficiency is considered constant and equal to 1 within the paper, but its influence can be introduced to enhance the model as described by [PERSON] [39].  
In the generic scheme of Figure 3, \(P_{load}=T_{load}\omega_{out}\) is the loading power (positive or negative) meanwhile the engine output power \(P_{e}=T_{e}\omega_{e}\) can be split into three contributions: the engine power \(P_{e1}\) that flows to the battery through the electric generators, the engine power \(P_{e2}\) that flows to the electric motors through the generators and the engine power \(P_{e3}\) that directly flows to the transmission output shaft (\(P_{e3}=P_{e}-P_{e1}-P_{e2}\)). A correct elaboration of engine power flow among \(P_{e1}\), \(P_{e2}\), and \(P_{e3}\) requires the calculation of the electric power produced by the generators \(P_{G}\), and the electric power absorbed by the motors \(P_{M}\), as described in Figure 4.  
If \(P_{G}>P_{M}\), the additional generated electric power is used to charge the battery: \(P_{e3}=P_{e}-\frac{P_{G}}{P_{G}}\), \(P_{e2}\eta_{G}=P_{M}\), \(P_{e1}\eta_{G}=P_{G}-P_{e2}\eta_{G}\), and \(P_{batt}=-P_{e1}\eta_{G}\eta_{batt}\). Otherwise, if \(P_{G}<P_{M}\), the battery energy is consumed: \(P_{batt}=(P_{M}-P_{G})/\eta_{batt}\), \(P_{e2}\eta_{G}=P_{G}\), \(P_{e1}\eta_{G}=0\), and \(P_{e3}=P_{e}-\frac{P_{G}}{\eta_{G}}\). \(P_{batt}\) and \(P_{fud}\) represent the absorbed battery and fuel power, respectively. The battery efficiency \(\eta_{batt}\) is evaluated by considering a simple circuit model (see the model described by [PERSON] et al. [17]):  
\[\eta_{batt}=1-\frac{R_{batt}I_{batt}^{2}}{V_{oc}I_{batt}} \tag{6}\]  
where \(I_{batt}\) is the current through the battery circuit:  
\[I_{batt}=\frac{V_{oc}-\sqrt{V_{OC}^{2}-4R_{batt}|P_{M}-P_{G}|}}{2R_{batt}} \tag{7}\]  
The resistance of the battery circuit \(R_{batt}\) and the battery open circuit voltage \(V_{oc}\) are considered constant quantities with the hypothesis of a constant State of Charge  
Figure 4: Absorbed and generated electric power \(P_{M}\) and \(P_{G}\) together with their correspondent efficiencies \(\eta_{G}\) and \(\eta_{M}\).  
Figure 3: Generic scheme of the power flow for a hybrid powertrain.  
(\(\text{SOC}=70\%\)) thus mostly relying on the electric energy stored in the battery (charge depleting).  
The power-weighted efficiency allows to normalize the efficiencies from different type of power sources, without penalizing the engine operation even if the maximum ICE efficiency \(\eta_{e,max}\) is much lower than \(\eta_{G,max}\), and \(\eta_{M,max}\).