# Battery Sizing for Mild P2 HEVs Considering the Battery Pack Thermal Limitations
## 3 Development of Supervisory Control Strategy
### ECMS Control Strategy without Thermal Limitations  
To achieve an effective reduction in fuel usage, the contribution of electric motor power should be defined in such a way that the engine operates in its higher efficiency zone. For this reason, the ECMS was applied to the MHEV backward model to determine the power split ratio between the engine and the EM. From the configuration of the MHEV  
Figure 9: Voltage loses \(v_{loss}\) obtained for an applied dynamic current profile. (**a**) dynamic current profile. (**b**) voltage losses. system, the rotational speeds of the power sources are set by vehicle velocity. Hence, the ECMS controller splits only the torque between the engine and EM. Objective function:  
The objective function for the ECMS is overall equivalent fuel consumption (\(J_{ecms}\)) which is evaluated as a sum of engine fuel consumption \(\dot{m}_{ice}\) and equivalent fuel consumption of the electric power usage \(\dot{m}_{apv}\) for the power drawn from the battery \(P_{bat}\)[10; 12]:  
\[J_{ecms}=\dot{m}_{ice}(P_{ice})+\dot{m}_{eqv}(P_{bat}) \tag{15}\]  
\(\dot{m}_{eqv}(P_{bat})\) equivalent fuel consumption of EM can be computed as the fuel consumption of the ICE to provide the same mechanical power generated by electric motor [12]:  
\[\dot{m}_{apv}(P_{bat})=P_{em}(T_{em},\omega_{em})\cdot s_{eqv}=T_{em}\cdot \omega_{em}\cdot s_{eqv} \tag{16}\]  
\(s_{eqv}\) is the battery equivalent fuel consumption factor, depending on battery discharging (\(P_{bat}>0\)) or charging (\(P_{bat}<0\)) modes as:  
\[s_{eqv}=\left\{ \begin{array}{cc}\frac{1}{LHV\cdot\dot{m}_{ice}\cdot\dot{m} \cdot\dot{\eta}_{inv}\cdot\dot{\eta}_{bat}}&(P_{bat}>0)\\ 0&(P_{bat}=0)\\ \frac{1}{LHV\cdot\dot{m}_{ice}}\cdot\dot{\eta}_{em}\cdot\dot{\eta}_{inv}\cdot \dot{\eta}_{bat}&(P_{bat}<0)\end{array} \right. \tag{17}\]  
where \(LHV\) is the lower heating value of the fuel, \(\dot{\eta}_{ice}\) is the average efficiency of the ICE, \(\dot{\eta}_{em}\) is the average efficiency of the EM and \(\dot{\eta}_{inv}\) is the average efficiency of the inverter. The mathematical model for ICE fuel consumption rate \(\dot{m}_{ice}\) at a given operation point \((T_{ice},\ \omega_{ice})\) was obtained by steady state curve fitting of the engine map data (see Figure 10). The regression coefficients of the polynomial function (see Equation (18)) were estimated accurately since the goodness of this fitting indicated R-square = 0.9961. \[\dot{m}_{ice}(T_{ice},\ \omega_{ice})=p_{00}+p_{10}\cdot\omega_{ice}+p_{01 }\cdot T_{ice}+p_{20}\cdot\omega_{ice}^{2}+\ldots \tag{18}\] \[+p_{11}\cdot\omega_{ice}\cdot T_{ice}+p_{02}\cdot T_{ice}^{2}+p_{ 21}\cdot\omega_{e}^{2}\cdot T_{ice}+\ldots\] \[+p_{12}\cdot\omega_{e}\cdot T_{ice}^{2}+p_{03}\cdot T_{ice}^{3}\]  
Figure 10: Points of the fuel rate as a function of engine torque and speed. The surface represents the curve fitting with a polynomial equation. Once the objective function is expressed as a function of EM torque and engine torque (independent variables), then constraints were constituted based on operation limits and satisfying the required torque demand of the vehicle. Constraints:  
As shown in Equation (19), a necessary condition is to provide the required torque and request that the sum of torque contributions of two sources at the output of the gearbox be equal to torque demand at the same level. Moreover, the ICE and EM torques must be within the corresponding operating envelope:  
\[\left\{ \begin{array}{c}T_{gb}=\left(T_{em}\cdot U_{pulley}+T_{ice}\right)/i_ {gb}\\ 0\leq T_{ice}\leq T_{ice,max}\\ -T_{em.max}\leq T_{em}\leq T_{em.max}\end{array} \right. \tag{19}\]  
where, \(U_{pulley}\)--gear ratio on the pulley that connects EM to the input shaft of the gearbox and \(i_{gb}\)--gear ratio of engaged gears in the gearbox. By minimizing the objective function (Equation (15)) over the range described in the constraints (Equation (18)), the optimum value of \((T_{ice},\ T_{em})\) can be estimated instantly, without considering the SOC level.