## 5 ECMS with Steady-State Map  
Equivalent consumption minimization strategy was applied to the system as a supervisory controller which defines the optimum ratio of energy split between EM and ICE. This controller is a real-time controller not requiring preliminary knowledge about the driving cycle. At each time step, the ECMS decides the power split depending on the current situation. Hence, it gives a local minimum because it minimizes the objective function at  
Figure 4: Instant fuel consumption rate from the steady-state map, the experiment, and the model including TCF. Figure 5: Overall fuel consumption of steady-state, experiment, and the model predictions. each instant. The objective function \(J_{ccms}\) is the sum of the fuel consumption rate of ICE \(\dot{m}_{ice}\) and the equivalent consumption rate of EM \(\dot{m}_{eqv}\) as [15; 17; 18]:  
\[J_{ccms}=\dot{m}_{ice}(P_{ice})+m_{eqv}(P_{bat}) \tag{5}\]  
where, \(\dot{m}_{eqv}(P_{bat})\) can be derived by calculating engine fuel used to generate the same power provided by EM considering all the efficiencies of the power flow path. \[\dot{m}_{eqv}(P_{bat})=P_{em}(T_{cm},\omega_{em})\cdot s_{eqv}=T_{em}\cdot \omega_{em}\cdot s_{eqv} \tag{6}\]  
Equivalent factor \(s_{eqv}\):  
\[s_{eqv}=\left\{ \begin{array}{ll}\frac{1}{LHV\cdot\dot{m}_{ice}\cdot\dot{m}_{ inc}\cdot\dot{m}_{tot}}&(P_{bat}\geq 0)\\ \frac{1}{LHV\cdot\dot{m}_{ice}\cdot\dot{m}_{tot}}\cdot\dot{m}_{tot}\cdot\dot{m} _{tot}&(P_{bat}<0)\end{array} \right. \tag{7}\]  
where \(LHV\) is the lower heating value of the fuel, \(\dot{m}_{ice}\) is the average efficiency of the ICE, \(\dot{m}_{em}\) is the average efficiency of the EM, \(\dot{m}_{bat}\) is the average efficiency of battery and \(\dot{m}_{totv}\) is the average efficiency of the inverter. To reduce calculation time the equivalent factor \(s_{eqv}\) can be evaluated by using mean efficiencies. However, the fuel economy model can be improved even more by using specific efficiencies of each operation point. The fuel consumption rate of the engine \(\dot{m}_{ice}(P_{ice})\) is approximated as a polynomial with independent variables of engine torque and speed \(T_{ice\_m}\) and \(\omega_{ice\_m}\) from the engine steady-state map of Figure 6.  
The experimental data are approximated by a polynomial fit carried out by means of MATLAB curve fitting toolbox with a good quality \(R^{2}=0.9961\)[18]. To have a fair comparison of the fuel consumption of the HEV system the SOC, of the battery should maintain at the same level at the beginning and the end of the driving cycle. This is usually referred to as the charge sustaining operation mode of HEV. Therefore, the equivalent fuel consumption of the EM is penalized by an S-shaped correction function that varies with SOC. The expression suggested in [14] was utilized in the objective function as shown in Equations (9) and (10). The ECMS controller with the penalization factor and the operational constraints are described in detail in Equations (8) and (11) [18]. Figure 6: Curve fitting of the engine fuel consumption rate by engine torque and rotation speed [18]. The objective function in the ECMS controller is described as:  
\[J_{ecms}=\dot{m}_{ice}(P_{ice})+PF_{soc}\cdot\dot{m}_{eqp}(P_{bat}) \tag{8}\]  
where \(PF_{soc}\) is the penalization factor on SOC [17]:  
\[PF_{soc}=1-0.15\cdot\ \overline{SOC}^{3} \tag{9}\]  
And normalized battery \(\overline{SOC}\)[17]:  
\[\overline{SOC}=\frac{2\cdot SOC-\left(SOC_{high}+SOC_{low}\right)}{SOC_{high }-SOC_{low}} \tag{10}\]  
The torque split logic between the electric motor and the internal combustion engine must satisfy the following constraints:  
\[ \begin{array}{c}T_{gb}=(T_{em}\cdot U_{pullcy}+T_{ice})/i_{gb}\\ 0\leq T_{ice}\leq T_{ice,max}\\ -T_{em,max}\leq T_{em}\leq T_{em,max}\end{array}  \tag{11}\]  
The cost function in Equation (8) is two-variables (\(P_{em}\) and \(P_{ice}\)) nonlinear optimization problem. However, the problem can be reduced to a single variable type by means of the first constraint described in Equation (11).