## 4 Energy Management Strategies
### Equivalent Consumption Minimization Strategy (ECMS)  
ECMS is a control strategy to search for an optimal power split between the ICE and battery while satisfying several equality and inequality constraints and control objectives such as SoC maintenance and fuel consumption minimization. It works based on equivalence fuel consumption factors derived as a feedback of instantaneous SoC-based functions grounded on Hamiltonian optimal control theory and [PERSON]'s minimum principle [54; 55; 56; 57].  
In ECMS, the main objective is to minimize the overall fuel consumption rate, which comprises the ICE fuel mass flow rate \(\overset{\bullet}{m}_{ICE}\), plus the equivalent fuel rates of electricity \(\overset{\bullet}{m}_{equatt}\) being calculated by the equivalence factors.  
\[\overset{\bullet}{m}_{f}=\overset{\bullet}{m}_{ICE}(t)+\overset{\bullet}{m}_ {qubit}(t) \tag{14}\]  
For calculation of the equivalent fuel drawn from/supplied to the battery, the equivalence factors play an important role since they directly affect the power split and consequently the fuel consumption. Optimal power sharing relies on finding the factors leading to a minimized fuel consumption. The equivalence factor \(K_{euf}\) can be instantaneously used as a control feedback, which is a function of SoC for optimal power split and battery charge maintenance [24]. This can be expressed mathematically based on [21; 58] for the formulation of the ECMS-based strategy as follows:  
\[\overset{\bullet}{m}_{qubit}(t)=\frac{K_{euf}P_{\text{hut}}\eta}{Q_{LHV}}; \text{ while charging} \tag{15}\]  
\[\overset{\bullet}{m}_{equatt}(t)=\frac{K_{euf}P_{\text{hut}}}{Q_{LHV}\eta}; \text{ while discharging} \tag{16}\]  
\[K_{euf}=\mu P_{P}P_{I} \tag{17}\]  
\[x_{1}(t)=\frac{SoC(t)-\frac{SoC_{\text{max}}+SoC_{\text{min}}}{2}}{\frac{SoC_ {\text{max}}-SoC_{\text{min}}}{2}}=\frac{2 SoC(t)-(SoC_{\text{max}}+SoC_{\text {min}})}{SoC_{\text{max}}-SoC_{\text{min}}} \tag{18}\]  
\[P_{P}(SoC)=1-x_{1}{}^{n},\ n=2k+1,\ k\in N \tag{19}\]  
\[x_{2}(t)=0.01(\frac{SoC_{\text{max}}+SoC_{\text{min}}}{2}-SoC(t))+0.99x_{2}(t- \Delta t) \tag{20}\]  
\[P_{I}(x_{2})=1+\tanh(nx_{2}) \tag{21}\]  
where \(Q_{LHV}\) represents the fuel lower heating value and \(\eta\) is the total drivetrain chain efficiency of the downstream components included in the backward modeling. Acting as the equivalence coefficient factor, \(K_{euf}\) is the conversion weight of the electricity into fuel. \(P_{P}\) and \(P_{I}\) are the gain multipliers controlling the stiffness and deviation of \(SoC\) around the nominal average of minimum and maximum values. Along this line, the ECMS provides instantaneous power sharing between ICE and the battery considering predefined \(SoC_{\text{min}}\) and \(SoC_{\text{max}}\) values, current SoC values \(SoC(t)\), time step \(\Delta t\), and feedback gain multiplier \(\mu\) as the fine-tune decision variable. To this end, ECMS updates the instantaneous equivalence factors by way of counteracting SoC deviations to maintain the SoC proportionally when it is reaching its maximum/minimum allowable values, supporting discharging/charging the battery, respectively. Figure 7 illustrates a simplified flowchart of input and output for the ECMS states.