# Energy Management Optimization of a Dual Motor Lithium Ion Capacitors-Based Hybrid Super Sport Car
## 3 Hybrid Control Strategies Design
### Equivalent Consumption Minimization Strategy
#### 3.2.3 Equivalent Fuel Consumption  
The ECMS aimed to identify the best power flow distribution between the energy converters at every integration step, such that the optimality criterion that has been chosen is achieved.  
At first, a global cost function was defined, that considered the usage of both the ICE and the EMs to power the vehicle. Their contribution was evaluated through the calculation of the equivalent consumption value, at every integration step, defined as follows:  
\[\dot{m}_{eq}=\dot{m}_{f}+\dot{m}_{bat} \tag{12}\]  
where \(\dot{m}_{eq}\) [g/s] is the equivalent fuel consumption flow, \(\dot{m}_{f}\) [g/s] is the fuel consumption and \(\dot{m}_{bat}\) [g/s] is the electrical energy equivalent consumption.  
\(\dot{m}_{f}\) can be determined through interpolation of FC maps during the simulation of the longitudinal dynamics model, while the \(\dot{m}_{bat}\) value will be calculated based on the electrical power request and to the cost function, which depends on the equivalent cost and on the system's working point.  
The Lower Heating Value (\(LHV\)) [J/g] will divide the power request to the battery to convert, using the equivalent factor \(s\), electrical energy into virtual fuel consumption.  
\[\dot{m}_{bat}=s\cdot\frac{P_{bat}}{LHV}, \tag{13}\]  
Consequently,  
\[\dot{m}_{bat}=\frac{s}{LHV}\cdot\sum_{i}P_{EM,\ i}\cdot(\gamma_{i}\cdot\frac{1 }{\eta_{EM,\ i}}+(1-\gamma_{i})\cdot\eta_{EM,\ i}), \tag{14}\]  
where \(\eta_{EM}\) is the efficiency of the EM, \(P_{EM}\) is the power request to the EM and \(\gamma\) is the factor that allows to properly evaluate if the EM is working as a generator (\(P_{EM}<0\)) or as an electric motor (\(P_{EM}>0\)), and it is defined as follows:  
\[\gamma=\frac{1+sign(P_{EM})}{2}, \tag{15}\]  
This formulation can be easily implemented into the longitudinal dynamics model.  
\begin{table}
\begin{tabular}{c c} \hline
**Split Factor** _hy_ & **Electric Traction Mode** \\ \hline \(hy=1\) & Front Traction \\ \(0<hy<1\) & Torque Split \\ \(hy=0\) & Rear Traction \\ \hline \end{tabular}
\end{table}
Table 3: Split factor _hy_ complete working range, it will be further limited due to the system’s limitations.  
The minimization of the equivalent fuel consumption \(\dot{m}_{eq}\) brings along the definition of the power flow distribution between ICE and EM, as the \(T_{ICE}\) and \(T_{HY}\) pair is defined.  
The general formulation of the minimization problem refers to any kind of energy storage system and can be found in [8,21]. Defining \(\xi\) as the SoC, \(u\) the control variable, \(Q_{bat}\) the battery charge capacity, \(l_{bat}\) the battery current:  
\[\dot{\xi}(t)=f(\xi,\ u,\ hy,\ t)=-\frac{l_{bat}(\xi,\ u,\ hy,\ t)}{Q_{bat}}, \tag{16}\]  
It is possible to define the Hamiltonian of the optimal control problem:  
\[H(\xi,\ u,\ hy,\ \lambda,\ t)=-\lambda(t)\cdot f(\xi,\ u,\ hy,\ t)+\dot{m}_{f}(u,\ t), \tag{17}\]  
\[\dot{\lambda}(t)=-\lambda(t)\frac{\partial f(\xi,u,hy,t)}{\partial\xi}, \tag{18}\]  
Thus, the Hamiltonian is the total equivalent fuel consumption. Introducing \(s(t)\):  
\[s(t)=-\lambda(t)\frac{LHV}{P_{bat}(t)}, \tag{19}\]  
At the end:  
\[H(\xi,u,\lambda,hy,t)=\dot{m}_{eq}(\xi,u,hy,s,t)=s(t)\cdot\frac{P_{bat}(t)}{ LHV}\cdot f(\xi,u,hy,t)\ldots+\dot{m}_{f}(u,t), \tag{20}\]  
The optimal control satisfies:  
\[u^{*}(t)=\underset{u}{\text{argmin}}H(\xi,u,\lambda,t), \tag{21}\]  
The optimal control depends on \(s(t)\), but its value is unknown a priori, thus the strategy is sub-optimal.