# Energy Management Optimization of a Dual Motor Lithium Ion Capacitors-Based Hybrid Super Sport Car
## 3 Hybrid Control Strategies Design
### Equivalent Consumption Minimization Strategy
#### 3.2.4 Battery Energy Cost Function (\(s\))  
The \(s\) factor indicates the cost of the electrical energy; it is dependent on the system's working conditions and can be formulated as follows [8,9]:  
\[s=\left\{1-k_{p}\cdot\left[\frac{SoC-\left(\frac{SoC_{max}+SoC_{min}}{2} \right)}{\left(\frac{SoC_{max}-SoC_{min}}{2}\right)}\right]^{3}\right\}\cdot \left[k_{a}\cdot\left(SoC_{target}-SoC\right)\right], \tag{22}\]  
The cost function was calculated at every integration step, and when its value is high, it makes it preferable to use the engine and recharge the battery, while if it is low, it makes the electric traction preferable.  
The curly brackets contain a penalty term that modifies the \(s\) value when the SoC is near to the maximum or minimum acceptable values, making the electrical energy cost, respectively, lower or higher. On the other hand, the second term is a proportional correction obtained considering the difference between the target value of the SoC, and the actual one.  
The parameters \(k_{p}\) and \(k_{a}\) are calibrated. Their value choice is decisive for the simulation results, as it directly impacts the cost function and consequently, the hybrid performance and the fuel consumption.  
For this application, the electrical energy is low-cost since the LiC energy storage is a high-power system, and it can be charged and discharged rapidly. The values of the calibrated parameters are chosen accordingly.  
The SoC target can be chosen depending on the hybrid vehicle mission and the technology that is used. The choice to work with a constant value is the simplest one, and it makes the strategy a Charge Sustaining (CS) one.  
Other possibilities that are commonly used for Li-ion Batteries-based vehicles are represented by the Charge-Depleting/Charge Sustaining (CD/CS), which firstly discharges the battery until a certain SoC value and then keeps the value around a target SoC, or the Charge-Blended (CB), which follows a SoC target that linearly decreases with the driven distance [8].  
The present work is based on a LiC-based super sport car, and an alternative SoC target is formulated similarly as it has been done in [1]. According to the previous work, the SoC target is a speed-dependent quantity related to the detachment speed value of the EM (in this application it has been set equal to 190 km/h, the detachment speed of the front EM).  
As explained in [1], the SoC target was high at low speeds, as the kinetic energy was low, and this solution guaranteed a high energy quantity stored in the electrical system. On the other side, the SoC target was low at high speeds since energy was already existing in the form of kinetic energy that could be rapidly recovered through the EM. Moreover, if we were to have high electrical energy at high speed, the energy content would end up unused once the detachment speed was reached.  
Differently from [1], in this study, the dependency of the SoC on speed was modeled as linear to better fit the complete SoC range.  
This alternative SoC target formulation, represented in Figure 11 for a WLTP 3b cycle, was thought to fit better the behavior of LiC that dispose of a high charge and discharge rate and which can perform a high number of cycles [22; 23].  
This kind of application aims to avoid unused energy and to maintain high energy content where it is mostly needed, ensuring the satisfaction of performance or drivability requests, important objectives for a super sport car. Different control functions could be implemented to uniquely guarantee improvements in performance and drivability, especially at low speeds.