# Energy Management Optimization of a Dual Motor Lithium Ion Capacitors-Based Hybrid Super Sport Car
## 2 Materials and Methods
### Hybrid Powertrain
#### 2.2.3 Lithium-Ion Capacitors  
The usage of LiCs as the main and only energy storage system was innovative. LiCs were derived from Electric Double Layer Capacitors (EDLC) [6], and they combined the activated Carbon cathode of an EDLC with the Li-doped Carbon anode of Lithium-Ion Batteries to ensure great power and good energy content.  
They were typically adopted for a few low energy demanding operations in automotive applications [3; 5; 6; 7].  
On the other side, Li-Ion Batteries were more commonly installed in hybrid electric vehicles and electric vehicles due to the high energy content that they can provide [12]. Typically, common batteries guarantee high energy values (meaning a high electric range in automotive applications) but cannot guarantee high power performance [2; 13].  
In Figure 4, a first approach to the description of the LiC behavior is shown. The scheme was simple, representing only a series resistance \(R_{s}\) and a capacitance \(C\)[2]. The leakage resistance \(R_{L}\) allowed to describe a more detailed model but could be omitted without losing too much in accuracy.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Configuration** & **Transmission Ratio** \\ \hline Front P4 & 4.8 \\ Rear P3 & 3.2 \\ Rear P2 & 2.84 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Transmission ratios for electric motor (EM) connections.  
Figure 4: Lithium-Ion Capacitor scheme.  
Defining \(Q_{SC}\) the charge stored within the LiC, and \(C\) the capacitance, the voltage \(V_{c}\) can be calculated as follows:  
\[V_{c}=\frac{Q_{SC}}{C}, \tag{3}\]  
According to [PERSON]'s rule, the terminal power is given by:  
\[P=i\cdot\left(R_{s}\cdot i+\frac{Q_{SC}}{C}\right), \tag{4}\]  
Thus, the current \(i\) can be determined as [14]:  
\[i=\frac{-\frac{Q_{SC}}{C}+\sqrt{\left(\frac{Q_{SC}}{C}\right)^{2}+4\cdot P \cdot R_{s}}}{2\cdot R_{s}}, \tag{5}\]  
Figure 5 shows the model of LiC that has been integrated in the vehicle model, where the power [W] is the input and the terminal voltage [V] and the current flow [A] are the outputs.  
As explained in [1; 15], the LiC model could be more complicated and detailed, reproducing a wider range of phenomena, but that level of detail was not strictly needed for automotive control-oriented applications. Moreover, according to previous works [1], the analysis of industrial capacitors from different suppliers showed that the choice to work with a single RC circuit branch was justified.  
A validation based on experimental tests has been run, setting the required power as input (Figure 6). The simulated terminal voltage was compared with the experimentally measured one in Figure 7.  
Figure 5: Lithium-Ion Capacitor model.  
Figure 6: Power input normalized to its maximum value.  
The voltage root mean square error was equal to 0.2 V, and the model can thus be considered reliable.  
Once the model was validated, the LiC configuration was chosen. The system will be a 60s1p (60 series cells and 1 parallel string) that stores 0.26 kWh and could reach over 130 kW of power, satisfying the power request of the EMs. The working voltage spans from 132 V to 228 V.