## 3 Materials and Methods
### Bidirectional DC-DC Converter  
In this section, a simple bidirectional DC-DC converter development for the energy storage systems was developed to provide the vehicle with the required voltage and current at the DC bus link. The DC-DC buck-boost converter equivalent circuit model was adopted from [39; 40]. The voltage assumed for LAB is 12.223 V and for LIB is 12.8 V. The alternator voltage in the ICEVs normally ranges from 12 V to 14.8 V, as described in [41]. Therefore, Figure 2 shows a simple bidirectional DC-DC converter developed. In Figure 2, during the DC-DC converter boost mode operation, the battery provides current to the DC-bus link and the battery voltage is stepped up to match the required DC-bus voltage of between 12 V and 14 V. The current is normally continuous and has a ripple that depends on the value of the inductance and switching frequency. When the S1 is ON (close), the S2 is OFF (close) because of the complementary mechanism between the signal which controls the two switches. The input voltage and inductor voltage are equal (i.e., \(V_{batt}=V_{L}\)) and can be expressed as in Equation (7) [37]:  
\[V_{Lmin}=V_{Batt}=L_{min}\bigg{(}\frac{di_{Lm}}{dt}\bigg{)}; \tag{7}\]  
where \(V_{Lmin}\) is the minimum inductor voltage, \(V_{Batt}\) stands for the battery voltage, and \(L_{min}\) is the minimum size of the inductor. Therefore, the minimum inductor current was determined as expressed in Equation (8) [37]:  
\[\Delta i_{Lmin(on)}=\frac{V_{L}}{L_{min}}.DT; \tag{8}\]  
where \(\Delta i_{Lmin(on)}\) is the change in minimum inductor current, \(V_{L}\) stands for the inductor voltage, \(DT\) is the duty cycle and the total sampling or operation period. Figure 2: A bidirectional DC–DC buck–boost converter equivalent circuit model. Thus, taking into consideration when S1 is OFF and switch S2 is ON, \(V_{Lm}\) for this state was determined as in Equations (9) and (10) [37]:  
\[V_{Lmin}=V_{batt}-V_{o}=L_{min}\frac{di_{Lm}}{dt}; \tag{9}\]  
\[\Delta i_{Lmin(off)}=\frac{V_{L}-V_{H}}{L}.(1-D)T; \tag{10}\]  
where \(V_{o}\) represent the output converter voltage, \(V_{L}\) and \(V_{H}\) is the low and high voltage of the inductor voltage. Hence, the converter parameters were determined by using Equations (11) and (12) as follows:  
\[D=1-\frac{V_{s}}{V_{o}}; \tag{11}\]  
where \(D\) is the duty cycle and ranges between 0-0.99, \(V_{s}\) representing the DC input converter voltage and \(V_{o}\) is the converter output voltage. Thus, the inductance and capacitance of the converter are as in Equations (12) and (13) [37]:  
\[L_{min}=\frac{D(1-D)^{2}\times V_{o}}{2f_{s}I_{o}}; \tag{12}\]  
\[C_{min}\geq\frac{I_{o}D}{f_{s}\times\Delta V_{o}}; \tag{13}\]  
where \(I_{o}\) is the CCC and \(f_{s}\) stands for the switching frequency and is considered as 25 kHz. While the \(C_{min}\) is the minimum capacitance, \(\Delta V_{o}\) is considered as 2% \(V_{o}\). Whereas during the buck mode operation, the battery receives charge from the alternator through the DC-bus link because the alternator produces more power than required and acts as a generator. The battery is charged and discharged as they prefer through the control. The converter steps down the alternator voltage from 14.5 V to the required battery charge voltage, which is 13.1 V for LAB and 14.2 V for LIB. Therefore, When S2 is ON, S1 is OFF during this mode of operation because of the complementary mechanism between the signals which control the two switches. Hence, the minimum inductor voltage was determined as in Equation (14) [37];  
\[ \begin{array}{c}V_{Lm}=V_{DC-Bus}-V_{Batt}\\ =L_{m}.\frac{di_{Lm}}{dt};\end{array}  \tag{14}\]  
where \(V_{DC-Bus}\) is the voltage at the DC-bus link.