*2.3. On-Board ESS*

The electric model of the ESS is reported in Figure 3, it includes the battery and supercapacitor pack, the DC/DC converter and the power flow controller. The on-board ESS is modelled as a pair of ideal current sources describing the battery-based and the SC-based energy storage system, respectively.

**Figure 3.** On-board Hybrid ESS electric model.

Figure 4a presents the equivalent circuit of the battery pack, which consists of an ideal voltage source that represents the open circuit voltage (OCV), which depends on battery state of charge (SOC); the series resistor *RINT* represents the internal resistance, whereas *rd* and *Cd* are the RC parallel circuit describing the charge transfer and double layer capacity, respectively. The set of equations that describes the electric model of the battery pack is reported in Equations (7) to (10): the first equation represents Kirchho ff's voltage law, the second one is the n-polynomial relationship between OCV and SOC. The third equation models the SOC update law according to the required current from the battery pack and the last one is the di fferential equation describing the RC parallel circuit [29].

$$V\_{BATT}(t) = OCV(t) - R\_{INT}I\_{BATT}(t) - \mu\_d(t) \tag{7}$$

$$\text{OCV}(\text{SOC}) = \beta\_n \text{SOC}^n + \beta\_{n-1} \text{SOC}^{n-1} + \beta\_0 \tag{8}$$

$$\text{SOC}(t) = \text{SOC}(t=0) - \frac{1}{3600 \cdot \text{C}\_{AH}} \int\_{0}^{t} I\_{\text{BAT}}(\tau) d\tau \tag{9}$$

$$r\mu\_d(t) + r\_d \mathbb{C}\_d \frac{d\mu\_d(t)}{dt} = r\_d I\_{BATT}(t) \tag{10}$$

where *ud(t)* is the *rdCd* parallel circuit voltage, β*0* ... β*n* are the interpolation coe fficients and CAH [Ah] is the battery pack capacity. The electrical model of the SC pack, shown in Figure 4b, consists of the capacitor *C,* modelling SC's capacity; an equivalent series resistance *RS* that describes the power loss during the charging and discharging operations; the self-charge resistance *RL* models the losses due to the leakage current, which is usually neglected [28,47]:

$$V\_{\rm SC}(t) = V\_{\rm C}(t) - R\_{\rm s}I\_{\rm SC}(t) \tag{11}$$

$$V\_{\mathcal{C}}(t) = V\_{\mathcal{SC}}(t=0) - \frac{1}{\mathcal{C}} \int\_{0}^{t} I\_{\mathcal{SC}}(\tau) d\tau \tag{12}$$

$$\text{SOC}\_{\text{SC}}(t) = \frac{1}{3} \left[ 4 \left( \frac{V\_{\text{C}}(t)}{V\_{\text{min}}} \right)^{2} - 1 \right] \tag{13}$$

**Figure 4.** Equivalent circuit of the battery pack (**a**) and SC pack (**b**).

It is highlighted that, in order to represent the equivalent circuit of a battery or supercapacitor pack, starting from the characteristics associated with the individual cells or modules, it is used an equivalent circuit: *n* RC blocks in series and *m* in parallel [48]. The DC/DC converters are modeled by its average efficiency η describing power losses. It operates as step-up or step-down converter according to the control characteristic. Given the power reference value *Pref* provided by the power flow control characteristic, the ESS current value to be delivered during the traction phase is computed by using the following equation:

$$I\_{ref} = \frac{1}{\eta} \frac{P\_{ref}}{V\_t} \tag{14}$$

where *Vt* is the battery pack or supercapacitor pack voltage.
