2.3.2. Supercapacitor Model

The supercapacitor model was developed as an equivalent circuit model. The model takes the impact of temperature into account. The resistance and capacitor are the functions of the temperature and current, and described as

$$Re\_{\rm sc} = f\_{\rm Re,sc}(Tem\_{\rm sc}, I\_{\rm sc}), \tag{11}$$

$$\mathbb{C}\_{\text{sc}} = f\_{\mathbb{C}, \text{sc}}(Tem\_{\text{sc}}, I\_{\text{sc}}), \tag{12}$$

where *Re*sc is the resistance of the supercapacitor, *Tem*sc is the temperature, *I*sc is the current, *C*sc is the capacitor, *f*Re,sc represents the resistance function, and *f*C,sc represents the capacitor function.

**Figure 4.** Efficiency map of a (**a**) driving motor and (**b**) generator.

The parameters of the circuit part can be calculated as

$$I\_{\rm sc} = \frac{\mathcal{U}\_{\rm sc} - \sqrt{(\mathcal{U}\_{\rm sc}^2 - 4\mathcal{R}\varepsilon\_{\rm sc}P\_{\rm sc})}}{2\mathcal{R}\varepsilon\_{\rm sc}},\tag{13}$$

$$\mathcal{U}\_{\rm sc}(n+1) = \mathcal{U}\_{\rm sc}(n) - I\_{\rm sc}dt / \mathcal{C}\_{\rm sc} \tag{14}$$

$$SOC = \left( \mathcal{U}\_{\rm sc}(n+1) - \mathcal{U}\_{\rm sc,min} \right) / \left( \mathcal{U}\_{\rm sc,max} - \mathcal{U}\_{\rm sc,min} \right), \tag{15}$$

where *U*sc is the voltage, and *U*sc,min and *U*sc,max are the minimum and maximum voltage of the supercapacitor, respectively.

The temperature of the supercapacitor is calculated according to the power loss of resistance and coulombic efficiency and heat transfer process. The temperature can be shown as

$$Term\_{\rm sc} = Tem\_{\rm sc,ini} + \int\_0^t \frac{P\_{\rm sc,h} - P\_{\rm sc,a}}{m\_{\rm sc}c\_{\rm sc}} dt,\tag{16}$$

$$P\_{\rm sc,h} = \begin{cases} I\_{\rm sc}^2 Re\_{\rm sc}^{\rm dis} & I\_{\rm sc} \ge 0\\ I\_{\rm sc}^2 Re\_{\rm sc}^{\rm chg} - I\_{\rm sc} \mathcal{U}\_{\rm sc} (1 - \eta\_{\rm coul}) & I\_{\rm sc} < 0 \end{cases} \tag{17}$$

$$P\_{\text{sc.a}} = \left(Term\_{\text{sc}} - Tem\_{\text{a}}\right) / Re\_{\text{h}} \tag{18}$$

where *Tem*sc is the supercapacitor temperature; *Tem*sc,ini is the initial temperature of the supercapacitor; *Tem*<sup>a</sup> is the ambient air temperature; *P*sc,h is the heat power; *P*sc,a is the heat power transferred to air; *m*sc is the mass; *c*sc is the specific heat capacity; *Re*dis sc and *Re*chg sc are discharge and charge resistance, respectively; ηcoul is the coulombic efficiency; and *Re*<sup>h</sup> is the heat resistance.
