*3.2. BES Converter and Its Mathematical Modeling*

In this study, the BES uses a bi-directional converter topology as shown in Figure 7. The control system determines the duty cycle of the gating signal for the switches. Assume *d*b(*t*) is the duty cycle of PWM of the BES converter and *v*b(*t*) is the BES voltage, then the average converter voltage is *<sup>v</sup>*bo(*t*) =*d*b(*t*)*v*b(*t*). If *<sup>u</sup>*b(*t*) =*v*bo(*t*) is the control input, then *<sup>d</sup>*b(*t*) = *<sup>u</sup>*b(*t*) *<sup>v</sup>*b(*t*) and

$$\frac{d\dot{u}\_{\rm bo}(t)}{dt} = -\frac{R\_{\rm b}}{L\_{\rm b}}\dot{u}\_{\rm bo}(t) + \frac{1}{L\_{\rm b}}u\_{\rm b}(t) - \frac{V\_{\rm bus}}{L\_{\rm b}},\tag{1}$$

where *i*bo(*t*), *R*b, *L*b, and *V*bus are the BES converter output current, filter resistance, filter inductance, and the voltage at the point of connection of the BES converter, respectively. In terms of power, (1) can be rewritten as

$$\frac{dP\_{\rm bo}(t)}{dt} = -\frac{R\_{\rm b}}{L\_{\rm b}}P\_{\rm bo}(t) + \frac{V\_{\rm bus}}{L\_{\rm b}}u\_{\rm b}(t) - \frac{V\_{\rm bus}^2}{L\_{\rm b}},\tag{2}$$

where *P*bo(*t*) is the output power from the BES system. The dynamics of the SOC depends on the amount of current flowing in/out of the battery and it is mathematically expressed as

$$\frac{d}{dt}\text{SOC}(t) = -\frac{P\_{\text{bo}}}{Qv\_{\text{b}}},\tag{3}$$

where *Q* and *v*<sup>b</sup> are the battery capacity (in As) and battery voltage (in V), respectively.

**Figure 7.** BES converter system.
