*2.3. Operational Model of EV*

For simplicity, a linear battery model can be applied to obtain the SOC behavior for each vehicle, which is given by:

$$SOC\_v[n+1] = SOC\_v[n] + \eta\_{EV} \frac{P\_{EV\_v}[n] \Delta t}{E\_{bat\_v}} \,\tag{1}$$

wherein *Ebatv* is the total battery capacity to each *v* vehicle, *ηEV* is the charging/discharging efficiency, and *SOCv* represents the resulting battery percentage one sample ahead [*n* + 1], based on charging power *PEVv* available on current sample *n*.

Fundamental restrictions are established for the *SOCv* in order to adjust the EV performance and limit the control behavior *PEVv* . Therefore, the SOC limits are given by:

$$0 \leqslant SOC\_p[n+1] \leqslant 1,\tag{2}$$

which restricts the control performance to:

$$P\_{\mathbb{E}V\_{v}} = \begin{cases} P\_{\mathbb{E}V\_{v'}} & \text{if } 0 \leqslant \text{SOC}\_{v}[n+1] \leqslant 1, \\ 0, & \text{otherwise.} \end{cases} \tag{3}$$

The EV connection status is defined as:

$$EV\_{\text{Com}} = \begin{cases} 1, \text{ if } EV\_{i,j}. \{ t\_0 \} \leqslant EV\_v(t\_m) \leqslant EV\_{i,j}. \{ t\_f \},\\ 1, \text{ if } SOC\_v[n+1] \leqslant 1, \\ 0, \text{ otherwise.} \end{cases} \tag{4}$$

wherein *EVi*,*j*.{*t*0} and *EVi*,*j*.{*tf* } are arguments of the EV acquisition structure and are related to initial and final charging instants. Still, *tm* indicates the current time inside the interval [*ti*, *tf* ]. Initially, *ti* is specified as *t*0.
