*3.4. Battery Degradation (V2G)*

Since V2G mode enables the vehicle to discharge, one of the aspects that must be taken into account is the battery degradation. This is an important issue for the user's profit. This paper deals with temperature effects and Depth Of Discharge (DOD), considering the scenarios analyzed by [20,21]. Thereby, the factor associated with temperature degradation is given by:

$$R\_{d\text{cg}} = \frac{r\_D E\_{bat\_v}}{L(t\_m)} \Delta t \tag{11}$$

Herein, *rD* is the Lithium battery price per kWh (in this paper, *rD* = 742.90 \$/kWh), and *L* is the battery lifetime in seconds [3].

In Equation (11), the battery lifetime *L*(*tm*) describes the charging power influence on temperature, as:

$$L(t\_m) = a \frac{b}{a} \frac{b}{T\_{amb} + 1000 R\_{th} P\_{EV\_{V2G}}^\*(t\_m)}\tag{12}$$

where *a* and *b* are constants obtained from curves that relate cycles/charging time versus 50% DOD, available in [20,21]. *Tamb* is the ambient temperature, and *Rth* is the thermal resistance. From this analysis, it is demonstrated that the higher the temperature applied on the battery, the lower is *L*.

The factor related to DOD degradation is considered only when discharging occurs, and it is given by:

$$E\_{\text{dig,soc}} = \varepsilon \left( \Delta SOC\_{\text{dod}} E\_{\text{flat}\_{V2G}} \right) \; , \tag{13}$$

wherein:

$$
\Delta \text{SOC}\_{dod} = \text{SOC}\_{\text{possible}} - \text{SOC}\_{EVV2G}(t\_m), \tag{14}
$$

The gain *ε* includes a relative DOD weight; Δ*SOCdod* is the difference between possible values of SOC tested in optimization (*SOCpossible*) and the calculated SOC (*SOCEVV*<sup>2</sup>*<sup>G</sup>* (*tm*)) according to the control law applied from the possible values of power (*Pevpossible*). The *SOCpossible* represents all quantized levels in the SOC state variable, while *Pevpossible* represents the quantized levels in the control law. Both are tested verifying all possibilities in a defined quantized space to achieve the minimum control law that results in the optimal state trajectory. The main idea of this factor is to balance the battery discharging, while minimizes the SOC difference.
