In recent years there has been a significant increase in the production of electric vehicles (EVs), in the global strive to reduce polluting gases produced by conventional fossil-fuel driven vehicles. Therefore, many optimization algorithms have been proposed for EV mobility and the charging of battery packs in the stations connected to power grids. However, there are situations in which experimental results are not sufficient, and simulations are needed. In this work, we address the effects of the charge demands of an EV fleet on the grid by considering the attitude of EV drivers, and especially their range anxiety. This influences their decision of when to recharge the battery pack. To this end, an agent-based model has been developed for the simulation of a power grid considering different scenarios based mainly on the state of charge (SOC) of battery packs at the time of the charging requests of EVs at service stations. The results indicate that in general a high battery SOC at the beginning of charging increases the probability of reaching higher power peaks on the grid.