In recent decades, scientific interest has grown in tackling the vehicle routing problem with a sustainable approach (Green VRP). There are numerous studies in the literature addressing environmental problems from the point of view of efficient planning that allows visualizing the benefits associated with the use of the new technologies in electric vehicles. This paper focuses on the electric vehicle routing problem and considers the batteries’ state of charge (SoC). The problem considers a set of customers, where each one has a specific demand and a time window. Deliveries are performed through a homogeneous fleet of electric vehicles with a fixed charging capacity and limited autonomy. In particular, when the vehicle is traveling, it consumes an amount of energy proportional to the distance it travels; therefore, it must visit battery recharging stations to continue and complete its route. The objective is to determine the performed routes with the minimum cost (time), while seeking to visit the recharging stations as many times as possible. In this way, overcharging and deep discharges are avoided by protecting the battery from degradation. In this paper, four models are proposed: the first model requires that the battery be fully charged in the stations; the second model allows partial recharging; the third formulation limits deep discharge; and a fourth formulation adheres to a limitation associated with overcharging and tries to keep the battery in its most comfortable place. The efficiency of the proposed formulations is tested in structured instances of different sizes. The results obtained show the efficiency of the formulations proposed for the electric vehicle routing problem when considering battery degradation.

This paper addresses the electric vehicle routing problem with time windows (E-VRPTW), considering the battery’s state of charge ( $$SoC$$SoC ) and the recharging process's linearity and non-linearity. We compare two proposed models: the first assumes a linear charging process, and the second evaluates the impact generated by including the non-linearity of the battery recharging process. The non-linear model considers the limitation of the state of charge and restricts the deep battery discharge during movement. Additionally, the effect of overload on the supplied energy process has been evaluated to extend the batteries' useful life. The models are tested on instances commonly used in the literature. The obtained results verify that including the non-linearity recharging process reduces the total time of the routes. Indeed, by accessing the upper sections of the recharge curve ( $$> 85\% \; SoC$$> 85 % S o C , the more significant degradation), autonomy is obtained to avoid unnecessary visits to stations. In addition, including the option to carry out a fast recharge could reduce the total time, even reducing the number of vehicles necessary to carry out the delivery tasks and the maximum time defined by each route.