This article addresses a relocation and recharging problem faced by modern carsharing operators who manage a fleet of electric vehicles. As customers utilize the fleet, batteries are depleted and vehicles are possibly left in low-demand locations. Consequently, carsharing operators need to arrange the charging of depleted batteries and the relocation of poorly positioned vehicles in order to better meet the demand of the customers. Most of these activities require the intervention of dedicated staff. This article provides a framework for planning recharging and relocation activities based on periodically routing and scheduling a number of dedicated staff as a result of updated system information. The periodic planning problem is formulated as a Mixed Integer Linear Program and solved in a rolling-horizon fashion. For the solution of the problem a fast Adaptive Large Neighborhood Search heuristic is proposed. Tests based on data for the city of Oslo show that the heuristic can deliver, in reasonable computational time, high quality solutions for instances compatible with real-life planning problems.