The Green Vehicle Routing Problem with Capacitated Alternative Fuel Stations assumes that, at each station, the number of vehicles simultaneously refueling cannot exceed the number of available pumps. The state-of-the-art solution method, based on the generation of all feasible non-dominated paths, performs well only with up to 2 pumps. In fact, it needs cloning the paths between every pair of pumps. To overcome this issue, in this paper, we propose new path-based MILP models without cloning paths, for both the scenario with private stations (i.e., owned by the fleet manager) and that with public stations. Then, a more efficient cutting plane approach is designed for addressing both the scenarios. Numerical results, obtained considering a set of benchmark instances ad hoc generated for this work, show both the efficiency and the effectiveness of this new cutting plane approach proposed. Finally, a sensitivity analysis, carried out by varying the number of customers to be served and their distribution, shows very good performances of the proposed approach.