Decompositions of the optimal dispatching problem of electric and electric-hybrid buses with energy constraints for Luxembourg City

We are a team of engineers working on a concrete project of Mobility in Luxembourg. We want to solve the problem of optimally determining the sequence of electric and hybrid electric buses, considering both service constraints (schedule adherence) and energy constraints (electric bus charging status, bus recharging scheduling in capacitated facilities) and at the same time ensure a high level of quality of service for the user satisfaction. The problem is formulated as a Mixed Integer Linear Program, with the objective of minimizing the total operational cost for the bus lines in question. System dynamics are captured by twenty sets of constraints, ranging from scheduling adherence to discharge-recharge dynamics. Individual operational costs at the bus level (cost of running an electric / non-electric bus per km, cost of recharging) and at the trip level (penalty due to failed schedule adherence) are fully parametrised, allowing for extensive sensitivity analysis. We investigate a real-life case study based in the city of Luxembourg, where the objective is to reach the all-electric mode for principal urban buses network. Through the model we investigate: the minimum amount of electric buses necessary to perform a day’s schedule for two currently partially electrified lines, without resorting to conventional internal combustion alternatives; the impact of electrifying two additional lines, specifically considering the trade-offs related to either adding new buses or new charging stations at the bus terminal. Finally, we studied how to best decompose the overall problem in several smaller problems, to be able to solve also realistic scenarios and using large real data sets from the Mobility Data owner of Luxembourg. We analysed and compared two kinds of decomposition: a bus line-based decomposition, and a time-based decomposition.eCoBu