Making electrical drives plannable



Many countries have committed to reducing their greenhouse gas emissions in the Paris Agreement on Climate Change. One element of this is the replacement of diesel buses with electric buses. That is why the use of e-buses in local public transport has increased steadily in recent years and will continue to do so. In addition, the use of e-buses reduces nitrogen oxide and fine dust emissions in city centers.


A single battery charge is not enough for the typical daily route of an electric bus. Therefore, the energy available in the battery must be taken into account when planning the bus rotations, so that it is guaranteed at all times that the battery is not completely discharged. The energy requirement of a vehicle does not only depend on the distance covered, but also e.g. also from the height profile of the route, the ambient temperature and the speed of the vehicle.

Also charging the battery is a complex process that depends on many factors and, above all, can only be carried out in locations with the appropriate infrastructure. Important factors here are the charging technology used, the charging current and the battery charging level. In addition, the capacity of a battery changes over time, as it decreases with the number of charging cycles performed.

A basic distinction is made between the following charging strategies:

  • Depot charging / overnight charging: the vehicles are charged during the shutdown in the depot,
  • in motion charging: the vehicles are charged while driving as a trolleybus,
  • Opportunity charging: the vehicles are charged during operation by fast chargers at bus stops or in parking facilities.
The aim of the optimization is to find cost-optimal bus rotations, analogous to VS-OPT, which are designed in such a way that the battery is not completely discharged by charging it throughout the day or by making the daily rotation so short that the battery does not run empty. During the charging processes, the capacities of the charging devices must be adhered to, since only a limited number of charging devices can be present or sometimes the total electric output of the charging devices is limited. If we only optimize electric busses, mainly costs are optimized, i.e. the number of buses and the length of the traveled route dominate the objective function. If diesel buses and electric buses are optimized together, the electric bus fleet can be specified to operate as many productive trips as possible.


Our mathematical model for electric bus optimization combines the multi-commodity flow model from VS-OPT for vehicle rotation optimization with a set partitioning model that contains one rotation for an electric bus per column. In the set partitioning model, the energy demand conditions for the electric buses are taken into account. The energy requirement per vehicle service element (i.e. part of a passenger or empty trip) is assumed to be constant. The calculation of this energy requirement and the charging curves are carried out by ebus Solutions GmbH, taking into account all relevant factors. The charging curves are piecewise linearly approximated for each charging point, i.e. for each location where charging is allowed.

The set partitioning model is built using a column generation method. The pricing problem is a shortest route problem with additional constraints for the energy consumption of the bus. With this model it is possible optimize the charging of the batteries in the depot as well as the (partial) charging of the batteries en route or on the route.

In the multi-commodity flow model, the fleet capacities are taken into account. These two problems are coupled analogously to the coupling of vehicle scheduling and duty scheduling at IS-OPT. This approach makes it possible to model the complexity of discharging and charging the batteries in detail and at the same time to optimize mixed fleets of practical size in an integrated manner.


ES-OPT is in use at BVG and is being further developed in cooperation with IVU and ebus Solutions GmbH. We examine research topics in connection with ES-OPT as part of the Modal research campus.

In addition, we support several transport companies in testing ES-OPT so that they can use their electric buses as efficiently as possible.