Mixed Integer Programming Formulation for the Energy-Efficient Train Timetables Problem

Abstract

Railway traffic is the biggest individual electricity consumer in Germany, amounting to 2% of the country’s total electricity usage. However, up to 20% of the annual electricity cost depends on the highest power value drawn within the billing period. In this paper, we optimize the timetables of railway traffic in order to avoid high peaks in power consumption, while preserving at the same time some usability and safety considerations. We propose an exact mixed integer programming model together with a systematic way of simplifying the model in order to obtain feasible solutions that are not far from the optimum. We also discuss two possible dynamic programming approaches that may be used for solving small instances with a specific structure. Our approach became Team Optimixtli’s winning entry in the Discrete Optimization Challenge: Energy-Efficient Train Timetables. This competition was part of the Open Research Challenge 2015 organized by the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) in Germany.

Appendix

In what follows, we display power consumption in two figures for each instance: the first for the given schedule, the second for our best result (Figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24).

Fig. 5

The original input for instance 1

Fig. 6

The best output for instance 1

Fig. 7

The original input for instance 2

Fig. 8

The optimal output for instance 2

As before, a red dot means power consumption, a blue dot means power generation. However, to avoid clutter, each dot represents an average power consumption over one minute. The vertical scale accommodates the maximum instantaneous power consumption as given in the current schedule,² while the horizontal scale accommodates the 17 fifteen-minutes intervals. This scale holds for both figures.

We also display in light red the average power consumption (and in light blue the average unused power generation) on each fifteen-minutes interval. In this case, the vertical scale accommodates the maximum average power generation on a fifteen-minutes interval as given in the current schedule. This scale also holds for both figures.

Fig. 9

The original input for instance 3

Fig. 10

The optimal output for instance 3

Fig. 11

The original input for instance 4

Fig. 12

The optimal output for instance 4

Fig. 13

The original input for instance 5

Fig. 14

The optimal output for instance 5

Fig. 15

The original input for instance 6

Fig. 16

The optimal output for instance 6

Fig. 17

The original input for instance 7

Fig. 18

The best output for instance 7

Fig. 19

The original input for instance 8

Fig. 20

The optimal output for instance 8

Fig. 21

The original input for instance 9

Fig. 22

The best output for instance 9

Fig. 23

The original input for instance 10

Fig. 24

The best output for instance 10