Coasting advice based on the analytical solutions of the train motion model

Supervision, data analysis and communication algorithms monitor trains, exploiting most of their available computational power. On-board eco-driving algorithms such as Driver Advisory Systems (DAS) are no exception, as the computational power available limits their complexity and features. This was the case of Roltijd, the in-house developed DAS based on coasting advice of NS, the main Dutch passenger railway undertaking. This platform calculated the coasting curves at every second by integrating the equations of motion numerically, assuming that the track is flat. However, generating more complex driving advice required replacing this coasting curve calculation by a more computationally-efficient algorithm. In this article we propose a new coasting advice algorithm based on the analytical solutions of the train motion model, assuming that gradients and speed limits are piecewise constant functions of the train location. We analyse the qualitative properties of these solutions using bifurcation theory, showing that bifurcations arise depending on the value of the gradient and the applied tractive effort. We validate the proposed algorithm, finding that our algorithm is accurate and can be 15 times faster than the previous method. This allowed NS to implement our algorithm on their trains, contributing daily to the sustainable mobility of 1.3 million passengers.