X. Luan, Y. Wang, B. De Schutter, L. Meng, G. Lodewijks, and F. Corman, "Integration of real-time traffic management and train control for rail networks - Part 2: Extensions towards energy-efficient train operations," Transportation Research Part B, vol. 115, pp. 72-94, Sept. 2018.
We study the integration of real-time traffic management and train control by using mixed-integer nonlinear programming (MINLP) and mixed-integer linear programming (MILP) approaches. In Part 1 of the paper, three integrated optimization problems, namely the PNLP problem (NLP: nonlinear programming), the PPWA problem (PWA: piecewise affine), and the PTSPO problem (TSPO: train speed profile option), have been developed for real-time traffic management that inherently include train control. A two-level approach and a custom-designed two-step approach have been proposed to solve these optimization problems. In Part 2 of the paper, aiming at energy-efficient train operation, we extend the three proposed optimization problems by introducing energy-related formulations. We first evaluate the energy consumption of a train motion. A set of nonlinear constraints is first proposed to calculate the energy consumption, which is further reformulated as a set of linear constraints for the PTSPO problem and approximated by using a piecewise constant function for the PNLP and PPWA problems. Moreover, we consider the option of regenerative braking and present linear formulations to calculate the utilization of the regenerative energy obtained through braking trains. We focus on two objectives, i.e., delay recovery and energy efficiency, through using a weighted-sum formulation and an ϵ-constraint formulation. With these energy-related extensions, the nature of the three optimization problems remains same to Part 1. In numerical experiments conducted based on the Dutch test case, we consider the PNLP approach and the PTSPO approach only and compare their performance with the inclusion of the energy-related aspects; the PPWA approach is neglected due to its bad performance, as evaluated in Part 1. According to the experimental results, the PTSPO approach still yields a better performance within the required computation time. The trade-off between train delay and energy consumption is investigated. The results show the possibility of reducing train delay and saving energy at the same time through managing train speed, by up to 4.0% and 5.6% respectively. In our case study, applying regenerative braking leads to a 22.9% reduction of the total energy consumption.