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 (Luan et al., 2018), three integrated optimization problems, namely the P_NLP problem (NLP: nonlinear programming), the P_PWA problem (PWA: piecewise affine), and the P_TSPO 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 P_TSPO problem and approximated by using a piecewise constant function for the P_NLP and P_PWA 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 P_NLP approach and the P_TSPO approach only and compare their performance with the inclusion of the energy-related aspects; the P_PWA approach is neglected due to its bad performance, as evaluated in Part 1. According to the experimental results, the P_TSPO 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.

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. Three innovative integrated optimization approaches for real-time traffic management that inherently include train control are developed to deliver both a train dispatching solution (including train routes, orders, departure and arrival times at passing stations) and a train control solution (i.e., train speed trajectories). Train speed is considered variable, and the blocking time of a train on a block section dynamically depends on its real speed. To formulate the integrated problem, we first propose an MINLP problem (P_NLP), which is solved by a two-level approach. This MINLP problem is then reformulated by approximating the nonlinear terms with piecewise affine functions, resulting in an MILP problem (P_PWA). Moreover, we consider a preprocessing method to generate the possible speed profile options for each train on each block section, one of which is further selected by a proposed MILP problem (P_TSPO) with respect to safety, capacity, and speed consistency constraints. This problem is solved by means of a custom-designed two-step approach, in order to speed up the solving procedure. Numerical experiments are conducted using data from the Dutch railway network to comparatively evaluate the effectiveness and efficiency of the three proposed approaches with heterogeneous traffic. According to the experimental results, the MILP approach (P_TSPO) yields the best overall performance within the required computation time. The experimental results demonstrate the benefits of the integration, i.e., train delays can be reduced by managing train speed.