More and more people in big cities choose urban rail transit as the main means of public transportation. With the increasing unbalanced passenger flow in time and space, the traditional operation mode with fixed train formation (or composition) is difficult to satisfy the varying passenger demands. This paper distinguishes different train formations in urban rail transit, and specifies the definition and the operation process for flexible composition of trains. An integrated train scheduling problem with flexible train composition is proposed, where the key constraints for practical train operation and the utilization of rolling stocks are considered. These constraints involve turnaround constraints, flexible train formation constraints, headway constraints and passenger flow constraints. The resulting problem is a mixed integer nonlinear programming problem, which can be transformed into a mixed integer linear programming problem and then be solved using existing optimization solvers, e.g., CPLEX. Based on the practical infrastructure and passenger demand data of the Beijing Daxing International Airport Express, a set of case studies is carried out to demonstrate the effectiveness of the presented model and solution approach. The computational results show that the train schedule with flexible train compositions can largely reduce the number of waiting passenger when compared with the train schedules with fixed train compositions and with multiple train compositions.@en

This paper introduces decomposition and distributed optimization approaches for the real-time railway traffic management problem considering microscopic infrastructure characteristics, aiming at an improved computational efficiency when tackling large-scale railway networks. Based on the nature of the railway traffic management problem, we consider three decomposition methods, namely a geography-based (GEO) decomposition, a train-based (TRA) decomposition, and a time-interval-based (TIN) decomposition, in order to partition the large railway traffic management optimization problem into several subproblems. In particular, an integer linear programming (ILP) model is developed to generate the optimal GEO solution, with the objectives of minimizing the number of interconnections among regions and of balancing the size of regions. The decomposition creates couplings among the subproblems, in terms of either capacity usage or transit time consistency; therefore the whole problem gets a non-separable structure. To handle the couplings, we introduce three distributed optimization approaches, namely an Alternating Direction Method of Multipliers (ADMM) algorithm, a priority-rule-based (PR) algorithm, and a Cooperative Distributed Robust Safe But Knowledgeable (CDRSBK) algorithm, which operate iteratively. We test all combinations of the three decomposition methods and the three distributed optimization algorithms on a large-scale railway network in the South-East of the Netherlands, in terms of feasibility, computational efficiency, and optimality. Overall the CDRSBK algorithm with the TRA decomposition performs best, where high-quality (optimal or near-optimal) solutions can be found within 10 s of computation time.

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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.

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Most of previous studies optimize maintenance time window scheduling problem under a given train schedule, leading to a relatively poor quality of maintenance time window schedule, increasing the influence on traffic assignment. In order to reduce the negative effects on maintenance schedule and improve the utilization of railway resources, we consider integrating maintenance time window scheduling and train timetabling. In this way, more reasonable maintenance time window schedule can be obtained. We propose a mixed integer programming model and in particular we focus on the characteristics of the problem, including the speed limits affected by maintenance tasks on a double-track railway line. The benefits of the proposed integrated optimization model are demonstrated by numerical experiments.

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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.

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Train dispatching is vital for the punctuality of train services, which is critical for a train operating company (TOC) to maintain its competitiveness. Due to the introduction of competition in the railway transport market, the issue of discrimination is attracting more and more attention. This paper focuses on delivering non-discriminatory train dispatching solutions while multiple TOCs are competing in a rail transport market, and investigating impacting factors of the inequity of train dispatching solutions. A mixed integer linear programming (MILP) model is first proposed, in which the inequity of competitors (i.e., trains and TOCs) is formalized by a set of constraints. In order to provide a more flexible framework, a model is further reformulated where the inequity of competitors is formalized as the maximum individual deviation of competitors’ delay cost from average delay cost in the objective function. Complex infrastructure capacity constraints are considered and modelled through a big M-based approach. The proposed models are solved by a standard MILP solver. A set of comprehensive experiments is conducted on a real-world dataset adapted from the Dutch railway network to test the efficiency, effectiveness, and applicability of the proposed models, as well as determine the trade-off between train delays and delay equity.@en

We address the problem of simultaneously scheduling trains and planning preventive maintenance time slots (PMTSs) on a general railway network. Based on network cumulative flow variables, a novel integrated mixed-integer linear programming (MILP) model is proposed to simultaneously optimize train routes, orders and passing times at each station, as well as work-time of preventive maintenance tasks (PMTSs). In order to provide an easy decomposition mechanism, the limited capacity of complex tracks is modelled as side constraints and a PMTS is modelled as a virtual train. A Lagrangian relaxation solution framework is proposed, in which the difficult track capacity constraints are relaxed, to decompose the original complex integrated train scheduling and PMTSs planning problem into a sequence of single train-based sub-problems. For each sub-problem, a standard label correcting algorithm is employed for finding the time-dependent least cost path on a time-space network. The resulting dual solutions can be transformed to feasible solutions through priority rules. Numerical experiments are conducted on a small artificial network and a real-world network adapted from a Chinese railway network, to evaluate the effectiveness and computational efficiency of the integrated optimization model and the proposed Lagrangian relaxation solution framework. The benefits of simultaneously scheduling trains and planning PMTSs are demonstrated, compared with a commonly-used sequential scheduling method.

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This work addresses a tactical railway traffic management problem focused on the opti-mization of train dispatching decisions and timing decisions related to short-term main-tenance works in a railway network subject to disturbed process times. This is modeled as a mixed-integer programming formulation in which the traffic flow and track mainten-ance variables, constraints and objectives are integrated under a stochastic environment. The resulting bi-objective optimization problem is to minimize the deviation from a sche-duled plan and to maximize the number of aggregated maintenance works. The two ob-jectives require to schedule competitive train operations versus maintenance works on the same infrastructure elements. Numerical experiments are performed on a realistic railway network. We measure the quality of the integrated solutions in terms of their robustness to random perturbations of the train travel times and of the maintenance works. Pareto op-timal methods are compared for the bi-objective problem. We also evaluate the impact of introducing routing stability constraints in order to force the trains to keep the same route among the different stochastic disturbed scenarios. The experiments show that forcing the routing stability reduces the routing flexibility and the ability to optimize the two perfor-mance indicators when dealing with stochastic disturbances of process times.@en