Real-time optimization strategy for single-track high-speed train rescheduling with disturbance uncertainties: A scenario-based chance-constrained model predictive control approach

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Abstract

To improve the operational efficiency of high-speed railway system with disturbance uncertainties, a real-time optimization rescheduling strategy is designed based on the updated information for single-track high-speed railway system in this paper. Based on the characteristics of high-speed railway lines, a mixed-integer linear optimization model is constructed, where the decision variables involve the arrival times, departure times, arrival orders, departure orders and dwelling plans. Furthermore, to satisfy real-time requirements and to enhance the robustness of solutions, a scenario-based chance-constrained model predictive control (SC-MPC) algorithm is designed for solving the train rescheduling problem. Under the designed algorithm, the original linear model is converted to a non-linear mixed-integer model. To reduce the computational burden, the nonlinear model is converted to a linear mixed-integer model by a linearization method. The proposed strategy is compared with several typical benchmark strategies via a case study on the Beijing-Shanghai high-speed railway line. The simulation results show that the train delays can be effectively reduced by the proposed strategy and the rescheduling timetable has a good robustness.

Introduction

High-speed railways offer a fast and comfortable transportation mode that enhances the efficiency of railway traffic and serve an increasing level of passenger flows. High-speed railways are under construction in many countries in recent years. At the same time, the punctuality and reliability of high-speed railways have become increasingly prominent. However, the operation of high-speed trains is inevitably affected by various perturbations, e.g., bad weather, maintenance operations, infrastructure failure and man-made operational mistakes (Cacchiani et al., 2014, Corman and Quaglietta, 2015, Zhan et al., 2016, Xu et al., 2017, Ghaemi et al., 2018, Zhang et al., 2019). Moreover, due to the high speed and short headway of high-speed trains, delays caused by a perturbation can be easily propagated to other trains, which may greatly decrease the comfort and satisfaction of passengers. Therefore, it is urgent to address the safety, reliability and a high level of punctuality for high-speed railways. Besides, considering that dwelling and running disturbances have strong uncertainties, it is necessary to investigate the robustness of the train rescheduling strategy. Stochastic chance-constrained model predictive control can improve the robustness of rescheduling timetable, where expected values of constraints/performance indices and convergence in probability are considered, by exploiting the available statistical information on the perturbation. Based on the above considerations, a real-time optimization strategy is designed to recover the operation of trains for a single-track high-speed railway line with disturbance uncertainties in this paper, where the strategy is obtained via a scenario-based chance-constrained model predictive control (SC-MPC) approach.

In recent years, train rescheduling has widely attracted the attention of scholars (Cacchiani et al., 2014). Generally, the literature on train rescheduling for perturbation management can be split into three categories: (1) train rescheduling for disturbances, (2) train rescheduling for disruptions, and (3) train rescheduling for both disturbances and disruptions. A ‘disturbance’ refers to a relatively small perturbation of a railway system, which may cause train delays that are usually handled only by rescheduling the timetable. A ‘disruption’ indicates a relatively large external incident, which has a strong impact on timetable and is usually associated with large delays or train cancelations. The duration of a ‘disruption’ is usually unknown when it occurs. The timetable may need to be rescheduled immediately when the new information of a disruption is available. Hence, the rescheduling timetable may be updated several times during the disruption management process.

A quantity of researchers have focused on ‘disturbance’ for train rescheduling problems. Törnquist and Persson (2007) proposed a mixed-integer linear programming (MILP) model for train rescheduling problem in a n-tracked network under disturbances, where the reordering and the rerouting of trains are considered. However, the approach in (Törnquist and Persson, 2007) cannot obtain good solutions within seconds for certain scenarios. To reduce the computation time, Törnquist (2012) presented a greedy algorithm which delivers good solutions in a shorter time for the railway disturbance management problem. In order to minimize the delay propagation and reduce the computation time, D’Ariano and Pranzo (2009) divided a long time horizon into several small time stages and the train rescheduling problem is then solved in a cascading manner. Kanai et al. (2011) introduced optimal delay management to minimize the dissatisfaction of passengers across the whole railway network and designed a combined simulation and tabu search optimization algorithm. To reduce the delay of trains, van den Boom et al. (2011) proposed a model predictive controller by using a railway traffic model with train position feedback information and designed an effective permutation-based algorithm to address running time adjustments for trains on the same railway track. Lamorgese and Mannino (2013) proposed a novel modeling method similar to Benders decomposition, which can reduce the computation time to a level acceptable to dispatchers. To solve the train dispatching problem on an N-track network, Meng and Zhou (2014) proposed an integer programming model that was divided by a Lagrangian relaxation technique into many small-scale optimization subproblems for each train. Dollevoet et al. (2014) designed an iterative optimization framework based on a macroscopic delay management model and a microscopic train scheduling model, which not only considered the network level passenger flow, but also considered the detailed scheduling of train paths on the local microscopic infrastructure of the station. Kersbergen et al. (2016) introduced a method to solve the railway rescheduling problem by a centralized model predictive control method and then developed several distributed model predictive control schemes for the rescheduling problem of whole railway networks. Based on a mixed-integer linear programming model, Cavone et al. (2017) proposed a real-time procedure with three steps for robust train rescheduling under disturbances. The first two steps provide a real-time rescheduling timetable, while the third step is performed off-line and offers improvements for rescheduling quality.

Many existing studies have been investigated on the train rescheduling regarding ‘disruption’. To address the blockage of a line over a long time, Hirai et al. (2009) used a petri-net model to investigate train stop deployment planning and formulated an MILP model with the objective of minimizing train delays and the number of stops outside stations. To solve the railway rescheduling problem after disruptions, Acuna-Agost et al. (2011) proposed a MILP model and designed a new method named SAPI (statistical analysis of propagation of incidents) to solve the train rescheduling problem efficiently. Cacchiani et al. (2012) studied the robust rolling stock circulation problem for large disruptions based on a two-stage optimization model, which is solved by LP-relaxation and benders heuristic algorithm. For the sake of reducing delays of all trains, Narayanaswami and Rangaraj (2013) proposed a model for both directions of a single track layout, where only the affected trains are included and the conflicts-resolving constraints are formulated. Based on a rolling horizon approach, Zhan et al. (2016) investigated a disruption management problem in a double-track high-speed railway line in which one track in a segment is temporarily unavailable, where an MILP model is proposed with an uncertain duration of the disruption. Ghaemi et al. (2018) proposed a framework with three models that involves the passenger assignment and short-turning strategy, to evaluate the influence of the estimated disruption duration on the optimization strategy.

Train rescheduling considering both ‘disturbance’ and ‘disruption’ is an interesting topic in recent years. Corman et al. (2014) studied a traffic management decision support system for a large-scale busy railway network under severe disturbances (including entrance delay and blocked track), which is supported by a new heuristic algorithm for local dispatching and coordination. Considering a set of expected disturbance and/or disruption scenarios, Corman and Quaglietta (2015) designed a novel traffic control framework, which combines the most advanced automatic rescheduling tool ROMA with the real railway traffic simulation environment EGTRAIN as an alternative to the real filed. Lamorgese and Mannino (2015) adopted an MILP approach inspired by Benders decomposition to create a master–slave problem using a microscopic model within the stations and a macroscopic model for the overall network. Dispatchers can interact with their system by confirming or modifying solutions and updating parameter settings such as slowdowns, interruptions, delays, cancellations. In addition, the performance of RECIFE-MILP was analyzed and compared for a large real-life network in (Pellegrini et al., 2016), where several scenarios representing realistic disturbances are tackled, i.e., train entrance delays in the control area, prolonged stops at stations, impracticability of track sections, and temporary speed limitation. Quaglietta et al. (2016) introduced a framework for the automatic real-time management of railway traffic, which was designed for standard and interoperable train operations across different European countries. For real-time railway traffic management in busy networks under strong traffic disturbances (such as multiple train delays and temporarily unavailable block sections), Samá et al. (2017) proposed an MILP model and designed a fast scheduling and routing metaheuristics. Moreover, Samà et al. (2017) analyzed the impact of solving the train routing selection problem at different levels, and the results show that the best approach depends on the type of disturbances and disruptions tackled. Xu et al. (2017) proposed a microscopic train rescheduling model that integrates speed and headway supervision with a quasi-moving block system, which is solved by using a two-step method to accelerate the solution process.

Most of the above literature dealt with the constant perturbations by considering the traffic prediction over the next hour. Moreover, to cope with the uncertainties of perturbation and better satisfy the real-time requirement, a scenario-based chance-constrained model predictive control (SC-MPC) algorithm is introduced in this study, which combines a model predictive control (MPC) algorithm and a scenario-based chance-constrained algorithm. The MPC algorithm is a model-based closed-loop control strategy that adopts a strategy of rolling optimization and thus repeatedly optimizes the calculation in real time, which not only reduces the computational scale but also improve the robustness of the control stretagy. This approach has been applied in many different systems. Lin et al. (2011) adopted an MPC approach to regulate and control urban traffic networks. Based on the demonstrated strengths of an MPC framework, Caimi et al. (2012) presented a dispatching assistant for a complex central railway station area, which aims to maximize passenger satisfaction. Schildbach and Morari (2015) presented a practical scenario-based MPC approach for time-varying linear systems with additive disturbances. Li et al. (2017) applied distributed MPC to design the intermodal freight transport planning. Portilla et al. (2016) designed an MPC approach to improve the efficiency of bicycling. Li et al., 2016, Li et al., 2017, Li et al., 2019, Wang et al., 2020 introduced MPC for the control of underground railway transportation. The robustness is a very desirable quality for control systems. Particularly, the robustness refers to an ability of a system to maintain some level of performance even under perturbation within certain parameters; that is, the control system has the ability to withstand the influence of uncertainty. In practice, a high-speed railway system is inevitably affected by various stochastic perturbations. Therefore, it is necessary to consider the robustness of the train rescheduling strategy. To further improve the robustness of rescheduling timetable, the combination of MPC and the min–max approach was often proposed, where the performance index to be minimized is computed over the worst possible disturbance realization. However, min–max policies are often computationally demanding, and the resulting control law is generally too conservative, as no statistical properties about the disturbance are taken into account (Bernardiniy and Bemporad, 2010). Alternatively, a different approach is addressed by stochastic chance-constrained MPC, where expected values of constraints/performance indices and convergence in probability are considered, by exploiting the available statistical information on the disturbance. Based on the stochastic chance constraints, Gavilan et al. (2012) solved spacecraft rendezvous problems with additive disturbances. Due to the finite horizon stochastic optimization problem, Grosso et al. (2014) applied a chance-constrained MPC approach to the management of drinking water networks. Yo et al. (2015) applied a chance-constrained MPC approach to determine the wind output ratio, which is bypassed through a thermal plant. Su et al. (2017) used scenario-based chance-constraint MPC to obtain the optimal long-term intervention plan for component wise for a railway infrastructure development. The work by Xu et al. (2019) was based on the chance-constraint MPC algorithm to study the max-plus linear system under random uncertainty distribution.

Based on the considerations mentioned above, this paper studies the real-time single-track high-speed train rescheduling problem with disturbance uncertainties. The aim of this paper is to minimize the deviation from the planned timetable through adjusting arrival times, departure times, arrival orders, departure orders and dwelling plans. The key contributions of this paper are presented in two folds.

  • (1)

    Compared to most of the previous literature with the pre-given constant disturbance information, this study essentially considered the uncertainties of disturbance and designed a scenario-based chance-constrained model predictive control (SC-MPC) approach to solve high-speed railway train rescheduling problem based on the updated information, which not only meets the real-time requirements with high quality, but also improves the robustness of the rescheduling timetable. In addition, with the consideration of statistical properties of the dwelling and running disturbances, the conservativeness of the designed approach is also reduced to a certain extent.

  • (2)

    With the designed SC-MPC algorithm, we need to transform the constraints with stochastic parameter into chance constraints. Since the chance constraints are nonlinear, the original mixed-integer linear model is converted to a mixed-integer non-linear model. Due to the nature of nonlinearity, it is not easy to solve the formulated optimization model in real time given the growth of scale of the problem. To improve the computational efficiency, the nonlinear model is then converted to a mixed-integer linear model via a linearization method, which can then be solved effectively by existing MILP solvers, such as IBM Cplex, Gurobi, FICO Xpress, and SCIP.

The remainder of this paper is organized as follows. In Section 2, the detailed problem statement and assumptions are given. Section 3 proposes a mathematical model for high-speed train rescheduling with disturbance uncertainties. In Section 4, a scenario-based chance-constrained model predictive control (SC-MPC) algorithm is introduced and a real-time optimization strategy for high-speed train rescheduling is designed. In Section 5, numerical experiments are conducted to illustrate the effectiveness of the proposed train rescheduling strategy. The paper is concluded in Section 6.

Section snippets

Problem description

The topology of a railway line is first introduced, as illustrated in Fig. 1. One operation direction of a high-speed railway line is considered in this paper, in which all the stations are modeled as corresponding nodes and numbered S1,S2,,S. Trains are numbered 1,2,,T. The operation zone of a train consists of stations and sections, where the train travels through. As Fig. 1 shows, the single railway line is divided into different operational zones to provide various operational services

Mathematical formulations

This section will present a macroscopic model with a rigorous mathematical formulation of the rescheduling problem with disturbance uncertainties. The following section is divided into the notations and parameters, system constraints, objective function and train rescheduling model.

Solution algorithm

A scenario-based chance-constrained model predictive control (SC-MPC) algorithm combines a model predictive control (MPC) algorithm and a scenario-based chance-constrained algorithm. Specifically, the uncertain constraints in the predictive optimization model are transformed into scenario-based chance constraints within the framework of the MPC algorithm. With the SC-MPC algorithm, the designed strategy not only satisfies the real-time requirements, but also deals with the randomness in the

Numerical experiments

To demonstrate the feasibility and effectiveness of the proposed train rescheduling strategy, numerical experiments are conducted in this section which are based on the actual operational data of one day of the Beijing-Shanghai high-speed railway. MATLAB R2016a with the IBM ILOG CPLEX 12.5.0 solver is used to solve the train rescheduling problem. The numerical experiments are performed on a personal computer environment (Microsoft Windows 7 operating system, Intel(R) Core(TM) i7-8550U CPU

Conclusion

This paper has studied a real-time optimal high-speed train rescheduling strategy with disturbance uncertainties. Based on the characteristics of high-speed railway system, we construct a linear mixed-integer optimization model for a high-speed railway line. With the need for real-time performance and robust solutions, a scenario-based chance-constrained model predictive control (SC-MPC) algorithm is designed to solve train rescheduling problem. However, with the designed SC-MPC algorithm, the

CRediT authorship contribution statement

Huimin Zhang: Data curation, Writing - original draft, Conceptualization, Software, Visualization. Shukai Li: Investigation, Methodology, Writing - review & editing, Validation. Yanhui Wang: Resources, Supervision, Project administration. Yihui Wang: Writing - review & editing, Validation. Lixing Yang: Methodology, Validation.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 71771017, 71825004, 72071016, U1734210) and the Research Foundation of State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (No. RCS2019ZJ001).

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