Train Departure Times Research Articles

Optimization on metro timetable considering train capacity and passenger demand from intercity railways

Purpose To evacuate passengers arriving at intercity railway stations efficiently, metros and intercity railways usually share the same station or have stations close to each other. When intercity trains arrive intensively, a great number of passengers will burst into the metro station connecting with the intercity railway station within a short period, while the number of passengers will decrease substantially when intercity trains arrive sparsely. The metro timetables with regular headway currently adopted in real-world operations cannot handle the injected passenger demand properly. Timetable optimization of metro lines connecting with intercity railway stations is essential to improve service quality. Design/methodology/approach Based on arrival times of intercity trains and the entire process for passengers transferring from railway to metro, this paper develops a mathematical model to characterize the time-varying demand of passengers arriving at the platform of a metro station connecting with an intercity railway station. Provided the time-varying passenger demand and capacity of metro trains, a timetable model to optimize train departure time of a bi-direction metro line where an intermediate station connects with an intercity railway station is proposed. The objective is to minimize waiting time of passengers at the connecting station. The proposed timetable model is solved by an adaptive large neighborhood search algorithm. Findings Real-world case studies show that the prediction accuracy of the proposed model on passenger demand at the connecting station is higher than 90%, and the timetable model can reduce waiting time of passengers at the connecting station by 28.47% which is increased by 5% approximately than the calculation results of the generic algorithm. Originality/value This paper puts forward a model to predict the number of passengers arriving at the platform of connection stations via analyzing the entire process for passengers transferring from intercity trains to metros. Also, a timetable optimization model aiming at minimizing passenger waiting time of a metro line where an intermediate station is connected to an intercity railway station is proposed.

Practical Descriptor Modeling of Traffic in Intersecting Metro Lines and Robust Controller Design for Train Synchronization at Transfer Stations

In this article, a discrete-event linear descriptor model for intersecting metro lines based on the deviations of train departure times from the nominal values is introduced. In intersecting metro lines, the delays are transmitted from one line to the other intersecting lines in transfer stations (TSs). Therefore, considering the effect of the TSs in traffic modeling leads to a more accurate model. The introduced model is an uncertain model, and we reformulate it as a linear uncertain descriptor model with additive uncertainty. A robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> controller is designed based on the bounded real lemma and linear matrix inequality to compensate the delays and regulate the time deviations in the presence of uncertainty. In addition to the aforementioned objective of the control system, another objective is synchronizing the trains at TSs of intersecting metro lines. The simulation results demonstrate the performance and effectiveness of the proposed automatic traffic regulation system using real data from two intersecting lines of the Tehran metro system.

Improving interregional rail timetabling through welfare assessment of departure time shifts

In order to improve current rail timetabling processes, the suitability of train departure times for passengers should be included in timetable assessment. To achieve this, one possibility is to calculate the change in consumer and producer surplus (i.e., the economic welfare) resulting from departure time shifts in rail timetables. However, existing methods for this calculation are quite limited. To fill this gap, we propose a new method in the current paper. This method enables comparing most scenarios involved in interregional rail timetabling in terms of economic welfare. To this end, our method takes advantage of schedule-based models that allow assessing the impact of departure time shifts on the demand and valuation of each possible route using the timetable. As a proof of concept, we illustrate this method on a case study on the busiest Swedish interregional line. This case study shows the potential of the method to deliver detailed calculations with analysis of equity effects. To conclude, the method presented in this paper improves on the current literature, and it can be used to improve timetable optimisation algorithms or to better resolve conflicts between train path requests.

Timetable Optimization for Metro Lines Connecting to Intercity Railway Stations to Minimize Passenger Waiting Time

In major metropolitan cities, metro and intercity rail services usually share stations or have stations in close proximity to facilitate smooth connections for passengers. A large number of passengers may arrive on metro platforms within a short period of time upon the arrivals of intercity trains, while the number may drop significantly afterward. This injected passenger demand is not coped well by the metro timetable with the commonly adopted regular train headway. Focusing on metro lines connecting to intercity railways, this study puts forward a timetable model to optimize train headways at each station, assuming all passengers board the first train available. The objective is to minimize the total waiting time of passengers at platforms throughout the metro line, without changing the number of vehicle trips. To calculate the passenger waiting time accurately, a mathematical model is proposed to obtain the number of transfer passengers arriving at metro platform in each time interval, based on arrivals of intercity trains and transfer facilities at the station. A genetic algorithm, combined with interior-point algorithm, is developed to obtain the solution of the proposed timetable model. Real-life case studies show that the optimized timetable reduces the total passenger waiting time when compared with the current timetable and the solution of another model optimizing train departure times at the first station only. In addition, the peak number of passengers congregating at metro platforms drops significantly, which relieves passenger crowding on the platform.

Discrete Optimization on Train Rescheduling on Single-Track Railway: Clustering Hierarchy and Heuristic Search

This paper focuses on discrete dynamic optimization on train rescheduling on single-track railway with the consideration of train punctuality and station satisfaction degree. A discrete dynamic system is firstly described to mimic train rescheduling, and a state transition function is specially designed according to the train departure event. The purpose of this function is to improve simulation efficiency by directly confirming the next discrete time. After the construction and analysis of optimization models to discrete dynamic system, a two-stage heuristic search strategy is developed, by using clustering hierarchy theory and stochastic search strategy, to obtain train departure time and arrival time before each state transition of the system. Finally, a numerical experiment is conducted to verify the proposed system, models, and the heuristic search strategy. The result shows that the discrete dynamic system, together with the state transition function and heuristic search strategy, shows better performance in simulation efficiency and solution quality.

Multiperiod Transfer Synchronization for Cross-Platform Transfer in an Urban Rail Transit System

Transfer synchronization is an important issue in timetable scheduling for an urban rail transit system, especially a cross-platform transfer. In this paper, we aim to optimize the performance of transfer throughout the daily operation of an urban rail transit system. The daily operation is divided into multiple time periods and each time period has a specific headway to fulfill time varied passenger demand. At the same time, the turn-back process of trains should also be considered for a real operation. Therefore, our work enhances the base of the transfer synchronization model taking into account time-dependent passenger demand and utilization of trains. A mixed integer programming model is developed to obtain an optimal timetable, providing a smooth transfer for cross-transfer platform and minimizing the transfer waiting time for all transfer passengers from different directions with consideration of timetable symmetry. By adjusting the departure time of trains based on a predetermined timetable, this transfer optimization model is solved through a genetic algorithm. The proposed model and algorithm are utilized for a real transfer problem in Beijing and the results demonstrate a significant reduction in transfer waiting time.

Coordinated Optimization of Departure Time Domains of Multiple Trains at a Station Based on Passenger Satisfaction

In order to optimize the passenger train working diagram and improve the passenger satisfaction towards a coordinated optimization problem of departure time domains of multiple passenger trains at a station, the optimization model is designed to maximize the overall passenger travel satisfaction based on the quantification of the level of passenger travel satisfaction. The simulated annealing algorithm is then used to solve the model. Finally, 23 pairs of passenger trains from Chengdu Railway Station are selected as examples to verify the feasibility and effectiveness of the algorithm. The result shows that the model and algorithm proposed in this paper can realize the coordinated optimization of the departure time of multiple trains at a station while considering the passengers’ travel satisfaction, which can provide an effective reference for the compilation of passenger train working diagram.

Scheduling synchronization in urban rail transit networks: Trade-offs between transfer passenger and last train operation

The scheduling synchronization problem in this paper is to obtain an optimal schedule by optimizing running time, departure time, dwell time and arrival time of the last train in urban rail transit networks. Operators are often faced with multiple conflicting requirements simultaneously such as high passenger’s service quality and less cost, smooth transfer events and less passenger’s travel time, and high accessibility and less operation time. Thus, researchers in this industry concentrate on utilizing operations and management decisions to solve the scheduling problem while balancing the service trade-offs. In this paper, we formulate a mixed integer programming approach for the last train schedule planning, and passengers benefit from smoother transfer in the form of maximizing the transfer synchronization events, while operators simultaneously can benefit with lower operation costs by minimizing the worst big difference between last trains. We propose an improved non-dominated sorting approach embedded in a genetic algorithm to obtain close-to-optimal solutions in a much shorter time for a sophisticated, real-world and large-scale Beijing subway network. Results demonstrate that significant service performance gains (76.33% for Just-missed, 32.01% for successful transfer, 15.39% for non-equity for last trains and 45.25% for variance indicators, etc.), which indicate the effectiveness of the proposed modeling framework and solution algorithm. The operator formulates an efficient schedule for the actual operation of the urban rail transit network by the proposed application method, and it also would be applicable to solving schedule problems among a large-scale network in other industries.

Robust optimization of train timetable and energy efficiency in urban rail transit: A two-stage approach

With the rapid development of urban rail transit systems in large cities, the travel demand of urban rail transit has increased and the total energy consumption has risen sharply. These situations will result in a huge operating cost, especially the energy consuming. In urban rail transit operations, some uncertain or stochastic factors frequently occur, which will lead to trains deviating from the schedule train timetable. Therefore, it is of great significance to study the optimal strategy to reduce the energy consumption and enhance the flexibility. Providing a robust train timetable is able to reduce operating costs and improve service quality. This paper proposes a two-stage robust optimization model for finding an optimal train timetable to save energy-efficient and reduce passenger waiting time. In the stage-one model, the objective is to reduce passenger waiting time, and the decision variable is the train departure time interval. In the stage-two model, the objective is to minimize the total energy consumption of all trains, and the decision variable is the train departure and arrival time at each station. Due to the complexity of the mathematical model, this paper proposed a nested heuristic genetic algorithm to solve it. An adaptive generation operator and a nested chromosome set generation strategy are included to solve the two-stage robust optimization model. The proposed model and algorithm is then validated by the practical data of Beijing Subway Yizhuang line. The proposed approach can be potentially applied to metro operation optimization of urban rail transit system.

Optimizing the Number of Express Freight Trains on a High-speed Railway Corridor by the Departure Period

In China, the transportation capacity of high-speed railways is gradually sufficient to provide services for high-value express freight besides meeting passenger demands. However, similar as the passengers, express freight demands fluctuate and show clear peak and trough periods daily. Therefore, optimizing running numbers of express freight trains on high-speed railway by periods is quite necessary to guarantee the revenues of railway industry and to meet the various requirements of all consignees simultaneously. First, a space-period-pattern three-dimensional network with virtual arcs is built to describe the departure period selection and the stopping or skip-stopping operations. Second, by constructing the arcs' impedance function, the user equilibrium principle is introduced to optimize the express freight flow distribution for each pattern in each period. To elaborate the comprehensive goal of balancing the relationship between profits and the flow distribution, a bi-level programming model is established. The upper model addresses the railway industry's maximum profits, and the lower model addresses the minimum and similar impedance values of the final express freight flow distribution. Finally, through the use of a hybrid algorithm that combines the heuristic genetic algorithm with the Frank-Wolfe algorithm, an experimental case of 10 stations on the Beijing-Xi'an high-speed railway corridor is used to validate the study. The results show that the express freight volume is reasonably distributed to each kind of virtual arc, the impedance value of each pattern is minimized and almost equalized, and the profits of railway industry are maximized by optimizing the number and departure time of trains by each pattern in each period on the basis of meeting all the express freight demands.

An Iterative Heuristic for Passenger-Centric Train Timetabling with Integrated Adaption Times

In this paper we present a method to construct a periodic timetable from a tactical planning perspective. We aim at constructing a timetable that is feasible with respect to infrastructure constraints and minimizes average perceived passenger travel time. In addition to in-train and transfer times, our notion of perceived passenger time includes the adaption time (waiting time at the origin station). Adaption time minimization allows us to avoid strict frequency regularity constraints and, at the same time, to ensure regular connections between passengers’ origins and destinations. The combination of adaption time minimization and infrastructure constraints satisfaction makes the problem very challenging. The described periodic timetabling problem can be modelled as an extension of a Periodic Event Scheduling Problem (PESP) formulation, but requires huge computing times if it is directly solved by a general-purpose solver for instances of realistic size. In this paper, we propose a heuristic approach consisting of two phases that are executed iteratively. First, we solve a mixed-integer linear program to determine an ideal timetable that mini- mizes the average perceived passenger travel time but neglects infrastructure constraints. Then, a Lagrangian-based heuristic makes the timetable feasible with respect to infra- structure constraints by modifying train departure and arrival times as little as possible. The obtained feasible timetable is then evaluated to compute the resulting average per- ceived passenger travel time, and a feedback is sent to the Lagrangian-based heuristic so as to possibly improve the obtained timetable from the passenger perspective, while still respecting infrastructure constraints. We illustrate the proposed iterative heuristic approach on real-life instances of Netherlands Railways and compare it to a benchmark approach, showing that it finds a feasible timetable very close to the ideal one.

Performance improvement of energy consumption, passenger time and robustness in metro systems: A multi-objective timetable optimization approach

As an effective transport mode to reduce the impact of urban mobility with great capacity and low pollution, metro systems have received rapid development all over the world in recent years. Since a well-designed timetable can improve the service level of passengers, increase the revenue of operating companies, and reduce the environmental pollution, the issue of timetable design/optimization is usually represented as a multi-objective decision problem. This paper develops a multi-objective integer programming model which integrates the energy consumption, the passenger waiting time, and the robustness by optimizing departure and arrival time of trains at each station. A non-dominated sorting genetic algorithm II (NSGA-II) is designed to find the optimal solution. Finally, a case study of Yizhuang line in Beijing metro system is conducted to illustrate the efficiency and effectiveness of the suggested timetable optimization approach.

Optimal Coordination of Last Trains for Maximum Transfer Accessibility with Heterogeneous Walking Time

Last train coordination aims to synchronize the arrival and departure times of the last feeder trains and the last connecting trains at transfer stations to improve the transfer accessibility of urban rail networks. This study focuses on the transfer accessibility between last trains with considering heterogeneous transfer walking time. Three mathematical models are developed on the last train timetable optimization. The first model fine-tunes the last train timetable under the given bound of the dwell time. The second one aims to allow the mutual transfers with the prolonged dwell time to maximize the transfer accessibility. A biobjective function is proposed to seek the trade-off between the maximal transfer accessibility and the minimal extension of dwell time. The third model considers the heterogeneity of transfer walking time that is represented as a random variable following a probability distribution. A discrete approximation method is proposed to reformulate the nonlinear model. The embedded Branch &amp; Cut algorithm of CPLEX is applied to solve the models. A real case on the Shenzhen metro network is conducted to demonstrate the performance of the models. The three models all provide better last train timetable than the current timetable in practice. The sensitivity analysis manifests that the third model are always advantageous in the optimization of successful transfer passengers.

Synchronized optimization of last trains’ timetables in mass rail transit networks based on extra dwell time

Synchronized optimization of last trains’ timetables is purposed to improve the connecting effects of last trains by coordinating the arrival and departure times of trains at transfer stations. Due to the great complexities of transfer relationships, it is more and more difficult to solve this problem. In this paper, in order to reduce the number of unable transfer passengers, an optimization model of last trains’ timetables is proposed by determining the departure time and extra dwell time. And we minimize the difference between last trains’ actual departure time and the optimized ones for the convenience of rolling stocks daily maintenance. Possible optimized solution is investigated through a case study in Beijing subway by using the tool of CPLEX. The results indicate that the volume of unable transfer passengers is reduced by 15.0%, and the number of coordinated trains is improved by 8.9%.

Minimizing Metro Transfer Waiting Time with AFCS Data Using Simulated Annealing with Parallel Computing

Coordinating train arrivals at transfer stations by altering their departure times can reduce transfer waiting time (TWT) and improve level of service. This paper develops a method to optimize train departure times from terminals that minimizes total TWT for an urban rail network with many transfer stations. To maintain service capacity and avoid operational complexity, dispatching headway is fixed. An integrated Simulated Annealing with parallel computing approach is applied to perform the optimization. To demonstrate model applicability and performance, the Shenzhen metro network is applied, where passenger flows (i.e., entry, transfer, and exit) at stations are approximated with the automatic fare collection system (AFCS) data. Results show that the total TWT can be significantly reduced.

Estimation of Passengers Left Behind by Trains in High-Frequency Transit Service Operating Near Capacity

Measuring rail system crowding is important to transit agencies. Crowding data has implications for safety, operations control, service planning, performance measurement, and customer information. This paper proposes a bi-level regression model that transit agencies can use to estimate the number of passengers left behind on a platform by high-frequency trains operating at capacity. Inputs to the model include the number of passenger arrivals between trains and train departure times, which are derived from automatic fare collection and vehicle location data. The data are used to calculate the proposed measure of cumulative capacity shortage, which is shown to have high correlation with the number of passengers left behind. A bi-level regression approach is introduced and applied to calibrate the model parameters based on manual counts of passengers left behind. A case study using data from the Chicago Transit Authority’s Blue Line demonstrates promising results, with an adjusted coefficient of determination of 0.81. The model could be used for post-hoc analysis of crowding performance or, in the context of real-time operations monitoring, for near-term predictions of passengers left behind.

Stochastic passenger train timetabling using a branch and bound approach

Abstract We investigate the integrated scheduling of freight and passenger trains in complex railway networks and the timetabling for passenger trains to improve the efficiency of the schedules. In the proposed model, the objective is to study the robust passenger train timetable considering the uncertainty in freight train departure times. The joint routing and scheduling are considered between the two types of trains. We propose a branch and bound framework with hybrid heuristics to solve the problem. On each search tree node, a Lagrangian method is first developed to solve for a relaxed passenger train schedule and then a feasibility recovery heuristic is applied. Then a labeling algorithm is proposed to jointly optimize the freight and passenger trains sequentially. Numerical experiments show the proposed solution approach outperforms other heuristics on actual railway networks.

Coordinated optimisation problem integrating EMU circulation and timetabling for high‐speed railway

In this study, the problem of electric multiple unit (EMU) circulation planning and train timetabling is studied through a coordinated process that provides a feedback mechanism to simultaneously minimise the number of EMUs, the number of EMU maintenance tasks and the train travel time. Based on an adjustable train departure time window as the key parameter in the problem, the approach mainly consists of two components: a column generation process, which searches for better EMU circulation routes; and a mathematical model, which produces timetables. These components interact according to the results of computational assessments until the solution reaches a certain level of optimality or the allotted computation time is exhausted. Finally, the authors test this approach using a small example to illustrate the effectiveness, and they also study a real-world case. A quantitative comparative analysis shows that long travel distances and travel times of trains significantly affect the number of EMUs used. The results indicate that the proposed model and algorithm can effectively address the coordinated optimisation problem of integrating EMU circulation planning and timetabling.

Markov chain model for delay distribution in train schedules: Assessing the effectiveness of time allowances

Train movements are subject to disturbances and disruptions, which may cause late departures and/or late arrivals at particular locations (e.g., terminals, stations, crossings) with respect to their pre-determined times. To prevent possible deviations from the scheduled times, time allowances are added to the process times (i.e., running times) as time supplements and to the time interval between successive trains (i.e., minimum headway) as buffer times. The amount of time allowances is decided during the timetable planning process and their contribution to the service quality is monitored in actual operation. The adequacy of time allowances results in punctual train operations, which is one of the primary interests of service users and providers. Because the sequence of train departure and arrival times can be viewed as a stochastic process, the performance of train operations may be analyzed and modeled using Markov chains. We modeled train departure and arrival delays at stations as states and analyzed the successive state changes along train paths in a single track railway. This allowed us to predict train states at certain event time steps and to estimate steady-state delay probabilities. The former may be used to reschedule train movements and the latter to measure timetable robustness.

A dynamic programming approach to synchronize train timetables

This article focuses on synchronizing timetables of train services at a rail transfer station. The main aim is to determine an optimal schedule of train services, given that the departure and arrival times of some particular trains are known. An exponential utility function is introduced to measure the synchronization levels between different train services. A nonlinear integer programming model is proposed to achieve the objective of a synchronized timetable. A dynamic programming approach is then designed to solve the developed model. Finally, a numerical example with real-world datasets is implemented to demonstrate the effectiveness of the proposed approaches.