Timetable Optimization Research Articles

A multi-objective programming model for timetables on corridors integrating macroscopic and microscopic approaches

This paper proposes an integrated framework for timetable design that meets the needs of service planners, who usually adopt a purely macroscopic point of view toward the demand requirements and conflicts that these timetables may cause at stations due to their particular configurations and structures. Service planners may not take these aspects into account, which require a microscopic analysis of the structure of the stations, considering possible conflicts at platforms and/or internal junctions. To this end, the proposed integrated model guarantees feasible timetables while balancing performance indexes for the interests of passengers and operators. Specified bounds are maintained for the differences between service planners optimal timetables and those that are operational at stations. The resulting problem is modeled as a multi-objective Mixed Integer Linear Programming problem. The model can be solved with sufficient accuracy and in a reasonable amount of computing time, as shown by tests on the Madrid-Zaragoza-Barcelona high speed railway network. In this case, we perform a Pareto frontier analysis and find that the problem is well-posed. In addition, the previous model forms the basis of a heuristic procedure for obtaining timetables that avoid, as much as possible, conflicts and concentrations of arrivals/departures at/from stations, thus allowing for improved timetable recoverability and robustness in case of unexpected disruptions.

Demand-driven timetable and stop pattern cooperative optimization on an urban rail transit line

ABSTRACTThis study proposes a modelling framework for the demand-driven train timetable and stop pattern cooperative optimization problem on an urban rail transit line. By embedding the train stop pattern into the timetable optimization process, we consider the minimization of total passenger travel time. A binary variable determination (BVD) method, which can transform complicated linear constraints into simple logical constraints, is proposed to calculate the large number of binary variables easily, and a genetic algorithm (GA) based on the BVD method is designed to solve the proposed model. A case study of the Batong line in the Beijing subway network is conducted to test the proposed model and algorithm. This study can provide beneficial advice for the operator to improve the operational service of urban rail transit lines.

Optimal High-Speed Railway Timetable by Stop Schedule Adjustment for Energy-Saving

Timetable optimization techniques offer opportunity for saving energy and hence reducing operational costs for high-speed rail services. The existing energy-saving timetable optimization is mainly concentrated on the train running state adjustment and the running time redistribution between two stations. Not only the adjustment space of timetables is limited, but also it is hard for the train to reach the optimized running state in reality, and it is difficult to get feasible timetable with running time redistribution between two stations for energy-saving. This paper presents a high-speed railway energy-saving timetable based on stop schedule optimization. Under the constraints of safety interval and stop rate, with the objective of minimizing the increasing energy consumption of train stops and the shortest travel time of trains, the high-speed railway energy-saving timetable optimization model is established. The fuzzy mathematics programming method is used to design an efficient algorithm. The proposed model and algorithm are demonstrated in the actual operation data of Beijing-Shanghai high-speed railway. The results show that the total operating energy consumption of the train is reduced by 3.7%, and the total travel time of the train is reduced by 11 minutes.

Last train timetable optimization considering detour routing strategy in an urban rail transit network

As an important means of transportation, urban rail transit provides effective mobility, sufficient punctuality, strong security, and environment-friendliness in large cities. However, this transportation mode cannot offer a 24-h service to passengers with the consideration of operation cost and the necessity of maintenance, that is, a final time should be set. Therefore, operators need to design a last train timetable in consideration of the number of successful travel passengers and the total passenger transfer waiting time. This paper proposes a bi-level last train timetable optimization model. Its upper level model aims to maximize the number of passengers who travel by the last train service successful and minimize their transfer waiting time, and its lower level model aims to determine passenger route choice considering the detour routing strategy based on the last train timetable. A genetic algorithm is proposed to solve the upper level model, and the lower level model is solved by a semi-assignment algorithm. The implementation of the proposed model in the Beijing urban rail transit network proves that the model can optimize not only the number of successful transfer directions and successful travel passengers but also the passenger transfer waiting time of successful transfer directions. The optimization results can provide operators detailed information about the stations inaccessible to passengers from all origin stations and uncommon path guides for passengers of all origin–destination pairs. These types of information facilitate the operation of real-world urban rail transit systems.

Models and a solution algorithm for planning transfer synchronization of bus timetables

Mixed-integer linear programming models and a heuristic algorithm are proposed for the transfer synchronization planning of bus timetables. The most important novelty of the proposed methodology is that the bus timetables and passenger choices of travel paths are simultaneously optimized in the model. The possibilities of transfer synchronization are fully explored in the timetable optimization because passengers select their paths to the destination in response to the timetables. Hence, bus timetables can be planned with high precision, and bus timetabling can be improved.

An Energy-Efficient Train Operation Approach by Integrating the Metro Timetabling and Eco-Driving

Energy-efficient train operation is regarded as an effective way to reduce the operational cost and carbon emissions in metro systems. Reduction of the traction energy and increasing of the regenerative energy are two important ways for saving energy, which is closely related to the train timetable and driving strategy. To minimize the systematic net energy consumption, i.e., the difference between the traction energy consumption and the reused regenerative energy, this paper proposes an integrated train operation approach by jointly optimizing the train timetable and driving strategy. A precise train driving strategy is presented and the timetable model considers the headway between successive trains, the distribution of the trip time, and passenger demand in this paper. In addition, a distributed regenerative braking energy model is proposed, based on which the integrated optimization model is formulated. Then, a two-level approach is proposed to solve the problem. At the driving strategy level, the train control problem is transferred into a multi-step decision problem and the Dynamic Programming method is introduced to calculate the energy-efficient driving strategy with the given trip time. As for the timetable level, the trip times and headway of trains are optimized by using the Simulated Annealing algorithm based on the results of dynamic programming method. The timetable optimization level balances the mechanical traction energy of multi-interstations and the amount of the reused regenerative energy such that the net mechanical energy consumption of the metro system is minimized. Furthermore, two numerical examples are conducted for train operations in the peak and off-peak hours separately based on the real-world data of a metro line. The simulation results illustrate that the proposed approach can produce a good performance on energy-saving.

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.

Timetable Optimization for Regenerative Energy Utilization in Subway Systems

In subway systems, kinetic energy can be converted into electrical one by using regenerative braking systems. If regenerative energy (RE) is fully used, the energy demands from power grid can be dramatically reduced. Since energy storage systems usually have a high cost, they are not considered in this work. Thus, RE has to be immediately utilized by accelerating trains; otherwise, it is wasted into heat via resistors. Timetable optimization methods are often used to coordinate accelerating and braking trains at a station, such that RE can be optimally used by the former. To improve RE utilization (REU) in a subway line, we propose a timetable optimization problem and establish its mathematical model. Many realistic constraints with the decision variables, i.e., headway time and dwell time, are considered. Then we design an improved artificial bee colony (IABC) algorithm to solve the problem. Several numerical experiments are conducted based on the actual data from a subway line in Beijing, China. The correctness of the mathematical model and effectiveness of IABC are shown by comparing it with commercial software CPLEX and a genetic algorithm, respectively. The impact of the decision variables on REU is analyzed, which helps to improve the timetable currently used in this subway line. We also test the robustness of the optimized timetable when certain disturbance takes place.

An Integrated Optimization Model for Energy Saving in Metro Operations

In metro train systems, energy-saving operations can include timetable optimization and speed control. Timetable optimization aims to promote the effective utilization of regenerative energy, and speed control aims to minimize the tractive energy consumption. To effectively reduce the net energy consumption (i.e., the difference between tractive energy consumption and regenerative energy utilization), this paper introduces an integrated optimization model that incorporates timetable and speed profile optimization. A cataclysmic genetic algorithm was developed to solve the proposed optimization model. Data were collected from the No. 2 metro line in Nanjing, China; the results showed that the proposed model could reduce the practical energy consumption by 17.5% in comparison with the original timetable. In addition, compared with the cooperative scheduling model that optimized the timetable and speed separately, the proposed model reduced the practical energy consumption by 8.9% on average. Finally, sensitivity analyses were also conducted to investigate the effects of the number of stations on the energy-saving operations.

Recuperation of Regenerative Braking Energy in Electric Rail Transit Systems

Electric rail transit systems are large consumers of energy. In trains with regenerative braking capability, a fraction of the energy used to power a train is regenerated during braking. This regenerated energy, if not properly captured, is typically dumped in the form of heat to avoid overvoltage. Finding a way to recuperate regenerative braking energy can result in economic as well as technical merits. In this comprehensive paper, the various methods and technologies that were proposed for regenerative energy recuperation have been analyzed, investigated and compared. These technologies include: train timetable optimization, energy storage systems (onboard and wayside), and reversible substations.

Integrated Optimization of Tram Schedule and Signal Priority at Intersections to Minimize Person Delay

Modern trams, as a rapidly developed high-volume transport model, have strict requirements on schedule, because the delay will reduce the attractiveness of public transportation to passengers. To improve punctuality and reliability, Transit Signal Priority (TSP) has been employed at intersections, which can extend or insert green phase to trams. However, extending or inserting the green phase for every tram might lead to heavy delays to crossing vehicles. To address this problem, this study developed an integrated optimization model on tram schedule and signal priority which can balance the delay between trams and other vehicles to minimize person delay. Three conditional strategies named early green, green extension, and phase insertion are proposed for the signal priority. Simultaneously, arrival time, departure time of trams at stations, and stop line are optimized as well. The proposed model is tested with a numerical case and a real-world case at Ningbo tramline in China. The results indicate that the integrated optimization can reduce the average delay of all passengers on trams and other vehicles, compared to timetable optimization only and TSP only. It is also found that the proposed model is able to adapt to the fluctuation in the ratio of tram passenger to auto vehicle user, compared with only minimizing tram passenger delay or auto vehicle user delay.

An Energy-Efficient Timetable Optimization Approach in a Bi-DirectionUrban Rail Transit Line: A Mixed-Integer Linear Programming Model

The quick growth of energy consumption in urban rail transit has drawn much attention due to the pressure of both operational cost and environmental responsibilities. In this paper, the timetable is optimized with respect to the system cost of urban rail transit, which pays more attention to energy consumption. Firstly, we propose a Mixed-Integer Non-Linear Programming (MINLP) model including the non-linear objective and constraints. The objective and constraints are linearized for an easier process of solution. Then, a Mixed-Integer Linear Programming (MILP) model is employed, which is solved using the commercial solver Gurobi. Furthermore, from the viewpoint of system cost, we present an alternative objective to optimize the total operational cost. Real Automatic Fare Collection (AFC) data from the Changping Line of Beijing urban rail transit is applied to validate the model in the case study. The results show that the designed timetable could achieve about a 35% energy reduction compared with the maximum energy consumption and a 6.6% cost saving compared with the maximum system cost.

Urban Rail Timetable Optimization to Improve Operational Efficiency with Flexible Routing Plans: A Nonlinear Integer Programming Model

At present, most urban rail transit systems adopt an operation mode with a single long routing. The departure frequency is determined by the maximum section passenger flow. However, when the passenger flow varies greatly within different sections, this mode will lead to a low load factor in some sections, resulting in a waste of capacity. In view of this situation, this paper develops a nonlinear integer programming model to determine an optimal timetable with a balanced scheduling mode, where the wasted capacity at a constant departure frequency can be reduced with a slight increase in passenger waiting time. Then, we simplify the original model into a single-objective integer optimization model through normalization. A genetic algorithm is designed to find the optimal solution. Finally, a numerical example is presented based on real-world passenger and operation data from Beijing Metro Line 4. The results show that the double-routing optimization model can reduce wasted capacity by 9.5%, with a 4.5% increase in passenger waiting time, which illustrates the effectiveness of this optimization model.

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 & 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.

A combined simulation-optimization approach for minimizing travel time and delays in railway timetables

Minimal travel time and maximal reliability are two of the most important properties of a railway transportation service. This paper considers the problem of finding a timetable for a given set of departures that minimizes the weighted sum of scheduled travel time and expected delay, thereby capturing these two important socio-economic properties of a timetable. To accurately represent the complex secondary delays in operational railway traffic, an approach combining microscopic simulation and macroscopic timetable optimization is proposed. To predict the expected delay in the macroscopic timetable, a surrogate function is formulated, as well as a subproblem to calibrate the parameters in the model. In a set of computational experiments, the approach increased the socio-economic benefit by 2–5% and improved the punctuality by 8–25%.

Robust timetable optimization for bus lines subject to resource and regulatory constraints

Timetables are typically generated based on passenger demand and travel time expectations. This work incorporates the travel time and passenger demand uncertainty to generate robust timetables that minimize the possible loss at worst-case scenarios. We solve the resulting minimax problem with a genetic algorithm that uses sequential quadratic programming to evaluate the worst-case performance of each population member. Our approach is tested on a bus line in Singapore demonstrating an improvement potential of ≃5% on service regularity and excessive trip travel times.

Optimal Control of Reversible Substations and Wayside Storage Devices for Voltage Stabilization and Energy Savings in Metro Railway Networks

Growing concerns about environmental issues dictate the necessity for improving the energy efficiency and total performance of metro railway systems. Proper management of the regenerative energy is a key element in this direction, and several strategies, such as timetable optimization, use of reversible substations, and installation of storage devices, have been proposed. This paper focuses on the optimal control of reversible substations and wayside storage devices for energy savings and voltage stabilization. The methods proposed for maximizing the recovered energy are presented together with their theoretical analysis. A novel optimization strategy, based on the linearization of the dc power flow equations, is analytically explained, while a successive approximation method is used. The proposed control strategy is evaluated through simulations on a modified version of the Thessaloniki metro railway network (Greece). Simulation results show that the system performance is significantly improved when reversible substations or energy storage are in operation.

A Dual-Objective Substation Energy Consumption Optimization Problem in Subway Systems

Maximizing regenerative energy utilization is an important way to reduce substation energy consumption in subway systems. Timetable optimization and energy storage systems are two main ways to improve improve regenerative energy utilization, but they were studied separately in the past. To further improve energy conservation while maintaining a low cost, this paper presents a strategy to improve regenerative energy utilization by an integration of them, which determines the capacity of each Wayside Energy Storage System (WESS) and correspondingly optimizes the timetable at the same time. We first propose a dual-objective optimization problem to simultaneously minimize substation energy consumption and the total cost of WESS. Then, a mathematical model is formulated with the decision variables as the configuration of WESS and timetable. Afterwards, we design an ϵ -constraint method to transform the dual-objective optimization problem into several single-objective optimization problems, and accordingly design an improved artificial bee colony algorithm to solve them sequentially. Finally, numerical examples based on the actual data from a subway system in China are conducted to show the effectiveness of the proposed method. Experimental results indicate that substation energy consumption is effectively reduced by using WESS together with a correspondingly optimized timetable. Note that substation energy consumption becomes lower when the total size of WESS is larger, and timetable optimization further reduces it. A set of Pareto optimal solutions is obtained for the experimental subway line—based on which, decision makers can make a sensible trade-off between energy conservation and WESS investment accordingly to their preferences.

Cyclic Timetable Scheduling Problem on High-speed Railway Line

Due to several obvious advantages both in transport marketing and train operation planning, the cyclic timetable has already applied in many high-speed railway (HSR) countries. In order to adopt the cyclic timetable in China's HSR system, a Mixed Integer Programmer (MIP) model is proposed in this paper involving many general constraints, such as running time, dwell time, headway, and connection constraints. In addition, the real-world overtaking rule that concerning a train with higher priority will not be overtaken by a slower one is incorporated into the cyclic timetable optimization model. An approach based on fixed departure is proposed to get a cyclic timetable with minimum total journey time within a reasonable time. From numerical investigations using data from Guangzhou-Zhuhai HSR line in China, the proposed model and associated approach are tested and shown to be effective.

Bi-level energy-efficient occupancy profile optimization integrated with demand-driven control strategy: University building energy saving

University (Educational) buildings comprise a considerable contribution of commercial buildings energy consumption. Optimizing operational measures are widely suggested for energy saving in university buildings. Hence, in this study, a bi-level energy-efficient occupancy profile optimization method using a metaheuristic algorithm, integrated with a demand-driven control strategy adjusted with dynamic set-point temperature is proposed to optimize the energy consumption within in a university departmental building. To proceed with the proposed energy saving strategies; first, building’s thermal behavior and AHU system performance are identified and modeled through Artificial Neural Networks (ANN), using the established data acquisition system’s experimentally measured data. Afterwards, the bi-level course timetable optimization has been introduced where, the first level obtained the optimal occupancy profile and the second level assigned the courses to the timeslots within the first level’s solution. Up to 1.23% of the energy was saved, solely by means of optimizing the timetable. Then, the demand-driven control strategy is implemented, and by predicting the heating demand, thermal comfort is maintained while the building energy use is decreased by 11.65%. Finally, the demand-driven control scheme is integrated with the optimized course timetable, which leads to an enhanced controller performance and achieving a saving potential of up to 18.97%.