Journal of Advanced Transportation

Volume 2018, Article ID 3690603, 26 pages

https://doi.org/10.1155/2018/3690603

## Metro Timetabling for Time-Varying Passenger Demand and Congestion at Stations

^{1}State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China^{2}Department of Civil & Environmental Engineering, University of Maryland, College Park, MD 20742, USA

Correspondence should be addressed to Hangfei Huang; nc.ude.utjb@00241141

Received 7 November 2017; Revised 4 March 2018; Accepted 11 March 2018; Published 3 July 2018

Academic Editor: Giulio E. Cantarella

Copyright © 2018 Keping Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For the train timetabling problem (TTP) in a metro system, the operator-oriented and passenger-oriented objectives are both important but partly conflicting. This paper aims to minimize both objectives by considering frequency (in the line planning stage) and train cost (in the vehicle scheduling stage). Time-varying passenger demand and train capacity are considered in a nonsmooth, nonconvex programming model, which is transformed into a mixed integer programming model with a discrete time-space graph (DTSG). A novel dwell time determining process considering congestion at stations is proposed, which turns the dwell times into dependent variables. In the solution approach, we decompose the TTP into a subproblem for optimizing segment travel times (OST) and a subproblem for optimizing departure headways from the shunting yard (OH). Branch-and-bound and frequency determining algorithms are designed to solve OST. A novel rolling optimization algorithm is designed to solve OH. The numerical experiments include case studies on a short metro line and Beijing Metro Line 4, as well as sensitivity analyses. The results demonstrate the predictive ability of the model, verify the effectiveness and efficiency of the proposed approach, and provide practical insights for different scenarios, which can be used for decision-making support in daily operations.

#### 1. Introduction

With increasing concerns about urban congestion and climate change, urban metro rail transportation receives increasing attention due to its high capacity, punctuality and sustainability. Metro timetabling is the problem of assigning precise utilization times for infrastructure resources to every train in the metro system [1]. In large cities, a metro system is often the busiest public transportation system. To meet travel demand and passenger satisfaction, a timetable should be as compact and flexible as possible. Meanwhile, since many metro systems need government subsidies to cover operating expenses, the planners focus more on operating cost than on passenger factors when designing a timetable [2]. This motivates some recent studies to combine these two conflicting objectives in the train timetabling problem [3, 4].

Two types of timetables are used in metro systems: cyclic and noncyclic. In a cyclic timetable, the departure times of trains are scheduled in equally spaced cycles (e.g., a half hour cycle). In a railway system, the cyclic timetables are assumed to be preferred by both operators and passengers because such timetables are easy to operate and remember. However, in a metro system, passengers usually arrive randomly at stations and wait for the next available train, without checking precise timetables beforehand. Besides, as pointed out by Robenek et al. [5], a cyclic timetable provides an inefficient operation as there is a mismatch between the supply (determined by the timetables) and the demand (characterized by the time-varying demand). On the other hand, a noncyclic timetable imposes no special rules on the departure times of trains [6]. The noncyclic timetables are more flexible regarding time-varying passenger demand, especially when the demand is large [2]. Thus, the noncyclic timetabling problem is worth exploring [2, 3, 7].

The planning process in public transportation can be generally split into three stages: line planning, timetabling, and vehicle scheduling [8]. The line planning stage determines the route/station layouts and frequency. Timetabling is based on the output of line planning, after which the vehicle scheduling can be designed. It is pointed out by Schöbel [8] that going through all these stages sequentially (i.e., independently) leads to unsatisfactory solutions. Since they are interrelated, important factors and objectives at the line planning stage and the vehicle scheduling stage should be accounted for in the timetabling problem.

This paper proposes a rolling optimization algorithm to obtain noncyclic timetables considering time-varying passenger demand and the effects of congestion at stations. It integrates the objectives in the line planning (frequency), timetabling (conflicting objectives including passenger wait/in-vehicle time and energy), and vehicle scheduling (train cost). The main contributions of this paper are as follows.

(1) This paper specifically models the dynamic evolution of passenger loads on trains at each station, by considering passenger arrival rates, limited train capacity and actual passenger alighting/boarding rates associated with congestion. This is an extension of existing studies [9]. Particularly, Niu and Zhou [10] and Niu et al. [11] propose the concept of “effective loading time” that represents the actual time interval within which the arriving passengers can board a train. The detailed boarding and alighting processes are not discussed in their model. Wang et al. [3] set a lower bound (which accounts for details including the number of boarding/alighting passengers and their boarding/alighting rates) for dwell times but consider the dwell times as variables to be optimized, rather than parameters that depend on arrival/departure times and passenger demand. Impacts of gradually increasing passenger demand are analyzed in Robenek et al. [2]; however, in modelling they do not consider the effects of passenger congestion.

(2) This paper integrates objectives from different planning stages in the train timetabling problem, for which solution approaches are scarce [8]. Particularly, Schöbel [8] considers the line planning, timetabling and vehicle scheduling in an integrated way, with passenger- and cost-oriented objectives. However, since his eigenmodel is general and resulting approaches are generic, important objectives described in this paper are not considered in his work, that is, passenger wait time, dwell time, and train capacity. These objectives are included in [2, 5], whereas the frequency and dwell times are fixed in their studies. Besides, energy is not considered in [2, 5].

(3) This paper proposes a novel and effective solution approach, which decomposes the master timetabling model into two subproblems. The first subproblem is solved by branch-and-bound and frequency determining algorithms, and the second one is solved by a rolling optimization algorithm which optimizes the departure headways from the shunting yard (which are equivalent to arrival times of trains at their first station) and considers interdependent variables in each rolling step. The design of decision variables in this paper is similar to [2, 5], but the models and solution approaches are different. Compared to Wang et al. [3], which addresses the similar timetabling problem with traditional approaches such as sequential quadratic programming and a genetic algorithm, the approach proposed in this paper performs better computationally in terms of run time and solution quality.

The remainder of this paper is as follows. Section 2 reviews relevant studies on the train timetabling problem and summarizes the research gap in the literature. Section 3 first describes the problem, introduces formulation of different cost functions, and constructs the train timetabling model. Then the discrete time-space graph is proposed to transform the model into a mixed integer program (MIP). Section 4 discusses the solution approach for the optimization model. Section 5 presents different numerical experiments and sensitivity analyses on a short metro line and on Beijing Metro Line 4, to demonstrate the predictive ability of the model as well as the effectiveness and efficiency of the proposed approach. Section 6 summarizes the works done in this paper and presents the limitations and potential topics in future studies.

#### 2. Literature Review

The train timetabling problem (TTP) aims to schedule trains to transport passengers (or goods) without conflicts, by specifying train arrival and departure times at stations. It is interrelated with other planning stages, that is, the line planning stage and the vehicle scheduling stage. We organize the related literature from the perspectives of operator-oriented objectives, passenger-oriented objectives, and their integration.

##### 2.1. Operator-Oriented Objectives

In the literature, operator-oriented objectives can be found only for the noncyclic version of the TTP [2]. Most studies focus on the operation indicators, for example, energy [17, 18], train delay [19–21], and train travel time [22–24].

The total train delay and train travel time are usually considered in the TTP for railway networks, which aims to find a feasible timetable by minimizing the profit loss resulting from changes to the ideal/planned timetable. Energy is the focus and main cost for metro system operators. Most energy is consumed in train operations [25, 26], and thus obtaining an energy-efficient train timetable receives considerable attention [17, 18, 27]. Note that energy is affected by the travel time in the segment, and the segment travel time is determined by the arrival and departure times at adjacent stations, that is, the timetable. Hence, the energy-efficient train timetabling problem may be extended to include other objectives. For example, Yang et al. [23] consider both energy and train travel time as optimization objectives. Some recent studies also account for regenerative braking. Yang et al. [28] develop a scheduling approach to coordinate arrivals and departures of trains within the same electricity supply zones, thereby effectively recovering the regenerative energy. Generally, these studies focus solely on the timetabling stage, where frequency (in line planning) is fixed and train cost (in-vehicle scheduling) is not considered.

The efficient use of railway rolling stock (vehicle scheduling) is an important objective pursued by a railway agency or company because of intensive capital investment in rolling stock [29]. To this end, Lai et al. [29] develop an optimization model to improve the efficiency of rolling stock usage considering necessary regulations and practical constraints, where a hybrid heuristic process is designed to improve solution quality and efficiency. Haahr et al. [27] use CPLEX and a column and row generation approach to assign rolling stock units to timetable services in passenger railways, prepare daily schedules, and check their real-time applicability by testing different disruption scenarios. These vehicle scheduling studies require the trips as input data. Based on Schöbel [8], in a metro system, every trip describes the operation of a train between the start and end time of the line at its first and its last stations (given from the timetable). The objective function then aims at minimizing the number of trains and the costs for their movements.

Particularly, Schmid and Ehmke [30] and Schöbel [8] demonstrate that the integration of timetabling and vehicle scheduling is more beneficial than the sequential planning process. However, the studies on the integration of timetabling and vehicle scheduling usually adopt relatively general and generic models, and the TTP model is often simplified [31]. For example, Cadarso and Marin [32] focus on shunting operations and their timetable is calculated by readjusting frequencies, whereas specific details such as the dwell times and segment travel times are not considered. This motivates us to consider train cost in the timetabling stage, while focusing on the specified TTP formulation.

In addition, the conflict-free timetable is also pursued by the operators [33, 34]. It is worth mentioning that the discrete time-space graph proposed in Caprara et al. [6] is a directed multigraph in which nodes correspond to arrivals/departures at a certain station and at a given time stamp. This graph is widely used to formulate the TTP and derive different integer programming models that correspond to specific objectives. For example, Caprara et al. [6] use the graph to derive an integer linear programming (ILP) model with Lagrangian relaxation, which is embedded within a heuristic algorithm. Cacchiani et al. [12] use the graph to formulate an ILP with linear programming (LP) relaxation. Cacchiani et al. [35] use the time-space graph-based LP relaxation of an ILP to derive a dual bound in the TTP for a set of stations in an urban area interconnected by tracks, thus aiming to resolve the conflicts and evaluate the capacity saturation.

##### 2.2. Passenger-Oriented Objectives

For a metro system, passenger-oriented timetables that consider reliability and reduction of passenger time are most desirable [11]. In some studies, the passenger demand is considered stable [36]. Assuming that passengers prefer easily memorable timetables, such timetables are usually cyclic. They are designed on the basis of a period event scheduling problem (PESP), aiming to minimize passenger travel and wait time [15, 37] and to maximize network stability [38]. However, as mentioned above, metro passengers arrive randomly and do not remember the timetable. Besides, cyclic timetables are less flexible than noncyclic ones in accounting for the passenger demand [2].

Recent studies consider time-varying demands due to the growing concerns for service level and congestion at stations. Such timetables are noncyclic, with objectives of improving passenger satisfaction [3, 7, 10]. Particularly, Barrena et al. [7] propose two nonlinear programming (NP) formulations to generalize noncyclic train timetables on a single line, which are solved by a fast adaptive large neighborhood search metaheuristic. The objective is to minimize passenger wait times at stations. Niu et al. [11] construct mathematically rigorous and algorithmically tractable nonlinear mixed integer programming (MIP) models for both real-time scheduling and medium-term planning applications, to jointly synchronize effective passenger loading time windows and train arrival and departure times at each station. Their work aims to minimize the total waiting times of passengers at stations. Robenek et al. [5] define a timetable as a set of departure times of every train from its origin station on every line and consider four attributes in passenger satisfaction: the in-vehicle-time, the waiting time at transfers, the number of transfers, and the schedule passenger delay. They describe the hybrid timetables through additional constraints that are imposed on the original passenger centric train timetabling problem [2], so that the passengers would obtain the same level of service as a cyclic timetable, with more flexibility. A specifically defined simulated annealing heuristic is proposed to solve the problem. Luan et al. [16] integrate the TTP and preventive maintenance time slots (PMTSs) planning problem on a general railway network. They propose an MILP model to jointly optimize train routes, orders and passing times at each station, and PMTSs. The objective is to minimize the sum of the absolute arrival time deviations of real trains at destinations between the ideal and actual timetables. A Lagrangian-based label correcting algorithm is designed for solving the time-dependent least cost path problem.

In addition, the delay management (DM) problem determines whether trains should wait for a delayed feeder train or should depart on time, while considering passenger-oriented objectives [13]. Some recent studies integrate the macroscopic DM with the microscopic TTP [39]. Particularly, Dollevoet et al. [14] formulate an integer programming (IP) model and propose an iterative optimization approach to solve such a bilevel problem that the macroscopic level is the delay management and the microscopic level is the train timetabling. The optimization approach repeats a process that uses DM to find a solution and uses TTP to validate it, until a feasible solution is found. It should be noted that the graph-based models are adopted in their work, which simplify the formulations: the TTP is formulated with the alternative graph [40], and the DM is formulated with an event-activity network [15].

##### 2.3. Integrated Objectives

Since the operator- and passenger-oriented objectives are partly conflicting, focusing on only one side would yield undesirable solutions; for example, the best possible service for passengers may also be the most costly alternative for the operator [2]. Thus, many TTP studies integrate these objectives jointly [22, 41].

Wang et al. [3] aim to minimize passenger travel time and energy in their nonsmooth nonconvex programming (NSNCP) model. They propose an event-driven model that involves arrival and departure events and passenger arrival rates change events and use nonlinear programming approaches and evolutionary algorithms to solve the problem.

Qi et al. [42] focus on a line planning problem with budget constraints and evaluate the station layout with train timetable indicators. The objectives include constructing cost for additional tracks, total travel time, and network capacity. A mixed integer linear programming (MILP) model is formulated to address the TTP, and then a bilevel programming (BP) model is designed to address the integrated problem. Both CPLEX and a local search-based heuristic are developed to solve the models.

Robenek et al. [2] model the PCTTP as an MILP model with objectives of maximizing the operator’s profit while maintaining passenger satisfaction. The model uses the output of the line planning problem as the input of the train timetabling problem, to address the gap found between them. CPLEX is used to construct a Pareto Frontier of the contradictory objectives.

Li et al. [43] combine dynamic train regulation and passenger flow control to minimize the timetable and the headway deviations for metro lines, thereby reducing the operator profit loss and the passenger delay. The model predictive control and a quadratic programming algorithm are proposed to solve the problem, and stability analysis is conducted to verify the system convergence.

However, the above studies do not consider the impacts of frequency. Schöbel [44] reviews some approaches to model and solve the line planning problem, which demonstrate the impacts of frequency to both operator- and passenger-oriented objectives. Particularly, Samanta and Jha [45] consider different objectives in the line planning stage which include minimizing user cost, operator cost, and location cost and maximizing the ridership or the service coverage of the line. The objectives are separately formulated with NP models, and a GA is developed to optimize the objective functions subject to constraints on the number and spacing of stations. Fu et al. [46] propose a BP model to determine line frequencies and the stopping patterns for a mix of fast and slow train lines, with the objectives of minimizing passenger travel time and the number of transfers and maximizing train capacity occupancy. The heuristics solve the problem by combining additional conflicting demand-supply factors. Cascetta and Coppola [47] specify both the frequency-based assignment models and the timetable-based models in the demand forecasting, to analyze their differences in modal split estimates and flows on individual trains. The results show that when passenger demand is time-varying or the timetable is noncyclic, the difference is significant. Thus, the integration of frequency and timetabling is worth investigating.

##### 2.4. Research Gap

Table 1 highlights some publications related to the TTP. All the graph-based models use integer programs, including the model in this paper. Although different objectives are considered, very few studies integrate the objectives in all three stages. Particularly, Wang et al. [3] only consider the objectives in one stage, that is, timetabling stage; Robenek et al. [2] integrate passenger satisfaction and operating profit (including train cost) but do not consider frequency and energy; Schöbel [8] takes into account objectives in all stages with an iterating framework; however, crucial factors such as passenger wait time, dwell time, and train capacity are missing. Kroon et al. [15] integrate the objectives in the TTP and vehicle scheduling but omit energy.