1. Introduction
Urban rail transit (URT) shows a trend of large-scale and rapid development for cities worldwide. Due to continuous expansion, high-intensity operation and aging equipment, unplanned service disruptions occur frequently, causing negative impacts on a large number of passengers as well as the transportation system, not limited to the URT system. What exactly is the influence of rail disruption on passengers and the transportation system? It is essential for transit agencies to identify the impacts and adopt appropriate measures to minimize the negative effects of rail disruptions.
The robustness (resilience, redundancy, vulnerability, etc.) of the URT service partly depends on the topological structure of the URT network, so some studies focused on the effects on the URT system under random disruptions based on graph theory or complex network theory [
1,
2,
3,
4,
5,
6]. The identical nodes or links were identified by analyzing the topological characteristics, such as node degree, betweenness, centrality, etc. For example, Yang et al. [
1] assessed the robustness of the Beijing URT network in face of unplanned disruptions. A new node importance evaluation index based on the weighted sum of betweenness and node degree was presented to measure node significance. Jian and Guan [
3] proposed a methodology to identify the most influential lines and measure the URT network vulnerability from a line operation perspective with consideration of the disruption probability. Chen et al. [
6] developed a demand-impedance indicator and an effective path betweenness index to evaluate the importance of rail stations.
The previous studies belong to the static impact analysis of the URT service disruptions. The dynamic passenger flow was not taken into consideration. In practice, once a disruption occurs, affected passengers would make travel adjustments, including mode shift, resetting original and terminal points, and travel route selection. The travel adjustment choice of the affected passengers at the micro level is finally reflected in the redistribution of passenger flow on the transportation network at the macro level. Thus, some studies were dedicated to understanding the mechanism of passengers’ behaviors and constructing passenger reassignment models under unplanned rail disruptions [
7,
8,
9,
10,
11,
12,
13,
14,
15]. Generally, these works were conducted by means of a passenger flow survey. For instance, Teng and Liu [
7] made a passenger behavior survey and a stated preference survey in Shanghai. They found that passengers’ primary choice in response to the interruption is to make a detour in the URT network and that the temporary shuttle bus routes and existing bus lines are also welcomed by passengers as alternative modes. Then a multinomial logit model was built to assign the passenger flow under section interruption in the URT system. An online survey of rail users was conducted by Currie and Muir [
8] to understand passenger behaviors, perceptions and priorities during unplanned rail disruptions. An interesting observation was obtained that over two-thirds of passengers choose the replacement buses as their substitute transit mode even though it takes a long time to wait for the bus service to operate. According to Zhu et al. [
9], income played a vital role in determining travel reactions to disruptions. Nguyen–Phuoc et al. [
10] conducted semi-structured interviews with 30 commuters from Melbourne, Australia, and found the most important factors influencing the mode shift when public transport ceases in the short term, e.g., car access, travel time, travel cost, trip importance, which were also observed by Adelé et al. [
11] and Li et al. [
12].
On the basis of survey results, logit models were commonly employed to make travel choice behavior analyses. Teng and Liu [
7] built a multinomial logit model to assign the passenger flow under section interruption in the URT system. Considering uncertain disruption duration, Li et al. [
14] developed a nested logit model to explore metro passengers’ travel plan choice behavior under unplanned service disruptions. Wang et al. [
15] established a nested logit model following random regret minimization principles to estimate passenger travel choice behaviors under metro emergency context. Dai et al. [
16] also proposed a nested logit model for the decision-making of affected passengers in a metro emergency evacuation.
Apart from survey-based approaches, some scholars used data-driven methods to explore the influence of unplanned rail disruptions on the URT passenger flows. The historical passenger flow data collected from the automated fare collection (AFC) system under normal states and disrupted states were compared and analyzed, hence capturing the passenger volume changes and estimating the impacts of rail disruptions [
17,
18,
19,
20,
21,
22]. For instance, Silva et al. [
17] used network-wide data obtained from smart cards in the London transport system to predict future traffic volumes and to estimate the effects of disruptions due to unplanned closures of stations or lines. Sun et al. [
18] introduced a Bayesian model to identify disruptions based on automated fare collection (AFC) data, and the effects of disruptions were modeled concerning the affected passengers and the length of delay in the URT network. Through analyzing mobile phone records before and after a collision accident of a metro line in Shanghai, Duan et al. [
19] studied the passenger evacuation process and the impacts of the accident on commuters. Based on smart card data, Eltved et al. [
20] proposed a method to analyze the effects of long-term disruptions on passenger travel behavior by distinguishing travel behavior changes prior to the disruption and after the disruption.
Deterministic methods or aggregated methods cannot reflect the heterogeneity of passengers’ behaviors under unplanned rail disruptions, while simulation is a better approach to model the heterogeneous passengers. On one hand, agent-based simulation was widely used [
23,
24,
25,
26,
27,
28,
29]. Leng and Corman [
24] applied agent-based micro-simulation (MATSim) to imitate large-scale passengers’ behaviors during public transport disruptions. A within-day replanning method was integrated into MATSim to study passengers’ route choices. The research revealed the importance of the issue time of information, which greatly impacts passenger satisfaction with public transport disruptions. A similar study was conducted by Muller et al. [
26], where MATSim was used to simulate agents’ choice to find an alternative route after a shutdown of an underground line. Based on simulation results with different policies, the effect of the disruption was analyzed, including travel time, number of line changes, and line usage. Li et al. [
27] developed a discrete- event simulation model with parallel computing to estimate the effects of metro emergencies. Zargayounaa et al. [
28,
29] presented a multi agent simulation model to evaluate the impact of real-time information provision on passenger travel times. Results showed that the impact was positive until a threshold was reached.
On the other hand, Monte Carlo (MC) is another simulation method to analyze the uncertainty of passengers’ behaviors. Seger and Kisgyörgy [
30] utilized MC simulation to quantify uncertainty in traffic assignment on a transport network. Similarly, the MC technique was applied by Bhat [
31] to evaluate the variations in travel mode choice in a multinomial logit model. To examine the day-to-day evolution of network traffic flow, Wang and Li [
32] constructed a mixed logit model and designed an MC simulation algorithm to solve the model. Huang et al. [
33] proposed an MC-based method to evaluate travel time reliability under the impact of traveler information.
However, there are some limitations in the previous studies. First, the existing works failed to provide a method to accurately identify and estimate the affected passenger flow. Second, concerning passengers’ modal choice behavior, the existing studies only considered passengers’ individual choices based on the utility of each alternative mode, such as cost, travel time, and transfer times. Actually, due to the limited capacity of transit modes, the choices of passengers who access the URT network in advance would affect those of passengers who enter the network later. However, the complex interactions among different passengers and the transit system were neglected in previous research. Third, most previous studies, model-based, or simulation-based, only performed the impact analysis of unplanned rail disruptions on affected rail passengers and rail networks, e.g., passenger volume, delay, travel time, and rail usage. As a matter of fact, the impacts are not limited to the interior of the rail system but extend to the whole transportation system. For instance, buses are a relatively popular choice for affected rail passengers as an alternative transportation mode to complete their travel [
8,
34]. When a large number of rail passengers shift to the bus system, the original bus passengers are certain to be affected. To the best of our knowledge, no other study focused on the impacts of unplanned rail disruptions on bus passengers.
To fill the research gap, this paper proposes a passenger behavior simulation framework under urban rail disruptions based on the Monte Carlo method and logit model. Affected rail passengers along with conventional bus passengers are modeled as agents that interact with each other and with the transport system. The heterogeneity of passengers’ travel choices can be simulated in a multi-modal public transit network, including the urban rail, bus, taxi, shared bikes and walking. On the basis of simulation, the impacts of the disruptions on rail passengers as well as bus passengers can be analyzed. The results can enable transit agencies to evaluate the passenger evacuation capacity of the present public transit system under unplanned rail disruptions. They can further assist transit management departments in taking appropriate measures to minimize the negative effects on commuters.
The rest of the paper is organized as follows.
Section 2 proposes a method to identify the affected passengers under unplanned rail disruptions.
Section 3 builds a multi-agent simulation framework to model the travel adjustment behavior of the affected passengers based on the logit model and Monte Carlo method. Then the impact evaluation indices of disruptions are introduced in
Section 4.
Section 5 discusses a case study. Finally, the conclusions and future research are summarized in
Section 6.
2. Identification of Affected Rail Passengers
This study focuses on unplanned URT disruptions that result in the closure of rail links between two adjacent turnover stations in both directions [
35]. Note that the interrupted section may contain transfer stations or not. The former case, i.e., containing transfer stations, is obviously a more complicated situation, which is studied in-depth in this paper. As shown in
Figure 1, Line 1 and Line 2 are both bidirectional rail lines. An unplanned disruption causes service breakdown on a segment of Line 1 including the transfer station A, while Line 2 still functions normally.
The passengers affected by the unplanned disruption refer to those whose space-time travel path in the URT network coincides with the occurrence place and duration of the disruption simultaneously. Under the scenario mentioned before, the affected passengers can be divided into two types in terms of whether they need to transfer in the interrupted line section. In this section, we analyze the time domain of the two types of affected passengers entering the URT system, respectively. On this basis, the affected passenger volume can be estimated using historical automatic fare collection (AFC) data. To identify the affected passengers by their travel route, it is assumed that passengers follow the shortest path to complete their trip.
2.1. Notations
: the set of rail stations, .
: the origin station of a passenger.
: the destination station of a passenger.
: the transfer station in the interrupted section.
: the two turnover stations adjacent to the interrupted section.
: the first interrupted station in the forward direction of a passenger’s path.
: the last interrupted station in the forward direction of a passenger’s path.
: the transfer station outside the interrupted section and nearest to the interrupted section in a passenger’s detour paths.
: the time of the occurrence of the rail disruption.
: the passenger tap-in time.
: the time duration of the rail disruption.
: the time of the recovery of the rail disruption, i.e., .
: the time spent by a passenger from the origin to via URT.
: the time spent by a passenger from to the destination via URT.
: the time duration from to the time at which the disruption information is obtained by passengers.
: the time spent by a passenger from to via URT.
2.2. Tap-In Time Domain of Affected Passengers
2.2.1. Affected Non-Transfer Passengers
Affected passengers who need not transfer in the interrupted section can be classified into four categories in terms of the spatial relationship between passengers’ origins and destinations and the interrupted section (see
Table 1). Based on the passenger tap-in time, we can identify whether a certain passenger is influenced by the disruption or not. With this regard, the affected tap-in time domain is determined for each category as follows.
Affected passengers of the first category are those whose origins and destinations are both in the interrupted section (i.e.,
and
). For passengers who enter the URT network before
, the affected time domain is
; for those who enter the network later than
and depart from the network earlier than
, the affected time domain is
; and for those who enter the network after
and leave the network after
, the affected time domain is
. Consequently, the affected tap-in time domain of the first category is
. Similarly, the affected time domain of the other three categories can be analyzed. The results are shown in
Table 1.
2.2.2. Affected Transfer Passengers
As analyzed to non-transfer passengers, affected passengers who need to transfer in the interrupted section can also be segmented into four categories. Note that the first category is inexistent for affected transfer passengers. The other three categories are explained in
Table 2.
2.3. Identification Method
2.3.1. Data Preparation
- (1)
The URT network diagram
We define the URT network as a directed graph where is the rail node set, corresponding to rail stations, and is the rail arc set, representing links between adjacent stations.
- (2)
The URT disruption data
The data on the rail disruption including the interrupted section and some time-related parameters, i.e., , , , are assumed to be known.
- (3)
The shortest paths of all the OD in the URT network
The Dijkstra algorithm is applied to acquire the shortest paths of all the OD in the URT network, which constitute the database of PathData. Correspondingly, the travel time of each shortest path is stored into the database. Based on the shortest path of each OD, we can determine its affected category and time domain that are also laid in PathData.
- (4)
The AFC data
After data cleaning, the AFC data of a certain day are stored in the database of OriginalURTData. Each piece of data represents a passenger trip, including the passenger ID, the tap-in station ID, the tap-in time, the tap-out station ID, and the tap-out time.
2.3.2. Identification Process
Based on the AFC data and the affected tap-in time domains, affected passengers can be identified. The basic idea is that for a piece of data, we clarify its affected category and compare its tap-in time with the corresponding affected time domain. However, it is tedious and time-consuming to make the comparison for every piece of AFC data, so we simplify the identification process in such a way that the data are sorted in the light of the tap-in time and then some data are excluded using the upper and lower limits of the time domains. The detailed process is listed as follows:
Step 1: Compare the affected tap-in time domains of all OD pairs in PathData. Find out the smallest lower limit and the largest upper limit .
Step 2: Sort the data in OriginalURTData according to the tap-in time. Store the data with the tap-in time within and into URTData.
Step 3: Divide the data in URTData into several groups based on the OD. As a result, data with the same OD are included in the same group. Then sort the data in each group according to the tap-in time.
Step 4: For data in each group, find the affected time domain in PathData. Then identify the data within the domain and store them into AffectedURTData.