1 Introduction

The urban rail transit (URT) system is gradually becoming the main way to alleviate urban traffic problems in large cities because of its advantages of large capacity [1], high reliability [2], and high speed [3]. According to a survey by Han et al. [4], as of 2021, there were 277 subway lines in operation in 50 cities in mainland China, with total operation of more than 9192.62 km, of which 23 cities had more than five subway lines. In large cities, with the expansion of the URT network and the increase in passenger flow scale, the URT network has become a closed complex network. It is crucial to ensure the URT network’s safe operation for maintaining the urban transportation system’s smooth and efficient operation. Emergencies such as large passenger flow shocks, equipment failures, and natural disasters can easily affect the stable operation of the URT network and even cause interruptions. In contrast to other modes of transportation, the URT system has the characteristics of high closure and low connectivity. Therefore, once an emergency occurs, it will affect not only the current train service but also the passenger travel of relevant sections and even the entire network. In 2019, a total of 1416 delays of 5 minutes or more occurred in China’s urban rail transit system, and a total of 8953 failures of trains exiting the main line were recorded [5]. In addition to causing substantial economic loss, emergencies significantly reduce the service level and passenger satisfaction with urban rail transit, which poses significant challenges to the operational safety of the URT system. Therefore, it is critically important to study the distribution of URT passenger flow distribution in the case of interruptions for assessing the vulnerability of the metro network [6], formulating a targeted plan [7], and improving service levels [8].

In recent years, with the expansion of network scale, the impact of emergencies on the stable operation of the URT network has attracted increasing attention among scholars. From a spatial perspective, interruption events will cause changes in the URT network topology, and the stability of critical nodes is crucial to the safe operation of the entire network. Therefore, many scholars focus on the influence of interruption on rail transit networks through graphs and complex network theories [9,10,11]. For example, Jenelius and Cats [12] proposed a methodology for assessing the value of new links for public transport network robustness, taking into account the disruptions of other lines and links as well as the new links themselves. Based on the complex network theory (CNT), Sun et al. [10] quantitatively analyzed the statistical topology parameters of the Beijing subway network and evaluated key stations through different evaluation indicators (such as node degree, betweenness degree, and strength). However, the above studies did not explain the impact of interruption on passenger travel decision-making and flow distribution. Lu [13] pointed out that critical stations were identified differently depending on the duration of different incidents and characteristics of the failed stations, and stations on network legs could be more important than those with redundant rail alternatives. Therefore, it is more practical to analyze the impact of interruption events on passenger flow distribution by combining the network topology and passenger volume characteristics.

In practice, the flow of passengers in the URT network is realized by trains. Once the interruption occurs, the train operation plan will change, and this change will spread to other parts of the network, affecting passengers’ travel decisions. Thus, it is also necessary to consider the impact of interruption on passenger travel decisions from the perspective of train operation, that is, the time dimension. Based on two surveys (a passenger behavior survey and a stated preference survey), Teng and Liu [14] developed a polynomial logit model to allocate the passenger flow of the rail transit system in the case of interruption. The results show that detours are the first choice for passengers in these cases. With automatic fare collection (AFC) data, Sun et al. [15] proposed a quantitative method based on Bayesian theory for estimating the effects of common disruptions (not catastrophes or large-scale emergencies) on passengers and stations from the spatial and temporal points of view. To analyze the impact of long-term interruption on passenger travel behavior, Eltved et al. [16] proposed a new method to analyze the changes in travel behavior before and after the interruption. Based on k-means clustering, passengers were originally divided into groups before and after the interruption. 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. The proposed methodology is unique in that historical disruption data are used to predict unseen scenarios by relying on simple physical assumptions of passenger flow and a system-wide model for origin–destination (OD) movement. The above studies attempt to use AFC data to mine the travel behavior of passengers when facing interruption, which provides a new idea for studying passenger flow distribution under interruption. However, the change in passenger travel time cost is affected by the interruption and many other factors, such as the headway and the number of people waiting on the platform. Therefore, relying solely on AFC data may fail to achieve ideal results.

Cong et al. [24] proposed an affected railway passenger identification method based on passenger check-in time to evaluate the impact of interruptions on the subway and original bus passengers, and used a multi-agent simulation system to analyze and model passenger response behavior. Unlike previous studies, this study uses simulation methods to depict passenger travel behavior, which can describe the impact of network dynamics such as interruption and congestion on passenger travel from a micro perspective. Simulation methods are widely used to study the distribution of normalized subway passenger flow [18,19,20,21,22]. They show the irreplaceable advantages of mathematical analysis methods, such as describing passenger behavior at the micro and individual levels.

Based on the above analyses, this paper proposes a simulation-based method to evaluate the impact of interruptions on URT passenger flow distribution. Specifically, we first analyze the impact of disruption on passengers in different travel states in detail, and then model the passenger response behavior. On this basis, we use the simulation method to simulate the interruption events and the interaction between trains and passengers in the network. Finally, by comparing and analyzing the changes in individual passenger behavior and network passenger flow distribution under two conditions of interruption and non-interruption, the influence of interruption events on passenger and network passenger flow distribution is explained.

To highlight the novelty of our study, some relevant studies are summarized in Table 1 and compared with our study in terms of input data (A), path utility (B), path assignment mode (C), passenger assignment mode (D), interruptions (E), and performance indicators (F).

Table 1 Characteristics of relevant works in comparison with our work
Full size table

The rest of this paper is organized as follows. Section 2 expounds on the influence of interruption events on passengers and the calculation framework of passenger flow distribution. Section 3 introduces the agent-based simulation model in detail. We apply the proposed model to the Chongqing Metro network in Sect. 4 as a case study. Finally, we summarize our research, summarize our main findings, and discuss future research directions in Sect. 5.

2 Problem Statement and Model Framework

This study focuses on the impact of two-way track interruption caused by disruptions in the distribution of subway passenger flow. The stations in the interruption area may be ordinary stations or transfer stations. During the interruption, there is no train service in the interruption area. When the service is restored, the train headway will quickly return to the normal level. In this case, the affected passengers usually adopt one of the following three schemes: (1) abandoning the urban rail transit system and adopting other transportation modes; (2) adjusting their path to reach or approach the destination quickly; (3) waiting for the service to resume. Since the AFC system does not record passengers’ travel intentions, this study takes the information recorded in the AFC system as their real travel intention, regardless of whether the passengers give up the trip or change their destination midway. In other words, this paper aims to study the changes in passenger travel behavior and passenger flow distribution left in the urban rail transit system under interruption. Taking the URT network containing three lines as an example, as shown in Fig. 1, they are Line 1, Line 2, and Line 3, respectively. The interruption area is located on Line 2, and the interruption time is from 7:00 to 8:00. After the interruption occurs, Line 2 is divided into two independent closed sections.

Fig. 1
figure 1

Diagram of URT network with interruption area

Full size image

During the interruption, the network topology changes. Passengers will now plan their paths according to the modified network. In this case, the passenger travel decision is similar to that under normal conditions, but the difference is that the travel path needs to be adjusted according to the interruption area. Specifically, (1) when the passenger’s departure station is in the interruption area, the passenger either abandons the URT trip or waits for the service to resume, depending on the cost of the two options, as shown in Fig. 2a. Generally speaking, passengers will not give up their destination, so the cost of passengers abandoning the trip is equal to the cost of choosing other transportation modes to reach the destination. The waiting cost is equal to the waiting time for service recovery plus the time of the remaining path. If the waiting cost is lower than the cost of abandoning the URT trip, the passenger will wait for service recovery; otherwise, they will turn to other modes of transportation. (2) When the passenger’s destination is in the interruption area, the passenger will re-plan the path with the destination’s alternative station as the target, as shown in Fig. 2b. An alternative station is close to the destination and not within the interruption area. If the service has not been restored when passengers arrive at the alternative station, passengers must choose between waiting for the service recovery and giving up the journey. (3) Passengers may detour when the planned path passes through the interruption area, as shown in Fig. 2c. In other words, passengers must choose between the cost of a detour and turning to different modes of transportation.

Fig. 2
figure 2

Diagram of passenger path selection during interruption

Full size image

Passengers need to adjust their travel paths in real time according to network topology changes. In other words, when the interruption occurs or service resumes, the network topology and the train schedule will change significantly. At this time, passengers who have not completed the trip need to adjust their travel plans according to their state. Specifically, according to their status, passengers can be divided into four categories: (1) passengers who have not entered the station, (2) passengers waiting at the platform, (3) onboard passengers, and (4) passengers who have completed the trip. It can be seen that except for the last one, the passengers may be affected by the change of interruption. When the interruption occurs, we can estimate the affected passenger group by their check-in time and planned path, the relationship between the current location and the interruption area, and other factors. However, when the service is resumed, it is difficult to judge whether the passenger departing during the interruption is affected, because the passenger’s planned path may not include the interruption area, as shown in Fig. 2c.

Therefore, it is necessary to re-plan the path for passengers based on their status and judge whether passengers are affected by comparing the new path with their original path. As shown in Fig. 3, the interruption has not yet been restored before passengers depart, so passengers choose to bypass Line 3 (as shown by the dotted line). Passengers will adjust their path according to their location when the service is restored. When the passengers are in area A or B, Line 1 is the best choice. When the passenger is at position C, they can turn back to Line 1 or travel along their planned path, both of which have advantages and disadvantages. Thus, it can be concluded that the interruption event will affect the passenger path decision and have a very complex spatiotemporal coupling with the passenger’s location.

Fig. 3
figure 3

Diagram of passenger path selection in different status when service resumes

Full size image

3 Simulation Modeling

To describe this coupling relationship and accurately calculate the impact of interruption events on passenger choice behavior and passenger flow distribution, this paper proposes a simulation-based method for passenger flow distribution calculation, as shown in Fig. 4. The method includes three main modules, namely the event module, path module, and network loading module. The role of the event module is to read network topology data and timetable data according to the interruption event and generate a simulation timestamp. The function of the path module is to generate a set of alternative paths based on network topology and interruption information and assign paths to passengers. When an interruption occurs or service resumes, the path is reassigned for passengers. The function of the network loading module is to load passengers to specific trains by controlling the interaction of passengers, trains, and platforms under the given passenger travel demand, path choices, and train capacity constraints according to the simulation timestamp. The loading process is realized through the interaction between three agents: the platform agent, the passenger agent, and the train service agent. As an intermediary between passenger and train agents, the platform agent does not involve state switching and is mainly used to manage the waiting passenger queue and evaluate the waiting time of the platform. For passenger agents, the main behaviors include waiting, boarding, onboard, and alighting. The main actions of the train agent include loading and unloading passengers.

Fig. 4
figure 4

Simulation method for URT passenger flow under interruption

Full size image

4 Notation

For readers’ convenience, the notations in this paper are summarized in Table 2.

Table 2 Notations in this paper
Full size table

4.1 Path Assignment

4.1.1 Path Generalized Utility

Taking the path with two journey segments as an example, without loss of generality, the travel process of passengers in the URT network is shown in Fig. 5. That is, passengers enter the URT system from the entrance of the origin station, walk to the departure platform to wait for the train, and then move forward with the train until arriving at their transfer station or destination station. If the arrival station is their destination, passengers will disembark and walk to the exit to complete the trip; otherwise, they will go to the departure platform of the next journey segment through the transfer channel and repeat the above process.

Fig. 5
figure 5

Passenger travel process diagram

Full size image

That is to say, the path travel time is equal to the sum of the access time, waiting time, in-vehicle time, transfer walking time, and egress time. Generally, walking and in-vehicle time are relatively fixed, while waiting time has strong time-varying characteristics, especially during peak hours. Therefore, it is necessary to calculate the real-time path travel time according to the time when passengers arrive at the platform. Assuming that the candidate path set of OD pair \((u,v)\) is \({R}^{u,v}\), where path \(r\in {R}^{u,v}\) includes \(m\in {M}_{r}^{u,v}\) transfer, \(|{M}_{r}^{u,v}|+1\) travel segments in total, the travel time of path \(r\in {R}^{u,v}\) at time \(t\) can be expressed as:

$${\tau }_{r}^{u,v}\left(t\right)=\sum_{m\in {M}_{r}^{u,v}}({\tau }_{u,v,r,m}^{a}+{\tau }_{u,v,r,m}^{b})+{\tau }_{u,v,r}^{c}(t)+{\tau }_{u,v,r}^{e}$$
(1)

where \({\tau }_{u,v,r,m}^{a}\) is the access walking time or transfer walking time of the \(m\) journey segment, and \({\tau }_{u,v,r}^{e}\) is egress walking time. The walking time can generally be estimated by the channel distance and the average walking speed of passengers. \({\tau }_{u,v,r,m}^{b}\) is the train running time in the \(m\) journey section, which is only related to the operation section and can be obtained from the timetable. \({\tau }_{u,v,r}^{c}(t)\) is the sum of the waiting time for each platform. It can be seen from Fig. 5 that the time when passengers arrive at each platform is different, and the waiting time at the platform is time-varying. Therefore, the total waiting time needs to be calculated according to the time passengers arrive at each platform, which can be expressed as the offset of the passengers’ check-in time. The offset time \({\Delta \tau }_{r,m}^{u,v}\left(t\right)\) is approximately equal to the sum of the time passengers spend in the journey segment before current platform.

$${\Delta \tau }_{r,m}^{u,v}\left(t\right)=\sum_{{m}^{^{\prime}}=0}^{m}({\tau }_{u,v,r,{m}^{^{\prime}}}^{a}+ {\tau }_{u,v,r,{m}^{^{\prime}}}^{b}+{\tau }_{u,v,r,{m}^{^{\prime}}}^{c}(t)), \forall r\in {R}^{u,v}, m\in {M}_{r}^{u,v}$$
(2)

Therefore, the total waiting time \({\tau }_{u,v,r}^{c}(t)\) can be expressed as the sum of time-varying waiting time when passengers arrive at each platform, as follows:

$${\tau }_{u,v,r}^{c}\left(t\right)=\sum_{m\in {M}_{r}^{u,v}}({\tau }_{u,v,r,m}^{c}(t+{\Delta \tau }_{r,m}^{u,v}\left(t\right))), \forall r\in {R}^{u,v}$$
(3)

Taking formula (3) into formula (1), the predicted travel time of path \(r\in {R}^{u,v}\) can be further expressed as:

$${\tau }_{r}^{u,v}\left(t\right)=\sum_{m\in {M}_{r}^{u,v}}({\tau }_{u,v,r,m}^{a}+{\tau }_{u,v,r,m}^{b}+{\tau }_{u,v,r,m}^{c}(t+{\Delta \tau }_{r,m}^{u,v}\left(t\right)))+{\tau }_{u,v,r}^{e}, \forall r\in {R}^{u,v}$$
(4)

In fact, not only will the path journey time affect the passengers’ travel decisions, but the number of transfers also cannot be ignored [25, 26]. Therefore, the generalized path cost considering the travel time and the number of transfers can be expressed as the sum of the weighted values of walking time, in-vehicle time, waiting time, and transfer punishment. The generalized cost of the predicted travel time of path \(r\in {R}^{u,v}\) at time \(t\) can be expressed as:

$${C}_{r}^{u,v}\left(t\right)=\alpha \sum_{m=1}^{|{M}_{r}^{u,v}|}\left({\tau }_{u,v,r,m}^{a}\right)+{\alpha \tau }_{u,v,r}^{e}+\alpha \sum_{m=1}^{|{M}_{r}^{u,v}|}\left({\tau }_{u,v,r,m}^{b}\right)+\alpha \sum_{m=1}^{|{M}_{r}^{u,v}|}\left({\tau }_{u,v,r,m}^{c}\left(t+{\Delta \tau }_{r,m}^{u,v}\left(t\right)\right)\right)+\gamma \left|{M}_{r}^{u,v}\right|, \forall r\in {R}^{u,v}$$
(5)

where \(\alpha\) is the weight of the path journey time, \(\gamma\) is the weight of transfer times, and the above can be calibrated based on the survey data; please refer to the research of Si et al. [26] for the detailed process. The waiting time of each time can be estimated by a method proposed in the literature [2] with the historical AFC data. In this study, the author takes a special OD pair as the object; the candidate path of this kind of OD pair is unique and without transfer. According to the passengers’ travel process, the travel time of this path can be divided into access time, waiting time, onboard time, and egress time. Based on this feature, the authors propose a data fusion method based on AFC data, network topology, and timetable data to calculate the waiting time distribution at the platform. Please refer to the literature [2] for details. Leurent et al. [27] reported similar studies on this issue.

4.1.2 Equilibrium-Based Pre-trip Path Assignment

Passengers are generally aware of the candidate paths between the origin and destination stations through maps, experience, and other information before traveling. Therefore, this paper assigns paths for passengers in the form of pre-trip. Given the low connectivity of the URT network, passengers usually do not switch their paths on the way, so this approach is in line with the actual situation. Since passengers cannot know the exact cost of each candidate path, they can only make decisions based on their perceived travel costs for each candidate path. According to the random utility theory, passengers always choose the path with the lowest perceived travel cost. Generally, the passenger’s perceived travel cost \({\widetilde{C}}_{r}^{u,v}\left(t\right)\) is expressed as a random variable, which is composed of the deterministic component \({C}_{r}^{u,v}\left(t\right)\) and the additive random error. That is:

$${\widetilde{C}}_{r}^{u,v}\left(t\right)={C}_{r}^{u,v}\left(t\right)+{\varepsilon }_{r}^{u,v},\forall r\in {R}^{u,v}$$
(6)

where \({\varepsilon }_{r}^{u,v}\) is the perception error of passengers and their expectation \(E\left({\varepsilon }_{r}^{u,v}\right)=0\).

Assuming that the random error term \({\varepsilon }_{r}^{u,v}\) for each candidate path is independent and follows the Gumbel distribution, the probability that passengers choose the path \(r\in {R}^{u,v}\) can be expressed by the multinomial logit (MNL) model. We use the logit model proposed in the literature [26] to calculate the probability of each path being selected. This logit model uses the relative difference of travel costs of different paths rather than the absolute difference to calculate the selection probability, namely:

$${p}_{r}^{u,v}(t)= \frac{\mathrm{exp}\left(-{C}_{r}^{u,v}\left(t\right)/{C}^{^{\prime}}\right)}{\sum_{{r}^{^{\prime}}\in {R}^{u,v}}\mathrm{exp}\left(-{C}_{{r}^{^{\prime}}}^{u,v}\left(t\right)/{C}^{^{\prime}}\right)}, \forall r\in {R}^{u,v}$$
(7)

where \({C}^{^{\prime}}=\mathrm{min}\{{C}_{r}^{u,v}\left(t\right)| r\in {R}^{u,v} \}\) is the minimum travel cost of all candidate paths.

Based on the selection probability of candidate paths, we use the roulette wheel selection method to assign paths for passengers. The path selection probability corresponds to different areas on the wheel: the greater the probability, the higher the probability that the path is assigned to passengers.

4.1.3 En-route Path Assignment

When the interruption occurs or service resumes, the network topology and train schedule will change, directly affecting the generalized cost of the path selected by passengers. Therefore, the affected passengers will adjust their planned path according to the information released by the operators. As mentioned earlier, the impact of interruption occurrence and service recovery on passengers differs. Figure 6 shows a flowchart of passenger path assignment under these two events.

Fig. 6
figure 6

Diagram of passenger path adjustment under different events

Full size image

It can be seen that when an interruption occurs, it must be determined whether the passenger’s travel is affected by the interruption according to their position. As mentioned above, when the passenger’s journey is blocked, the passenger needs to make a choice between selecting an alternative station and waiting for service recovery. The factor that affects passenger decision-making is travel costs. Teng and Liu [14] pointed out that relative speed or time is the main factor that affects passengers switching traffic modes. Therefore, this study chooses travel time as the cost and assumes that the maximum waiting time for service recovery that passengers can tolerate is \({w}_{i}^{max}\). Considering that the running speed of the ground bus is generally 20 km/h, and the average running speed of the subway is about 45 km/h, the time cost of taking the bus is about twice that of the subway. Therefore, \({w}_{i}^{max}\) can be regarded as the time required for passengers to travel from the current station to the destination. The waiting time \({w}_{i}\) is equal to the time for service recovery minus the time for passengers to obtain interruption information. Passenger waiting time \({w}_{i}\) is equal to the service recovery time minus the time passengers arrive at the interruption station along with the planned path. If \({w}_{i}<{w}_{i}^{max}\), passengers will wait for service recovery; otherwise, they will choose to travel from an alternative station. Therefore, the decision-making rules of passengers can be expressed by the following formula:

$$\delta =\left\{\begin{array}{ll}1&\quad if \,{w}_{i}^{max}>{w}_{i} \\ 0&\quad else\end{array}, \forall r,{r}^{^{\prime}}\in {R}^{{u}^{^{\prime}},v}\right.$$
(8)

Similarly, when passengers need to detour due to interruption, the travel cost of the detour path should not exceed twice the current path of passengers.

When the service resumes, it is necessary to determine whether passengers are affected, by comparing the new and planned paths. Generally speaking, the basis of the passenger flow assignment model is user-rational [28]. That is to say, passengers change their path only when the benefits obtained by switching paths is higher than that of traveling along the planned path. The switching rule can be expressed as:

$$\theta =\left\{\begin{array}{cc}1& if\, {C}_{r}^{{u}^{^{\prime}},v}\left(t\right)> \lambda {C}_{{r}^{^{\prime}}}^{{u}^{^{\prime}},v}\left(t\right)\\ 0& else\end{array}, \forall r,{r}^{^{\prime}}\in {R}^{{u}^{^{\prime}},v}\right.$$
(9)

where \({C}_{r}^{{u}^{\mathrm{^{\prime}}},v}\left(t\right)\) is the cost for passengers to continue traveling along the planned path \(r\) from current station \({u}^{\mathrm{^{\prime}}}\), \({C}_{{r}^{\mathrm{^{\prime}}}}^{{u}^{\mathrm{^{\prime}}},v}\left(t\right)\) is the cost of traveling from the current site \({u}^{\mathrm{^{\prime}}}\) along the new path \({r}^{\mathrm{^{\prime}}}\), λ is the benefit from switching paths, generally \(\lambda >1\).

4.2 Network Loading

As mentioned above, passengers and trains are considered independent individual agents. As the simulation clock advances, they update their respective states in turn, and finally, the passenger flow distribution calculation of the URT network is realized. This section will detail the interaction between all agents.

4.2.1 The Schedule-Based Network and Time Discretization

A schedule-based URT network can be represented by a weighted directed graph, \(G=(N,A,L)\), where \(N\) is the set of nodes, \(A\) is the set of arcs, and \(L\) is the set of subway lines. The entire analysis period is \(T\). The node represents the station or platform. The platforms on the same line are numbered and sequenced. The platforms in the upstream direction are ranked first, and then the platforms in the downward direction. Taking a line with three stations, for example, the station set is \(\{ A, B, C \}\), and the platforms are numbered as {1,2,3,4,5,6}, so \(( l, n )\) can uniquely represent a platform, \(l\in L, n\in N\). The network has three kinds of arcs: walk arc, boarding arc, and transit arc. Figure 5 gives the space–time displacement interpretation of various arcs. The starting node of a walking arc is the station, and the end node is the departure platform; the starting and ending nodes of the boarding arc are the same, and the passengers only have time displacement. The starting node of the transit arc is the departure platform, and the end node is the arrival platform on the travel section. It can be found that any arc contains two nodes, and for any two adjacent arcs, the end node of the former is the same as the start node of the latter.

It is assumed that the trains always run according to the timetable in a URT network with a directed weighted graph structure. Then, for any line \(l \in L\), the timetable is defined as follows. Let \({n}_{l}\in N\) denote the number of stations on line \(l\), and let \({j}_{l}\in J\) denote the number of trains on line \(l\) during the study period; \({h}_{l,n,j}\) denotes the time when the \(j\)th train on the \(l\)th subway line leaves station \(n\). Note that all stations of each line are sorted in upward and downward directions. All trains on each line are sorted in ascending order according to their departure time.

$${h}_{l,n,j}<{h}_{l,n,j+1}, \forall l\in L, n=\mathrm{1,2}\dots ,{n}_{l}, j=\mathrm{1,2},\dots ,{j}_{l}$$
(10)

As the boarding and alighting of passengers occur during the train stops at the station, passengers may board the current train as long as they arrive at the platform before the train leaves the platform, so this paper discretizes the network based on the train departure time.

4.2.2 Passenger Agent Generation

This study uses AFC data to generate passenger agents. Specifically, the origin station, destination station, and check-in time of passenger agents are derived from AFC data, and the checkout time is obtained from the proposed simulation model. The model accuracy can be verified by comparing the passengers' travel time recorded by AFC with that obtained by the proposed model.

4.2.3 Passenger Queue at Platform

The platform is an important node in the process of passenger travel. In practice, the numbers of waiting passengers and passengers stranded on the platform are important indicators that directly reflect the network performance [29, 30]. In the simulation model, the role of the platform is indispensable as an important node for the interaction between passengers and trains. Assuming that passengers will queue at the platform according to the arrival time sequence after arriving at the platform, as shown in Fig. 7, whether passengers can board the current train depends on the number of passengers waiting at the platform and the available capacity of the train. When the number of passengers is greater than the available capacity of the train, the passengers arriving at the platform later are detained on the platform, waiting for the next train. The time spent by passengers from arriving at the platform to leaving the platform by train is the passengers’ waiting time. All waiting times from the current platform can be used to evaluate the degree of congestion at the platform [29] and the dynamic characteristics of the network [2, 31].

Fig. 7
figure 7

Diagram of passenger queue at platform

Full size image

4.2.4 Interaction of Boarding and Alighting

Figure 8 shows the flowchart of passengers boarding and alighting the train when the train arrives at the platform. When a train arrives, the train first unloads passengers who take the current station as the destination or transfer station and updates the train status (for example, train available capacity and passengers onboard). If the current station is the destination of the alight passengers, the passengers will be added to the egress arc. After passengers exit, their checkout time will be recorded. If the current station is a transfer station for passengers, the passengers will be added to the corresponding transfer walking arc according to the passengers’ planned path. When the passengers arrive at the next platform, they are added to the waiting queue of the corresponding platform. It should be noted that this study focuses on the change in passenger flow. The train agent is only regarded as the carrier carrying passengers. Therefore, we only focus on the interaction between the train and the waiting passengers when the train stops at the station, which is the key to the change in the train load and the waiting passengers on the platform. The movement process of the train on the way, such as acceleration and deceleration, is omitted.

Fig. 8
figure 8

Flowchart of passenger-train interaction

Full size image

5 Case Study

In this section, we take the data for the Chongqing Metro on a specific working day in 2018 as an example to analyze the accuracy and applicability of the proposed model. We first present two indicators to investigate the model accuracy based on the passenger travel data recorded by the AFC system. Then, taking the passenger flow distribution under the normal condition as the benchmark, the influence of interruption events on passenger flow distribution is compared and analyzed in terms of boarding, alighting, entering, exiting, and so on. We finally attempt to explain this phenomenon from the impact of interruption events on passenger path choice behavior. The proposed simulation method is coded in Java on a Windows 10 PC with 2.6GHz CPU and 16.0 GB RAM.

5.1 Data Description

In 2018, Chongqing Metro consisted of 178 stations and seven lines, totaling 216.9 km. The interruption event studied occurred between Lijia and Hongqihegou of Line 6, as shown in Fig. 9, represented by a white cross on a red background. Lijia, Ranjiaba, Dalongshan, and Hongqihegou are four transfer stations, which are also alternative stations for other non-transfer stations in the interrupted section. We set the travel cost between the non-transfer station in the interrupted area and its alternative station as twice the train operation time. The interruption lasted 52 minutes, from 10:58 to 11:50. During the interruption, the relevant stations in the interrupted section were closed, and there was no train in operation. Considering the warmup and cooldown time, we set the research time as 6:00–15:00; generally, a passenger’s travel time will not exceed 2 hours. Most passengers use smart cards or mobile phones to pay for tickets, so the AFC data record the passengers’ travel data such as the origin station, check-in time, destination station, and checkout time. Generally speaking, the punctuality rate of subway service is relatively high, so the train schedule can be inferred according to the published operation plan and interruption events. The network topology distance data are sorted according to the field survey results. Please refer to Table 3 for other parameter settings. The parameters related to passenger path assignment are calibrated based on the survey data using the methods mentioned in the literature [26].

Fig. 9
figure 9

Diagram of Chongqing Metro

Full size image
Table 3 Parameter setting
Full size table

5.2 Model Verification

Inspired by the literature [21], we used two indicators to verify the accuracy and effectiveness of the model. The average relative deviation between the simulated travel time and the actual travel time for each tested passenger is calculated by (11). Another is the average relative deviation between the simulated checkout passenger volume and the volume from the AFC data, as shown in (12).

$${d}_{i}^{c}=\frac{|{c}_{i}-{c}_{i}^{^{\prime}}|}{{c}_{i}}\times 100\%, i\in I$$
(11)

where \({c}_{i}\) is the travel time of passenger \(i\) obtained from the AFC data, and \({c}_{i}^{^{\prime}}\) is the simulated travel time.

$${d}_{n}^{q}=\frac{|{q}_{n}-{q}_{n}^{^{\prime}}|}{{q}_{n}}\times 100\%,n\in N$$
(12)

where \({q}_{n}\) is the checkout passenger volume of station \(n,n\in N\) during interruption, and \({q}_{s}^{^{\prime}}\) is the checkout passenger volume obtained by our simulation model.

Figure 10 presents the \({d}_{i}^{c}\) distributions of 106,852 passengers selected from the passengers who enter the URT system between 9:58 and 12:20, which is 30 minutes before the interruption and 30 minutes after the service recovery. We can observe that most passengers’ deviation is less than 30%. The random variables of the path utility function and the individual difference in the passenger’s walking speed are the main reasons for the deviation. The weighted average relative deviation \(avg({d}_{i}^{c})\) of all test passengers is 12.7%. Considering that the average travel time of these passengers is about 25 minutes, the weighted average deviation is about 3.18 minutes, which is an acceptable value in practice.

Fig. 10
figure 10

Distribution of travel time deviation for tested passengers

Full size image

Figure 11 provides the \({d}_{s}^{q}\) distributions of checkout passenger volume during the interruption. We can see that the deviations for most stations are less than 15%, and for 97.8% of stations, the deviations are less than 30%. Therefore, it can be concluded that the proposed models have acceptable accuracy and are suitable for calculating the passenger flow distribution in URT networks under interruption.

Fig. 11
figure 11

Distribution of outbound passenger volume deviation during the interruption

Full size image

5.3 Analysis of Passenger Flow Distribution Changes with Typical Date Data

To analyze the impact of interruption on passenger flow, we choose AFC data for typical working days as the input of the simulation model. The interruption time is set as 10:58–11:50. Table 4 shows the main statistical indicators of the affected passengers in two periods: 8:58–12:20 and the interruption period. It can be seen that 14,817 passengers were affected by the interruption and 5287 passengers failed to reach their planned destination. Among them, the check-in time of 10,192 affected passengers is between 10:58–11:50, accounting for 15.11% of the inbound volume in this period. From the perspective of spatial distribution, 3032 OD pairs were affected, accounting for 18.26% of the total number of OD pairs. There are 117 OD pairs with more than 30 people affected, mainly in Line 6 and Line 3. It can be seen that the interruption had a severe impact on the URT network and the passenger flow distribution.

Table 4 Performance indicators of passenger flow
Full size table

Next, based on the simulation results without interruption, the impact of the interruption on the URT passenger flow distribution and passenger behavior is analyzed in terms of the inbound volume, outbound volume, transfer volume, etc.

Figure 12 shows the top 20 stations where passengers were most likely to give up traveling. The horizontal axis is the origin station for passengers. The green curve indicates the proportion of passengers who give up traveling relative to the inbound volume. It can be seen that these stations are concentrated in the interruption area (station names with "*") and Line 6. The number of passengers giving up traveling at Huahuiyuan Station is the largest, reaching 62.8% of the inbound volume. Fewer passengers from Lijia and Hongqihegou stations give up traveling, because these two stations are located at the ends of the interruption area. Passengers will be affected only when their destinations are situated in the interruption area. Dalongshan and Ranjiaba are two transfer stations in the interruption area, connecting Line 5 and Line 6. According to the network topology, Line 5 is interrupted and isolated from the other lines when the interruption occurs. Therefore, the number of passengers who give up traveling from these two stations is also significant, exceeding 40%.

Fig. 12
figure 12

Top 20 stations where passengers give up traveling

Full size image

Taking three typical OD pairs from Huahuiyuan to Beibei, Ranjiaba, and Xiaoshizi to illustrate passengers’ travel behavior in the interrupted area, Fig. 13 shows the travel behavior distribution of passengers under interruption. The horizontal axis is the passengers’ check-in time at Huahuiyuan station, and the vertical axis is their planned destination. The red square indicates that passengers will exit the URT system before arriving at their destination, and the green triangle indicates that passengers will wait for service recovery. It can be seen that within 30 minutes after the interruption, almost all the check-in passengers exited the URT system. Among them, passengers heading to Ranjiaba did not stop leaving the system until 10 minutes before the service recovery, while passengers heading to Beibei and Xiaoshizi tended to spend more time waiting for the service recovery. The network topology shows that Huahuiyuan and Ranjiaba are both in the interrupted area, and the distance is short. Therefore, in most cases, the cost of waiting for service recovery is higher than the cost of taking other means of transportation. However, Xiaoshizi and Beibei are outside the interrupted area and far from Huahuiyuan station, so waiting for service recovery is a better choice. Therefore, passengers heading to these two stations tend to stay when the service is about to resume.

Fig. 13
figure 13

Distribution of passengers’ behavior heading to different stations

Full size image

Figure 14 presents the top 20 stations with significant changes in outbound volume. The normal outbound passenger flow (as shown in the blue column in the figure) is derived from the simulation results without interruption. The outbound volume under interruption (as shown in the purple column in the figure) of most stations in the interruption area has decreased because the interruption prevented passengers from arriving at these stations. On the contrary, the outbound volume of Lijia station and Hongqihegou station has increased dramatically, and about 90% of these outbound passengers are those who disembark at stations other than the planned ones. In other words, affected by the interruption, passengers who planned to go to these stations are forced to disembark at Lijia station and Hongqihegou station. Two factors cause passenger flow changes in Ranjiaba and Dalongshan. On the one hand, as the transfer hub of Line 5 to other lines, many passengers from Line 5 are forced to disembark at these two stations due to the interruption. On the other hand, some passengers heading to these two stations are blocked at Lijia and Hongqihegou stations.

Fig. 14
figure 14

Comparison of outbound volume

Full size image

To further analyze the impact of interruption on passengers from the time dimension, we selected three groups of OD pairs to compare the planned destination and actual terminal of passengers, as shown in Fig. 15. The three groups of OD pairs are (1) Xiaoshizi to Hongqihegou, (2) Xiaoshizi to the stations in the interrupted area, and (3) Xiaoshizi to the stations of Line 5. The horizontal axis is the passengers’ check-in time at Xiaoshizi station, and the vertical axis is their planned destination. At the beginning of the interruption, the passengers in groups 2 and 3 disembarked at Hongqihegou to end their journey, which increased the outbound volume of Hongqihegou station. From 11:24, an increasing number of passengers were able to reach the destination because the service was almost restored when these passengers arrived at Hongqihegou. In this case, the cost of turning to other forms of transportation is higher than the cost of waiting for service recovery, so passengers increasingly choose to stay in the URT system. This may cause many passengers to be detained at Hongqihegou station simultaneously, impacting the platform order. The above analysis helps us understand how to improve the service level by formulating plans. For example, we can effectively evacuate the passenger flow stranded at Hongqihegou station by adding some shuttle buses between Ranjiaba and Hongqihegou station.

Fig. 15
figure 15

Comparison of passengers’ planned destination and actual terminal

Full size image

Figure 16 presents the top 10 stations with significant changes in transfer volume. During the interruption, the number of passengers transferring at Hongqihegou decreased by about 4000, ranking first in the whole network. When the interruption occurs, passengers heading to the stations in the interrupted area and Line 5 are forced to give up traveling, which leads to a sharp decrease in the transfer volume of passengers passing through the Hongqihegou station. At the same time, the outbound volume of Hongqihegou jumped from 1800 to 3500, an increase of about 90%, as shown in Fig. 12. These two changes indicate that many passengers are forced to give up their travel due to the interruption. In comparison, the change in transfer volume at Lijia station is much smaller. This is because the number of stations with Lijia station as the transfer hub is far less than that of Hongqihegou station.

Fig. 16
figure 16

Comparison of transfer volume

Full size image

The transfer volume of Yuelai station demonstrates the most significant growth rate, up to 350%. In addition, the transfer volume of ChongqingBS, Lijia, and Hongtudi stations increased to varying degrees. Passenger detours may cause an increase in transfer volume at the above stations. To further explore the reasons, we select several groups of OD pairs to analyze the travel behavior of passengers under interruption.

Figure 17 shows the distribution of passenger travel behavior from Xiaoshi to Lijia, Caijia, and Beibei stations between 10:28 and 12:00. The horizontal axis indicates when passengers access the URT system at Xiaoshizi station. The vertical axis is the passengers’ planned destination. "A" means that passengers are blocked in the interruption area and give up their travel, as shown in Fig. 18a. "B" means that passengers are forced to switch to the detour path, as shown in Fig. 18b. "C" indicates that the passenger arrives at the destination along the planned detour path, as shown in Fig. 18c. "D" means that the passengers switch to the shortest path midway from the detour path, as shown in Fig. 18d, and "E" indicates that passengers travel along the shortest path after the service is resumed. It can be seen that passengers departing at around 10:30 were forced to exit the URT system because these passengers were just in the interruption area when the interruption occurred. When the interruption occurs, those passengers do not have an alternative path to their destination and have to wait a long time for the service recovery, so exiting the URT system and turning to other modes of transportation is a better option. For passengers departing between 10:33 and 11:50, they either switch to the detour path from the planned path (as shown in Fig. 18b) or travel according to the detour path (as shown in Fig. 18c). It should be noted that although the passengers’ planned path is still a detour path when the service is close to recovery, they may switch to the shortest path midway, as shown in Fig. 18d. That is, service recovery will also affect the travel behavior of passengers.

Fig. 17
figure 17

Distribution of passengers’ travel behavior

Full size image
Fig. 18
figure 18

Diagram of passengers’ detour behavior under interruption

Full size image

Therefore, we can conclude that the detour caused by the interruption will increase the transfer volume of Hongtudi, ChongqingBS, Yuelai, and Lijia stations. In other words, these stations bear part of the passenger volume that should be transferred at Hongqihegou Station.

In general, the travel behavior of passengers is very complex, and studying the distribution of subway passenger flow under interruption is challenging. The above analysis shows that the simulation model can estimate the subway passenger flow distribution under interruption at the aggregation level and explain the reasons for passenger flow changes from the non-aggregate level by generating passenger trajectories. Therefore, although the calculation accuracy of the simulation model is still affected by such factors as parameter setting, it is still an effective means to evaluate the changes in passenger flow.

6 Conclusion

This paper provides a detailed analysis of the impact of interruptions on passenger path choice behavior and proposes an event-driven simulation model to calculate the URT passenger flow distribution under interruption events. Taking the simulation results without interruption as a benchmark, the changes in passenger flow are compared and analyzed using Chongqing Metro data. At the individual level, the simulation results show that not only that the interruption occurrence and duration will affect the passenger’s travel decision, but also that the service recovery will affect the passenger’s travel path, which has not been mentioned in previous studies. At the aggregate level, it focuses on analyzing the changes in important indicators such as the transfer volume and the number of boarding and alighting passengers at some stations. The output of multidimensional results is of great significance for the operating company to evaluate the impact of interruption events on the URT network and passenger flow and to formulate emergency plans.

Since the operator will provide guidance information for passengers when the interruption occurs, we assume that passengers are fully informed and rational—that is, they always choose the shortest path to travel. However, passengers’ travel decisions are affected by many factors. Therefore, this assumption does not fully consider the heterogeneity of passenger path choice. Some studies on passenger compliance with guidance information indicate that some passengers may not comply with guidance information. For example, Small [32] pointed out that passengers’ perception of the value of time strongly depends on very specific factors. Therefore, in the future, it may be an interesting research direction to consider passenger heterogeneity or compliance in studying interrupted passenger flow distribution. In terms of passenger flow distribution results, the OD travel volume obtained at the aggregation level has high accuracy, but the weighted average error of passenger travel time at the individual level also reached 12.7%. Although the absolute error of travel time is about 3.18 minutes, it cannot be ignored. This error is affected by many factors. In the future, we will consider improving the calculation accuracy by refining the passenger travel process and collecting more data. In addition, the path utility of this paper only considers the travel time and transfer times. An increasing number of studies show that congestion and crowding will also affect the path choice of passengers. Therefore, congestion/crowding factors in the path utility function is another interesting topic to consider.