Abstract

This study focuses on the large passenger flow control problem, after an operation interruption occurs, to develop a methodology that can efficiently control the passenger inflows of multiple stations and avoid overcrowding inside stations. An early-warning model for irregular large-scale passenger flows (ILSPF) and a dynamic ILSPF control model are proposed. The early-warning model is developed to predict passenger flows in the future with historical data and detect when to start control measures in actual time. The ILSPF cooperative control model focuses on cooperatively controlling the passenger inflows of multiple stations to ensure passenger safety in vehicles and stations, as well as maximize the number of passengers transported and minimize the passengers’ total waiting times. An improved particle swarm optimization algorithm was designed to determine an optimal solution, and a case study on the Chengdu metro in China was carried out to examine the performance of the model. The obtained results verify the effectiveness of the model and algorithm and prove that ILSPF control can regulate the passenger inflow demand, better match the passenger demand and capability on the line, increase the total number of passengers transported, and balance the proportion of passenger boarding at each station.

1. Introduction

Recently, urban rail transit (URT) preferred by more and more passengers and government word-wide because it is safe, rapid, and efficient. However, as passenger demand continues to increase, many URT systems are facing increasing pressure as well as challenges. In some metro systems, passengers even need to queue and wait for a long time outside the station during peak hours because of the insufficient transportation capacity of metro systems [1]. Even worse, emergent events, such as operational disruptions and facility failures, may cause passengers to gather in halls and on platforms in a short time, consequently resulting in large-scale passenger flows (LSPF) in stations. Unlike regular large-scale passenger flows (RLSPF), which usually occur during peak hours, irregular large-scale passenger flows (ILSPF) suddenly and dynamically change over time, which may cause accidents, pose hazards to passengers as well as put enormous pressure on URT operators. Thus, it is important to find efficient control solutions to eliminate the negative impacts of the ILSPF.

In general, there are two strategies for dealing with LSPF. One is to increase the transportation capacity of the URT system to meet the travel demands of passengers. The other is to control passenger inflows to respect the limitation of the transportation capacity. The second strategy relies on controlling the passenger inflow speed to adjust the number of passengers in the station halls and platforms. Commonly used measures include closing fare gates, closing turnstiles, setting up barriers, and setting up one-way channels. With the increase in passenger flow volume in URT systems, passenger flow control has become a regularly used strategy on a daily basis in many cities. Moreover, different cities usually have their own predesigned regulations to handle regular large-scale passenger flows. For example, for the Beijing metro system, the Beijing Municipal Administration of Quality and Technical Supervision issued “Regulations on the Operational Safety Management of Urban Rail Traffic” in 2013 [2], which states that when passenger flow exceeds 70% of the station capacity, passenger inflow control measures should come into effect to avoid overcrowding. However, the regulations may not apply to other URT systems because they have different features of passenger flows, networks, infrastructures, etc., To improve passenger satisfaction and reduce operational costs, increasing attention is being paid to building efficient passenger flow control methods.

Research on passenger flow control has been conducted both at microscopic and macroscopic levels. At the microscopic level, scholars have studied pedestrian behavior and evacuations within stations. Stubenschrott et al. [3] presented a dynamic pedestrian route choice model based on continuous observations of perceived time estimations. Wan et al. [4] developed a method for simulating crowd evacuations in subway stations by using the basic theory of the social force model combined with the Gaussian puff model. Kim et al. [5] studied the influence of subway station congestion on pedestrian path selection by using a logit model. Chen et al. [6] proposed a multiagent-based model for pedestrian simulations in subway stations. Zhang et al. [7] proposed a linear-programming-based feedback control model to compute optimal passenger inflows under different facilities and velocities to improve the level of service. These studies, however, primarily focused on the relationship between individuals and station facilities, as well as the method of passenger flow control in a single station, with little attention paid to cooperative passenger flow control for multiple stations.

At the macroscopic level, studies have focused on passenger flow control at different stations of the URT network. A passenger flow network was developed for a subway system to find links between stations, and a corresponding optimal passenger flow control method was established [8, 9]. The problem of passenger flow control on URT networks is similar to that of ramp metering on highways, which aims to adjust the number of vehicles entering a segment of a highway in real time to avoid highway congestion during peak hours. The linear programming method is commonly used in the ramp metering problem with the objective of maximizing the total inflow volume under highway capacity [1015]. Similar to the ramp metering problem, the purpose of passenger flow control for a URT network is to transport most passengers, while the security and level of service are kept within an acceptable level using certain measures [16]. Guo et al. [17] built a model for cooperative passenger inflow control in the entire network with constraints on the capacity of the station, and experiments were conducted on a simple network with many-to-one OD (origin-destination) flow. Yao et al. [18] developed a passenger flow control model that aims to minimize passenger delays and improve railway capacity usage, and an empirical case of the Beijing subway network in the morning peak time was presented. Jiang et al. [19] developed a mathematical model for limited boarding and stop-skipping in which the objective was to maximize passenger profit during peak times. Yin et al. [20] constructed a single-line equilibrium passenger flow control model that minimized total passenger delay. Some scholars have combined passenger flow control methods with other methods to develop composite strategies. Yang et al. [21] tackled the issue of metro system congestion and developed a compound strategy integrating passenger flow control and a bus-bridging service. Gong et al. [22] proposed an integrated optimization approach combining train timetabling and collaborative passenger flow control with the aim of minimizing the indicators associated with passenger service imbalance and train loading capacity utilization.

From the abovementioned studies, it can be concluded that most of the studies have focused on regular passenger flow control problems, such as peak hours, whereas relatively few studies have considered irregular passenger flow. RLSPF often occurs during peak hours. It usually repeats every working day and is easy to predict. URT operators often adopt predefined passenger flow control measures as well as predefined start and end times to control their operations. In contrast, ILSPF occurs shortly after disruption of metro traffic and is difficult to predict. Because of its unpredictability, it is difficult to estimate the start time and procedures of control measures in ILSPF.

To fill this knowledge gap, this study focuses on the ILSPF control problem after an operational interruption occurs. Our aim is to develop a methodology that can be applied to efficiently control passenger inflows of multiple stations to avoid overcrowding inside the stations. An early-warning module for the ILSPF and dynamic ILSPF control models are proposed. The early-warning module predicts passenger flows in the future using historical data and determines whether passenger flow control should come into effect. The ILSPF control model aims to reduce passenger delays by controlling passenger inflows at multiple stations. To handle unpredicted and irregular passenger flows, on the one hand, we discretize the time into small time intervals so that the proposed passenger flow control method can adjust its control variables according to real-time passenger flows at each time interval. The proposed method on the other hand, takes into account the cooperation of multiple stations; that is, we work on the passenger flow control of multiple stations, instead of a single station, to achieve an optimal balance of the passenger flows in all involved stations. An improved particle swarm optimization (PSO) algorithm is designed to solve the proposed cooperative control model, which is computationally simple and provides fast convergence, as well as a strong global search ability, for quickly solving the complex solution problem in this study. The proposed method was tested using data from Chengdu Metro Line No. 1. The case study results demonstrate the efficiency of the proposed method.

In summary, the main contributions of this study are listed as follows:(1)An early-warning model is proposed that monitors the real-time passenger flow, predicts the passenger flow according to historical data, and alarms station operators to take actions when the predicted passenger flow exceeds the critical limits.(2)A cooperative control model for the ILSPF is proposed, which aims to increase the number of passengers transported and reduce passengers’ waiting times. The control model coordinates passenger inflows of multiple stations instead of focusing on a single station to balance passenger flow pressures at different stations.(3)An improved PSO algorithm is designed to solve the cooperative control model, which is computationally simple and offers fast convergence as well as a strong global search ability for quickly solving the complex problem posed in this study.

The remainder of this paper is organized as follows. We first present a detailed problem description and then introduce the proposed model of ILSPF cooperative control as well as the solution method. Then, a case study is designed to demonstrate the proposed control method for ILSPF. Finally, Section 4 concludes the study with several future research directions.

2. Materials and Methods

2.1. Problem Description

Figure 1 presents a typical URT line with stations and L sections. Let the names of the stations be 1, 2, …, i, …, I. Let the segment between adjacent stations be called a section. Thus, in total, there are I-1 sections. Section l represents the section between stations i and i + 1. All train services start their trips from station 1 and head towards station . The train service repeats at regular headway times and stops at each station for passenger boarding and alighting. If the number of passengers who want to enter the station exceed the station capacity and the number of passengers who want to board the train exceed the train capacity, overcrowding and consequent large-scale passenger flows will occur. To avoid overcrowding, passenger flow control measures are proposed to limit the number of passengers entering the stations. Commonly used control measures include setting railings, closing entrances, closing fare gates, etc., (Figure 2).

This study focuses on the ILSPF control problem due to various operational interruptions caused by emergent events, such as operational accidents and facility failures. The control approach proposed in this paper aims to dynamically adjust passenger inflows of multiple stations to minimize passenger waiting times and maximize the number of passengers transported by the URT system. It coordinates passenger inflows of multiple stations instead of focusing on a single station, to balance passenger numbers at different stations. Figure 3 presents an example in which the ILSPF occurs at station C. Therefore, the train is fully occupied by passengers waiting at station C, and passengers in downstream stations D–G cannot get onboard (Figure 3(a)). This could lead to crowding at stations D–G and the dissatisfaction of passengers who fail to get onboard. In Figure 3(b), a certain measure for passenger flow control is implemented at station C to reduce the number of boarding passengers. Thus, the train capacity is not fully used and can be reserved for passengers at downstream stations. Accordingly, the propagation of large-scale passenger flows from station C to other stations is avoided. However, it is also important to avoid leaving too many passengers waiting at station C, resulting in an overcrowded station C. In summary, we propose a cooperative passenger flow control method to achieve a balance in the number of passengers at different stations.

This work also aims to develop a model to control passenger flows dynamically; that is, the start and end times of the control are determined based on the real-time passenger flow status, and the control variables are adjusted dynamically over time. We split the time of control into small intervals, and the length of each interval is represented by (e.g.,  = 15 min). When an emergent event occurs, an early-warning module is used to predict passenger flows on a further horizon and check whether the ILSPF occurs. In the event that an ILSPF is detected, the ILSPF control module will come into effect, which will control the passenger inflows of multiple stations at every time interval. The procedure for passenger flow control for the ILSPF is shown in Figure 4. The time interval within the early-warning process is represented by the mth time interval, whereas the time interval in the entire passenger flow control process is represented by the nth time interval.

2.2. Notations

The notations used in the following text are given in Table 1.

2.3. Early-Warning for ILSPF

The ILSPF early-warning model determines whether passenger flows in the stations reach the level of a large passenger flow. If a large passenger flow is detected, a warning message about the ILSPF is sent to the URT operators, and a corresponding ILSPF control solution must be generated as soon as possible.

The early-warning process concludes with a prediction and an alarm. The prediction here refers to the prediction of future passenger flow based on historical data. The causes of large passenger flows can generally be divided into two categories. One is the sudden increase in passenger flow, under which the total amount of passenger flow has changed. Another is the change in the URT network, or train organization, caused by emergency events, leading to congestion at some stations. This study focuses on the second situation. It is assumed that the total amount of passenger flow remains unchanged; thus, we can use historical ticket data to predict the number of passengers entering and leaving stations during each time interval.

As shown in Figure 5, the prediction is based on the current traffic state (number of passengers in sections and platforms at the current timestamp, and ) and the predicted number of passengers getting in and out of all the stations ( and , ). The two parameters were constructed from historical data. Based on this information, the number of passengers in sections () and on platforms () can be predicted. The prediction model is as follows:

It is assumed that all passengers waiting on a platform can get onboard. After the prediction, we determine whether the predicted passenger flow reaches the ILSPF level using two indicators. The first indicator is the number of passengers waiting on the platform in each time interval . The second indicator is the number of passengers transported by trains within each section in each time interval . If the two variables exceed the platform capacity or section capacity, it is necessary to issue a warning message about the ILSPF to URT operators to take actions in controlling passenger inflows to avoid overcrowding in stations.

2.3.1. Platform Capacity Restriction

The number of passengers standing on the platform is limited owing to the platform layout and its effective area. The number of passengers on the platform should not exceed its capacity. Taking the platforms in station i as an example, let denote the effective area of the platform and denote the maximum passenger density on the platform. The platform capacity can then be estimated as:

Most URT services follow the “alighting first, boarding second” rule, so, the most crowded moment on the station platform is when the alighting passengers have just gotten off the train but have not left the platform, and the boarding passengers have not yet boarded the train. In other words, the sum of passengers waiting on the platform and getting off the train cannot exceed the platform capacity:Where, represents the number of passengers waiting on the platform at station within the th time interval, and is the number of passengers getting off trains at station within the th time interval. refers to the number of trains expected to arrive at station during the th time interval (often more than one train arrives at station within a given time interval).

2.3.2. Section Capacity Restriction

The number of passengers passing through a rail section within a time interval is limited owing to the constraints of vehicle capacity and train frequency. We introduce variable to represent the maximum capacity of section l in the mth time interval, which can be calculated aswhere, C is the standard vehicle capacity, is the number of trains passing through section l during the mth time interval, is the maximum load factor (130%). The load factor is used to indicate the congestion level of the vehicle. It is equal to the ratio of the number of passengers on the train to the standard capacity of the train. For passenger safety, section flow should not be greater than section capacity.

2.4. Multi-Station Cooperative Control Model

Once a warning message about the ILSPF is sent to URT operators, an ILSPF control solution needs to be generated as soon as possible to help URT operators better control passenger inflows in stations. Here, ILSPF control refers to controlling the entrance rate of passengers to avoid overcrowding on trains and platforms. This study introduces an ILSPF cooperative control model that coordinates the passenger inflows of multiple stations to maximize the number of transported passengers and minimize the passengers’ total waiting times.

2.4.1. Constraints

In Figure 6, represents the number of passengers waiting outside station i during the nth control time interval, and is the number of passengers allowed to enter station i during the nth control time interval. is equal to

Equation (7) states that the number of passengers who must enter station i during the nth control time interval consists of two parts: represents the number of passengers arriving at station i during the nth control time interval. represents the number of passengers who cannot enter station i during the (n − 1)th control time interval. In the first control time interval, the number of passengers waiting outside the stations was assumed to be zero.

The number of passengers who are allowed to enter the station is less than the number of passengers waiting to enter the station, that is,

Variable is introduced to represent the control rate of passenger inflow at station I:

Often, each station has a maximum control rate, , to avoid long queues outside stations. The passenger inflow at each station should be greater than the number of passengers entering the station for the given maximum control rate. That is

The number of passengers waiting on platforms can be calculated aswhere, is the number of passengers waiting on the platforms at station i during the nth control time interval, and is the number of passengers transported at station i during the nth control time interval. The number of passengers transported depends on the number of passengers on the platforms and the remaining capacity of the train service. That iswhere, is the remaining capacity of the train service at station during the nth control time interval, which is calculated aswhere, represents the maximum section capacity of the th section during the nth time interval, and represents the section flow of section during the nth time interval. Similar to equation (6), the section flow cannot exceed the maximum section capacity, that is, in equation (14) depends on the number of boarding passengers at the previous stations. Parameter is introduced to represent the percentage of passengers whose destination is the ith station among the boarding passengers at station j. It is assumed that the values of for different OD values are constant. For the ith station, the number of passengers getting off at the ith station during the nth time interval is

To ensure passenger safety, the number of passengers waiting on platforms cannot exceed the maximum capacity of the platform. That is

Equation (17) is similar to equation (4), which restricts the number of passengers waiting on the platforms. The differences between equations (4) and (17) are as follows: in equation (4) is obtained from the historical AFC data, whereas, in equation (17) is computed using equation (16).

2.4.2. Objective Function

The model aims to minimize the total waiting time of passengers while trying to maximize the number of passengers transported to reduce crowding in stations by using limited facilities, equipment, and the URT service efficiently. The objective functions were defined considering the above aspects.

(1) Minimize Passenger Waiting Times.

The first objective is to minimize the total waiting time of passengers caused by control measures. That iswhere, represents the number of passengers who cannot enter and must wait outside station during the nth control time interval owing to the implementation of the control measures. is the length of the control time interval.

(2) Maximize the Number of Transported Passengers.

The second objective is to maximize the number of passengers transported. The objective function is written as follows:where, is the number of passengers on trains passing through section l during the nth control time interval.

2.5. Solution Algorithm

PSO was adopted to solve the proposed control model. Eberhart and Kennedy [23] compared PSO with other optimization algorithms such as Simulated Annealing, Genetic Algorithms, and Tabu Search and concluded that PSO is computationally simple and offers fast convergence and a strong global search ability, which is suitable for quickly solving complex problems. The basic principle of the PSO algorithm is as follows. A group containing a certain number of particles moves in the search space, and each particle represents a potential solution to a specific optimization problem. The position of each particle in the group is updated by tracking the individual and group extreme values. In the update process, the fitness value is calculated at each iteration. Usually, the objective function of the problem is used as the fitness function, and the individual and group extreme value positions are updated by comparing them with the fitness values of the new particles.

In particular, an improved particle swarm algorithm was used in this study. Suppose that the number of particles in a multidimensional search space is Y. A D-dimensional vector can be used to represent the position of the particle: particle position represents the fitness value, calculated from the objective function of the specific problem. Each particle can be represented by its current speed, individual extreme position, or group extreme position. represents the velocity, represents the individual extreme value, and represents the group extreme value of the particle, respectively. During each iteration, the particles update their position and velocity using their individual and group extreme values. The evolution process obeys the following equation.where, is the inertia weight that represents the search capacity. Time is the current number of iterations. and are the acceleration factors that represent the learning experiences of the particles. and are uniform random numbers in the range [0, 1], and the velocity of the particle, .

As there are decision variables in the model, the spatial dimension of the solution is . The passenger flow control scheme can be expressed as . The decision variable is the optimal passenger inflow of each station at every time interval, and the range of the solution is .

The calculation steps of the algorithm are as follows:

Step 1. Initialize parameters
The particle population number Y is initialized to 50, iteration number time to 800 (after 800 iterations of experiments, the objective function value is convergent), inertia weight to 0.8, and learning factor . The initial values of the position x and velocity of the particle swarm are set by generating a random matrix with values in the range [1, 100000] and [−1, 1], respectively:

Step 2. Calculate fitness value
Constraints (7–17) are converted into a penalty function. The objective function (18, 19) with the penalty function in the proposed cooperative control model is chosen as the fitness function in the algorithm. The fitness function is used to calculate the fitness function value fitness of each particle. When the fitness value is the maximum, the optimal value of a single particle is initialized to , and the optimal value of the global population is initialized to .

Step 3. Search for the optimal position of individual particles
The fitness function value fitness of particle position is calculated during time + 1 iterations. This value is compared with the historical optimal fitness function value fitness . If the latter is better than the former, the individual particle optimal position is updated. Otherwise, .

Step 4. Search for the optimal position of all particles
The fitness function value is compared with the global optimal fitness function value . If the is better than , the global particle optimal position is updated; that is, , otherwise, .

Step 5. Update the speed and position of the particles according to equations (22) and (23) and continue the iterative calculation.

Step 6. When the set number of iterations, time = 800, is reached, stop the calculation process, and set the solution of the last step of the iteration as the optimization objective. Otherwise, set time = time + 1 and enter the next cycle.
The steps involved in using the PSO algorithm to solve the proposed control problem are shown in Figure 7.

3. Results and Discussion

3.1. Case Scenario

The proposed model and algorithm were demonstrated and analyzed through a case study. The example is based on Line No. 1 of the Chengdu Metro (Figure 8) in China. The Chengdu Metro consists of 12 lines with 373 stations, including 46 transfer stations. Line No. 1 has a total length of 41 km and 35 underground stations. The passenger flow volume of the Chengdu Metro network reached a new record of 7,224,300 on April 30, 2021. An emergency event, such as an operational accident or facility failure, has a huge impact on the network, given the large passenger volume.

Here, we take the down direction of Line No. 1 from Weijianian to Science City as an example. The map of Line No. 1 is shown in Figure 8. Suppose that an accident occurs on Line 1 at 09: 00, resulting in a longer running interval between two trains. Some information on Line No. 1 is presented in Table 2.

3.2. Parameter Settings

In practice, it will take some time to set up the control facility. Since it is difficult to change measures frequently within a short period, the time interval was set to 15 min. In the case study, the total time for implementing the control measures was set at 2 h. Tables 3 and 4 show passenger inflows and outflows at different stations in Line No. 1. The first columns in the tables list the station names, while the rest indicate the passenger inflows and outflows within the nth time interval.

The effective area of the platform is related to its layout. The effective area of each platform in Line No. 1, determined through a field investigation, is listed in Table 5.

The train capacity C was calculated according to the standard train load (6 B marshalling), and its value was 1259. denotes the maximum number of passengers accommodated in a train, generally taken to be 1.3 times the standard capacity; that is,  = C × 130% = 1637.

4. Results and Analysis

After the accident occurred, passenger flows in each time interval were predicted to determine whether they reached the standard of control. As shown in Table 6, in the third time interval, the load factors from sections 115 to 117 exceeded 130%. Therefore, control measures for passenger flow must be implemented.

For this case study, the PSO code was programed to solve the model, and the optimal value of the passenger inflows for each station in each time interval was obtained. Figure 9 presents the convergence curve of the optimal values over iterations. The solution to passenger flow control (the control rates at different stations at different control time intervals) is shown in Table 7. The results show that there are mainly 10 control stations, from Luomashi to Financial City. Because the flow control intensity was limited, the maximum control rate was 50%, and some control rates at Tianfu Square and Sichuan Gymnasium reached 50%.

The numbers of boarding passengers at each time interval are listed in Table 8. The total number of passengers transported and the passenger waiting times are listed in Table 9. According to statistics, the total demand for passenger flows is 112301 people. Without control, 99811 passengers were transported, while the rest were stranded. With the proposed control measures, all passengers would be transported within two hours. In addition, the average waiting time of passengers would be reduced from 8.76 minutes (without control) to 7.63 minutes (with control).

The equalization coefficient θ of the boarding ratios at the stations was used to evaluate the effects of the strategy. This is the mean square error of the ratio of boarding passengers to waiting passengers at each station. The smaller the value, the more balanced is the proportion of boarding passengers at each station. θ can be calculated as:

The results are presented in Table 10 and Figure 10. The equalization coefficients of the boarding ratios without control were low in the first four periods. The passenger flow then increased, and the degree of congestion increased accordingly. Without control, passengers at upstream stations would quickly saturate the capabilities of the trains and sections, resulting in passengers being stranded at stations. After control, there were fewer stranded passengers, the overall fluctuation of the equalization coefficients was smaller, and the ratio of boarding passengers was more balanced.

In summary, this case study demonstrates that in the event of an emergency, such as facility failure, the proposed model effectively improves the utilization of the transport capacity, alleviates the problem of passengers being stranded, balances the passenger flow pressures at different stations, and consequently ensures the safety and security of the URT operations. The evaluation and analysis also show that the proposed model can achieve system optimization, thus verifying the effectiveness of the proposed methodology.

5. Conclusions

In a URT system, emergent events such as operational accidents, facility failures, and extreme weather may result in ILSPF events that are different from their counterparts during peak hours. This study focuses on modeling cooperative control for ILSPF in a URT system. An early-warning module for the ILSPF and a dynamic ILSPF control model are proposed. The early-warning module predicts passenger flows in the future using historical data and determines whether passenger flow control should come into effect. The ILSPF control model aims to reduce passenger delays by controlling passenger inflows at multiple stations. A case study on Line No. 1 of the Chengdu Metro in China demonstrated the performance of the proposed methods.

This study predicts the ILSPF using historical data by assuming that passengers do not abandon their travel by the URT system and shift to other transport modes when emergent events occur. This assumption and passenger travel behaviors in the case of operational disruptions will be investigated in the future. In addition, the judgment condition for starting the passenger inflow control is simple. More reasonable conditions for starting passenger inflow control, considering the duration of the emergent event, the volume of passenger flow, etc., are topics for further research. [24].

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study was financially supported by the National Natural Science Foundation of China (71701152, 72071147), Research Program of Science and Technology Commission in Shanghai (18510745800), and Shanghai Sailing Program (21YF1450200). The authors acknowledge Chengdu Metro Co., Ltd. for providing raw data during the research.