1. Article information
"Collaborative passenger flow control for oversaturated metro lines: a stochastic optimization method" is an article published in Transportmetrica A: Transport Science in 2021.
2. Summary
With the rapid growth of travel demand in urban areas, large passenger flow has become a common phenomenon in the subway systems of some large cities. In order to ensure the safety of the subway system and improve the operational efficiency, this paper proposes an effective method to formulate a robust passenger flow control strategy for subway lines, specifically considering the stochastic and dynamic passenger flow. By discretizing the time horizon into a series of time intervals, we propose an integer linear programming model whose goal is to minimize the expected passenger waiting time on a subway line. To solve the proposed model, a heuristic algorithm integrating Lagrangian relaxation method and CPLEX solver is designed to search for high-quality solutions to the research problem. Finally, two sets of numerical experiments, including a small-scale case and a real instance, verify the performance of the proposed method.
3. Introduction
This paper makes the following contributions in the field of subway passenger flow management:
(1) Use scene-based time-varying data to capture the dynamics and randomness of passenger flow; each scene corresponds to a set of single-day dynamic passenger flow data. Using these data, a robust passenger flow control method for different scenarios is proposed. The goal of this is to ensure that the resulting robust policy can counteract the diversity of passenger demand and ensure safe operation of the platform.
(2) A new integer linear programming model is proposed to characterize the problem of interest. The objective of the objective function is to minimize the total waiting time of passengers in the station hall and platform. In addition, some system constraints are considered, such as constraints related to train loading capacity, coupling of control strategies in different scenarios, and passengers getting on and off the train, etc.
(3) A heuristic algorithm based on Lagrangian relaxation is proposed to efficiently solve the model. Specifically, through dual coupling constraints, the main model is decomposed into a series of scene-related passenger flow control models, and the CPLEX solver is used to solve them. Next, the upper and lower bounds of the main model are iteratively updated using a subgradient algorithm. Finally, the effectiveness of the method is verified by two sets of experiments.
4. Problem description and assumptions
In this study, we consider a one-way subway line with n stations, where 1 and n stations denote the origin and destination stations, respectively. The trains involved run according to the given timetable during the scheduled operation. For modeling purposes, we assume that the trains in service can satisfy all passenger demands during the considered time frame (i.e., there are no passengers stranded at each station at the end of the operating period). Additionally, a set of historically recorded footfall data can be used in the decision-making process, where each set of day-specific footfall data corresponds to a random scenario with a predefined probability. Due to the dynamic changes in actual passenger demand, the passenger arrival rate is usually time-dependent (i.e., the arrival rate is large during peak hours and relatively small during off-peak hours). Therefore, this set of historical data can be used to describe the dynamics and randomness associated with actual passenger flow. It is worth noting that the proposed model can be applied to two-way subways when we take passenger travel demand in both directions as input data. However, since passenger congestion always occurs in one direction during a fixed time period (i.e., morning and evening rush hours), we model the problem as a one-way subway line.
In practice, passengers arrive at each station in a sequential manner within the time frame. Therefore, the passenger arrival rate at each station can be expressed as a continuous function of time. In order to simplify the modeling process, we specifically discretize the planning time range into a finite number of time steps with a time step of δ, using T={t0, t0 + δ,..., t0+Nδ}, where two adjacent time steps The length of each time interval between is δ. It can be seen that when the time length δ is small, the arrival rate of passengers in each section can be approximately regarded as a constant. Next, for the convenience of description, we denote the total number of arriving passengers whose journey is i→j within the time interval (t−δ, t] under the scenario ω as λi,j,t(ω). Similarly, the number of passengers allowed to enter the platform The number of passengers (that is, the control rate) can also be expressed in a similar way, that is, by βi,j,t(ω).In this way, we can roughly calculate the number of passengers arriving and boarding until each time step t∈T Quantity. Mathematically, the passenger arrival rate is time-related, and in practice it changes continuously. Figure 2 uses a discrete time step in the form of δ=1 min, showing the number of arriving passengers at different time intervals (the figure is Beijing The number of arriving passengers at the Tuqiao Station on the Batong Line of the Metro during the morning rush hour on weekdays in 2016).
As mentioned before, in order to capture the stochastic characteristics of passenger arrival rate in the subway system, we use the historical passenger flow data of different days to construct stochastic scenarios. Specifically, the set of considered scenarios can be represented by Ω = {ω1, ω2, ..., ω|Ω|}, where each sample ω ∈ Ω is associated with a probability pω. For each scenario, the passenger arrival rate is usually dynamic due to the diversity of passenger demand. Therefore, using this type of data representation, our goal is to handle stochastic passenger flows in order to generate robust passenger flow control strategies under different scenarios.
In the process of passenger flow control, passengers with different OD trips should be distinguished through the entrance facilities in the station hall. To achieve this, each station has developed detailed passenger entry principles, as described by Shi et al (2018). For clarity, Figure 3 shows the passenger flow control environment. It can be seen that several entrance facilities {k+1, k+2, ..., n} are set up in the station hall. Each facility corresponds to a designated terminal, and passengers can only enter the platform through the designated facility according to their destination. That is to say, each entrance facility is od-oriented, and the smart card should be used normally between the entrance facility and its connected destination station (otherwise an additional penalty will be paid). In fact, with the development of intelligent technology, this operation setting can easily meet the practical application. For example, when arriving at station k, passengers whose destination is n−1 (arrival rate λk,n−1,t) need to line up in the station hall first; then, according to the control strategy βk,n−1,t, passengers are allowed to pass through the entrance facility n−1 enters the platform. Once the number of passengers arriving during peak hours exceeds the remaining capacity of the train, some passengers will be stranded in the station concourse to avoid piling up on the platform. In addition, we require that once the train leaves the current station, no passengers will be stranded on the platform. By using this method, a large number of arriving passengers can be confined outside the boarding area. Over time, according to the control strategy, crowded passengers are gradually allowed to enter the platform and board the train. Since arriving passengers are prohibited from congregating on the platform, the operation safety of the subway system can be guaranteed.
Assume that the travel needs of passengers are random (for example, the travel needs of passengers are different on different days). Figure 4 shows the generation of robust control strategies for different scenarios based on OD centering dynamic passenger travel. Obviously, there are 4 stations along the subway line, and 4 trains in service are considered to meet the travel needs of passengers. Suppose the train capacity is 100 (unit: person), and the headway between adjacent trains is set to 5 minutes. Each train needs to stay at each station for 0.5 minutes, and the travel time between two adjacent stations is also set at a fixed 2 minutes. The passenger travel conditions of OD pairs in the two scenarios are shown in Figure 4(a, b), where the data in the green and blue boxes represent the number of passengers arriving in Scenario 1 and Scenario 2, respectively. The probability of each scenario is set to ps = 0.5. We assume that in both cases, passengers with trip S1→S4 arrive earlier than other passengers. Furthermore, the passengers involved can arrive at the departure station before the departure of the first train, and all passengers can be transported to their destinations.
In the case of no passenger flow control, passengers in S1→S4 section take the first train according to the first-in-first-out principle in both cases. Therefore, since train 1 is fully loaded at station 1, all passengers on trips S1→S2 and S1→S3 have to wait for trains 2 and 3 at station 1, which usually increases the total waiting time for passengers. In this case, the total waiting time of passengers in Scenario 1 and Scenario 2 is 4125 and 4450 min, respectively. Therefore, the total expected passenger wait time is 0.5 * 4125 + 0.5 * 4450 = 4287.5 minutes. Note that this study aims to study robust passenger flow control strategies in different scenarios. By implementing the control strategy for the two scenarios, as shown in Figure 4(c), passengers on the S1→S4 journey are controlled to wait in the station hall, and passengers on the S1→S2 and S2→S4 journeys enter the station to wait. At this time, the total waiting time corresponding to scene 1 and scene 2 is 3125 and 3500 minutes respectively. Therefore, the expected total waiting time can be calculated as 0.5 * 3125 + 0.5 * 3500 = 3312.5 min. Compared with the case without passenger flow control, the robust passenger flow control strategy can further reduce passenger waiting time by 22.7%. Therefore, optimizing passenger flow control strategies is usually an important way to improve transportation efficiency. In addition, from the results we can find that under the robust control strategy, the 3rd train can only carry the passengers of the S1→S4 section, even though the scenario 2 still has the remaining capacity (for example, 10). But the robust control strategy is not the best solution for all scenarios, but a better solution, which is more practical than passenger flow control generated for each scenario. Through the coordinated passenger flow control of the whole line, the rapid transportation of downstream passengers by prohibiting upstream passengers from entering the platform can avoid excessive backlogs of downstream passengers.
Next, in order to formulate a rigorous optimization model for the considered problem, first, we make the following assumptions.
Assumption 3.1 : The train schedule is given in advance, and all trains on the subway line run exactly according to the train schedule. The total passenger travel demand for each scenario can be met by trains in operation within the considered time frame.
Assumption 3.2 : Passengers arriving at each station first need to queue outside the entrance facility. A set of entrance facilities is set up in the station hall to connect different destinations to distinguish passengers of different OD pairs.
Hypothesis 3.3 : The number of passengers arriving at the station during peak hours is much greater than the number of passengers arriving at non-peak hours, and exceeds the total passenger capacity of trains during peak hours. In our study, it is assumed that all passengers in the platform area can board the next incoming train and that no passengers leave the subway system during the run.
Assumption 3.4 : The walking time from the entrance facility to the platform is not considered. This means that passengers can reach the boarding area as soon as they are allowed onto the platform.
5. Summary
Considering the dynamic and random nature of passenger flow, the robust passenger flow control strategy on saturated subway lines is studied using scene passenger flow data. To describe the problem mathematically, an integer linear optimization model is formulated by discretizing the considered time horizon into a sequence of time steps. In the model, two sets of decision variables are considered, where (1) βi,j,t(ω) are the robust control variables in different scenarios, (2)γi,j,t(ω) are the robust control variables in each scenario Based on the actual number of incoming passengers and system constraints such as train capacity, with the goal of minimizing the expected passenger waiting time, a robust passenger flow control optimization model is constructed to generate high-quality passenger flow control strategies for the entire subway line. The dual decomposition method is used to solve the model. By dualizing hard constraints, the main model is decomposed into several tractable subproblems, and a heuristic tuning algorithm combined with a CPLEX solver is designed to solve the proposed model. Finally, the performance of the proposed method is verified by two sets of numerical experiments. The experimental results verify that the proposed robust optimization model can prevent passengers from gathering on the platform in different scenarios, reduce the waiting time of passengers to a greater extent, and improve the safety and effectiveness of the subway system operation.
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