Discrete Dynamics in Nature and Society

Volume 2018, Article ID 5961090, 12 pages

https://doi.org/10.1155/2018/5961090

## A Passenger Flow Control Method for Subway Network Based on Network Controllability

^{1}College of Applied Science, Jiangxi University of Science and Technology, Ganzhou 341000, Jiangxi Province, China^{2}School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China^{3}State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China^{4}School of Electrical Engineering and Automation, Jiangxi University of Science and Technology, Ganzhou 341000, Jiangxi Province, China

Correspondence should be addressed to Li Wang; nc.ude.utjb@ilgnaw

Received 1 March 2018; Revised 14 June 2018; Accepted 5 August 2018; Published 4 September 2018

Academic Editor: Juan L. G. Guirao

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

#### Abstract

The volume of passenger flow in urban rail transit network operation continues to increase. Effective measures of passenger flow control can greatly alleviate the pressure of transportation and ensure the safe operation of urban rail transit systems. The controllability of an urban rail transit passenger flow network determines the equilibrium state of passenger flow density in time and space. First, a passenger flow network model of urban rail transit and an evaluation index of the alternative set of flow control stations are proposed. Then, the controllable determination model of the urban rail transit passenger flow network is formed by converting the passenger flow distribution into a system state equation based on system control theory. The optimization method of passenger flow control stations is established via driver node matching to realize the optimized control of network stations. Finally, a real-world case study of the Beijing subway network is presented to demonstrate that the passenger flow network is controllable when driver nodes compose 25.3% of the entire network. The optimization of the flow control station, set during the morning peak, proves the efficiency and validity of the proposed model and algorithm.

#### 1. Introduction

Due to their large size, fast speed, and safety, urban rail transit systems have become the backbone of city transportation. In recent years, the volume of passenger flow has increased rapidly. Congestion of passenger flow is very high, especially during the morning and evening rush hours, which is a severe challenge for the operational safety of urban rail transit. With network integration of urban rail transit, traditional passenger flow control methods cannot accommodate the increasingly large-volume, line-intensive, complex organizational conditions in modern transit systems. The strategy of flow control optimization for urban rail transit network controllability provides a new perspective for network control.

Along with the increasing in urban rail transit passenger volume, research on subway passenger flow control and related topics has attracted the interest of many scholars. Xu et al. [1] proposed a flow control method and analyzed the relationship between station capacity and demand based on queuing network theory. Cortés [2, 3] developed a strategy to control public transport lines using stop-station waiting and interchange station operation by minimizing the waiting time and uniform time interval. D. Felipe et al. [4] proposed a new mathematical programming model by minimizing the time delay of a bus. The model controls the number of passengers boarding the bus to minimize the delay time. In conclusion, current passenger flow control methods focus on a single station, line, or local network. Few works have considered the use of optimization flow control methods to control the overall stability of the network. In addition, existing passenger flow control is mainly based on static relationships between stations and does not consider the timing sequence. This paper optimizes the current flow control method via network controllability according to the characteristics of urban rail transit network and distribution.

The study of network controllability began relatively recently, and we can divide the main research methods into three categories: flock control, traction control, and structural control. The early theory of complex network control was initiated by the study of large-scale system flocking control. Flocking control is the analysis of emerging behavior based on simulations of biological groups in nature. Most flocking control studies are based on the Boids model [5], in which an individual is defined as a node in a cluster system, and the connections between individual are defined as edges. Tanner and Olfai et al. [6, 7] introduced a discontinuous control method based on this model and an algorithm to control the state of change.

Pinning control is representative of complex network control. Wang et al. [8, 9] combined pinning control and flocking control and applied pinning control to a scale-free dynamic network. The results showed that pinning control with a high degree of nodes requires fewer controllers than the conventional pinning control. Chen et al. [10] studied the pinning control of complex dynamic networks and the controllability of directed networks and proposed the theory of “network of networks”. Fu [11] demonstrated that the preferential pinning strategy of stochastic pinning is superior to the preferential pinning strategy of clustered complex networks. A new pinning strategy based on the cluster degree was proposed, and the results indicated that the new cluster pinning strategy was superior to the RP strategy when there were fewer pinning nodes.

Liu [12] studied the controllability of directed networks in 2011 and applied the judgment of the state-space equation of control theory to network controllability for the first time. In addition, the directed network was transformed into a binary graph, and the maximum matching was calculated. Liu's research represented a new starting point for network controllability and laid the foundation for subsequent studies by others. A great deal of subsequent work has begun to focus on the impact of network topology on the controllable performance of network structure [13–18]. Based on Liu's research, the relationship between the controllability and energy consumption of different types of networks was analyzed from the perspective of energy consumption [19]. Nepusz [20] converted the network to an edge-based model by considering the dynamics of the edges of the network. Lombardi [21] applied a controllable matrix to the network. The value of the matrix element was the path gain from the input signal to the node. Chen et al. [22] evaluated changes and control costs of network controllability under cascade failure conditions. The number of driver nodes of a random network and scale-free network were calculated in cascade failures. A minimum structure perturbation method was proposed to optimize the controllability of the network [23]. The minimum number of edges required for controllable optimization was equal to the minimum number of conversion edges, and a network with positive correlation facilitated optimal control.

With the in-depth study of controllability of complex networks, many studies have applied control methods to the control judgment and optimization of real networks. Meng [24] studied the controllability of a railway train service network and defined the driver nodes based on immune transmission and cascade failures. An improved theoretical model for the control of complex network and a dual graph of train service network were constructed. Ravindran [25] identified driver nodes with a maximum matching algorithm and classified the nodes. The key regulatory genes in the cancer signal network were identified by controllable analysis. The topology and controllability of the U.S. power grid were analyzed by Li [26], and a new method was proposed to quantify the probability of the intermittent node becoming the driver node.

Previous research on network controllability has mainly been based on the general characteristics of complex networks. In recent years, a few studies have examined controllable analysis of real networks. However, these studies have mainly focused on the analysis of complex topological properties. There is no specific strategy for the optimization of network controllability. Most studies have ignored the function attributes of the nodes and edge weights in the actual network, making it impossible to propose effective control methods and coping strategies for specific issues.

This paper analyzes the topological characteristics of an urban rail transit passenger flow network. Then, a controllability model of the passenger flow network is constructed based on traditional control theory. An improved controllability determination method for uncontrollable networks is proposed, and the minimum number of driver nodes in the controllability passenger flow network is calculated. The method of flow control optimization is built based on driver node matching, and the specific flow control station set for controllability of the passenger flow network is presented. The method is validated based on actual passenger flow data for the Beijing subway network. In the actual flow control process, the passenger flow will change. The set of flow control stations is obtained at different time periods. When the passenger flow is relatively stable, the flow control stations tend to be fixed.

#### 2. Controllability Model of the Urban Rail Transit Passenger Flow Network

##### 2.1. Basic Indicators of the Passenger Flow Network

The model of the passenger flow network must be built based on the rail infrastructure line. We define the station as a node and the rail connecting two adjacent stations as an edge. The nodes and edges constitute a physical network of urban rail transit. We then superimpose passenger flow on the orbital transport physical network, which can be extended to a passenger flow network of urban rail transit. The station is defined as a node of the network. If there is passenger flow between two stations, there is an edge between the two stations. The transferred passenger flow is the weight of the edge. This is the passenger flow network of urban rail transit.

A complex network generally has a high number of nodes, a large degree of distribution, high concentration, and so on. The passenger flow has the characteristic of strong fluidity. Therefore, the whole network cannot be controlled effectively solely by determining the flow control node from the aggregation number of passenger flow. This paper analyzes the basic indices of the passenger flow network, thus laying the foundation for study of the controllability of the passenger flow network.

*(**1) Degree*. The degree is defined as the number of connections between node and other nodes. The greater the degree is, the more connections between nodes and the more important the nodes are in the network. The degree is given bywhere is a variable from 0 to 1 representing the connection between nodes.

The degree value of the passenger flow network of urban rail transit reflects the accessibility of network nodes. A larger value indicates more transfer choices for a station. Conversely, a smaller degree value indicates weaker accessibility of a station. Passengers may need to make a high number of transfers before arriving at their destination.

*(**2) Node Strength*. If is the connection weight of nodes and , the node strength of network node is defined by

The node strength is the sum of the node weights. The node strength of the passenger flow network of urban rail transit reflects the passenger demand of the station. The greater the node strength is, the larger the passenger flow of the station is.

*(**3) Clustering Coefficient*. The clustering coefficient reflects the node aggregation of the network. It assumes that the number of connection edges of node is . The maximum number of edges of node number is . The clustering coefficient of node is defined as follows:

The clustering coefficient of the urban rail transit network reflects the connection of transfer passenger flow between stations. The larger the aggregation coefficient is, the higher the connection degree between stations is.

*(**4) Average Path Length of the Network*. Distance between nodes and is defined as the number of edges of the shortest path between two nodes. The average path length of network is defined as follows:where is the number of network nodes.

The average path length of the urban rail transit passenger network reflects the number of passing stations from the origin to the destination. This parameter is an indicator of the connectivity of the passenger flow network of urban rail transit.

##### 2.2. Controllability Determination Method of the Passenger Flow Network

###### 2.2.1. System Controllable Determination Theory

If there is a segmented continuous input , the system can proceed from an initial state to any specified terminal state in a finite time interval . It is then said that the state is controllable. If all states of the system are controllable, it is said that the system is fully controllable.

The input-output model of a linear time-invariant system can be represented as follows [27]:where the vector captures the state of a system of nodes at time .

The input signal .

is the state matrix: .

is the input matrix: .

The controllability determination model of an urban rail transit passenger flow network belongs to the linear time-invariance system model. The model has two properties: linear and time invariance. The objective of this paper is to optimize the flow control of the urban rail transit network. In this paper, the time interval of flow control of the passenger flow network is discretized. In subdivided period, the topology link of the passenger flow network is invariant. Therefore, the passenger flow network system in one time interval is constant. Second, the input of the system is the set of flow control stations. The network state is the result of the interaction among flow control stations. Therefore, the optimization of flow control of the urban rail transit network in this paper satisfies additivity.

###### 2.2.2. Controllability Analysis of the Passenger Flow Network of Urban Rail Transit

After the network operation of rail transit, the change and rule of passenger flow are more complicated than those of the single or simple network structures due to the greater number of flow and transfer opportunities. An urban rail transit network is a control system. The external flow control measures are the input signals, and the OD passenger flows of the network are the state variables. Under normal conditions, urban rail transit networks are within the controllable range. However, in the morning and evening peak hours or under large passenger flow conditions, due to the reduced levels of path service and the mismatch between passenger flow and section capacity, the entire network is in disequilibrium. At the system level, this is an uncontrollable state. To maintain the system in a state of controllability, corresponding flow control measures are taken to reduce flow aggregation.

During the statistical period, is the number of passengers in station of the line and is equivalent to the number of people in the station plus the difference between people inbound and outbound plus the difference in transfer passenger flow, which can be shown as follows:where is the passenger flow of station during period ; is the number of passengers inbound during period ; is the number of passengers outbound during period ; is the transfer passenger inflow of station during period ; is the transfer passenger outflow of station during period .

Passenger flow control is mainly the control of inbound passengers. There is no fundamental reduction in passenger demand, and the distribution of passenger flow demand is adjusted. The passenger flow of the network achieves a relatively stable state of time and space distribution.

###### 2.2.3. Controllability Determination Model of the Urban Rail Transit Network

According to the urban rail transit network topology and passenger flow characteristics, the controllable model of the urban rail transit network is presented inwhere is the element of state matrix in time , is the origination station, and is the destination station. is the number of passengers from to in time . is the number of passengers from to in time . When , ; , , .

is the passenger flow state level of station in time . According to the national standard subway design specification (GB50157-2013) of China and the Transit Capacity and Quality of Service Manual (2nd Edition) of TCRP, a single facility is divided into four levels (Table 1).