Hindawi Discrete Dynamics in Nature and Society Volume 2018, Article ID 5961090, 12 pages https://doi.org/10.1155/2018/5961090

Research Article A Passenger Flow Control Method for Subway Network Based on Network Controllability

Lu Zeng,1,2 Jun Liu ,2 Yong Qin,3 Li Wang ,2 and Jie Yang4

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

Correspondence should be addressed to Li Wang; [email protected]

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

AcademicEditor:JuanL.G.Guirao

Copyright © 2018 Lu Zeng et al. Tis 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.

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

1. Introduction relationship between station capacity and demand based on queuing network theory. Cortes´ [2, 3] developed a strategy to Due to their large size, fast speed, and safety, urban rail transit control public transport lines using stop-station waiting and systems have become the backbone of city transportation. interchange station operation by minimizing the waiting time In recent years, the volume of passenger fow has increased rapidly. Congestion of passenger fow is very high, especially and uniform time interval. D. Felipe et al. [4] proposed a new during the morning and evening rush hours, which is a severe mathematical programming model by minimizing the time challenge for the operational safety of urban rail transit. delay of a bus. Te model controls the number of passengers With network integration of urban rail transit, traditional boarding the bus to minimize the delay time. In conclusion, passenger fow control methods cannot accommodate the current passenger fow control methods focus on a single increasingly large-volume, line-intensive, complex organiza- station, line, or local network. Few works have considered tional conditions in modern transit systems. Te strategy of the use of optimization fow control methods to control fow control optimization for urban rail transit network con- the overall stability of the network. In addition, existing trollability provides a new perspective for network control. passenger fow control is mainly based on static relationships Along with the increasing in urban rail transit passenger between stations and does not consider the timing sequence. volume, research on subway passenger fow control and Tis paper optimizes the current fow control method via related topics has attracted the interest of many scholars. Xu network controllability according to the characteristics of et al. [1] proposed a fow control method and analyzed the urban rail transit network and distribution. 2 Discrete Dynamics in Nature and Society

Te study of network controllability began relatively network and defned the driver nodes based on immune recently, and we can divide the main research methods into transmission and cascade failures. An improved theoretical three categories: fock control, traction control, and structural model for the control of complex network and a dual graph control. Te early theory of complex network control was of train service network were constructed. Ravindran [25] initiated by the study of large-scale system focking control. identifed driver nodes with a maximum matching algorithm Flocking control is the analysis of emerging behavior based and classifed the nodes. Te key regulatory genes in the on simulations of biological groups in nature. Most focking cancer signal network were identifed by controllable analysis. control studies are based on the Boids model [5], in which an Te topology and controllability of the U.S. power grid were individual is defned as a node in a cluster system, and the analyzed by Li [26], and a new method was proposed to connections between individual are defned as edges. Tanner quantify the probability of the intermittent node becoming and Olfai et al. [6, 7] introduced a discontinuous control the driver node. method based on this model and an algorithm to control the Previous research on network controllability has mainly state of change. been based on the general characteristics of complex net- Pinning control is representative of complex network works. In recent years, a few studies have examined con- control. Wang et al. [8, 9] combined pinning control and trollable analysis of real networks. However, these studies focking control and applied pinning control to a scale-free have mainly focused on the analysis of complex topological dynamic network. Te results showed that pinning control properties. Tere is no specifc strategy for the optimization with a high degree of nodes requires fewer controllers than of network controllability. Most studies have ignored the the conventional pinning control. Chen et al. [10] studied function attributes of the nodes and edge weights in the actual the pinning control of complex dynamic networks and the network, making it impossible to propose efective control controllability of directed networks and proposed the theory methods and coping strategies for specifc issues. of “network of networks”. Fu [11] demonstrated that the Tis paper analyzes the topological characteristics of an preferential pinning strategy of stochastic pinning is superior urban rail transit passenger fow network. Ten, a control- to the preferential pinning strategy of clustered complex lability model of the passenger fow network is constructed networks. A new pinning strategy based on the cluster degree based on traditional control theory. An improved control- was proposed, and the results indicated that the new cluster lability determination method for uncontrollable networks pinning strategy was superior to the RP strategy when there is proposed, and the minimum number of driver nodes in were fewer pinning nodes. the controllability passenger fow network is calculated. Te Liu [12] studied the controllability of directed networks method of fow control optimization is built based on driver in 2011 and applied the judgment of the state-space equation node matching, and the specifc fow control station set for of control theory to network controllability for the frst time. controllability of the passenger fow network is presented. In addition, the directed network was transformed into a Te method is validated based on actual passenger fow data binary graph, and the maximum matching was calculated. for the Beijing subway network. In the actual fow control Liu’s research represented a new starting point for network process, the passenger fow will change. Te set of fow controllability and laid the foundation for subsequent studies control stations is obtained at diferent time periods. When by others. A great deal of subsequent work has begun to the passenger fow is relatively stable, the fow control stations focus on the impact of network topology on the controllable tend to be fxed. performance of network structure [13–18]. Based on Liu’s research, the relationship between the controllability and 2. Controllability Model of the Urban Rail energy consumption of diferent types of networks was Transit Passenger Flow Network analyzed from the perspective of energy consumption [19]. Nepusz [20] converted the network to an edge-based model 2.1. Basic Indicators of the Passenger Flow Network. Te by considering the dynamics of the edges of the network. model of the passenger fow network must be built based Lombardi [21] applied a controllable matrix to the network. on the rail infrastructure line. We defne the station as a Te value of the matrix element was the path gain from node and the rail connecting two adjacent stations as an the input signal to the node. Chen et al. [22] evaluated edge. Te nodes and edges constitute a physical network of changes and control costs of network controllability under urbanrailtransit.Wethensuperimposepassengerfowon cascade failure conditions. Te number of driver nodes of a the orbital transport physical network, which can be extended random network and scale-free network were calculated in to a passenger fow network of urban rail transit. Te station cascade failures. A minimum structure perturbation method is defned as a node of the network. If there is passenger was proposed to optimize the controllability of the network fow between two stations, there is an edge between the two [23]. Te minimum number of edges required for controllable stations. Te transferred passenger fow is the weight of the optimization was equal to the minimum number of conver- edge. Tis is the passenger fow network of urban rail transit. sion edges, and a network with positive correlation facilitated A complex network generally has a high number of nodes, optimal control. a large degree of distribution, high concentration, and so on. With the in-depth study of controllability of complex Te passenger fow has the characteristic of strong fuidity. networks, many studies have applied control methods to the Terefore, the whole network cannot be controlled efectively control judgment and optimization of real networks. Meng solely by determining the fow control node from the aggre- [24] studied the controllability of a railway train service gation number of passenger fow. Tis paper analyzes the Discrete Dynamics in Nature and Society 3 basic indices of the passenger fow network, thus laying the 2.2. Controllability Determination Method of the Passenger foundation for study of the controllability of the passenger Flow Network fow network. 2.2.1. System Controllable Determination Teory. If there is (1) Degree. Te degree is defned as the number of con- a segmented continuous input �(�), the system can proceed nections between node � and other nodes. Te greater the from an initial state �(�0) to any specifed terminal state �(��) degree is, the more connections between nodes and the more in a fnite time interval [�0,��]. It is then said that the state important the nodes are in the network. Te degree is given is controllable. If all states of the system are controllable, it is by said that the system is fully controllable. Te input-output model of a linear time-invariant system � =∑� � �� can be represented as follows [27]: �≤� (1) �=��̇ (�) +��(�) (5) where ��� is a variable from 0 to 1 representing the connection � between nodes. where the vector �(�) = (�1(�), ...... , ��(�)) captures the state Te degree value of the passenger fow network of urban of a system of � nodes at time �. � rail transit refects the accessibility of network nodes. A larger Te input signal �(�) = (�1(�), ..., ��(�)) ,�≤�. �×� value indicates more transfer choices for a station. Con- � is the state matrix: �∈� . �×M versely, a smaller degree value indicates weaker accessibility � is the input matrix: �∈� . of a station. Passengers may need to make a high number of Te controllability determination model of an urban rail transfers before arriving at their destination. transit passenger fow network belongs to the linear time- invariance system model. Te model has two properties: � � (2) Node Strength.If �� is the connection weight of nodes linear and time invariance. Te objective of this paper is to � � � and ,thenodestrength � of network node is defned by optimize the fow control of the urban rail transit network. � � In this paper, the time interval of fow control of the �� = ∑ ∑��� (2) passenger fow network is discretized. In subdivided period, �=1 �=1 the topology link of the passenger fow network is invariant. Terefore, the passenger fow network system in one time Te node strength is the sum of the node weights. Te interval is constant. Second, the input of the system is the node strength of the passenger fow network of urban rail set of fow control stations. Te network state is the result transit refects the passenger demand of the station. Te of the interaction among fow control stations. Terefore, the greater the node strength is, the larger the passenger fow of optimization of fow control of the urban rail transit network the station is. in this paper satisfes additivity. (3) Clustering Coefcient. Te clustering coefcient refects the node aggregation of the network. It assumes that the 2.2.2. Controllability Analysis of the Passenger Flow Network number of connection edges of node � is ��.Temaximum of Urban Rail Transit. Afer the network operation of rail number of edges of node number �� is ��(�� −1)/2.Te transit, the change and rule of passenger fow are more clustering coefcient �� of node � is defned as follows: complicated than those of the single or simple network structures due to the greater number of fow and transfer 2� � opportunities. An urban rail transit network is a control �� = (3) �� (�� −1) system. Te external fow control measures are the input signals, and the OD passenger fows of the network are the Te clustering coefcient of the urban rail transit network state variables. Under normal conditions, urban rail transit refects the connection of transfer passenger fow between networks are within the controllable range. However, in the stations. Te larger the aggregation coefcient is, the higher morning and evening peak hours or under large passenger the connection degree between stations is. fow conditions, due to the reduced levels of path service and the mismatch between passenger fow and section capacity, (4) Average Path Length of the Network.Distance��� between the entire network is in disequilibrium. At the system level, nodes � and � is defned as the number of edges of the shortest this is an uncontrollable state. To maintain the system in a path between two nodes. Te average path length of network state of controllability, corresponding fow control measures � is defned as follows: are taken to reduce fow aggregation. 1 During the �+Δ� statistical period, �� (�+Δ�) is the number �= ∑��� (4) (1/2) � (�−1) of passengers in station � of the line and is equivalent to the number of people in the station plus the diference between where � is the number of network nodes. people inbound and outbound plus the diference in transfer Te average path length of the urban rail transit passenger passenger fow, which can be shown as follows: network refects the number of passing stations from the origin to the destination. Tis parameter is an indicator of �� (�+Δ�) =�� (�) +�� (�) −�� (�) +�� (�) −�� (�) the connectivity of the passenger fow network of urban rail (6) ∀ >0; transit. t 4 Discrete Dynamics in Nature and Society

where ��(�) is the passenger fow of station � during period �; ��(�) is the number of passengers inbound � during period �; ��(�) is the number of passengers outbound � during period �; ��(�) is the transfer passenger infow of station � during period �; ��(�) is the transfer passenger outfow of station � during x period �. 4 Passenger fow control is mainly the control of inbound passengers. Tere is no fundamental reduction in passenger x2 x5 x6 demand, and the distribution of passenger fow demand x is adjusted. Te passenger fow of the network achieves a 1 relatively stable state of time and space distribution. u2 u3 x3 u 2.2.3. Controllability Determination Model of the Urban Rail 1 Transit Network. According to the urban rail transit network Figure 1: Schematic diagram of a controllable network. topology and passenger fow characteristics, the controllable model of the urban rail transit network is presented in actual passenger fow demand per unit time. Te larger the � � � � fow control rate is, the larger the fow control intensity is. �̇ (�) = ∑∑��� (�) �� (�) + ∑∑��� (�) �� (�) �≤� (7) Te Kalman controllability rank is used to determine the �=1�=1 �=1�=1 condition. Te determination model of the controllability of the urban rail transit network is given by where ��� (�) is the element of state matrix � in time �, � is the (� = (�, ��, �2�, ,��−1�)) origination station, and � is the destination station. ��� (�) is the rank ...... (8) number of passengers from � to � in time �. ���(�) is the number If rank(�) = �, the passenger fow network of urban rail of passengers from � to � in time �.When���(�) > �, ��� =1; transit is controllable. If rank(�) < �,thepassengerfow ���(�) < �, ��� (�) = ��� (�)/�, ∀��� (�) ∈ [0, 1]. network of urban rail transit is uncontrollable. ��(�) is the passenger fow state level of station � in time �. According to the national standard subway design specif- cation (GB50157-2013) of China and the Transit Capacity and 3. Optimization Method of Flow Control of Quality of Service Manual (2nd Edition) of TCRP, a single an Urban Rail Transit Network facility is divided into four levels (Table 1). Based on the congestion risk assessment standard for 3.1. Optimization Graph Method of Network Structure. In Beijing urban rail transit stations, the risk level is classifed complex network control, Liu [12] proposed a network con- as heightened risk, high risk, general risk, and risk free trol method based on minimizing node ��. Te simulation according to the evaluation score. Given the limited and results showed that the number of driver nodes is mainly unobstructed equipment and facilities, the risk level has little determined by the degree distribution of the network. Sparse efect on passenger travel and station operation and is not the heterogeneous networks tend to be more difcult to control, basis for classifying the risk rating. Terefore, the congestion while dense homogeneous networks require fewer driver level of the whole station can be obtained from the number nodes to be controlled. Sparsity means that the average degree of survey points, such as platforms, passages, security, up and of the network is much smaller than the maximum possible down staircases, and entrances. Te specifc principles are as connectivity � (number of network nodes), such as in an follows: urban rail transit network. A heterogeneous network con- (A.1) If there are more than 5 A or 8 B, the station is siders not only the topology structure but also the attribute classifed as a high-risk station (I level). information of nodes and edges in the network. (A.2) If there are 2-5 A or 5-8 B, the station is classifed as Combining the network structure with graph theory, the a heightened-risk station (II level). matching method of the node and edge is used to determine (A.3) If there are 1-2 A or 1-5 B, the station is classifed as whether the network can be controlled. If the input signal can a general-risk station (III level). reach all paths by input signals �1,�2,and�3, the system is (A.4) If there are 0 A or 0 B, the station is classifed as a controllable (Figure 1). How can an uncontrollable network risk-free station (IV level). be converted into a controllable network, and how can the According to the overall congestion risk assessment minimum input signal be determined? Tis is a matching results, the value of ��(�) is 1, 2, 3, 4. problem. Te nodes in the network can be matched by the ��� (�) is the element of input matrix � in time � and binary graph method. A matching edge means that any two corresponds to the number of fow control stations. If the directed edges do not have a common vertex (head or tail station implements fow control measures in time t, ��� (�) = 1; node). Te matching node is the head node of the matching otherwise, ��� (�) = 0. edge. ��(�) is the fow control intensity of station � in time Te determination of the minimum input for a directed �, ∀��(�) ∈ [0, 1]. Te fow control intensity of the station network can be converted to a maximum matching problem represents the ratio of the passenger fow in the station to the to solve the network (Figure 2). According to the system Discrete Dynamics in Nature and Society 5 1.33 ≥ 0.3-0.5 0.5-1.33 Speed (m/s) Check-in area 1 20 Indefnite ≤ 1-13 ≥ 13-20 number Average queue /p) 2 0.7 0.2 ≥ ≤ 0.2-0.3 0.3-0.7 Per capita area(m 49 ≤ 65-81 49-65 Passage rate (p/m/min) Unit width fow 76 47 Indefnite ≥ ≤ 47-69 69-76 (m/min) Average speed ) 2 1.6 0.7 ≥ ≤ 1.1-1.6 0.7-1.1 (p/m Flow density d d d d -1/2 -3/4 Table 1: Facility equipment passenger fow risk rating scale. 1/4 3/4 d d ≤ ≥ length/m Platform Average queue 2 0.7 0.2 ≥ ≤ area/m Per capita state Severe congestion Congestion 0.2-0.3 1/2 More smooth 0.3-0.7 1/4 Unobstructed Passenger fow D B C Passenger fow status level A 6 Discrete Dynamics in Nature and Society

b1

x1

a21 1000 a 2 3 0 21 00 x2 C = [B, AB, A B, A B] = b1 0 0 a21a32 0

a32 a42 0 0 a21a42 0

x x 3 4 rank(C) = 3 < 4 uncontrollable (a) (b)

Figure 2: Schematic diagram of uncontrollable system structure. structure (Figure 2(a)), the edges between nodes correspond Step 3. If �(�) = Γ,thereisnogreatermatch;return. � to state matrix � in the equation (Figure 2(b)). Otherwise, �0 ∈�(�). �� 3.2. Optimization Method of Flow Control Based on Driver Step 4. If 0 is matched by M,gotoStep6.Otherwise,forman � � � Node Matching. In this paper, the driver node matching augment path �(�0,�0), �(�0,�0)∈�, �=�Δ�(�0,�0). method is improved based on the Hopcrof–Karp algorithm Priority is given to the adjacent nodes connected to �0,which [28]. If there is no shared head node or tail node on all edges has a large value of integrated index values in the nodes, of the network, the network achieves maximum matching. If including the nodes of ��� ≥�and �� ≥�. the network does not match exactly, the value of the driver � � node � is equal to the number of nonmatching nodes. Step 5. Since �0 has been matched by �,thereisanedge � �� �� � For a nonmatching node, a given input signal can reach all (�0,�0 ) of �, � = � ∪ {�0 }, Γ = Γ ∪ {�0};gotoStep1. matching nodes to control the entire network. Even with diferent initial searches of matched edges, the number of Step 6. Determine whether the driver node is a transfer node; minimum driver nodes is fxed. For the network of rail transit if it is, keep the connected transfer node. passenger fow, it is necessary to identify the driver nodes of the fow control station to optimize the passenger fow Step 7. If �� > min ��, delete the head node of passenger network and enable network control. fow that is larger or equal to � and smaller than (� + �1), Tis paper proposes an optimization method of fow deleted by �1successive iterations. Delete the node for which control based on driver node matching according to the passenger fow entering the station is smaller than �, deleted network topology and the passenger fow characteristics of by d2 successive iterations. When �� = min ��,anewsetof urban rail transit. Te method is described as follows. fow control stations is generated. Te method defnes an urban rail transit network �= (�,�),where� is the station set; � is the section of fow Te above search process occurs in a period of �.With transfer; ��� is the passenger fow from station j to station the implementation of the fow control strategy and the � decrease in passenger fow, the above process can be repeated. �; � isthedegreevalue� of station �; �� is the passenger � Reducing or replacing the current fow control node can fow entering the station; � isthedegreethreshold;� is the formulate the fow control program for subpeak conditions. threshold of passenger fow; and � is the passenger fow threshold entering the station. If �can be divided into two mutually disjoint subsets (�, �), the two nodes associated 4. Case Study with each edge of the network belong to the two diferent 4.1. Analysis of the Characteristics of the Beijing Urban Rail subsets (�, �). �=(�,�)is then converted to a bipartite Transit Network. Based on the train operation schedule and �(�) = (�+ ,�− ,Γ) �+ ={�+,�+, ...., �+ } graph � � ,where � 1 1 � , passenger fow data for Beijing urban rail transit in 2015, �− ={�−,�−, ...., �− } Γ={(�−,�−)|� =0}̸ � 1 1 � ,and � � �� represent we extract the connections between stations in the urban the edge set. rail transit passenger network. Te Beijing urban rail transit passenger fow network has 269 stations and 18860 edges, and Step 1. Select an initial matching edge ��; � fall vertices of � the average degree is 70.1. are matched by �,then� is completely matched; return. Te degree distribution of the passenger fow network of Tis result indicates that the network is currently under Beijing urban rail transit is shown in Figure 3(a). Te number control. Otherwise, a breadth-frst search is performed for all of stations is high for degree values of 24, 36, 44, 68, and unmatched vertices as the source, and the distance from each 88. Most stations have low and relatively concentrated dis- node to the source node is marked. tributions, and 90% of the node values are within 100. Based � Step 2. Find the vertex �0 unmatched by M;thatis�� ≥�, on the cumulative degree distribution shown in Figure 3(b), and mark �={�0}, Γ = .̸⊂ the Beijing urban rail transit network is more consistent Discrete Dynamics in Nature and Society 7

40 1 35 0.9 0.8 30 0.7 25 0.6 20 0.5 0.4 15 0.3 Number of stations of Number 10 0.2 Cumulative degree distribution degree Cumulative 5 0.1 0 0 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 Degree Degree (a) Degree distribution (b) Cumulative degree distribution 0.02 1

0.018 0.9 0.016 0.8 0.014 0.7 0.012 0.6 0.01 0.5 0.008 0.4 0.006 0.004 0.3 Cumulative strength distribution 0.002 0.2

0 coefcient clustering of distribution Cumulative 0.1 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4 strength x 10 clustering coefcient (c) Cumulative strength distribution (d) Cumulative distribution of the clustering coefcient

0.35

0.30

0.25

0.20

0.15

Distance distribution Distance 0.10

0.05

0.00 1 2345 Shortest path (e) Shortest-distance distribution

Figure 3: Characteristics of the Beijing urban rail transit passenger fow network. with the power law distribution and is thus a scale-free the station strength is greater than 15 000, and 88.2% of network. Te station strength indirectly refects the service the station strength is smaller than 10 000. Tese results capacity of the station. Te cumulative strength distribution indicate that the intensity distribution of the station node is for the Beijing urban rail transit passenger fow network is extremely uneven. Te stations carrying large passenger fow shown in Figure 3(c). According to the statistics, 4.5% of are usually transfer nodes, which enhances the accessibility 8 Discrete Dynamics in Nature and Society

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34 − − − − − − − − Beijingzhan 1 2 3 4 1 2 3 4 (a) (b) (c) Tiantongyuanbei 1 + + + + + + + + Tiantongyuan 2 1 2 3 4 1 2 3 4 Tiantongyuannan 3 − − − − − − − − Lishuiqiao 4 1 2 3 4 1 2 3 4 (d) (e) (f)

Xidiaoyutai 1 + + + + + + + + + + 1 2 3 4 5 1 2 3 4 5 4 2 Gongzhufen 5 Wanshoulu Junshibowuguan − − − − − − − − − − 3 Lianhuaqiao 1 2 3 4 5 1 2 3 4 5 (g) (h) (i)

Figure 4: Example of method for determining the controllability of the local passenger fow network. of the network. Te average clustering coefcient of the We selected the Tiantongyuanbei, Tiantongyuan, Beijing urban rail transit network is calculated to be 0.563, Tiantongyuannan, and Lishuiqiao stations of of the indicating high clustering. Te cumulative distribution of Beijing subway (Figure 4(d)). Te passenger fow network the clustering coefcient for the Beijing urban rail transit of the four station nodes is converted into a binary graph passenger fow network is shown in Figure 3(d). Each station (Figure 4(e)). Te maximum matching edge of this small has a high connection degree with adjacent stations. When network is shown as the red matching edge in the binary the clustering coefcient of the station is 1, the station has graph (Figure 4(f)). V2, V3,andV4 are the matching nodes, and a low degree value. Terefore, the relationship between the V1 is the nonmatching node. Terefore, the driver node �� is degree value and the clustering coefcient is negative. Te V1. average distance of the urban rail transit network indirectly We selected the Junshibowuguan and Wanshoulu stations refects the transfer times of passengers. Te shortest distance of and the Xidiaoyutai, Gongzhufeng, and Lianhuaqiao distribution of the Beijing urban rail transit passenger fow stations of of the Beijing subway (Figure 4(g)). Te network is shown in Figure 3(e). Te average shortest path passenger fow network of the fve station nodes is converted length of the passenger fow network is 1.69. Tus, passengers into a binary graph (Figure 4(h)). Te maximum matching reach their destinations on average by 1.69 times. Te transfer edge of this small network is shown as the red matching rate is up to 84% within three times. edge in the binary graph (Figure 4(i)). V2, V3,andV5 are the matching nodes, and V1 and V4 are the nonmatching nodes. 4.2. Controllability of a Simple Passenger Flow Network. In Terefore, the driver nodes �D are V1 and V4. this paper, we selected a small network of the Beijing subway in 2015 to verify the controllability of the network structure. Te example shows that there is no cross road loop or We do not consider the volume of passenger fow. An edge is straight line and that there are relatively few required driver defned if there is passenger fow between stations. nodes. Trifurcated and cross structures are very common We selected the Dongdan and Jianguomen stations of line among subway passenger fow networks. Tese structures 1, the of , and the Chongwenmen usually have a high number of nonmatching nodes and station of line 5 (Figure 4(a)). Te passenger fow network require a high number of driver nodes. of the four stations is converted into a binary graph (Fig- + − + − 4.3. Controllability Example for a Passenger Flow Network ure 4(b)). V1 and V1 are the Dongdan station; V2 and V2 are the + − of Beijing Urban Rail Transit. Te Beijing subway had 18 Jianguomen station; V3 and V3 are the Chongwenmen station; + − operational lines (including 17 lines and 1 airport line) in 2015. and V4 and V4 are the Beijing railway station. Te arrow represents the direction of passenger fow. In this case, we Tere were 315 stations, including the repeated calculation consider only unidirectional passenger fow. Te maximum for transfer stations, except terminal 2 and terminal 3 (269 matching edge of this small network is shown as the red stations, excluding the repeated calculation for transfer sta- matching edge in the binary graph (Figure 4(c)). According tions). to the results of the maximum matching edge, V1, V2, V3,and Each station of the network is numbered by line, for V4 are the matching nodes of the network. Te number of example, line 1: Pingguoyuan 0101, Gucheng 0102, Bajiaoy- nonmatching nodes is 0, and the driver node �� is 0. ouleyuan 0103, Babaoshan 0104, Yuquanlu 0105, Wukesong Discrete Dynamics in Nature and Society 9

Table 2: Information for the Beijing Urban Railway Network.

Line Line number Number of stations Station number Line 1 01 23 0101-0123 Line 2 02 18 0201-0218 /Daxing line 04 35 0401-0435 Line 5 05 23 0501-0523 06 26 0601-0626 07 19 0701-0719 08 17 0801-0817 09 13 0901-0913 Line 10 10 45 1001-1045 13 16 1301-1316 14 17 1401-1417 15 19 1501-1519 Batong 13 1601-1613 Fangshan 11 1701-1711 Changping line 18 7 1801-1807 Yizhuang 13 1901-1913

0106; line 2: 0201, Chegongzhuang 0202, Fucheng- Matrix � is an input matrix that consists of fow control men 0203, etc. Te frst two digits are the line number, and stations. Te fow control stations are shown in Table 3. Te the last two are the station number, as shown in Table 2. matrix removes the evening peak fow control stations, such Te edges between nodes are directed, and the direction as Wudaokou, Liangmaqiao, Jintaixizhao, , is the same as the transport direction of passenger fow. Fuxingmen, Yonganli, Fengtaikejiyuan, and Dongwuyuan. Te passenger fow data used in this case are the early 100⋅⋅⋅⋅⋅⋅00 peak passenger fow data of the Beijing subway network on [ ] December 14, 2015. According to the passenger fow OD data, [010⋅⋅⋅⋅⋅⋅00] [ ] the state matrix a is 18 860∗18 860. [ ] [ . .] [00d . .] 0 65 178 137 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 [ ] [ ] [. . . ] [51 0 48 49 ⋅⋅⋅ ⋅⋅⋅ 88] �=[. . . ] [ ] [. . . d 10] (11) [ ] [ ] [47 15 0 16 114 ⋅ ⋅ ⋅ 45] [ ] [ ] [. . . ] [ ] [. . . d 00] [77 46 31 0 158 ⋅ ⋅ ⋅ 53] [ ] �=[ ] [ ] [ ] (9) [000⋅⋅⋅⋅⋅⋅00] [ . . . . . ] [ . . .⋅⋅⋅d . . ] [ ] [000⋅⋅⋅⋅⋅⋅01] [ ] [ . . . . ] [ . . . ⋅⋅⋅ ⋅⋅⋅ d . ] According to Kalman’s controllable rank determination, the rank(c)=12365. Tis value indicates that the system is [ 0 132 46 133 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 ] uncontrollable. Te subway network has a large number of Based on the comparison of the passenger fow OD transfer stations. Because the transfer stations connect two data for the early peak (7:00-9:00) and ordinary periods, we or more lines, there are more nonmatching nodes around selected a threshold � of 69. Te matrix can be transformed transfer stations according to the maximum matching theory. as shown below: Te number of driver nodes is insufcient, and the network is uncontrollable. 00.9411⋅⋅⋅⋅⋅⋅0 In this paper, an optimization method for fow control [ ] [0.74 0 0.70 0.71 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 ] based on driver nodes is proposed to optimize an uncontrol- [ ] [ ] lable passenger fow network. Te ratio of driver nodes to the [0.68 0.22 0 0.23 1 ⋅ ⋅ ⋅ 0.65] [ ] nodes during early peak is shown in Figure 5. [ ] [ 1 0.67 0.45 0 1 ⋅ ⋅ ⋅ 0.77] When the ratio of driver nodes is close to 0.253, the �=[ ] [ ] (10) number of driver nodes is 65, and the Beijing urban rail [ . . . . . ] [ . . . ⋅⋅⋅ d . . ] transit network is controllable during early peak. From the [ ] [ ] perspective of network controllability, it is suggested that the [ . . . . ] [ . . . ⋅⋅⋅ ⋅⋅⋅ d . ] Beijing urban rail transit increase the number of driver nodes to improve controllability and optimize the fow control 0 1 0.67 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 [ ] strategy. 10 Discrete Dynamics in Nature and Society

Table 3: Information on Beijing Urban Railway passenger control stations.

Line Flow control stations Line 1 Sihuidong, Gucheng, Pingguoyuan, Sihui, Babaoshan, Bajiaoyouleyuan, Gongzhufeng, Fuxingmen, Yonganli Chuanmeidaxue, Shuangqiao, Guanzhuang, , Tongzhoubeiyuan, Guoyuan, Jiukeshu, Liyuan Line 2 Chaoyangmen Line 4 Gongyixiqiao, Jiaomenxi, Beijingnan, , Dongwuyuan Tiantongyuanbei, Tiantongyuan, Tiantongyuannan, Lishuiqiao, Lishuiqiaonan, Beiyuanlubei, Datunludong, Ciqikou, Line 5 Huixinxijiebeikou, Huixinxijienankou, Puhuangyu, Dongdan, Tiantandongmen, Liujiayao, Songjiazhuang Line 6 Hujialou, Jintailu, Shilipu, Qingnianlu, Talianpo, Huangqu, Changying, Caofang, Wuzixueyuanlu Line 7 Ciqikou Line 9 Beijingxizhan, Liuliqiaodong, Fengtaikejiyuan Line 10 Jinsong, Shuangjing, Liangmaqiao, Sanyuanqiao, Guomao, Jintaixizhao Line 13 Shangdi, Huoying, Huilongguan, Longze, Wudaokou Line 14 Jintailu Changping line Xierqi, Zhuxinzhuang, Shengmingkexueyuan, Shahe, Shahegaojiaoyuan, Nanshao Jiugong Daxing line Xihongmen

70

60

50

40 Nd 30

20

10

0 0 50 100 150 200 250 300 N

Figure 5: Ratio of driver nodes to nodes of a controllable passenger Figure 6: Control stations of Beijing urban rail transit under the fow network. network controllability condition.

We attempt to retain the original fnite-fow station can be eliminated under the controllability condition: the during the entire deletion process, and the updated topology Datunludong station and the Tiantandongmen station. diagram of fow control stations in Beijing urban rail transit Te fow control stations are more concentrated in or near is shown in Figure 6. stations with large passenger fow. Te fow control of nearby Te resulting fow control scheme is shown in Figure 6. stations is also intended to relieve the station with large pas- Te three red nodes are the newly added fow control stations: senger fow. Current methods of fow control are restricted to Jianguomen, Chongwenmen, and Xizhimen. Te two blue experience. Te fow control stations are basically fxed in the nodes are the original fow control stations that were deleted. same time interval. With the implementation of fow control Te newly added fow control stations are transfer stations measures, the passenger fow of some stations is reduced. that have 4 or 5 degrees. Tese transfer stations have large Tere may be a subpeak state, and the implementation of fow passenger fow, and the internal structures are highly compli- control measures should be a dynamic process. cated. Among these stations, Xizhimen is a transfer station with three subway lines and, consequently, complicated 5. Conclusions transfer of passenger fow. Flow control can be implemented inside the station or in security. Te proposed method in Tis paper proposed an optimization method for fow control this paper is repeated. When the fow control measures are of urban rail transit based on the state-space equation implemented afer four time cycles, two fow control stations and the driver node-matching algorithm. In addition, the Discrete Dynamics in Nature and Society 11 characteristics of the optimization control of this method References were analyzed by taking the Beijing rail transit network as an example. Te conclusions are as follows. [1]X.Y.Xu,J.Liu,H.Y.Li,andJ.Q.Hu,“Analysisofsubway (1) Te analysis of the characteristics of the complex station capacity with the use of queueing theory,” Transportation Research Part C: Emerging Technologies,vol.38,no.1,pp.28–43, network based on the passenger fow indicated that the 2014. passenger fow of the Beijing urban rail transit network has [2] C. E. Cort´es, D. S´aez, F. Milla, A. N´u˜nez, and M. 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