Exploring the Relationship between ... - ScienceDirect.com

Available online at www.sciencedirect.com

ScienceDirect

Available online at www.sciencedirect.com Procedia Computer Science 00 (2016) 000–000

ScienceDirect

www.elsevier.com/locate/procedia

Procedia Computer Science 109C (2017) 600–607

The 8th International Conference on Ambient Systems, Networks and Technologies (ANT 2017)

Exploring the Relationship between Heterogeneity of Vehicle Distribution and the Macroscopic Fundamental Diagram under Segment Disruption Conditions Xiaobo Zhua,*, Xiaongguang Yanga, Yuntao Guob a The

The Key Laboratory of Road and Traffic Engineering (Ministry of Education), Tongji University, 4800 Cao’an road, Jiading District, Shanghai, 201804, China b Lyles School of Civil Engineering/NEXTRANS Center , Purdue University, 3000 Kent Avenue, West Lafayette, IN 47906, USA

Abstract The framework of a well-defined Macroscopic Fundamental Diagram (MFD) relating network-wide flow and density was proposed under the homogeneity assumption that congestion is distributed evenly which is not universal in practice. This study aims to explore deep details of the relation between heterogeneity of vehicle distribution and the MFD under segment disruption conditions. By plotting MFDs and calculating variance, kurtosis and skewness of segment traffic density for different network structures, details of the relation are revealed. Results show that under segment disruption conditions, if the network flow reaches the maximum level at the same density for different network structures, a negative correlation between heterogeneity of vehicle distribution and maximum flow can be established. Similar relation can be observed under other states except free flow state. Nevertheless, heavy congestion may affect this relation because of spillback. Overall, the results are significant for traffic control to maintain networks at a high-efficiency level under segment disruption conditions. © 2016 The Authors. Published by Elsevier B.V. 1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Conference Program Chairs. Peer-review under responsibility of the Conference Program Chairs. Keywords: heterogeneity of vehicle distribution; Macroscopic Fundamental Diagram; segment disruptions; network traffic flow; traffic management

* Corresponding author. Tel.: +86-13621611062. E-mail address: [email protected] 1877-0509 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Conference Program Chairs.

1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Conference Program Chairs. 10.1016/j.procs.2017.05.364

2

Xiaobo Zhu et al. / Procedia Computer Science 109C (2017) 600–607 Zhu, Yang, Guo/ Procedia Computer Science 00 (2017) 000–000

601

1. Introduction The network-wide relationship between traffic flow, density and speed, which is referred as the Macroscopic Fundamental Diagram(MFD) or the Network-wide Fundamental Diagram(NFD), has been theoretically established and verified by both simulation and empirical data in many recent studies1-3. Daganzo and Geroliminis2, 3 found that traffic in large urban road network can be modeled at an aggregate level if congestion distributes evenly in the network. They used simulated data from a part of San Francisco road network and collected data in downtown Yokohama to verify the existence of MFD relating accumulation (the accumulated number of vehicles in the network) and the network-wide average speed or flow. The MFD provides a macroscopic view of network performance and a new tool for traffic control to improve network mobility and relieve congestion. In principle, network flow can be maintained at or near the maximum level by appropriate traffic control and management based on the well-defined MFD4. Several studies used MFD to analyze traffic state of urban network and to develop network-level traffic management strategies5-7. However, the framework of the well-defined MFD was proposed under the homogeneity assumption that congestion is distributed evenly which is difficult to be satisfied in practice. The spatial heterogeneity of congestion may affect the shape and even the existence of the MFD. Therefore, there is a need to deeply understand the relationship between spatial heterogeneity of vehicle distribution and the MFD, which is significant to urban traffic control and management. Some previous studies tried to explore how the heterogeneity affects the shape and scatter of an MFD. The effects of various of heterogeneity, such as networks types (freeway networks and arterial networks) and loop detector locations (distance to intersections), on network flow represented by the MFD for a network has been investigated using field data or simulation experiments4,8. It is shown that heterogeneity has strong impacts on an MFD, even the existence of it. Some studies focused on the relation between heterogeneity of congestion or vehicle distribution and network flow for a network9, 10. It is observed that the increasing heterogeneity of density which is applied to estimate vehicle distribution may cause decrease of network performance since a high heterogeneity of vehicle distribution increases the probability of spillover. In addition, different shapes of the MFD were revealed and the heterogeneity was considered as an important factor that causes the hysteresis phenomena in MFDs which shows different network average flows between traffic transitions from stable to unstable states and the return transitions 1113 . Since congestion cannot solve instantaneous over all locations, recovery from congestion will increase the spatial heterogeneity of density and thus decrease the network production, which forms hysteresis loops. Although previous studies investigated how heterogeneity of density and network structure like link length and traffic signal affect the scatter and shape of the MFD, few of them explored the effects of vehicle distribution on network flow under segment disruption conditions caused by catastrophic or short-term events like traffic accidents and maintenance activities which are common in urban road networks. In these conditions, network structure is changed and some vehicles must reroute because of segment disruptions. Therefore, understanding the relation between heterogeneity of vehicle distribution and network flow under segment disruption conditions is significant not only for network control but also for information provision or guidance. This paper aims to explore details of the relation between heterogeneity of vehicle distribution and network flow represented by the MFD under segment disruption conditions. Simulation experiments on the Dallas Fort Worth network with high and dynamic demand are applied to collected traffic related data. Variance of segment density is employed to evaluate the heterogeneity of vehicle distribution for the network. Ten segments are randomly selected to simulate segment disruption conditions by removing one of them at a time. For different network structures caused by segment disruptions, the relation between heterogeneity of vehicle distribution and maximum network flow are investigated by relating the maximum network flow to variance of segment density and average density. In addition, in other traffic states under segment disruption conditions, including free flow state, optimal flow state, congestion state and gridlock state, the relation between heterogeneity of vehicle distribution and network flow for different network structures are also investigated in this study. The reminder of this paper is organized as follows. Section 2 presents the method and measures used to explore the relation between heterogeneity of vehicle distribution and network flow represented by the MFD. Numerical experiments are provided in section 3. Under segment disruption conditions, the relations between heterogeneity of vehicle distribution and network flow in different traffic states are investigated using simulation experiments on Dallas Fort Worth network in this section. Finally, conclusions are presented in section 4.

602


3

2. Methodology This section describes the theory of the MFD and measures used to estimate the spatial distribution of segment density for different traffic states and different network constructions caused by segment disruption. 2.1. The MFD for a network The MFD for a network relates network-wide flow and density, as described by Daganzo and Geroliminis2, 3. Define i as segment number, li, ki and qi as its length, density and traffic flow respectively in a particular time interval; and define qw as weighted average flow, and ka as average density. These two macroscopic variables can be calculated as: (1) (2) The numerator of qw is travel production, the total distance travelled from all the vehicles travelling in a segment3. By collecting these traffic variables in every time interval, the MFD linking network-wide average flow and density for a road network can be plotted and will show the overall network performance. 2.2. Measurements of network distribution Some measures of statistics are applied in this study to describe the characteristics of segment density and capacity distributions, including variance, kurtosis and skewness14, 15. The variance describes the heterogeneity of density while kurtosis and skewness describe the characteristics of vehicle or capacity distributions. The variance of density is calculated using Equation 3, as shown in the following. (3) Where Sk2 is the variance of density, n is the number of segments. Other parameters are the same as above. Kurtosis and skewness are calculated using Equations 4 and 5. Kurtosis describes the tail of a distribution. The higher the kurtosis is, the fatter the tail of data is, which means the mass of data are near the mean value. Skewness describes the asymmetry of a distribution about its mean. If more data concentrate on the right of the mean, the value of skewness would be negative. Conversely, positive skewness indicates that more data concentrate on the left of the mean of a distribution. (4) (5) Where bk and Ske is the kurtosis and skewness of density distribution, respectively, and Sk is the standard deviation of density. Other parameters are the same as above. In this study, the measurements defined above are applied to analyze the characteristics of vehicle distribution. By calculating and plotting the variance of density and network average flows under different states and different network constructions, the relations between them can be established. In next section, numerical experiments are presented to investigate the relations between the heterogeneity and network average flow. 3. Numerical experiments The variables in Equations 1 and 2 can be determined by in field 3, 12 or simulation experiments4, 13. In this study, the numerical experiments were conducted using DYNASMART-P 1.3.0, and traffic data for all segments are obtained from output files of the software.

4


603

3.1. Network and experiment design Dallas Fort Worth network is employed to obtain required traffic related data (density and flow) in Equations 1 and 2. As illustrated in Fig. 1a, the modeled network in DYNASMART-P contains 13 zones, 180 nodes and 455 segments. Ten segments (numbered in Figure 1a) are selected randomly to simulate segment disruption conditions by removing one of them at a time. These segments are divided into two experiment groups. Group one includes segments 1 to 6 which are in the same zone, while group two includes segment 2 and segments 7 to 10 which are in different zones. In each experiment, the MFD is plotted and measures of vehicle heterogeneity are calculated. The origin-destination (OD) demand matrix is time-dependent, and specified in 10-minute intervals, shown in Fig. 1b. To obtain the complete shape of MFD and make as more as vehicles output the network, the demand is high and the entire analysis period is 7 hours, from 3:30 p.m. to 10:30 p.m., including a 30 minutes’ warm-up period (3:30 p.m. to 4:00 p.m.) and a 3.5 hours’ peak period (4:00 p.m. to 7:30 p.m.). Since traffic demand is higher than capacity on part of segments in the network, different traffic states will appear (e.g. free flow, congestion and gridlock). To simplify the process of dynamic traffic assignment (DTA), only passenger cars are included in the experiments.

Fig. 1 (a) Modelled network of Dallas Fort Worth in DYNASMART-P and experimental segments; (b) the OD demand.

3.2. The relation between heterogeneity of vehicle distribution and the MFD The initial network (without removing any segment) is set as the base case. As shown in Fig. 2a, four network states can be observed from the base MFD of the base case, including the free flow state, the optimum flow state (network flow maintains near the maximum level), the congestion state and the gridlock state which is a state of the network with minimal throughput. The evolutions of variance of density and network flow are illustrated in Fig. 2b. In the warm-up period (first 30 minute), both network average flow and variance of density increase, since the average density is low and the network is in free flow state. After network flow reaches the maximum level (at about 4:10 p.m.) and becomes congested, variance of density increases notably while network flow decreases. There is a negative relation between heterogeneity of vehicle distribution and network in this period. Nevertheless, once the network is heavily congested, a slight decrease of variance of segment density can be observed. Since more and more vehicles accumulated in the network, the network becomes very congested and evenly gridlock because of spillback. Therefore, the densities of segments become closer and accordingly variance of density decreases after the network is heavily congested. Under this condition, both variance of density and network flow decrease. Some of results accord with findings of previous studies8, 10, which indicates that the simulation experiments in this study are reasonable. Figs. 3a to 3d show segment density distributions at four randomly selected average densities in different states, representing free flow, maximum flow, congested flow and gridlock (7 vehicles/km, 19 vehicles/km, 52 vehicles/h and 91 vehicles/km, respectively). The full density range is discretized from 0 to 150 vehicles/km in 51 groups; one group for empty segments with zero density and 50 groups of equal range. The group 51 is the maximum segment density which is about 150 vehicles/km in this study.

604


5

The evolution of heterogeneity of vehicle distribution is clearly illustrated by these Figures. In free flow state, segment densities concentrate on low values (kurtosis is 14.23), shown in Fig. 3a. As more traffic generates in the network, both the kurtosis (from 14.23 to -1.72) and skewness (from 2.97 to -0.33) are lower and lower, which indicates the tail of density distribution becomes fat and more densities move to high values. When the network average flow reaches the maximum level, most densities are less than 40 vehicles/km, while congestion only concentrates on few segments, illustrated in Figure 3b. As more segments are congested, a bi-modal distribution is observed, shown in Figs. 3c and 3d. In gridlock state (Fig. 3d), the skewness of density distribution decreases to a negative value (-0.33), which means the mass of segments concentrate on high densities. There are more than 170 segments blocked while about 65 segments are empty, which shows the notable heterogeneity of vehicle distribution.

Fig. 2 (a) The Base MFD; (b) The evolutions of the variance and network flow..

Fig. 3 Segment density distributions in different traffic states: a) free flow; b)maximum flow; c) congested flow; d) gridlock.

3.3. Effects of segment disruption on the MFD Under segment disruption conditions, updated MFDs for networks affected by segment disruptions are plotted and compared, shown in Figure 4. The experiment number is in accordance with the disrupted segment number. The overall trends of updated MFDs in both experiments groups are similar to the base MFD, as illustrated in Fig. 4. Under the free flow state, there is not much difference between these MFDs. Compared to the base MFD, not all the maximum flows of updated MFDs decrease when a segment is removed. For example, when segment 5 or 8 is removed, the maximum flow increases slightly. Under the congested and gridlock states, all the updated MFDs vary

6


605

significantly. When segment 9 or 10 is removed, the performance of the updated network becomes worse significantly and goes in gridlock at a lower density compared to the base MFD. It means that spillover has strong impacts on network flow under gridlock state if segment 9 or 10 is removed. On the contrary, when segment 5 or 8 is removed, network flow of the updated network stays near the maximum for a longer period, and about 5000 vehicles more can output compared to the initial network. It indicates that the overall network performance may be better because of rerouting traffic caused by segment disruption, which is consistent with the Braess’ Paradox8.

Fig. 4 (a) The update MFDs in group one; (b) the updated MFDs in group two

3.4. Relations between heterogeneity and the MFD under segment disruption conditions 3.4.1. Heterogeneity of vehicle distributions and maximum flows for updated networks The variance of density and maximum flows for different networks of two experiment groups in which disrupted segments are in the same zone are shown in Table 1. It seems there is no relation between them if network construction is affected by segment disruptions. For example, if segment 5 is removed, network flow of updated network is the highest while the variance of density is not the lowest compared to other experiments. While comparing experiments 2 and 3, the lower variance of density indicates the higher maximum flow, which cannot be established for all the experiments. Table 1. Indexes under maximum flow states of different networks in group one Experiments

Variance of density

(disrupted segment)

Group one

Group two

Maximum flow

Average density

(vehicles/h)

(vehicles/km)

Basic case

1245

883

14.7

Segment 1

2450

855

18.8

Segment 2

2977

859

21.1

Segment 3

3199

830

20.9

Segment 4

2117

876

17.7

Segment 5

2321

918

19.3

Segment 6

2048

887

17.1

Segment 2

2977

859

20.1

Segment 7

2588

825

18.9

Segment 8

3751

903

25.6

Segment 9

1680

820

15.0

Segment 10

1597

895

16.1

Nevertheless, if more indexes are considered, like the average density at which network flow reaches the maximum value, more details about the relations can be revealed. As shown in Table 1, if the network flows for different network constructions reach the maximum levels at a similar average density, the variances and maximum flows are negatively correlated, such as experiments 2 and 3. The same conclusion can be obtained by comparing results of experiments 4 and 6 or experiments 1 and 5. These results indicate that for different network


606

7

constructions caused by disrupted segments in the same zone, a negative correlation between heterogeneity of vehicle distribution and network maximum flows can be established only if network flows reach the maximum levels at the same density. Similar results can be found in experiment group two. 3.4.2. Heterogeneity of vehicle distributions and flows at the same densities for different networks The relation between heterogeneity of vehicle distributions and maximum network flows under segment disruption conditions are explored in the previous section. In this section, the relations between vehicle distributions and network flow under other traffic states (free flow state, congestion state and gridlock state) are explored. The two experiments groups are also applied. Table 2 show the variance of density and network flows at the average densities analyzed in the base case (7, 19, 52 and 91 vehicles/km, respectively) for experiment one and two, respectively. In both groups, At each density except 7 vehicles/km, the larger variance of density results in the lower network flow for different network constructions. Nevertheless, when density is 91 vehicles/km, the negative relation between heterogeneity of vehicle distribution and network flow is not strict, especially for group two. Since the initial network and updated networks are all heavily congested at this density, the variance of density would decrease because of spillover, which affects the relation between heterogeneity of vehicle distribution and network flow. For example, when segment 9 or 10 is removed, since network becomes heavily and evenly congested because of spillback, both network flow and variance of density are lower than other experiments. Therefore, heavy congestion caused by segment disruptions can break the negative relationship. Table 2 Indexes under free flow, congestion and gridlock states of networks in group one Average density (vehicles/km) 7

Experiments

Group two

Network flow

52

91

Network flow (vehicles/h)

Variance of density

Network flow

(vehicles/h)

Variance of density

(vehicles/h)

Variance of density

(vehicles/h)

Variance of density

Basic case

666

251

779

2439

446

9791

102

11647

Segment 1

561

378

842

2319

459

9636

32

11966

Segment 2

585

405

778

2530

486

9454

41

11715

Segment 3

583

417

773

2806

284

10071

39

11954

Segment 4

601

277

836

2465

425

9795

27

11930

Segment 5

558

384

873

2184

440

9648

49

11742

Segment 6

691

425

789

2482

364

9843

125

11612

Segment 2

521

363

812

2307

376

9878

14

11263

Segment 7

585

405

778

2530

486

9454

41

11015

Segment 8

466

371

818

2200

431

9712

19

11239

Segment 9

531

396

743

2532

459

9480

10

11037

Segment 10

533

340

861

1982

375

10141

6

10736

(disrupted segment)

Group one

Network flow

19

Results of these two experiment groups indicate that under segment conditions, the relation between heterogeneity of vehicle distribution and network flow is insecure. Under free flow state, optimal flow state and congestion state, the increase of variance of density would cause the decrease of network flow at the same density if network construction is affected by segment disruption. Nevertheless, if segment disruption causes heavy congestion in a network, this negative relation would be broken. 4. Conclusions This study revealed some details about the relation between heterogeneity of vehicle distribution and network flow represented by the MFD under segment disruption conditions. Two groups of simulations are applied on Dallas

8


607

Fort Worth network to obtain the traffic related data for the MFD and vehicle distribution under a high and dynamic demand condition. The characteristics of vehicle distribution are estimated by measurements of statistics, including variance, kurtosis and skewness of segment density. By comparing the changes of network flow for different network constructions caused by segment disruptions, the relations between heterogeneity of vehicle distribution and the MFD under different traffic states are investigated. Results show that there is a negative relation between heterogeneity of vehicle distribution and maximum network flow for different networks under segment disruption conditions, only when flows reach the maximum level at the same density. If the characteristics of vehicle distribution is drastically affected by rerouting vehicles and the network flows reach the maximum levels at completely different densities, the negative relation would not exist. For other traffic states except free flow state, similar results can be obtained. Only if the average density is the same, a negative correlation between heterogeneity of vehicle distribution and network flows for different network constructions caused by segment disruptions can be established. Nevertheless, if the network is heavily congested, this relation would not be strict since spillover affects the heterogeneity of vehicles distribution. Results in this study are significant for traffic control and management to improve the efficiency of network and maximize network flow, as well as for emergency response to evacuate rerouting traffic. Nevertheless, exploring more details about how heterogeneity of vehicle distribution affects network flow under other states (e.g. recovery period) is promising for daily traffic control and management. Acknowledgements This study is supported by the National Nature Science Foundation of China (No. 51238008). It is also based on research supported by the NEXTRANS Center, the USDOT Region 5 University Transportation Center at Purdue University. The authors are solely responsible for the contents of this paper. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Daganzo CF. Urban Gridlock: Macroscopic Modeling and Mitigation Approaches. Transportation Research Part B, 2007, 41, pp. 49-62. Geroliminis N, Daganzo CF. Macroscopic Modeling of Traffic in Cities. Transportation Research Board 86th Annual Meeting. CD-ROM. Transportation Research Board of the National Academies, Washington, D.C., 2007. Geroliminis N., Daganzo CF. Existence of Urban-scale Macroscopic Fundamental Diagrams: Some Experimental Findings. Transportation Research Part B, 2008, 42(9), pp. 759-770. Mahmassani HS, Saberi M, Zockaie A. Urban Network Gridlock: Theory, Characteristics, and Dynamics. Transportation Research Part C, 2013, 36, pp. 480-497. Yildirimoglu M, and Geroliminis N. Approximating Dynamic Equilibrium Conditions with Macroscopic Fundamental Diagrams. Transportation Research Part B, 2014, 70, pp. 186-200. Muhlich N, Gayah VV, and Menendez M. An Examination of MFD Hysteresis Patterns for Hierarchical Urban Street Networks using Micro-simulation. Transportation Research Board 94th Annual Meeting. CD-ROM. Transportation Research Board of the National Academies, Washington, D.C., 2015. Zheng N, Rerat G, Geroliminis N. Area-based Pricing Scheme for Multimodal Systems and Heterogeneous Users in an Agent-based Environment. Transportation Research Board 94th Annual Meeting. CD-ROM. Transportation Research Board of the National Academies, Washington, D.C., 2015. Buisson C, Ladier C. Exploring the Impact of Homogeneity of Traffic Measurements on the Existence of Macroscopic Fundamental Diagrams. Transportation Research Record: Journal of the Transportation Research Board, No. 2124, Transportation Research Board of the National Academies, Washington, D.C., 2009, pp. 127–136. Mazloumian A, Geroliminis N, Helbing D. The Spatial Variability of Vehicle Densities as Determinant of Urban Network Capacity. Philosophical Transactions of the Royal Society A Mathematical Physical & Engineering Sciences, 2010, 368(1928), pp. 4627-47. Geroliminis N, Sun J. Properties of A Well-defined Macroscopic Fundamental Diagram for Urban Traffic. Transportation Research Part B, 2011, 45, pp. 605-617. Gayah VV, Daganzo CF. Clockwise Hysteresis Loops in the Macroscopic Fundamental Diagram: An Effect of Network Instability. Transportation Research Part B, 2011, 45, pp. 643-655. Saberi M, Mahmassani HS. Exploring the Properties of Network-wide Flow-Density Relations in A Freeway Network. Transportation Research Record: Journal of the Transportation Research Board, No. 2315, Transportation Research Board of the National Academies, Washington, D.C., 2012, pp. 153-163. Geroliminis N, Boyaci B. The Effect of Variability of Urban Systems Characteristics in the Network Capacity. Transportation Research Part B, 2012, 46(10), pp. 1607-1623. Rice JA. Mathematical Statistics and Data Analysis. Elsevier, Inc., Netherlands, 2006. Guo, Y., and Peeta, S. Rail–Truck Multimodal Freight Collaboration: Truck Freight Carrier Perspectives in the United States. Journal of Transportation Engineering, Vol. 141, No. 11, 2015, pp. 1-11.