Macroscopic Traffic Flow Characterization at Bottlenecks
Vol. 6 No. 7 (2020): July
Research Articles
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Traffic congestion is a significant issue in urban areas. Realistic traffic flow models are crucial for understanding and mitigating congestion. Congestion occurs at bottlenecks where large changes in density occur. In this paper, a traffic flow model is proposed which characterizes traffic at the egress and ingress to bottlenecks. This model is based on driver response which includes driver reaction and traffic stimuli. Driver reaction is based on time headway and driver behavior which can be classified as sluggish, typical or aggressive. Traffic stimuli are affected by the transition width and changes in the equilibrium velocity distribution. The explicit upwind difference scheme is used to evaluate the Lighthill, Whitham, and Richards (LWR) and proposed models with a continuous injection of traffic into the system. A stability analysis of these models is given and both are evaluated over a road of length 10 km which has a bottleneck. The results obtained show that the behavior with the proposed model is more realistic than with the LWR model. This is because the LWR model cannot adequately characterize driver behavior during changes in traffic flow.
Iftikhar, A., Khan, Z. H., Gulliver, T. A., Khattak, K. S., Khan, M. A., Ali, M., & Minallahe, N. (2020). Macroscopic Traffic Flow Characterization at Bottlenecks. Civil Engineering Journal, 6(7), 1227–1242. https://doi.org/10.28991/cej-2020-03091543
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[14] A. Farina, A. Graziano, F. Mariani, M. C. Recchioni and F. Zirilli, "Homogeneous and heterogeneous traffic of data packets on complex networks: The traffic congestion phenomenon.” Communications and Network 4 (May 2012):157–182. doi:10.4236/cn.2012.42021.
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[19] H. Yu and M. Kristic, "Traffic congestion control for Aw–Rascle–Zhang model.” Automatica 100 (September 2019):38–51. doi: 10.1016/j.automatica.2018.10.040.
[20] J. Z. Chen, Z. Shi and Y. Hu, "Numerical solutions of a multi-class traffic flow model on an inhomogeneous highway using a high-resolution relaxed scheme.” Frontiers of Information Technology & Electronic Engineering 13 (February 2017):1338–1355. doi: 10.1631/jzus.C10a0406.
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[23] T. Nagatani, "The physics of traffic jams.” Ph.D. dissertation, Department of Mechanical Engineering, Shizuoka University, Shizuoka, Japan, 2003.
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[26] E. A. Edison, "Modelling vehicle traffic flow with partial differential equation.” B.Sc. thesis, Department of Mathematics, Presbyterian University College, Abetifi, Ghana, 2016.
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[29] Z. H. Khan, T. A. Gulliver, H. Nasir, A. Rehman and K. Shahzada, "A macroscopic traffic model based on driver physiological response.” Journal of Engineering Mathematics 10 (June 2019):1–12. doi: 10.1007/s10665-019-09990-w.
[30] I. Yperman, S. Logghe and B. Immers, "The link transmission model: An efficient implementation of the kinematic wave theory in traffic networks.” Advanced OR and AI Methods in Transportation 39 (January 2005):122–130.
[31] A. Spiliopoulou, M. Kontorinaki, M. Papageorgiou and P. Kopelias, "Macroscopic traffic flow model validation at congested freeway off-ramp areas.” Transportation Research Part C 41(January 2015):18–29. doi: /10.1016/j.trc.2014.01.009.
[32] A. Ali, L. S. Andallah and Z. Hossain, "Numerical solution of a fluid dynamic traffic flow model associated with a constant rate inflow.” American Journal of Computational and Applied Mathematics 5 (January 2015):18–26.
[33] D. M. Causon and C. G. Mingham, Introductory Finite Difference Methods for PDEs. New York, NY: Ventus Publishing, (January 2010):223–227.
[34] W. L. Jin, "Nonstandard second-order formulation of the LWR model.” Transportmetrica B: Transport Dynamics 7 (January 2016):1338–1355. doi: 10.1080/21680566.2019.1617.
[35] R. Marzouga, H. Echab and H. Ez-Zahraouy, "Car accidents induced by a bottleneck.” European Physical Journal B 90 (December 2017):1–7. doi: 10.1140/epjb/e2017-70695-5.
[36] M. Badhrudeen, V. Ramesh and L. Vanajakshi, "Headway analysis using automated sensor data under Indian traffic conditions.” Transportation Research Procedia 17 (December 2016):331–339. doi: 10.1016/j.trpro.2016.11.103.
[37] H. Khan, Zawar, Waheed Imran, Sajid Azeem, Khurram S. Khattak, T. Aaron Gulliver, and Muhammad Sagheer Aslam. "A Macroscopic Traffic Model Based on Driver Reaction and Traffic Stimuli.” Applied Sciences 9, no. 14 (July 17, 2019): 2848. doi:10.3390/app9142848.
[38] W. Imran, Z. H. Khan, T. A. Gulliver, K. S. Khattak and H. Nasir, "A macroscopic traffic model for heterogeneous flow.” Chinese J. Physics 63 (January 2020):419–435. doi: 10.1016/j.cjph.2019.12.005.
[39] Z. H. Khan, T. A. Gulliver, K. Azam and K. S. Khattak, "Macroscopic model on driver physiological and psychological behavior at changes in traffic,” J. Eng. Appl. Sci. 38 (May 2019):1–9. doi: 10.25211/jeas.v38i2.3150.
[40] Z. H. Khan, T. A. Gulliver, K. S. Khattak and A. Qazi, "A macroscopic traffic model based on reaction velocity.” Iranian Journal of Science and Technology, Transactions of Civil Engineering 38 (May 2019):1–9. doi: 10.1007/s40996-019-00266-y.
[41] Z. H. Khan, A. Sullehri, T.A. Gulliver and W. Imran, "On the route choice.” Pakistan Journal Of Science 38 (December 2019):212–215.
[42] Z. H. Khan, S. A. A. Shah and T. A. Gulliver. "A macroscopic traffic model based on weather conditions.” Chinese Physics B 27 (July 2018):1–8. doi: 10.1088/1674-1056/27/7/070202.
[2] About traffic indices at this website. Available online: https://www.numbeo.com/traffic/indices_explained.jsp (accessed on 22 April 2019).
[3] TomTom traffic index. Available Online:http:/www.tomtom.com/en_gb/gntrafficindex/. (Accessed on 11 Sep 2018).
[4] Congestion costs U.K. nearly £8 billion in 2018. Available Online:http://inrix.com/press-releases/scorecard-2018-uk/. (Accessed on: 10 Apr 2019).
[5] P. Varaiya, "What we've learned about highway congestion.” Access Magazine 27 (September 2005):1–9. doi: https://escholarship.org/uc/item/31j7k3b4.
[6] Z. H. Khan, "Traffic modelling for intelligent transportation systems.” Ph.D. dissertation, University of Victoria, Victoria, BC, 2016.
[7] Z. H. Khan and T. A. Gulliver, "A macroscopic traffic model for traffic flow harmonization.” European Transport Research Review 10 (June 2018):1–12. doi: 10.1186/s12544-018-0291-y.
[8] Daganzo, Carlos F. "A Behavioral Theory of Multi-Lane Traffic Flow. Part I: Long Homogeneous Freeway Sections.” Transportation Research Part B: Methodological 36, no. 2 (February 2002): 131–158. doi:10.1016/s0191-2615(00)00042-4.
[9] Y. Han and S. Ahn, "Stochastic modeling of breakdown at freeway merge bottleneck and traffic control method using connected automated vehicle.” Transportation Research Part B 107 (February 2018):146–166. doi: 10.1016/j.trb.2017.11.007.
[10] T. T. Qiao, H. H. Jun, S. C. Wong, G. Z. You and Z. Ying, "A new macro model for traffic flow on a highway with ramps and numerical tests.” Communications in Theoretical Physics 51 (January 2009):71–78. doi: 10.1088/0253-6102/51/1/15.
[11] C. A. O'Flaherty, Transport Planning and Traffic Engineering, second edition. (January 2006).
[12] Khan, Zawar H., and T. Aaron Gulliver. "A Macroscopic Traffic Model Based on Anticipation.” Arabian Journal for Science and Engineering 44, no. 5 (January 30, 2019): 5151–5163. doi:10.1007/s13369-018-03702-9.
[13] Shvetsov, Vladimir, and Dirk Helbing. "Macroscopic Dynamics of Multilane Traffic.” Physical Review E 59, no. 6 (June 1, 1999): 6328–6339. doi:10.1103/physreve.59.6328.
[14] A. Farina, A. Graziano, F. Mariani, M. C. Recchioni and F. Zirilli, "Homogeneous and heterogeneous traffic of data packets on complex networks: The traffic congestion phenomenon.” Communications and Network 4 (May 2012):157–182. doi:10.4236/cn.2012.42021.
[15] W. Jin, "Traffic flow models and their numerical solutions.” M.S. thesis, University of California Davis, Davis, CA, 2000.
[16] F. van Wageningen-Kessels, H. van Lint, K. Vuik and S. Hoogendoorn, "Genealogy of traffic flow models.” European Journal of Transport Logistics 4 (January 2015):445–473. doi: 10.1007/s13676-014-0045-5.
[17] J. N. Bosire, J. K. Sigey, J. A. Okelo and J. Okwoyo, "Zhang's second order traffic flow models and its application to the Kisii-Kisumu highway within Kisii country.” International Journal of Science and Research 4 (April 2015):3341–3344.
[18] H. M. Zhang, "Driver memory, traffic viscosity and a viscous vehicular traffic flow model.” Transportation Research Part B: Methodological 37 (October 2001):27–41. doi:10.1016/s0191-2615(01)00043-1.
[19] H. Yu and M. Kristic, "Traffic congestion control for Aw–Rascle–Zhang model.” Automatica 100 (September 2019):38–51. doi: 10.1016/j.automatica.2018.10.040.
[20] J. Z. Chen, Z. Shi and Y. Hu, "Numerical solutions of a multi-class traffic flow model on an inhomogeneous highway using a high-resolution relaxed scheme.” Frontiers of Information Technology & Electronic Engineering 13 (February 2017):1338–1355. doi: 10.1631/jzus.C10a0406.
[21] D. Helbing and M. Treiber, "Gas-kinetic-based traffic model explaining observed hysteretic phase transition.” Physical Review Letters 81 (October 1998):3042–3045. doi: 10.1103/PhysRevLett.81.3042.
[22] A. Kotsialos, M. Papageorgiou, C. Diakaki, Y. Pavlis and F. Middelham, "Traffic flow modeling of large-scale motorway networks using the macroscopic modeling tool METANET.” IEEE Transactions on Intelligent Transportation Systems 3 (December 2002):282–292. doi: 10.1109/TITS.2002.806804.
[23] T. Nagatani, "The physics of traffic jams.” Ph.D. dissertation, Department of Mechanical Engineering, Shizuoka University, Shizuoka, Japan, 2003.
[24] D. Vikram, P. Chakroborty and S. Mittal, "Exploring the behavior of LWR continuum models of traffic flow in presence of shock waves.” Procedia - Social and Behavioral Sciences 104 (December 2013):412–421. doi: 10.1016/j.sbspro.2013.11.134.
[25] P. Zhang, R. Liu, S. C. Wong and S. Dai, "Hyperbolicity and kinematic waves of a class of multi-population partial differential equation.” Europe Journal of Applied Mathematics 17 (June 2006):171–200. doi: 10.1017/S095679250500642X.
[26] E. A. Edison, "Modelling vehicle traffic flow with partial differential equation.” B.Sc. thesis, Department of Mathematics, Presbyterian University College, Abetifi, Ghana, 2016.
[27] N. H. Gartner, C. J. Messer and A. K. Rathi, Traffic Flow Theory: A State of the Art Report. Committee on Traffic Flow Theory and Characteristics. (January 2001).
[28] H. Greenberg, "An analysis of traffic flow.” Operations Research 7 (August 1958):79–85. doi: 10.1287/opre.7.1.79.
[29] Z. H. Khan, T. A. Gulliver, H. Nasir, A. Rehman and K. Shahzada, "A macroscopic traffic model based on driver physiological response.” Journal of Engineering Mathematics 10 (June 2019):1–12. doi: 10.1007/s10665-019-09990-w.
[30] I. Yperman, S. Logghe and B. Immers, "The link transmission model: An efficient implementation of the kinematic wave theory in traffic networks.” Advanced OR and AI Methods in Transportation 39 (January 2005):122–130.
[31] A. Spiliopoulou, M. Kontorinaki, M. Papageorgiou and P. Kopelias, "Macroscopic traffic flow model validation at congested freeway off-ramp areas.” Transportation Research Part C 41(January 2015):18–29. doi: /10.1016/j.trc.2014.01.009.
[32] A. Ali, L. S. Andallah and Z. Hossain, "Numerical solution of a fluid dynamic traffic flow model associated with a constant rate inflow.” American Journal of Computational and Applied Mathematics 5 (January 2015):18–26.
[33] D. M. Causon and C. G. Mingham, Introductory Finite Difference Methods for PDEs. New York, NY: Ventus Publishing, (January 2010):223–227.
[34] W. L. Jin, "Nonstandard second-order formulation of the LWR model.” Transportmetrica B: Transport Dynamics 7 (January 2016):1338–1355. doi: 10.1080/21680566.2019.1617.
[35] R. Marzouga, H. Echab and H. Ez-Zahraouy, "Car accidents induced by a bottleneck.” European Physical Journal B 90 (December 2017):1–7. doi: 10.1140/epjb/e2017-70695-5.
[36] M. Badhrudeen, V. Ramesh and L. Vanajakshi, "Headway analysis using automated sensor data under Indian traffic conditions.” Transportation Research Procedia 17 (December 2016):331–339. doi: 10.1016/j.trpro.2016.11.103.
[37] H. Khan, Zawar, Waheed Imran, Sajid Azeem, Khurram S. Khattak, T. Aaron Gulliver, and Muhammad Sagheer Aslam. "A Macroscopic Traffic Model Based on Driver Reaction and Traffic Stimuli.” Applied Sciences 9, no. 14 (July 17, 2019): 2848. doi:10.3390/app9142848.
[38] W. Imran, Z. H. Khan, T. A. Gulliver, K. S. Khattak and H. Nasir, "A macroscopic traffic model for heterogeneous flow.” Chinese J. Physics 63 (January 2020):419–435. doi: 10.1016/j.cjph.2019.12.005.
[39] Z. H. Khan, T. A. Gulliver, K. Azam and K. S. Khattak, "Macroscopic model on driver physiological and psychological behavior at changes in traffic,” J. Eng. Appl. Sci. 38 (May 2019):1–9. doi: 10.25211/jeas.v38i2.3150.
[40] Z. H. Khan, T. A. Gulliver, K. S. Khattak and A. Qazi, "A macroscopic traffic model based on reaction velocity.” Iranian Journal of Science and Technology, Transactions of Civil Engineering 38 (May 2019):1–9. doi: 10.1007/s40996-019-00266-y.
[41] Z. H. Khan, A. Sullehri, T.A. Gulliver and W. Imran, "On the route choice.” Pakistan Journal Of Science 38 (December 2019):212–215.
[42] Z. H. Khan, S. A. A. Shah and T. A. Gulliver. "A macroscopic traffic model based on weather conditions.” Chinese Physics B 27 (July 2018):1–8. doi: 10.1088/1674-1056/27/7/070202.
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