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Reformulating the Hoogendoorn–Bovy predictive dynamic user-optimal model in continuum space with anisotropic condition

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  • Du, Jie
  • Wong, S.C.
  • Shu, Chi-Wang
  • Zhang, Mengping

Abstract

Hoogendoorn and Bovy (2004) developed an approach for a pedestrian user-optimal dynamic assignment in continuous time and space. Although their model was proposed for pedestrian traffic, it can also be applied to urban cities. The model is very general, and consists of a conservation law (CL) and a Hamilton–Jacobi–Bellman (HJB) equation that contains a minimum value problem. However, only an isotropic application example was given in their paper. We claim that the HJB equation is difficult to compute numerically in an anisotropic case. To overcome this, we reformulate their model for a dense urban city that is arbitrary in shape and has multiple central business districts (CBDs). In our model, the minimum value problem is only used in the CL portion, and the HJB equation reduces to a Hamilton–Jacobi (HJ) equation for easier computation. The dynamic path equilibrium of our model is proven in a different way from theirs, and a numerical algorithm is also provided to solve the model. Finally, we show two numerical examples under the anisotropic case and compare the results with those of the isotropic case.

Suggested Citation

  • Du, Jie & Wong, S.C. & Shu, Chi-Wang & Zhang, Mengping, 2015. "Reformulating the Hoogendoorn–Bovy predictive dynamic user-optimal model in continuum space with anisotropic condition," Transportation Research Part B: Methodological, Elsevier, vol. 79(C), pages 189-217.
    • Handle: RePEc:eee:transb:v:79:y:2015:i:c:p:189-217
    DOI: 10.1016/j.trb.2015.06.005
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    1. Ramezani, Mohsen & Haddad, Jack & Geroliminis, Nikolas, 2015. "Dynamics of heterogeneity in urban networks: aggregated traffic modeling and hierarchical control," Transportation Research Part B: Methodological, Elsevier, vol. 74(C), pages 1-19.
    2. Friesz, Terry L. & Han, Ke & Neto, Pedro A. & Meimand, Amir & Yao, Tao, 2013. "Dynamic user equilibrium based on a hydrodynamic model," Transportation Research Part B: Methodological, Elsevier, vol. 47(C), pages 102-126.
    3. Huang, Ling & Wong, S.C. & Zhang, Mengping & Shu, Chi-Wang & Lam, William H.K., 2009. "Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm," Transportation Research Part B: Methodological, Elsevier, vol. 43(1), pages 127-141, January.
    4. Kuwahara, Masao & Akamatsu, Takashi, 2001. "Dynamic user optimal assignment with physical queues for a many-to-many OD pattern," Transportation Research Part B: Methodological, Elsevier, vol. 35(5), pages 461-479, June.
    5. Haddad, Jack & Ramezani, Mohsen & Geroliminis, Nikolas, 2013. "Cooperative traffic control of a mixed network with two urban regions and a freeway," Transportation Research Part B: Methodological, Elsevier, vol. 54(C), pages 17-36.
    6. Richard Bellman, 1957. "On a Dynamic Programming Approach to the Caterer Problem--I," Management Science, INFORMS, vol. 3(3), pages 270-278, April.
    7. Geroliminis, Nikolas & Sun, Jie, 2011. "Properties of a well-defined macroscopic fundamental diagram for urban traffic," Transportation Research Part B: Methodological, Elsevier, vol. 45(3), pages 605-617, March.
    8. Geroliminis, Nikolas & Daganzo, Carlos F., 2008. "Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings," Transportation Research Part B: Methodological, Elsevier, vol. 42(9), pages 759-770, November.
    9. Tong, C. O. & Wong, S. C., 2000. "A predictive dynamic traffic assignment model in congested capacity-constrained road networks," Transportation Research Part B: Methodological, Elsevier, vol. 34(8), pages 625-644, November.
    10. Daganzo, Carlos F. & Gayah, Vikash V. & Gonzales, Eric J., 2011. "Macroscopic relations of urban traffic variables: Bifurcations, multivaluedness and instability," Transportation Research Part B: Methodological, Elsevier, vol. 45(1), pages 278-288, January.
    11. Bar-Gera, Hillel & Boyce, David, 2006. "Solving a non-convex combined travel forecasting model by the method of successive averages with constant step sizes," Transportation Research Part B: Methodological, Elsevier, vol. 40(5), pages 351-367, June.
    12. Ji, Yuxuan & Geroliminis, Nikolas, 2012. "On the spatial partitioning of urban transportation networks," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1639-1656.
    13. Daoli Zhu & Patrice Marcotte, 2000. "On the Existence of Solutions to the Dynamic User Equilibrium Problem," Transportation Science, INFORMS, vol. 34(4), pages 402-414, November.
    14. Zhang, Xiaoning & Huang, Hai-Jun & Zhang, H.M., 2008. "Integrated daily commuting patterns and optimal road tolls and parking fees in a linear city," Transportation Research Part B: Methodological, Elsevier, vol. 42(1), pages 38-56, January.
    15. Gayah, Vikash V. & Daganzo, Carlos F., 2011. "Clockwise hysteresis loops in the Macroscopic Fundamental Diagram: An effect of network instability," Transportation Research Part B: Methodological, Elsevier, vol. 45(4), pages 643-655, May.
    16. Wong, S. C., 1998. "Multi-commodity traffic assignment by continuum approximation of network flow with variable demand," Transportation Research Part B: Methodological, Elsevier, vol. 32(8), pages 567-581, November.
    17. Terry L. Friesz & David Bernstein & Tony E. Smith & Roger L. Tobin & B. W. Wie, 1993. "A Variational Inequality Formulation of the Dynamic Network User Equilibrium Problem," Operations Research, INFORMS, vol. 41(1), pages 179-191, February.
    18. Yildirimoglu, Mehmet & Geroliminis, Nikolas, 2014. "Approximating dynamic equilibrium conditions with macroscopic fundamental diagrams," Transportation Research Part B: Methodological, Elsevier, vol. 70(C), pages 186-200.
    19. Lo, Hong K. & Szeto, W.Y., 2005. "Road pricing modeling for hyper-congestion," Transportation Research Part A: Policy and Practice, Elsevier, vol. 39(7-9), pages 705-722.
    20. Geroliminis, Nikolas & Boyacı, Burak, 2012. "The effect of variability of urban systems characteristics in the network capacity," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1607-1623.
    21. Daganzo, Carlos F., 2007. "Urban gridlock: Macroscopic modeling and mitigation approaches," Transportation Research Part B: Methodological, Elsevier, vol. 41(1), pages 49-62, January.
    22. Han, Ke & Friesz, Terry L. & Yao, Tao, 2013. "Existence of simultaneous route and departure choice dynamic user equilibrium," Transportation Research Part B: Methodological, Elsevier, vol. 53(C), pages 17-30.
    23. Buckley, D. J., 1979. "Traffic assignment in a two-dimensional continuous representation of a traffic network with flow-dependent speeds," Transportation Research Part B: Methodological, Elsevier, vol. 13(2), pages 167-179, June.
    24. Lo, Hong K. & Szeto, W. Y., 2002. "A cell-based variational inequality formulation of the dynamic user optimal assignment problem," Transportation Research Part B: Methodological, Elsevier, vol. 36(5), pages 421-443, June.
    25. Du, Jie & Wong, S.C. & Shu, Chi-Wang & Xiong, Tao & Zhang, Mengping & Choi, Keechoo, 2013. "Revisiting Jiang’s dynamic continuum model for urban cities," Transportation Research Part B: Methodological, Elsevier, vol. 56(C), pages 96-119.
    26. Daganzo, Carlos F. & Geroliminis, Nikolas, 2008. "An analytical approximation for the macroscopic fundamental diagram of urban traffic," Transportation Research Part B: Methodological, Elsevier, vol. 42(9), pages 771-781, November.
    27. Leclercq, Ludovic & Geroliminis, Nikolas, 2013. "Estimating MFDs in simple networks with route choice," Transportation Research Part B: Methodological, Elsevier, vol. 57(C), pages 468-484.
    28. Daganzo, Carlos F & Geroliminis, Nikolas, 2008. "An analytical approximation for the macropscopic fundamental diagram of urban traffic," Institute of Transportation Studies, Research Reports, Working Papers, Proceedings qt4cb8h3jm, Institute of Transportation Studies, UC Berkeley.
    29. Hoogendoorn, Serge P. & Bovy, Piet H. L., 2004. "Dynamic user-optimal assignment in continuous time and space," Transportation Research Part B: Methodological, Elsevier, vol. 38(7), pages 571-592, August.
    30. Amin Mazloumian & Nikolas Geroliminis & Dirk Helbing, "undated". "The Spatial Variability of Vehicle Densities as Determinant of Urban Network Capacity," Working Papers CCSS-09-009, ETH Zurich, Chair of Systems Design.
    31. Szeto, W. Y. & Lo, Hong K., 2004. "A cell-based simultaneous route and departure time choice model with elastic demand," Transportation Research Part B: Methodological, Elsevier, vol. 38(7), pages 593-612, August.
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