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Approximate network loading and dual-time-scale dynamic user equilibrium

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  • Friesz, Terry L.
  • Kim, Taeil
  • Kwon, Changhyun
  • Rigdon, Matthew A.

Abstract

In this paper we present a dual-time-scale formulation of dynamic user equilibrium (DUE) with demand evolution. Our formulation belongs to the problem class that Pang and Stewart (2008) refer to as differential variational inequalities. It combines the within-day time scale for which route and departure time choices fluctuate in continuous time with the day-to-day time scale for which demand evolves in discrete time steps. Our formulation is consistent with the often told story that drivers adjust their travel demands at the end of every day based on their congestion experience during one or more previous days. We show that analysis of the within-day assignment model is tremendously simplified by expressing dynamic user equilibrium as a differential variational inequality. We also show there is a class of day-to-day demand growth models that allow the dual-time-scale formulation to be decomposed by time-stepping to yield a sequence of continuous time, single-day, dynamic user equilibrium problems. To solve the single-day DUE problems arising during time-stepping, it is necessary to repeatedly solve a dynamic network loading problem. We observe that the network loading phase of DUE computation generally constitutes a differential algebraic equation (DAE) system, and we show that the DAE system for network loading based on the link delay model (LDM) of Friesz et al. (1993) may be approximated by a system of ordinary differential equations (ODEs). That system of ODEs, as we demonstrate, may be efficiently solved using traditional numerical methods for such problems. To compute an actual dynamic user equilibrium, we introduce a continuous time fixed-point algorithm and prove its convergence for effective path delay operators that allow a limited type of nonmonotone path delay. We show that our DUE algorithm is compatible with network loading based on the LDM and the cell transmission model (CTM) due to Daganzo (1995). We provide a numerical example based on the much studied Sioux Falls network.

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Bibliographic Info

Article provided by Elsevier in its journal Transportation Research Part B: Methodological.

Volume (Year): 45 (2011)
Issue (Month): 1 (January)
Pages: 176-207

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Handle: RePEc:eee:transb:v:45:y:2011:i:1:p:176-207

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Related research

Keywords: Dynamic user equilibrium Differential variational inequalities Differential algebraic equations Dual-time-scale Fixed-point algorithm in Hilbert space;

References

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  1. Yu Nie & H. Zhang, 2010. "Solving the Dynamic User Optimal Assignment Problem Considering Queue Spillback," Networks and Spatial Economics, Springer, vol. 10(1), pages 49-71, March.
  2. Friesz, Terry L. & Mookherjee, Reetabrata, 2006. "Solving the dynamic network user equilibrium problem with state-dependent time shifts," Transportation Research Part B: Methodological, Elsevier, vol. 40(3), pages 207-229, March.
  3. Carey, Malachy, 1992. "Nonconvexity of the dynamic traffic assignment problem," Transportation Research Part B: Methodological, Elsevier, vol. 26(2), pages 127-133, April.
  4. Wu, J. H. & Chen, Y. & Florian, M., 1998. "The continuous dynamic network loading problem: a mathematical formulation and solution method," Transportation Research Part B: Methodological, Elsevier, vol. 32(3), pages 173-187, April.
  5. Li, Jun & Fujiwara, Okitsugu & Kawakami, Shogo, 2000. "A reactive dynamic user equilibrium model in network with queues," Transportation Research Part B: Methodological, Elsevier, vol. 34(8), pages 605-624, November.
  6. Daganzo, Carlos F., 1995. "The cell transmission model, part II: Network traffic," Transportation Research Part B: Methodological, Elsevier, vol. 29(2), pages 79-93, April.
  7. Nie, Yu (Marco) & Zhang, H.M., 2008. "A variational inequality formulation for inferring dynamic origin-destination travel demands," Transportation Research Part B: Methodological, Elsevier, vol. 42(7-8), pages 635-662, August.
  8. Huang, Hai-Jun & Lam, William H. K., 2002. "Modeling and solving the dynamic user equilibrium route and departure time choice problem in network with queues," Transportation Research Part B: Methodological, Elsevier, vol. 36(3), pages 253-273, March.
  9. Nie, Yu (Marco), 2010. "Equilibrium analysis of macroscopic traffic oscillations," Transportation Research Part B: Methodological, Elsevier, vol. 44(1), pages 62-72, January.
  10. 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.
  11. Ban, Xuegang (Jeff) & Liu, Henry X. & Ferris, Michael C. & Ran, Bin, 2008. "A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations," Transportation Research Part B: Methodological, Elsevier, vol. 42(9), pages 823-842, November.
  12. 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.
  13. 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.
  14. Larry Samuelson, 1998. "Evolutionary Games and Equilibrium Selection," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262692198, December.
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Citations

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Cited by:
  1. 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.
  2. Chung, Byung Do & Yao, Tao & Friesz, Terry L. & Liu, Hongcheng, 2012. "Dynamic congestion pricing with demand uncertainty: A robust optimization approach," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1504-1518.
  3. Zhong, R.X. & Sumalee, A. & Friesz, T.L. & Lam, William H.K., 2011. "Dynamic user equilibrium with side constraints for a traffic network: Theoretical development and numerical solution algorithm," Transportation Research Part B: Methodological, Elsevier, vol. 45(7), pages 1035-1061, August.

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