20 research outputs found
Lagrangian-based Hydrodynamic Model: Freeway Traffic Estimation
This paper is concerned with highway traffic estimation using traffic sensing
data, in a Lagrangian-based modeling framework. We consider the
Lighthill-Whitham-Richards (LWR) model (Lighthill and Whitham, 1955; Richards,
1956) in Lagrangian-coordinates, and provide rigorous mathematical results
regarding the equivalence of viscosity solutions to the Hamilton-Jacobi
equations in Eulerian and Lagrangian coordinates. We derive closed-form
solutions to the Lagrangian-based Hamilton-Jacobi equation using the Lax-Hopf
formula (Daganzo, 2005; Aubin et al., 2008), and discuss issues of fusing
traffic data of various types into the Lagrangian-based H-J equation. A
numerical study of the Mobile Century field experiment (Herrera et al., 2009)
demonstrates the unique modeling features and insights provided by the
Lagrangian-based approach.Comment: 17 pages, 7 figures, current version submitted to Transportation
Research Part
A Link-based Mixed Integer LP Approach for Adaptive Traffic Signal Control
This paper is concerned with adaptive signal control problems on a road
network, using a link-based kinematic wave model (Han et al., 2012). Such a
model employs the Lighthill-Whitham-Richards model with a triangular
fundamental diagram. A variational type argument (Lax, 1957; Newell, 1993) is
applied so that the system dynamics can be determined without knowledge of the
traffic state in the interior of each link. A Riemann problem for the
signalized junction is explicitly solved; and an optimization problem is
formulated in continuous-time with the aid of binary variables. A
time-discretization turns the optimization problem into a mixed integer linear
program (MILP). Unlike the cell-based approaches (Daganzo, 1995; Lin and Wang,
2004; Lo, 1999b), the proposed framework does not require modeling or
computation within a link, thus reducing the number of (binary) variables and
computational effort.
The proposed model is free of vehicle-holding problems, and captures
important features of signalized networks such as physical queue, spill back,
vehicle turning, time-varying flow patterns and dynamic signal timing plans.
The MILP can be efficiently solved with standard optimization software.Comment: 15 pages, 7 figures, current version is accepted for presentation at
the 92nd Annual Meeting of Transportation Research Boar
Existence of optima and equilibria for traffic flow on networks
This paper is concerned with a conservation law model of traffic flow on a
network of roads, where each driver chooses his own departure time in order to
minimize the sum of a departure cost and an arrival cost. The model includes
various groups of drivers, with different origins and destinations and having
different cost functions. Under a natural set of assumptions, two main results
are proved: (i) the existence of a globally optimal solution, minimizing the
sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium
solution, where no driver can lower his own cost by changing his departure time
or the route taken to reach destination. In the case of Nash solutions, all
departure rates are uniformly bounded and have compact support.Comment: 22 pages, 5 figure
Dynamic Congestion and Tolls with Mobile Source Emission
This paper proposes a dynamic congestion pricing model that takes into
account mobile source emissions. We consider a tollable vehicular network where
the users selfishly minimize their own travel costs, including travel time,
early/late arrival penalties and tolls. On top of that, we assume that part of
the network can be tolled by a central authority, whose objective is to
minimize both total travel costs of road users and total emission on a
network-wide level. The model is formulated as a mathematical program with
equilibrium constraints (MPEC) problem and then reformulated as a mathematical
program with complementarity constraints (MPCC). The MPCC is solved using a
quadratic penalty-based gradient projection algorithm. A numerical study on a
toy network illustrates the effectiveness of the tolling strategy and reveals a
Braess-type paradox in the context of traffic-derived emission.Comment: 23 pages, 9 figures, 5 tables. Current version to appear in the
Proceedings of the 20th International Symposium on Transportation and Traffic
Theory, 2013, the Netherland
Existence of simultaneous route and departure choice dynamic user equilibrium
This paper is concerned with the existence of the simultaneous
route-and-departure choice dynamic user equilibrium (SRDC-DUE) in continuous
time, first formulated as an infinite-dimensional variational inequality in
Friesz et al. (1993). In deriving our existence result, we employ the
generalized Vickrey model (GVM) introduced in and to formulate the underlying
network loading problem. As we explain, the GVM corresponds to a path delay
operator that is provably strongly continuous on the Hilbert space of interest.
Finally, we provide the desired SRDC-DUE existence result for general
constraints relating path flows to a table of fixed trip volumes without
invocation of a priori bounds on the path flows.Comment: 21 page
A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks
In this paper we propose a LWR-like model for traffic flow on networks which
allows one to track several groups of drivers, each of them being characterized
only by their destination in the network. The path actually followed to reach
the destination is not assigned a priori, and can be chosen by the drivers
during the journey, taking decisions at junctions.
The model is then used to describe three possible behaviors of drivers,
associated to three different ways to solve the route choice problem: 1.
Drivers ignore the presence of the other vehicles; 2. Drivers react to the
current distribution of traffic, but they do not forecast what will happen at
later times; 3. Drivers take into account the current and future distribution
of vehicles. Notice that, in the latter case, we enter the field of
differential games, and, if a solution exists, it likely represents a global
equilibrium among drivers.
Numerical simulations highlight the differences between the three behaviors
and suggest the existence of multiple Wardrop equilibria
A variational approach for continuous supply chain networks
We consider a continuous supply chain network consisting of buffering queues and processors first proposed by [D. Armbruster, P. Degond, and C. Ringhofer, SIAM J. Appl. Math., 66 (2006), pp. 896–920] and subsequently analyzed by [D. Armbruster, P. Degond, and C. Ringhofer, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), pp. 433–460] and [D. Armbruster, C. De Beer, M. Fre- itag, T. Jagalski, and C. Ringhofer, Phys. A, 363 (2006), pp. 104–114]. A model was proposed for such a network by [S. G ̈ottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559] using a system of coupling ordinary differential equations and partial differential equations. In this article, we propose an alternative approach based on a variational method to formulate the network dynamics. We also derive, based on the variational method, a computational algorithm that guarantees numerical stability, allows for rigorous error estimates, and facilitates efficient computations. A class of network flow optimization problems are formulated as mixed integer programs (MIPs). The proposed numerical algorithm and the corresponding MIP are compared theoretically and numerically with existing ones [A. Fu ̈genschuh, S. Go ̈ttlich, M. Herty, A. Klar, and A. Martin, SIAM J. Sci. Comput., 30 (2008), pp. 1490–1507; S. Go ̈ttlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559], which demonstrates the modeling and computational advantages of the variational approach