1,471 research outputs found
On the global stability of departure time user equilibrium: A Lyapunov approach
In (Jin, 2018), a new day-to-day dynamical system was proposed for drivers'
departure time choice at a single bottleneck. Based on three behavioral
principles, the nonlocal departure and arrival times choice problems were
converted to the local scheduling payoff choice problem, whose day-to-day
dynamics are described by the Lighthill-Whitham-Richards (LWR) model on an
imaginary road of increasing scheduling payoff. Thus the departure time user
equilibrium (DTUE), the arrival time user equilibrium (ATUE), and the
scheduling payoff user equilibrium (SPUE) are uniquely determined by the
stationary state of the LWR model, which was shown to be locally,
asymptotically stable with analysis of the discrete approximation of the LWR
model and through a numerical example. In this study attempt to analytically
prove the global stability of the SPUE, ATUE, and DTUE. We first generalize the
conceptual models for arrival time and scheduling payoff choices developed in
(Jin, 2018) for a single bottleneck with a generalized scheduling cost
function, which includes the cost of the free-flow travel time. Then we present
the LWR model for the day-to-day dynamics for the scheduling payoff choice as
well as the SPUE. We further formulate a new optimization problem for the SPUE
and demonstrate its equivalent to the optimization problem for the ATUE in
(Iryo and Yoshii, 2007). Finally we show that the objective functions in the
two optimization formulations are equal and can be used as the potential
function for the LWR model and prove that the stationary state of the LWR
model, and therefore, the SPUE, DTUE, and ATUE, are globally, asymptotically
stable, by using Lyapunov's second method. Such a globally stable behavioral
model can provide more efficient departure time and route choice guidance for
human drivers and connected and autonomous vehicles in more complicated
networks.Comment: 17 pages, 3 figure
A bi-level model of dynamic traffic signal control with continuum approximation
This paper proposes a bi-level model for traffic network signal control, which is formulated as a dynamic Stackelberg game and solved as a mathematical program with equilibrium constraints (MPEC). The lower-level problem is a dynamic user equilibrium (DUE) with embedded dynamic network loading (DNL) sub-problem based on the LWR model (Lighthill and Whitham, 1955; Richards, 1956). The upper-level decision variables are (time-varying) signal green splits with the objective of minimizing network-wide travel cost. Unlike most existing literature which mainly use an on-and-off (binary) representation of the signal controls, we employ a continuum signal model recently proposed and analyzed in Han et al. (2014), which aims at describing and predicting the aggregate behavior that exists at signalized intersections without relying on distinct signal phases. Advantages of this continuum signal model include fewer integer variables, less restrictive constraints on the time steps, and higher decision resolution. It simplifies the modeling representation of large-scale urban traffic networks with the benefit of improved computational efficiency in simulation or optimization. We present, for the LWR-based DNL model that explicitly captures vehicle spillback, an in-depth study on the implementation of the continuum signal model, as its approximation accuracy depends on a number of factors and may deteriorate greatly under certain conditions. The proposed MPEC is solved on two test networks with three metaheuristic methods. Parallel computing is employed to significantly accelerate the solution procedure
Continuous formulations and analytical properties of the link transmission model
The link transmission model (LTM) has great potential for simulating traffic
flow in large-scale networks since it is much more efficient and accurate than
the Cell Transmission Model (CTM). However, there lack general continuous
formulations of LTM, and there has been no systematic study on its analytical
properties such as stationary states and stability of network traffic flow. In
this study we attempt to fill the gaps. First we apply the Hopf-Lax formula to
derive Newell's simplified kinematic wave model with given boundary cumulative
flows and the triangular fundamental diagram. We then apply the Hopf-Lax
formula to define link demand and supply functions, as well as link queue and
vacancy functions, and present two continuous formulations of LTM, by
incorporating boundary demands and supplies as well as invariant macroscopic
junction models. With continuous LTM, we define and solve the stationary states
in a road network. We also apply LTM to directly derive a Poincar\'e map to
analyze the stability of stationary states in a diverge-merge network. Finally
we present an example to show that LTM is not well-defined with non-invariant
junction models. We can see that Newell's model and LTM complement each other
and provide an alternative formulation of the network kinematic wave model.
This study paves the way for further extensions, analyses, and applications of
LTM in the future.Comment: 27 pages, 5 figure
A kinematic wave theory of capacity drop
Capacity drop at active bottlenecks is one of the most puzzling traffic
phenomena, but a thorough understanding is practically important for designing
variable speed limit and ramp metering strategies. In this study, we attempt to
develop a simple model of capacity drop within the framework of kinematic wave
theory based on the observation that capacity drop occurs when an upstream
queue forms at an active bottleneck. In addition, we assume that the
fundamental diagrams are continuous in steady states. This assumption is
consistent with observations and can avoid unrealistic infinite characteristic
wave speeds in discontinuous fundamental diagrams. A core component of the new
model is an entropy condition defined by a discontinuous boundary flux
function. For a lane-drop area, we demonstrate that the model is well-defined,
and its Riemann problem can be uniquely solved. We theoretically discuss
traffic stability with this model subject to perturbations in density, upstream
demand, and downstream supply. We clarify that discontinuous flow-density
relations, or so-called "discontinuous" fundamental diagrams, are caused by
incomplete observations of traffic states. Theoretical results are consistent
with observations in the literature and are verified by numerical simulations
and empirical observations. We finally discuss potential applications and
future studies.Comment: 29 pages, 10 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
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