85 research outputs found
Control of a lane-drop bottleneck through variable speed limits
In this study, we formulate the VSL control problem for the traffic system in
a zone upstream to a lane-drop bottleneck based on two traffic flow models: the
Lighthill-Whitham-Richards (LWR) model, which is an infinite-dimensional
partial differential equation, and the link queue model, which is a
finite-dimensional ordinary differential equation. In both models, the
discharging flow-rate is determined by a recently developed model of capacity
drop, and the upstream in-flux is regulated by the speed limit in the VSL zone.
Since the link queue model approximates the LWR model and is much simpler, we
first analyze the control problem and develop effective VSL strategies based on
the former. First for an open-loop control system with a constant speed limit,
we prove that a constant speed limit can introduce an uncongested equilibrium
state, in addition to a congested one with capacity drop, but the congested
equilibrium state is always exponentially stable. Then we apply a feedback
proportional-integral (PI) controller to form a closed-loop control system, in
which the congested equilibrium state and, therefore, capacity drop can be
removed by the I-controller. Both analytical and numerical results show that,
with appropriately chosen controller parameters, the closed-loop control system
is stable, effect, and robust. Finally, we show that the VSL strategies based
on I- and PI-controllers are also stable, effective, and robust for the LWR
model. Since the properties of the control system are transferable between the
two models, we establish a dual approach for studying the control problems of
nonlinear traffic flow systems. We also confirm that the VSL strategy is
effective only if capacity drop occurs. The obtained method and insights can be
useful for future studies on other traffic control methods and implementations
of VSL strategies.Comment: 31 pages, 14 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
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
Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model
The Aw-Rascle-Zhang (ARZ) model can be interpreted as a generalization of the
Lighthill-Whitham-Richards (LWR) model, possessing a family of fundamental
diagram curves, each of which represents a class of drivers with a different
empty road velocity. A weakness of this approach is that different drivers
possess vastly different densities at which traffic flow stagnates. This
drawback can be overcome by modifying the pressure relation in the ARZ model,
leading to the generalized Aw-Rascle-Zhang (GARZ) model. We present an approach
to determine the parameter functions of the GARZ model from fundamental diagram
measurement data. The predictive accuracy of the resulting data-fitted GARZ
model is compared to other traffic models by means of a three-detector test
setup, employing two types of data: vehicle trajectory data, and sensor data.
This work also considers the extension of the ARZ and the GARZ models to models
with a relaxation term, and conducts an investigation of the optimal relaxation
time.Comment: 30 pages, 10 figures, 3 table
Steady-state traffic flow on a ring road with up- and down- slopes
This paper studies steady-state traffic flow on a ring road with up- and
down- slopes using a semi-discrete model. By exploiting the relations between
the semi-discrete and the continuum models, a steady-state solution is uniquely
determined for a given total number of vehicles on the ring road. The solution
is exact and always stable with respect to the first-order continuum model,
whereas it is a good approximation with respect to the semi-discrete model
provided that the involved equilibrium constant states are linearly stable. In
an otherwise case, the instability of one or more equilibria could trigger
stop-and-go waves propagating in certain road sections or throughout the ring
road. The indicated results are reasonable and thus physically significant for
a better understanding of real traffic flow on an inhomogeneous road
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