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

Hybrid Optimal Control for Time-Efficient Highway Traffic Management

This article examines the hybrid traffic control problem to minimize total travel time (TTT) of a highway network through traffic management infrastructures, including dynamic speed limit signs, ramp metering, and information board.We first build the traffic flow model based on the Moskowitz function for each highway link to predict traffic status within a control horizon. The traffic density is predicted based on the flow dynamic model and corrected periodically by measured traffic flow data. The minimum TTT traffic control problem is then formulated as a mixed-integer quadratic programming problem with quadratic constraints. Numerical simulation of a real world highway network is provided to demonstrate significant reduction of TTT and alleviation of traffic congestion compared to results obtained from ALINEA and PI-ALINEA methods

The Variational Formulation of a Non-equilibrium Traffic Flow Model: Theory and Implications

AbstractThe analysis and numerical solution of non-equilibrium traffic flow models in current literature are almost exclusively carried out in the hyperbolic conservation law framework, which requires a good understanding of the delicate and non-trivial Riemann problem for conservation laws. In this paper, we present a novel formulation of certain non-equilibrium traffic flow models based on their isomorphic relation with optimal control problems. This formulation extends the minimum principle observed by the LWR model. We demonstrate that with the new formulation, generic initial-boundary conditions can be conveniently handled and a simplified numerical solution scheme for non-equilibrium models can be devised. Besides deriving the variational formulation, we provide a comprehensive discussion on its mathematical properties and physical implications

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

The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes

We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function $u_T$ and a time horizon $T>0$, we aim to construct all the initial conditions for which the viscosity solution coincides with $u_T$ at time $T$. As it is common in this kind of nonlinear equations, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of $u_T$, is the one obtained by reversing the direction of time in the equation, considering $u_T$ as terminal condition. In this case, we use the notion of backward viscosity solution, that provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, that relates the reachability of the target with its semiconcavity properties. Then, for the case when $u_T$ is reachable, we construct the set of all initial conditions for which the solution coincides with $u_T$ at time $T$. Note that in general, such initial conditions are not unique. Finally, for the case when the target $u_T$ is not necessarily reachable, we study the projection of $u_T$ on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem, and can be interpreted as the semiconcave envelope of $u_T$, i.e. the smallest reachable target bounded from below by $u_T$

Studies on Complex and Connected Vehicle Traffic Networks

Transportation networks such as road networks are well-known for their complexity. Its users make choices of route, which mode to take, etc.; these users then interact with each other, producing emergent dynamics such as traffic jams on roads. These localized multi-user emergent physical phenomena then interact with similar group movements occurring in other locations, creating more complex network-scale dynamics. These patterns of hierarchical levels of organization and emergent phenomena at each level are typical of so-called "complex systems." In addition, the increasing adoption of information-technology systems like connected and autonomous vehicles is creating new challenges in modeling transportation networks, as new emergent behaviors become possible, but also provide new sources of information and possibilities for traffic operations management.The complexity of transportation networks precludes the use of a single all-encompassing theory for all situations at all scales. This dissertation describes several analyses into understanding and controlling emergent dynamics on road traffic networks. It is broken into three parts. The first part proposes models for several new phenomena at the "macroscopic," group-of-vehicles to group-of-vehicles, level. In particular, we solve a problem of modeling arbitrary road junctions with populations of behaviorally-heterogenous vehicles, where the vehicle flows are modelled by a continuum-approximation, partial-differential-equation-based model. We also present several new modeling constructions for a particular complex road network topology: freeways with managed lanes. It has been noted that these managed lane-freeway networks induce new emergent behaviors that are not present in traditional freeways; we propose modeling techniques for several of them, and fit them into traditional modeling paradigms.The second part presents several contributions for estimating the state of the macro-scale traffic dynamics on the road network, based on the micro-scale data of global navigational satellite system readings of the speed and position of individual vehicles. These contributions are extensions of the particle filtering mathematical framework. First, we demonstrate the use of a Rao-Blackwellized particle filter in assimilating vehicle-local speed measurements to better estimate the macroscopic density state of a freeway. Then, we propose new "hypothesis-testing" particle filters that can be used to reject outlier or otherwise malign measurements in a principled statistical manner.The third and final part presents two items on applying deep neural networks to transportation system problems at smaller scales. Both items make use of neural attention, which is a neural network design technique that allows for the integration of structural domain knowledge. First, we demonstrate the applicability of this technique towards estimating aggregate traffic states at the lane level, and present evidence that designing the neural network architecture to encode different types of lane-to-lane relationships (e.g., upstream lane vs neighboring lane) greatly benefits statistical learning. Then, we apply similar methods to an autonomous vehicle coordination problem in a deep reinforcement learning framework, and show that an attention-based neural network that allows each vehicle to attend to the other vehicles enables superior learning compared to a naive, non-attention-based architecture, and also allows principled generalization between varying numbers of vehicles