Order Macroscopic Traffic Models Research Articles

Generalization Second Order Macroscopic Traffic Models via Relative Velocity of the Congestion Propagation

This paper presents a generalized second-order hydrodynamic traffic model. Its central piece is the expression for the relative velocity of the congestion (compression wave) propagation. We show that the well-known second-order models of Payne–Whitham, Aw–Rascal and Zhang are all special cases of the featured generalized model, and their properties are fully defined by how the relative velocity of the congestion is expressed. The proposed model is verified with traffic data from a segment of the Interstate 580 freeway in California, USA, collected by the California DOT’s Performance Measurement System (PeMS).

Multiscale Control of Generic Second Order Traffic Models by Driver-Assist Vehicles

We study the derivation of generic high order macroscopic traffic models from a follow-the-leader particle description via a kinetic approach. First, we recover a third order traffic model as the hydrodynamic limit of an Enskog-type kinetic equation. Next, we introduce in the vehicle interactions a binary control modelling the automatic feedback provided by driver-assist vehicles and we upscale such a new particle description by means of another Enskog-based hydrodynamic limit. The resulting macroscopic model is now a Generic Second Order Model (GSOM), which contains in turn a control term inherited from the microscopic interactions. We show that such a control may be chosen so as to optimise global traffic trends, such as the vehicle flux or the road congestion, constrained by the GSOM dynamics. By means of numerical simulations, we investigate the effect of this control hierarchy in some specific case studies, which exemplify the multiscale path from the vehicle-wise implementation of a driver-assist control to its optimal hydrodynamic design.

The Aw–Rascle Traffic Model: Enskog-Type Kinetic Derivation and Generalisations

We study the derivation of second order macroscopic traffic models from kinetic descriptions. In particular, we recover the celebrated Aw-Rascle model as the hydrodynamic limit of an Enskog-type kinetic equation out of a precise characterisation of the microscopic binary interactions among the vehicles. Unlike other derivations available in the literature, our approach unveils the multiscale physics behind the Aw-Rascle model. This further allows us to generalise it to a new class of second order macroscopic models complying with the Aw-Rascle consistency condition, namely the fact that no wave should travel faster than the mean traffic flow.

Derivation of a first order traffic flow model of Lighthill-Whitham-Richards type

In this paper, we introduce a new type of macroscopic traffic model which, includes a delay term in the flux function. We start from a delayed microscopic model and use well-known techniques to derive a corresponding first order macroscopic traffic model. This leads to a delayed hyperbolic partial differential equation converging to the classical Lighthill-Whitham-Richards (LWR) model for diminishing delay. Furthermore, we present numerical results to show the impact of the delay, and to verify the new model we found. To do so, we adapt known methods to analyze the derived equations numerically.

Boundary coupling of microscopic and first order macroscopic traffic models

We couple, at a fixed interface, the microscopic Follow the Leader model and the macroscopic Lighthill–Whitham–Richards model, both used for describing vehicular traffic. The coupling is obtained by suitable boundary conditions. We prove existence of solutions for the corresponding Cauchy problem, by using the wave-front tracking method and classical results on ordinary differential equations.

Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models

Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.

Calibration of second order traffic models using continuous cross entropy method

Second order macroscopic traffic flow models are often used to replicate non-linear traffic flow phenomena such as phantom traffic jams or traffic instabilities. In contrast to the (first order) Lighthill–Whitham–Richards (LWR) traffic model, which assumes an equilibrium speed-density relationship (or so-called fundamental diagram), the second order model uses one more dynamic equation to describe the evolution of the speed, therefore allows the speed to fluctuate around the equilibrium diagram. In general, in the second order model, a given model parameter set may exhibit traffic instabilities due to a small initial traffic perturbation (e.g. lane-changing or sudden deceleration). Therefore, small changes of parameter set in second order models will lead to completely different model performance, which consequently leads to a complex calibration effort and hence prohibits its real-life application as compared to the LWR model. So far relatively few calibration results for general macroscopic traffic flow models have been reported. To contribute to the state-of-the-art, this paper puts forward an effort to find global optimal parameters of a second order macroscopic traffic model using a stochastic optimization approach, namely cross entropy method (CEM). Basically, the CEM is set up to solve combinatorial optimization problems so the main novelty of this paper is to apply the CEM to solve continuous multi-extremal optimization problems in transportation through the use of the Kernel density estimation method. Numerical studies are carried out to show that the Kernel-based CEM can search for the global optimal model parameters in a second order model and is a promising method for the calibration of traffic models in general.