Payne-Whitham Model Research Articles

Energy and mobility impacts of connected autonomous vehicles with co-optimization of speed and powertrain on mixed vehicle platoons

Intersections are known to be traffic bottlenecks where a significant amount of energy consumption could be caused due to deceleration/acceleration in the presence of red signals. With an increased level of connectivity and automation of intelligent transportation systems, connected autonomous vehicles (CAVs) are expected to be able to proactively adjust their driving strategies subject to constraints imposed by the predicted future traffic. As a result, many potential benefits can be achieved, such as improved energy efficiency, enhanced traffic safety, among many others. Notably, the way CAVs are controlled affects the following legacy vehicles (LVs) due to complex traffic dynamics. In this article, we are particularly interested in studying the energy and mobility impact of CAVs with an improved traffic prediction method on mixed vehicle platoons at various market penetration rates. Leveraging traffic prediction, CAVs are controlled with co-optimization of their speed and gear position. Specifically, a traffic prediction framework in a rolling horizon fashion is employed based upon a modified Payne–Whitham (PW) model capable of handling mixed traffic consisting of CAVs and LVs. The prediction error of the modified PW model is reduced by 53.62% compared to that of the standard PW model under test scenarios. According to the predicted traffic conditions, speed and gear position of CAVs are co-optimized with the primary goal of minimizing energy consumption when driving on a signalized arterial. The energy benefits achieved by CAVs and the impact of CAVs on LVs behind are studied comprehensively for mixed vehicle platoons. The lead LV follows a real-world speed profile collected on TH-55 in Minnesota. Numerical results show that energy benefits achieved by the vehicle platoon range from 2% to 16%, and a 1% to 5% reduction in travel time for LVs behind CAVs is also observed, at different penetration rates of CAVs in various traffic scenarios. In addition, it is observed that CAVs using the proposed eco-driving approach appear to have a positive impact on the LVs behind in terms of energy consumption, regardless of the driving styles of the LVs ahead.

Macroscopic Traffic-Flow Modelling Based on Gap-Filling Behavior of Heterogeneous Traffic

Traffic-flow modelling has been of prime interest to traffic engineers and planners since the mid-20th century. Most traffic-flow models were developed for the purpose of characterizing homogeneous traffic flow. Some of these models are extended to characterize the complex interactions involved in heterogeneous traffic flow. Existing heterogeneous traffic-flow models do not characterize the driver behavior leading to gap filling in heterogeneous traffic conditions. This study aimed at explaining the gap-filling behavior in heterogeneous traffic flow by using the effusion model of gas particles. The driver’s behavior leading to gap filling in heterogeneous traffic was characterized through developing analogies between the traffic flow and the Maxwell–Boltzmann equation for effusion of gases. This model was subsequently incorporated into the Payne–Whitham (PW) model by replacing the constant anticipation term. The proposed model was numerically approximated by using Roe’s scheme, and numerical simulation of the proposed model was then carried out by using MATLAB. The results of the proposed and PW models were therefore compared. It is concluded that the new model proposed in this study not only produces better results compared to the PW model, but also better captures the expected reality. The main difference between the behavior of the two models is that the effect of bottleneck in the density of traffic is propagated in the form of a shockwave travelling backwards in time in the new model, while the PW model does not exhibit this effect.

A Macroscopic Traffic Model Based on Reaction Velocity

In this paper, a new macroscopic traffic model is presented which characterizes driver response during transitions. The commonly employed Payne–Whitham (PW) model characterizes this response using a constant regardless of the reaction time and interaction and distance between vehicles. As a consequence, the resulting traffic behavior can be unrealistic. In this paper, a model is proposed which considers the reaction time, transition distance, and velocity profile based on the traffic density. This produces smooth changes in density and velocity during transitions compared to the PW model. Results are presented for a bottleneck along a circular road which show that the proposed model provides more realistic behavior than the PW model.

A macroscopic traffic model based on driver physiological response

In this paper, a new traffic model is presented, which characterizes spatial changes in traffic density to align the traffic flow with forward conditions. The commonly employed Payne–Whitham (PW) model adjusts the traffic with a constant speed (velocity) regardless of the conditions, which is unrealistic. With the proposed model, the spatial density alignment is a function of the driver physiological response. As a consequence, traffic aligns quickly for an aggressive driver and slowly for a sluggish driver. Results are presented for abrupt changes in traffic density caused by a bottleneck on a circular road, which shows that the proposed model provides more realistic behavior than the PW model.

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.

Essence of conservation forms in the traveling wave solutions of higher-order traffic flow models

This paper shows the essence of conservation forms when applying the weak solution theory to solve the traveling wave solution of a wide cluster in the Payne-Whitham (PW) model. The consideration of the conservation form for the acceleration equation is an important ingredient in the development of higher-order traffic flow models, but it is largely ignored in the research community. To fix the idea, we define two conservation forms for the same PW model, and consequently derive two solutions with different sets of characteristic parameters of the wide cluster. The analytical results are in good agreement with those that are obtained from numerical simulations. Moreover, these two solutions are also shown to be asymptotic to those of the well-known Kühne and Kerner-Konhaüser models with a viscosity term. More importantly, the careful treatment of the conservation form for the acceleration equation closes the important gap in the literature. Without the conservation form, the solution obtained depends very much on the design of numerical schemes, and can be quite arbitrary and may not adequately conform to the physically relevant properties.

Nonequilibrium Continuum Traffic Flow Model with Frozen Sound Wave Speed

Results are presented from a recent study on a variation of a new non-equilibrium continuum traffic flow model in which traffic sound speed is constant. Hence this model is called the frozen-wave model. This model resembles the Payne–Whitham model but avoids the “back-traveling” of the latter. For this frozen-wave model, the Riemann problem is analyzed for its homogeneous system, two numerical solution methods are developed to solve it, and numerical simulations are carried out under both stable and unstable traffic conditions. These results show that under stable conditions, the model behaves similarly to the Payne–Whitham model. However, under unstable traffic conditions, it has nonphysical solutions or no solutions when a vacuum problem occurs. This study, on the one hand, provides a more complete picture of the properties of this frozen-wave model and reduces the risk of improper applications of it. On the other hand, it also highlights the need to adopt a density-dependent sound speed.

The formation and structure of vehicle clusters in the Payne–Whitham traffic flow model

In this paper, we study the Payne–Whitham (PW) model as a hyperbolic system of conservation laws with relaxation and are interested in its solution patterns when it is unstable. With numerical simulations, we observed the formation of clusters in the solutions. A cluster, which consists of a shock and a transition layer, is a “strong” traveling wave in the sense of Liu (Commun. Math. Phys. 108 (1987) 153), and its traveling speed is equal to that of the corresponding shock. The time-asymptotic structure of a transition layer is described by a first-order ordinary differential equation. We also found that, due to the instability of the PW model, we could not predict the number of clusters or the position, height, or width of each cluster. Our study also reveals the intricate relationship between the structure of vehicle clusters and the parameter c 0.

Driver memory, traffic viscosity and a viscous vehicular traffic flow model

This paper studies the linkage between microscopic car-following and macroscopic fluid-like behavior of traffic flow. We find that driver memory in car-following leads to viscous effects in continuum traffic flow dynamics. This linkage is further exploited to develop a second-order continuum model with viscosity. Further, we show that this viscous model contains practically every well-known continuum model as a special case (e.g., the Lighthill–Whitham–Richards model and the Payne–Whitham model), has a stable wave structure of first- and second-order waves, and controls the extent of non-anisotropic and diffusive influences through a dimensionless parameter called anisotropic factor.

Effects of Relaxation and Anticipation on Riemann Solutions of Payne-Whitham Model

The solutions of Riemann problems of a particular higher-order model—the Payne-Whitham (PW) model—are studied using Roe’s flux splitting scheme as presented by Leo and Pretty. Despite numerous works on higher-order models, little is known about Riemann solutions of these models and how relaxation and anticipation affect these solutions. Riemann solutions of the PW model are computed and compared with those of the Lighthill-Whitham-Richards (LWR) model having the same initial (density) data. It was found that faster relaxation forces the PW model to behave much like the LWR model, that strong anticipation has a stabilizing effect on traffic, and that shock waves travel at different speeds in the PW model than they do in the LWR model. These findings provide a basic checklist for experimental validation of PW-like higher models.