A macroscopic traffic model based on transition velocities
Introduction
Realistic modeling of traffic on a road is important for mitigating traffic problems such as congestion. In particular, it is required to characterize the alignment with forward traffic conditions. Further, it is essential that model parameters such as speed (velocity) stay within the maximum and minimum values. The traffic density should evolve according to changes in speed, and a density distribution with a low variance is expected at smaller speeds and vice versa. It is clear that traffic cannot align to forward traffic conditions instantaneously. The time required for traffic alignment is known as the transition time, . The distance required for alignment is the transition distance, , which is covered during the transition time. A traffic flow model should realistically characterize the evolution of the traffic density during transitions on a road.
There are three main types of traffic models. Macroscopic models consider the collective flow of vehicles, whereas microscopic models are used to examine the temporal and spatial behavior of drivers based on the influence of vehicles in their proximity. Mesoscopic models share the properties of macroscopic and microscopic traffic models as the time-space traffic flow behavior is modeled using probability distributions and queuing theory. In this case, vehicles are modeled at an individual level and the aggregate behavior is approximated. Thus, small groups of vehicles and their interactions are considered. Macroscopic models are the most commonly employed because of their low complexity and good overall performance.
In macroscopic models, speed and density are used to determine the cumulative behavior of traffic. The density ρ is the average number of vehicles on a road segment per unit length, and the traffic flow is the product of velocity υ and density ρ, and so is measured in terms of vehicles per unit time. Lighthill, Whitham, and Richards [1,2] developed a macroscopic traffic flow model (known as the LWR model), which is based on the equilibrium flow of vehicles. They assumed vehicles align their speed in zero time [3], and ignored transitions in the traffic flow.
Some of the deficiencies of the LWR model are overcome by the Payne model [4], which is a two-equation model. The first equation is based on conservation of vehicles on a road while the second models the acceleration behavior of traffic based on driver anticipation, relaxation, and traffic inertia. Driver anticipation results from a presumption of changes in the forward traffic while relaxation is the tendency of traffic to align its speed to a desirable level [5]. Inertia encompasses the spatial and temporal changes in acceleration. Witham independently developed a similar model [6] which is known as the Payne-Witham (PW) model. This model is based on the assumption that vehicles on a road have similar behavior. Smooth speeds and density distributions are assumed [7], i.e. the traffic speed and density vary continuously in space and time. Unfortunately, this can result in unrealistic speed and density behavior for abrupt changes (discontinuities) in the traffic flow [3]. Del Castillo et al. [8] improved the PW model by incorporating the anticipation and reaction time for small changes in density and velocity. The anticipation term characterizes driver behavior such as the response to changes in the forward traffic. However, the anticipation term is too large and so does not accurately model the physical behavior of traffic [7]. Aw and Rascle [9] provided an improved traffic model in which driver perception of temporal and spatial changes in the traffic is assumed to be an increasing function of density. However, the velocity profile of the traffic is not considered. It is assumed that greater braking and acceleration occur for a higher forward density regardless of the speed profile of the traffic.
Some of the deficiencies of the PW model are overcome by the Zhang model [10] which is based on changes in the equilibrium velocity distribution. The PW model has been criticized for having a traffic density which moves in a direction opposite to the flow at the tail of the density distribution. The Zhang model replaces the speed constant in the PW model with the derivative of the equilibrium velocity distribution to eliminate this behavior. The PW model was also improved by Berg, Mason, and Woods [11]. Their model is based on the headway (distance) between vehicles. At abrupt changes in traffic flow, the spatial alignment of traffic results in large density variations which evolve in space and time. A noise term based on the density is used to reduce these variations and smooth the traffic flow so that it is more realistic. Other macroscopic traffic models incorporate a similar term based on the second derivative of the speed or density.
The PW model can produce unrealistic (often oscillatory), behavior at traffic discontinuities due to an inadequate characterization of spatial changes in the traffic density during transitions. Traffic alignment is assumed to occur with a constant speed, which can result in the speed at discontinuities exceeding the maximum or going below zero, which is impossible. In this paper, a new model is proposed in which the spatial alignment of the traffic density is based on the velocity. In particular, an anticipation term is introduced which depends on the average traffic velocity at a transition, the maximum velocity, and the transition distance. To investigate the behavior of the Zhang, PW and proposed models, an inactive bottleneck on straight and circular roads is considered. An inactive bottleneck is defined as congestion resulting from either a large density or slow moving vehicles, and thus creates a transition in the traffic.
The rest of the paper is organized as follows. Section 2 presents the PW, Zhang, and proposed models. In Section 3, the Roe decomposition technique is described, and this is used in Section 4 to evaluate and compare the performance of the three models. Some concluding remarks are given in Section 5.
Section snippets
Traffic flow models
Payne [4] and Whitham [6] independently developed a two-equation model for traffic flow which is known as the Payne-Whitham (PW) model. The first equation models traffic conservation on a road with a constant number of vehicles while the second models traffic acceleration. This model can be expressed aswhere is the velocity (speed) constant, τ is the relaxation time to align velocities, and is the relaxation term for velocity alignment. The
Roe Decomposition
The proposed, Zhang and PW models are discretized using the Roe decomposition technique [[16], [17], [18], [19], [20], [21]] to evaluate their performance. This technique can be used to approximate a nonlinear system of equationswhere G denotes the vector of data variables, denotes the vector of functions of the data variables, and is the vector of source terms. The subscripts t and x denote the partial derivatives with respect to time and distance, respectively. Eq. (8) can be
Simulation results
The performance of the proposed, PW and Zhang models is evaluated in this section using the parameters given in Table 3. Non-reflective boundary conditions are used for the first example to evaluate the traffic evolution on a straight road for 1.2 s with the proposed and PW models. The second example employs periodic boundary conditions for traffic on a circular road. The proposed and PW models are evaluated for 6 s with m/s, whereas the Zhang model is evaluated for 0.17 s with m/s.
Conclusion
It was shown that the PW model can produce unrealistic traffic behavior at density discontinuities. In particular, speed oscillations were observed which go beyond the maximum and minimum values, and changes in both speed and density were very rapid. These unrealistic results are due to the PW model aligning the traffic with a constant speed regardless of the transitions on the road. The Zhang model also produces oscillatory behavior at traffic discontinuities which is unrealistic. This is
Conflict of interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgement
This project was supported by the Higher Education Commission, Pakistan under the establishment of the National Center of Big Data and Cloud Computing at UET, Peshawar, Pakistan.
Zawar H. Khan received the PhD degree in Electrical Engineering from the University of Victoria, Victoria, BC, Canada, in 2016. He is currently an Assistant Professor with the Department of Electrical Engineering, University of Engineering and Technology Peshawar. His research interests include intelligent transportation systems.
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2022, Alexandria Engineering JournalCitation Excerpt :Thus, a traffic model should consider driver response. Traffic characterization can be based on microscopic and macroscopic vehicle behavior [14]. The microscopic approach considers individual vehicle spatial and temporal movements and their interactions.
Zawar H. Khan received the PhD degree in Electrical Engineering from the University of Victoria, Victoria, BC, Canada, in 2016. He is currently an Assistant Professor with the Department of Electrical Engineering, University of Engineering and Technology Peshawar. His research interests include intelligent transportation systems.
T. Aaron Gulliver received the Ph.D. degree in Electrical Engineering from the University of Victoria, Victoria, BC, Canada in 1989. From 1989 to 1991 he was employed as a Defence Scientist at Defence Research Establishment Ottawa, Ottawa, ON, Canada. He has held academic appointments at Carleton University, Ottawa, ON, Canada and the University of Canterbury, Christchurch, New Zealand. He joined the University of Victoria in 1999 and is a Professor in the Department of Electrical and Computer Engineering. In 2002, he became a Fellow of the Engineering Institute of Canada, and in 2012 he was elected a Fellow of the Canadian Academy of Engineering. His research interests include information theory and communication theory, algebraic coding theory, multicarrier systems, smart grid, intelligent transportation, underwater acoustics, and security.