LWR Model for Traffic Flow Mixed with CACC Vehicles

In the future, road traffic will incorporate a random mix of manual vehicles and cooperative adaptive cruise control (CACC) vehicles, where a CACC vehicle will degrade to an adaptive cruise control (ACC) vehicle when vehicle-to-vehicle communications are not available. This paper proposes a generalized framework of the Lighthill-Whitham-Richards (LWR) model for such mixed vehicular flow under different CACC penetration rates. In this approach, the kinematic wave speed propagating through the mixed platoon was theoretically proven to be the slope of mixed fundamental diagram. In addition, the random degradation from CACC to ACC was captured in mathematical expectation for traffic scenarios where the CACC only monitors one vehicle ahead. Three concrete car-following models, the intelligent driver model (IDM) and CACC/ACC models validated by Partners for Advanced Transit and Highways (PATH) program, were selected as examples to investigate the propagation of small perturbations and shock waves. Numerical simulations were also performed based on the selected car-following models. Moreover, the derived mixed LWR model was applied to solve some traffic flow problems. It indicates that the proposed LWR model is able to describe the propagation properties of both small perturbations and shock waves. The mixed LWR model can also be used to solve some practical problems, such as the queue caused by a traffic accident and the impact of a moving bottleneck. More importantly, the proposed generalized framework admits other CACC/ACC/regular car-following models, including those developed from further experiments.

A Macroscopic Traffic Model Based on the Safe Velocity at Transitions

The increasing volume of vehicles on the road has had a significant impact on traffic flow. Congestion in urban areas is now a major concern. To mitigate congestion, an accurate model is required which is based on realistic traffic dynamics. A new traffic model is proposed based on the conservation law of vehicles which considers traffic dynamics at transitions. Traffic alignment to forward conditions is affected by the time and distance between vehicles. Thus, the well-known Lighthill, Whitham, and Richards (LWR) model is modified to account for traffic behavior during alignment. A model for inhomogeneous traffic flow during transitions is proposed which can be used to characterize traffic evolution. The performance of the proposed model is compared with the LWR model using the Greenshields and Underwood target velocity distributions. These models are evaluated using the Godunov technique and numerical stability is guaranteed by considering the Courant, Friedrich, and Lewy (CFL) condition. The results obtained show that the proposed model characterizes the flow more realistically, and thus can provide better insight into traffic behavior for use in controlling congestion and pollution levels, and improving public safety. Doi: 10.28991/cej-2021-03091710 Full Text: PDF

Simulasi Numerik Model Matematika Arus Lalu Lintas Berbasis Fungsi Velositas Underwood

Abstrak.Model matematika arus lalu lintas pertama kali dikembangkan oleh Lighthill, Whitham dan Richards pada tahun 1956 yang dikenal dengan model (LWR). Dalam model LWR, fungsi kecepatan adalah unsur yang terpenting. Dalam makalah ini digunakan fungsi kecepatan underwood karena memiliki tingkat kesesuaian yang terbaik dibadingkan dengan fungsi kecepatan lainnya. Metode beda hingga implisit digunakan untuk menemukan solusi numerik model LWR dengan model kecepatan Underwood. Konvergensi metode beda hingga implisit dibuktikan dengan menggunakan teorema Ekuivalensi Lax. Simulasi numerik jalan raya satu lajur sepanjang 10 km dilakukan selama 1 jam menggunakan metode beda hingga implisit berdasarkan data awal dan batas yang dibuat secara artifisial. Simulasi numerik dilakukan dengan dua parameter berbeda. Hasil eksperimen menujukkan bahwa semakin tinggi rata-rata kepadatan kendaraan pada suatu laju mengakibatkan rata-rata kecepatan kendaraan akan berkurang. Kata kunci: Metode Beda Hingga Implisit, Model LWR, Arus Lalu Lintas, Fungsi Felositas Underwood, Simulasi Numerik.Kata kunci : Abstract. Mathematical traffic flow model was first developed by Lighthill, Whitham and Richards in 1956, known as (LWR) model. In LWR model, velocity function was most important. In this paper, Underwood velocity function was used. Implicit finite difference method used to found the numerical solution of LWR model with Underwood velocity model. Convergence the implicit finite difference method proved using the Lax equivalence theorem. The numerical simulation of 10 km highway of single lane was performed for 1 hours using the implicit finite difference method based on artificially generated initial and boundary data. Numerical simulation performed with two different parameters. An experimental result for the stability condition of the numerical scheme was also presented. Density, velocity, and fluks for 1 hours was experimental result of numerical simulation.Keywords: Implicit finite difference method, Lax equivalence theorem, LWR model, Traffic flow, Under-wood velocity Function, Numerical simulation.

Macroscopic Traffic Flow Characterization for Stimuli Based on Driver Reaction

The design and management of infrastructure is a significant challenge for traffic engineers and planners. Accurate traffic characterization is necessary for effective infrastructure utilization. Thus, models are required that can characterize a variety of conditions and can be employed for homogeneous, heterogeneous, equilibrium and non-equilibrium traffic. The Lighthill-Whitham-Richards (LWR) model is widely used because of its simplicity. This model characterizes traffic behavior with small changes over a long idealized road and so is inadequate for typical traffic conditions. The extended LWR model considers driver types based on velocity to characterize traffic behavior in non lane discipline traffic but it ignores the stimuli for changes in velocity. In this paper, an improved model is presented which is based on driver reaction to forward traffic stimuli. This reaction occurs over the forward distance headway during which traffic aligns to the current conditions. The performance of the proposed, LWR and extended LWR models is evaluated using the first order upwind scheme (FOUS). The numerical stability of this scheme is guaranteed by employing the Courant, Friedrich and Lewy (CFL) condition. Results are presented which show that the proposed model can characterize both small and large changes in traffic more realistically. Doi: 10.28991/cej-2021-03091632 Full Text: PDF

A Fast Lax–Hopf Algorithm to Solve the Lighthill–Whitham–Richards Traffic Flow Model on Networks

Efficient and exact algorithms are important for performing fast and accurate traffic network simulations with macroscopic traffic models. In this paper, we extend the semianalytical Lax–Hopf algorithm in order to compute link inflows and outflows with the Lighthill–Whitham–Richards (LWR) model. Our proposed fast Lax–Hopf algorithm has a very low computational complexity. We demonstrate that some of the original algorithm’s operations (associated with the initial conditions) can be discarded, leading to a faster computation of boundary demands/supplies in network simulation problems for general concave fundamental diagrams (FDs). Moreover, the computational cost can be further reduced for triangular FDs and specific space–time discretizations. The resulting formulation has a performance comparable to the link transmission model and because it solves the original LWR model for a wide range of FD shapes, with any initial configuration, it is suitable to solve a broad range of traffic operations problems. As part of the analysis, we compare the performance of the proposed scheme with that of other well-known computational methods.

Macroscopic Traffic Flow Characterization at Bottlenecks

Traffic congestion is a significant issue in urban areas. Realistic traffic flow models are crucial for understanding and mitigating congestion. Congestion occurs at bottlenecks where large changes in density occur. In this paper, a traffic flow model is proposed which characterizes traffic at the egress and ingress to bottlenecks. This model is based on driver response which includes driver reaction and traffic stimuli. Driver reaction is based on time headway and driver behavior which can be classified as sluggish, typical or aggressive. Traffic stimuli are affected by the transition width and changes in the equilibrium velocity distribution. The explicit upwind difference scheme is used to evaluate the Lighthill, Whitham, and Richards (LWR) and proposed models with a continuous injection of traffic into the system. A stability analysis of these models is given and both are evaluated over a road of length 10 km which has a bottleneck. The results obtained show that the behavior with the proposed model is more realistic than with the LWR model. This is because the LWR model cannot adequately characterize driver behavior during changes in traffic flow.

Using High-Density Rain Gauges to Validate the Accuracy of Satellite Precipitation Products over Complex Terrains

Topography and precipitation intensity are important factors that affect the quality of satellite precipitation products (SPPs). A clear understanding of the accuracy performance of SPPs over complex terrains and its relationship with topography is valuable for further improvement of product algorithms. The objective of this study is to evaluate three SPPs—the Climate Prediction Center morphing method bias corrected product (CMORPH CRT), Global Precipitation Measurement Integrated MultisatellitE Retrievals (IMERG), and Tropical Rainfall Measuring Mission 3B42V7 (TRMM 3B42V7) against a high-density network of 104 rain gauges over the Taihang Mountains from 1 January 2016 to 31 December 2017, with special focus on the reliability of products’ performance at different elevation and precipitation intensity. The results show that three SPPs slightly overestimate daily precipitation, compared to rain gauge observations, with bias ratios (β) from 1.02 to 1.06 over the entire regions. In terms of accuracy, 3B42 slightly outperforms CRT and IMERG over the Taihang Mountains. As for different elevation ranges, three SPPs show better performance in terms of accuracy in low and moderate elevation (0–500 m) regions. Similar performances of precipitation detection capability can be found for three products over the whole areas, with detection scores ranging from 0.53 to 0.58. Better precipitation detecting performance of three SPPs was discovered in high-elevation (>1000 m) regions. We adopted a linear regression (LR) model and Locally Weighted Regression (LWR) model in an attempt to discover the linear/non-linear relationships between SPPs’ performances and topographic variations. In the accuracy statistical metrics, the errors of 3B42 and CRT showed significantly positive correlations (p < 0.01) with elevation variations. The critical success index for three products gradually increased with elevation variation based on the LR model. The correlation coefficient and probability of detection for three products showed significant non-linear trends in the LWR model. The probability distribution function for the three products in different elevation regions is similar to that over the entire regions. Three SPPs slightly overestimated the frequency of heavy rain events (6.9 < precipitation intensity (PI) ≤ 19.6 mm/d); CRT and 3B42 tended to underestimate the frequency of no rain events (PI < 0.1 mm/d), while IMERG generally overestimated the frequency of no rain events. Our results not only give a detailed assessment of mainly current SPPs over the Taihang Mountains, but also recommend that further improvement on retrieval algorithm is needed by considering topographical impacts for SPPs in the future.