Fundamental diagram of traffic flow

The fundamental diagram of traffic flow is a diagram that gives a relation between road traffic flux (vehicles/hour) and the traffic density (vehicles/km). A macroscopic traffic model involving traffic flux, traffic density and velocity forms the basis of the fundamental diagram. It can be used to predict the capability of a road system, or its behaviour when applying inflow regulation or speed limits. There is a connection between traffic density and vehicle velocity: The more vehicles are on a road, the slower their velocity will be. To prevent congestion and to keep traffic flow stable, the number of vehicles entering the control zone has to be smaller or equal to the number of vehicles leaving the zone in the same time. At a critical traffic density and a corresponding critical velocity the state of flow will change from stable to unstable. If one of the vehicles brakes in unstable flow regime the flow will collapse. The primary tool for graphically displaying information in the study traffic flow is the fundamental diagram. Fundamental diagrams consist of three different graphs: flow-density, speed-flow, and speed-density. The graphs are two dimensional graphs. All the graphs are related by the equation “flow = speed * density”; this equation is the essential equation in traffic flow. The fundamental diagrams were derived by the plotting of field data points and giving these data points a best fit curve. With the fundamental diagrams researchers can explore the relationship between speed, flow, and density of traffic. The speed-density relationship is linear with a negative slope; therefore, as the density increases the speed of the roadway decreases. The line crosses the speed axis, y, at the free flow speed, and the line crosses the density axis, x, at the jam density. Here the speed approaches free flow speed as the density approaches zero. As the density increases, the speed of the vehicles on the roadway decreases. The speed reaches approximately zero when the density equals the jam density.

The influence of trip length distribution on urban traffic in network-level models

Mikhail Murashkin

Modeling urban traffic on the level of network is a wide research area oriented to the development of ITS. In this thesis properties of models based on MFD (Macroscopic Fundamental Diagram) are studied. The idea behind MFD is to say that the state of the traffic inside an urban zone is fully determined by its accumulation (the number of traveling vehicles) and that the dynamics of accumulation is caused by the inflow of vehicles (flow of vehicles that enter the zone or start their trips from inside). Nowadays, two different philosophies of modeling dynamics of accumulation exist in the literature. The first one (outflow-MFD) postulates that the outflow of vehicles depends on accumulation. The second one (speed-MFD) postulates that the space-mean speed of vehicles depends on accumulation. The second philosophy already has strong empirical support based on observations of traffic inside many big (kilometer-scale) urban areas around the world. Thus, different speed-MFD models are of great scientific interest. The thesis is mainly devoted to the comparison of so-called PL model (which assumes the existence of both speed-MFD and outflow-MFD) and TB model (which assumes the equality of speeds of vehicles and explicitly assumes the existence of trip length distribution). It was shown that PL model cannot accurately describe the dynamics of accumulation after the jump of inflow. In this case TB model is more preferable. Moreover, it was proven that PL model is a specific case of TB model for the exponential trip length distribution. This makes TB model more attractive than PL model for the practical usage. TB model can be formulated mathematically either as integral equation or nonlocal PDE. Thus, the main drawback of TB model that can be an obstacle in practice is its computational complexity. In this thesis it was proposed to approximate TB model with a simpler model which does not require precise information about the trip length distribution. This so-called M model operates with only the mean and the standard deviation of distribution and has a form of ODE. The analytical comparison between PL, TB and M models proved that M model is much closer to TB model than PL model in the case of constant speed-MFD. The more realistic case of decreasing speed-MFD was studied through the numerical tests and also showed the same effect. Thus, the main conclusion of the study is that M model has practical potential as an elegant and computationally cheap approximation of TB model. Also, given that in the case of constant speed TB model is a type of LTI system, it can be expected that M model might be useful for a wide range of problems that are not related to transportation. However, in this thesis the conclusion about the small difference between M and TB models was made for the inflows that are typical for the transportation field. More precisely, only peak hour shaped (smooth and with small jumps) functions were studied. Despite the simple form of M model it was found that there exists even more simple ODE approximation of TB model. This so-called

\alpha

model works quite well for both smooth and jumping inflows except the case of a short period of time following the jump of inflow. Thus, it might be another good alternative to TB model. The main advantage of

\alpha

model in the case of realistic speed-MFD function is its convex formulation which allows

\alpha

model to be efficiently used inside optimization frameworks.

EPFL2021

Perimeter Control and Route Guidance of Multi-Region MFD Systems With Boundary Queues Using Colored Petri Nets

Nikolaos Geroliminis, Anastasios Kouvelas, Hui Fu

Perimeter control based on Macroscopic Fundamental Diagram (MFD) aims to meter the number of transferring vehicles at the periphery of the protected urban region in order to obtain the desired number of vehicles in that region. The advantage of perimeter control is less computational effort, while its drawback is that it may create long queues and delays at the perimeter of the controlled area. For capturing boundary queue dynamics, an enhanced accumulation-based MFD model is proposed using colored Petri Nets by considering transfer flows, boundary queues and travel delays simultaneously. The gated intersections and related road segments on the border of a protected region are modeled as so-called boundary buffers. Based on the enhanced MFD model, anintegrated perimeter control framework is proposed with consideration of travel time and queuing time in buffers. In this framework, the controllers between peripheral and protected region are optimized using model predictive control theory. Then, internal flow controllers are adopted to homogenize traffic density among subregions, and route guidance is also used to balance the number of queuing vehicles among boundary buffers. Simulation results verify the effectiveness of the proposed integrated perimeter control. Furthermore, the impacts of buffer storage capacity on region heterogeneity and trip completion rates are also investigated in this paper.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2021

Empirical investigation of the emission-macroscopic fundamental diagram

Nikolaos Geroliminis, Emmanouil Barmpounakis, Eric Justin Gonzales, Martí Montesinos Ferrer

Vehicle emissions are a major contributor of air pollution in urban areas, with severe consequences for public health and the environment. This paper focuses on analyzing vehicle emissions in a multimodal urban context to better understand its determinants and investigate relations between emissions and congestion patterns. Thousands of naturalistic trajectories collected with a swarm of drones during the pNEUMA experiment are analyzed and EPA's microscopic emission model, project level MOVES, is implemented to estimate vehicular emissions. The method is developed and illustrated with a CO2 analysis. The results empirically show the aggregated relationships between congestion and vehicular emissions, while accounting for the individual characteristics of each trajectory. These systematic relationships on a network scale with the use of empirical data, allow us to extend the macroscopic fundamental diagram (MFD) to the emissions-MFD (e-MFD). The spatiotemporal distribution of emissions is also studied, highlighting areas with high concentration over the urban area.

PERGAMON-ELSEVIER SCIENCE LTD2021

The influence of trip length distribution on urban traffic in network-level models

Mikhail Murashkin

\alpha

\alpha

model in the case of realistic speed-MFD function is its convex formulation which allows

\alpha

model to be efficiently used inside optimization frameworks.

EPFL2021