Fundamental diagram of traffic flow
It can be used to predict the capability of a road system, or its behaviour when applying inflow regulation or speed limits.
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.
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.
As the density increases, the speed of the vehicles on the roadway decreases.
Currently, there are two types of flow density graphs: parabolic and triangular.
Academia views the triangular flow-density curve as more the accurate representation of real world events.
The intersection of freeflow and congested vectors is the apex of the curve and is considered the capacity of the roadway, which is the traffic condition at which the maximum number of vehicles can pass by a point in a given time period.
The flow density diagram is used to give the traffic condition of a roadway.
With the traffic conditions, time-space diagrams can be created to give travel time, delay, and queue lengths of a road segment.
The diagram is not a function, allowing the flow variable to exist at two different speeds.
Once the optimum flow is reached, the diagram switches to the congested branch, which is a parabolic shape.
The parabola suggests that the only time there is free flow speed is when the density approaches zero; it also suggests that as the flow increases the speed decreases.
A macroscopic fundamental diagram (MFD) is type of traffic flow fundamental diagram that relates space-mean flow, density and speed of an entire network with n number of links as shown in Figure 1.
The maximum capacity or “sweet spot” of the network is the region at the peak of the MFD function.
The MFD function can be expressed in terms of the number of vehicles in the network such that:
The study[1] revealed that city sectors with approximate area of 10 km2 are expected to have well-defined MFD functions.
Most beneficially though, the MFD function of a city network was shown to be independent of the traffic demand.
Thus, through the continuous collection of traffic flow data the MFD for urban neighborhoods and cities can be obtained and used for analysis and traffic engineering purposes.
In turn, using congestion pricing, perimeter control, and other various traffic control methods, agencies can maintain optimum network performance at the "sweet spot" peak capacity.
Agencies can also use the MFD to estimate average trip times for public information and engineering purposes.
Keyvan-Ekbatani et al.[2] have exploited the notion of MFD to improve mobility in saturated traffic conditions via application of gating measures, based on an appropriate simple feedback control structure.
They developed a simple (nonlinear and linearized) control design model, incorporating the operational MFD, which allows for the gating problem to be cast in a proper feedback control design setting.
This allows for application and comparison of a variety of linear or nonlinear, feedback or predictive (e.g. Smith predictor, internal model control and other) control design methods from the control engineering arsenal; among them, a simple but efficient PI controller was developed and successfully tested in a fairly realistic microscopic simulation environment.
