Macroscopic Traffic Flow Modeling： I am interested in fluid-dynamics traffic flow models, which describe the collective vehicle dynamics in terms of some average quantities, such as traffic density u(x,t) (# of vehicles per unit length of road), flow rate Q(x,t) (# of vehicle passing through a fixed position per unit time), and average velocity v(x,t) (flow rate/density). This modeling approach results in conservation laws that are appropriate in describing various important phenomenon in vehicular flow.
Generic Second Order Models： One fundamental equation in macroscopic traffic flow model is based on the conservation of number of vehicles. First order traffic flow models, such as the LWR model, is generated from this scalar conservation law, and assume that traffic flow rate depends only on traffic density. In contrast, second order traffic models include another evolution equation in addition to the conservation of mass. The second equation is built on different principles, e.g., Payne-Whitham models momentum, Aw-Rascle-Zhang says that a quantity w = v+h(u), i.e., adding a hesitation equation (h) motivated from gas dynamics to velocity function, is conserved along vehicle trajectories.
Fundamental diagram of the LWR model together with sensor data.
The main difference of a first order model and a second order model is that LWR type models fix a uniform equilibrium flow-density relationship Q(u) that is the fundamental diagram, and force all drivers to possess the same property, all vehicles behave to adjust to the equilibrium state. In contrast, second order models assign properties to different classes of drivers, each class holds a distinct flow-density diagram. In this sense, they possess a family of flow rate curves. By observing the figures, one sees that the fundamental diagram data collected by sensors is set-valued in congested traffic state. Thus, a second order model is more appropriate to capture this dynamics in congestion.
A family of flow vs. density curves of the ARZ model, with fundamental diagram data.
A generic framework for the second order models is motivated by the ARZ model. Under this framework, a conservation of mass is always involved. In addition, an identification of a property of drivers “I” (generic invariance) is the key. Thus, the second equation is the translation of the fact that drivers do not change their property, i.e., the quantity “I” is preserved along vehicle trajectories. This gives rise to an convection equation for variable “I”.
A generalized Aw-Rascle-Zhang model.
Generic Second Order Model From Historical Data： The objective of my research is to develop more realistic second order traffic flow models. Therefore, we have employed an empirical approach, which is incorporated with intensive investigations of historical traffic data sets. For example, one can generate more realistic flow rate curves by a least squares fitting with data. New models should also be able to correct several shortcomings of existing models, e.g., different flow-rate diagrams should have a uniform stagnation traffic density (the traffic density that vehicles completely stopped). A new model, that is called generalized Aw-Rascle-Zhang (GARZ) model, is proposed to address some inconsistent aspects of classical ARZ model. In particular, a systematic approach is outlined to develop second order models from historic fundamental diagram data.
In practice, the choice of generic invariant (the “property” of drivers) is kind of arbitrary, e.g., in the GARZ model, we use the quantity of an empty road velocity “w”. This quantity describes the maximum traffic velocity of a vehicle with a locally empty road, i.e., there is no interaction between this vehicle and others.
A collapsed version of the GARZ model.
A Phase-Transition-Like Collapsed GARZ Model: Based on Kerner’s three-phase-traffic theory, there is a clear phase transition between free flow and congested flow. Thus, phase transition models, such as Colombo’s model, is presented to capture transitions between different traffic states. In our research, we present a model under the generic framework for ARZ type models that can capture the effect of phase transition. We let the flow rate curves in the free flow regime coincide in the GARZ model. This gives a collapsed version of the GARZ model. A fundamental difference between the new model and the classical phase transition model is that there is no real phase transition. In viewing of flow rate curves, we see that classical phase transition models cut a spread of flow rate curves with a single free flow diagram. In contrast, the collapsed model shrinks these intersections into a single point. Thus, the free diagram is connected with the family of congested diagram with a single point.
The new model advantages over phase transition models in the aspects of mathematical analysis. For example, we avoid analyzing the complicated Riemann solution in the occurrence of phase transition. When it comes to ARZ models, we persist the good features of phase transition models, such as the absence of vacuum problem in dealing with Riemann problem.
