Research On Traffic Flow Model Based On Hamilton-jacobi Equations

Traffic flow models have been developed and used to understand,describe,and predict traffic flow since the beginning of the 1950 s.Many traffic planning and management technologies,such as traffic control and numerical traffic simulation,rely heavily on traffic flow theory development.In recent years,with the expansion of economic and urban scale,the road between and within cities is becoming more complicated.The traffic flow model research is also developed with various road conditions(such as speed bumps,traffic signals,road bifurcation,and merging,etc.).Using a macroscopic traffic flow model to simulate local perturbation,road bifurcation,and merging in the road network is easy to calculate large-scale traffic conditions.On the other hand,with the rapid increase in car ownership,only increasing infrastructure measures and planning and managing the road network are not enough to alleviate the congestion problem,so intelligent networked vehicles have emerged.However,it will take time to implement intelligent connected vehicles fully.This period will be a stage where intelligent connected vehicles and manually driven vehicles coexist.The research on the homogeneous traffic flow between intelligent connected vehicles and manually driven vehicles can provide scientific methods and support for traffic planning and management.This article first studies the macroscopic traffic flow model of the road network based on the Hamilton-Jacobi equations.Then we discuss the properties of Hamilton function and apply it to the fundamental diagram model of heterogeneous traffic flow mixed with intelligent network vehicles and manual driven vehicles.The contents of this paper are as follows.(1)Traffic flow for local perturbation.The simplest nonlinear micro behavior is that there is a particular area on a straight road,which makes the vehicles pass through the area decelerate.In this part,we add a local perturbation into the micro car-following model to get a micro model that can describe the deceleration phenomenon.Then,by introducing the cumulative distribution function and making the coordinate transformation,the Lagrange coordinates in the micro car-following model are transformed into Euler coordinates.After rescaling,we propose that the solution of this system converges to the solution of a macroscopic homogenized Hamilton-Jacobi equation,which can be seen as an LWR(Lighthill-Whitham-Richards)model with a junction condition at the origin.We design a numerical method to solve this system and verify it through mathematical analysis and numerical experiments.Finally,we apply the validated model to the control of traffic lights by numerical experiment and prove that the proposed model can simulate the queuing phenomenon at the traffic lights.(2)Traffic flow for bifurcation.We extend the macroscopic traffic flow model with a local perturbation to a more complex situation and get the macroscopic traffic flow model,which can describe the road bifurcation.The model is composed of a set of HamiltonJacobi equations describing the traffic flow on the roads and a junction condition describing the traffic flow at the node.The modeling framework continues to use the ideas of the previous chapter: We establish the micro traffic flow model of bifurcation behavior,and then get the corresponding macro traffic flow model through some transformations.Comparing this model with the macro model of local perturbation,we find that the two models have the same equation for the roads,but the junction condition of this model is more complicated.This model can simulate the traffic flow when a road is divided into N roads.N can be 1,2,3,....Finally,we use numerical simulation to verify the validation of the model.(3)Traffic flow for merging and M-in-N-out network.We build a macroscopic traffic flow model that simulates merging and M-in-N-out network nodes.Both models are composed of a set of Hamilton-Jacobi equations describing the road and the junction conditions describing the nodes.In the merging model,we propose different junction conditions for two different types of merge modes: one is merging as a zipper,and the other is a First-In-First-Out model.In the M-In-N-Out model,we construct the junction conditions that simulate the general traffic flow networks.We point out that all the nodes mentioned above are a particular case of the M-in-N-out model.In addition,we propose an efficient numerical algorithm for solving the Hamilton-Jacobi equations with junction conditions and prove that it can keep the conservation,positive definiteness,and boundedness of the original partial differential equations.Finally,we take a complex traffic crossing of Chaoyangmenwai Street in Beijing as an example to introduce the numerical simulation technology of Hamilton-Jacobi equations and verify the validation of the model.(4)Fundamental diagram for heterogeneous traffic flow.When there are Intelligent Connected Vehicles in a road,the traditional traffic flow will become a heterogeneous traffic flow mixed with Intelligent Connected Vehicles and manual driving vehicles.The fundamental diagram model will inevitably be changed.In this part,we introduce the continuum hypothesis,which is different from the traditional understanding of the equilibrium state,and build a new fundamental diagram model of heterogeneous traffic flow.The fundamental diagrams under different Intelligent Connected Vehicles penetration rates are drawn based on the long-time numerical simulation on the ring road.The equilibrium state of mixed traffic flow is analyzed.We point out that in the congestion area,the fundamental diagram of heterogeneous traffic flow under the equilibrium state may have multiple branches.Finally,we compare the fundamental diagram obtained by numerical simulation with that obtained by analytical solution and point out that the latter overestimates the traffic efficiency by mistake.We also analyze the reasons.(5)We discuss the case of multi-branch in the fundamental diagram of the congestion region.First,we solve each traffic flow state's stability and the law of transition between different traffic flow states.Then we load and unload continuously in the equilibrium traffic flow,analyze its evolution process,and give the idea of a single-valued solution of the fundamental diagram in multi-branches.Finally,we apply the new fundamental diagram model to predict the velocity of small perturbation wave and shock wave.We compare the results with the results of real car experiment and previous work.Results show that the error is less than one percent,far smaller than the previous results.