Analysis and Modeling of Some Traffic Characteristics using Lattice Hydrodynamic Approach

Abstract

Mathematical models can be used to study various non-equilibrium systems that are of current interest. Among them, traffic flow is the most important one. In this light,

present research contributes to the modeling and analysis of some real traffic charac- teristics using lattice hydrodynamic approach. Under lattice hydrodynamic approach,

space is discretized in the form of lattice sites. The model is formed by the set of

continuity equation and a momentum equation consisting of global variables that de- scribe the aggregate behavior of traffic flow at each site. Traffic problems are defined

by considering road design such as curved road and curved road with slope as well as driver’s behavior both for one and two lane. Furthermore, the effect of heterogeneous traffic on unidirectional single lane highway is presented. The most common feature of overtaking has also been incorporated in lattice model using feedback control for

one dimensional system. Moreover, the extensions are made for two dimensional sys- tem by incorporating the effect of bidirectional flow. All the models are analyzed

theoretically by linear as well as nonlinear analysis and theoretical investigations are validated through numerical simulations. With the aim to investigate, how different factors affect the stability of traffic flow, models are analyzed and programming is done on MATLAB software. The effect of different parameters representing different traffic phenomena has been presented through graphs. The thesis is presented in six chapters: In Chapter 1 background of the proposed study followed by its need is given. Then the objective is defined along with the justification of the study. Further, the tools and techniques adopted in the study are discussed. Briefly, the scope and limitation of the study are proposed. Chapter 2 is devoted to the problem of traffic due to road geometry. We propose an

improved lattice model for curved road by including the impact of driver’s characteris- tics on a unidirectional one lane road. Parameters representing the influence of curved

road and driver’s behavior are added to Nagatani’s model [1]. With the aim to derive

the stability conditions for the new model, linear analysis is conducted and to ana- lyze the evolution of density waves near critical point, nonlinear analysis is performed.

Phase plot in the (ρ, a) plane, presenting the neutral and coexisting curves are plotted

which reflect the influence of the curved road angle and the intensity of driver’s at- tribute on dynamics of traffic. It is found that the curved road angle negatively impact

the flow stability and aggressive drivers help in overcoming the congestion. In order torepresent traffic jams in terms of density waves, kink solution is obtained by deriving mKdV equation in the neighborhood of critical point. Theoretical investigations are confirmed through numerical simulations and then the summary of results as well as future scope is given. Chapter 3 deals with the problem of traffic on curved road with slope by incorporating driver’s anticipation effect. As in real traffic situations, design of roads affect flow conditions, thus an important aspect of road designs, i.e., curved road having slope is incorporated in the lattice model. As in Chapter 2, description for curved road is given that have negative impact on stability of flow. But what if curved road also have slope? Clearly it gives rise to more serious congestion. So, the Chapter is devoted to the model for curved road with slope on two lane highway along with anticipation effect of drivers. The effects of different slopes on traffic flow phenomena are presented and are examined theoretically via linear and nonlinear analysis. Phase plot, showing the impact of different angles of slopes is presented. Simulation is performed and the results agree with the theoretical analysis. At last, the summary of results is given with possible future scope. The influence of heterogeneous vehicles on traffic flow is explored in Chapter 4. All lattice models presented before deals with homogeneous traffic flow, but in real traffic situations there exists different vehicles that affect vehicular flow differently. Thus by using lattice approach, a new model is presented by considering two distinct kinds of vehicles. One type of vehicles are small and the other one are taken to be large. The model is developed by including the parameters representing the impacts of heterogeneous mix as well as optimal current difference effect for one lane road. For the purpose to check the influence of bigger and smaller vehicles on stability of flow, the model is analyzed theoretically via linear as well as nonlinear analysis. Two phase plots are presented; one for a parameter representing heterogenous mix and other represent five distinct regions corresponding to the fraction of mixed vehicles. It is shown that bigger vehicles cause congestion and smaller vehicles are liable to reduce congestion. In addition, mKdV equation representing kink antikink density waves is obtained near critical point. Then simulation is performed and the summary of results is given. In Chapter 5, basic lattice model is extended by taking the passing effect using feedback control. Control theory is used and the stability of the system is checked by using Hurwitz criteria and H∞ norm of transfer functions are obtained. The efficiency of control signal in minimizing the jamming transitions due to passing is shown through Bode plot. The results show that the control parameter helps in reducing congestionand makes the flow stable even at higher rate of passing. In order to justify the theoretical results numerical simulation is conducted. In the end of the Chapter, conclusion of results is given. Lattice model for bidirectional flow on a square lattice is presented in Chapter 6. Because in real traffic situations vehicles can move in any possible direction and the flow varies according to the route choice of drivers. Thus, the effect of bidirectional flow on a square lattice is analyzed with optimal current difference effect. Theoretical investigations are done using linear and nonlinear analysis. Results are presented by taking different fractions of cars moving in different directions. Results show that the

flow will be uniform in case of an equal fraction of vehicles otherwise unequal frac- tions of vehicles give rise to congestion. The results also show that optimal current

difference help in making the flow more stable. The results are found compatible with

the characteristics of real network situations. The validity of theoretical investiga- tions is checked through numerical simulation and the last section refers to concluding

remarks. As in the present thesis, traffic flow models describing some features of real road situations using lattice approach are presented and traffic jams are represented in terms of kink density waves. Therefore, possible future scope based on the significance of lattice approach is also given in the end.