Mathematical Investigation of Vehicular Traffic Features

Abstract

The aim of the current thesis is to develop and analyze the mathematical models for the purpose of better understanding the dynamics of traffic flow. In this view, lattice hydrodynamic approach is very popular nowadays and the present research contributes to the modeling and analysis of some real traffic characteristics by using lattice hydrodynamic approach. In this model, the idea of discrete lattice is proposed to describe the density wave profile. In this modeling, to describe the density waves, the model space is discretized into lattice sites. This model is composed by the set of two equations, i.e., continuity equation and flow evolution equation with global variables (density, velocity and flow) that describe the aggregate features of traffic flow. In this work, traffic problems are defined considering different important aspects of traffic dynamics like average flow, predictive effect, driver’s behavior and multiphase transition on one-dimensional single lane highway. Further, single-lane lattice model is extended for two-lane incorporating lane changing phenomenon with empirical lane changing rates and the effect of driver’s behavior has been analyzed. Furthermore, in this thesis a new multi-lane model is purposed by considering optimal current difference. Later, the multi-lane model is investigated with delayed-feedback control. 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 phenomenon has been presented through plots. The whole work is divided into six chapters: Chapter 1 provides an overview and introduction to the traffic flow theory. Different modeling approaches describing important features of traffic flow are discussed with brief details. In addition, the limitations or the complexities of the existing approaches are also mentioned. The major and the most important contribution in the development of traffic flow theory is the representation of traffic jams in terms of density waves. Therefore, theoretical studies conducted in recent past are also discussed. In the similar fashion, lattice models came into the existence which incorporates the properties of both microscopic and macroscopic models. Moreover, lattice model is able to overcome the limitations of existing models because it can completely analyze the microscopic details of traffic using global variables. Further, the recent developments in traffic flow theory using lattice approach are discussed. Chapter 2 aims to explore the impact of the predictive effect and the forward average flow on current traffic dynamics in the lattice hydrodynamic model. The predictive effect and the effect of average flow with different sites are investigated theoretically with the help of linear stability as well as nonlinear stability analysis. The stability condition is obtained by using stability analysis. The modified Korteweg-de Vries (mKdV) equation is formulated through nonlinear analysis to describe the propagating behavior of traffic density waves near the critical point. It is observed that both the factors (predictive effect and forward average flow) on curv rent traffic dynamics play an important role to enhance the stability of traffic flow. Finally, numerical simulation verifies the theoretical results which confirm that the traffic jam can be suppressed more effectively on current traffic dynamics by taking predictive effect into the traffic system with the consideration of the average flow on forwarding sites. The effect of multi-phase optimal velocity (OV) on a lattice model accounting for driver’s characteristics in a unidirectional traffic system is investigated in Chapter 3. From theoretical analysis, it is found that the presence of aggressive drivers enlarges the stability region on the phase diagram in density-sensitivity phase plane. As the number of stages in multi phase transition is closely related to the number of critical points, two stage (three phase) OV function is considered and the simulation is carried out to find the effect of sensitivity and drivers behavior on traffic dynamics. Further, with the variation of traffic density, multiple phase transition is reported which not only depends on sensitivity but also is strongly influenced by the driver’s characteristics. Finally, the numerical simulations is performed which verifies the theoretical findings. In real traffic, driver’s behavior influences lane changing and hence traffic dynamics. Motivated by the impact of driver’s behavior on lane changing phenomenon, in Chapter 4 a lattice model is examined by considering driver’s behavior with empirical lane changing rate (ELCR). To analyze the two lane traffic system more effectively, the lane changing rate (LCR) is assumed to be dependent on density and their relationship is considered based on available empirical data. Theoretical analysis is performed to study the effect of small amplitude perturbation as well as long wavelength perturbation on traffic characteristics. Stability condition is obtained via linear stability analysis and the modified Korteweg-de Vries (mKdV) equation is formulated through nonlinear analysis to describe the propagating behavior of traffic density wave near the critical point. It is found that empirical lane changing has a nontrivial influence on traffic dynamics in terms of stabilizing/destabilizing traffic flow. All the theoretical results are verified with numerical simulations on a hypothetical circular road. To investigate the traffic dynamics and emergence of congestion in a traffic network, a multi-lane lattice hydrodynamic model is developed in Chapter 5 and analyzed for unidirectional traffic system incorporating the optimal current difference effect. Analytically, the applicability of the proposed model is investigated against small perturbation via linear stability analysis. It is shown that considering the multi-lane highways with optimal current difference help in improving traffic stability. Nonlinear analysis is performed to derive the mKdV equation and traffic characteristics are obtained in terms of density waves. Numerical tests are conducted and found consistent with theoretical results. Chapter 6 is devoted to investigate the effect of delayed feedback control on a multi-lane system. An extended lattice hydrodynamic model is derived on a multilane road which includes more comprehensive information. The stability condition is obtained via control method (based on the Hurwitz criteria and the H∞- norm). The Bode-plot of transfer function shows that the stability region enhances when delayed-feedback controller on multi-lane system is considered. To describe the vi propagating behavior of traffic density wave near the critical point, the modified Korteweg-de Vries (mKdV) equation is formulated through nonlinear analysis. It is concluded that considering the delayed-feedback control in the multi-lane system contributes to mitigating traffic jams. All the theoretical results are verified in both transient and steady state with numerical simulations. The overall summary of this study and few significant directions for the future scope based on the significance of lattice approach are given in Chapter 7 . vii