Macroscopic modeling of lane-changing for two-lane traffic flow

Journal of Advanced Transportation, Fall, 2009 by Tie-Qiao Tang, S.C. Wong, Hai-Jun Huang, Peng Zhang

1. Introduction

An increasing numbers of traffic flow models are being developed to study the physical mechanisms of lane-changing and the complex traffic phenomena that result from this activity. Some of these models focus on the rules of lane-changing and the resulting physical phenomena that lane-changing causes, such as phase transitions, overtaking, and various types of density waves (Chowdhury et al., 1997; Nagel et al., 1998; Kurata and Nagatani, 2003; Nagai et al., 2005; Tang et al., 2006; Tang et al., 2007a,b,c; Lan and Chang, 2005; Ngoduy, 2006, 2008; Webster et al., 2008), whereas others study the qualitative properties of lane-changing using the kinematic wave theory (Daganzo, 1997; 2002a,b; Daganzo et al., 1997; Greenberg et al., 2003; Wei et al., 2000; Holland and Woods, 1997; Laval, 2003; 2004; Laval and Daganzo, 2003; 2006; Jin and Zhang, 2003a,b,c, 2004; Jin and Jayakrishnan, 2005; Jin, 2006). Some useful empirical evidence has also been produced (Mauch and Cassidy, 2002; Ahn, 2005; Cassidy and Bertini, 1999; Cassidy and Rudjanakanoknad, 2005). However, a clear understanding of the physical mechanisms of lane-changing and the complex traffic phenomena to which it gives rise remains elusive.

Various traffic models, such as cellular automaton models, car-following models, and kinematic wave models, have recently been developed to describe the physical mechanism of lane-changing and explain its effects on various complex traffic phenomena. The Lighthill-Whitham-Richards (LWR) model has been extended (Lighthill and Whitham, 1995; Richards, 1956) to incorporate lane-changing in several studies (Daganzo, 1977; Daganzo et al., 1997; Holland and Woods, 1997; Javal, 2003, 2004; Munjal and Pipe, 1971; Manjal et al., 1971; Michalopoulos et al., 1984), but they have been subject to criticism because they treat lane-changing vehicles as a fluid that can accelerate instantaneously and are thus unable to represent car-following interaction. To overcome this problem, Laval and Daganzo (2003, 2006) proposed a hybrid method that combines the advantages of microscopic and macroscopic models. In his kinematic wave model, Jin (2006) considered the effects of lane-changing on the effective traffic density. However, kinematic wave models do not explicitly ensure consistency between lane-changing and car-following behavior, and are therefore unable to effectively describe the consequences of lane-changing for non-equilibrium traffic phenomena, such as small disturbance instability, stop-and-go traffic, local clusters, and phase transitions.

To cope with the drawbacks of the kinematic wave models, several dynamic models have been proposed to study the complex traffic phenomena that are caused by lane-changing (Tang and Huang, 2004, 2005; Huang et al., 2006). These models can reproduce certain non-equilibrium traffic properties, but are unable to give analytical results for lane-changing because the flow transition functions are discontinuous with respect to the traffic density. Moreover, these dynamic models consist of four partial differential equations that are so complex that it is very difficult to derive the structure of the solution. Tang et al. (2005, 2007a,b) developed a car-following model of two-lane traffic flow and analyzed lane-changing behavior, but obtained only simulation results. The models that are currently available to study lane-changing also generally fail to represent the inter-relationship between the actual flow rate and lane-changing.

In this paper, we develop a dynamic model of lane-changing for two-lane traffic flow. First, we study the relationship between lane-changing and car-following behavior. We present a car-following model of lane-changing, transform it into a dynamic model of lane-changing by using the relationship between the microscopic and macroscopic variables, and analyze the effects of lane-changing on the eigenvalues of the system. Second, we study the linear stability of our model and further discuss the analytical effects of lane-changing on the stable region of traffic flow. Finally, we solve the Riemann problems with appropriately set initial and boundary conditions to show the time-space evolution of the traffic density and actual flow rate when a small disturbance occurs, and to demonstrate the quantitative relationship between various kinds of flow rates and the traffic density, including the actual flow rate, the lane-changing rate, and the difference between the potential and actual flow rates.

The remainder of this paper is organized as follows. Section 2 presents a dynamic model of lane-changing that is consistent with car-following behavior. Section 3 discusses the linear stability of the model and analyzes the effects of lane-changing on the stable region of traffic flow. Section 4 gives the numerical results, and Section 5 concludes the paper.

2. Dynamic model of lane-changing for two-lane traffic flow

We develop a dynamic model of lane-changing that consists of two equations, rather than the usual four (two conservation equations and two acceleration equations) that are used in existing two-lane traffic flow models (Tang and Huang, 2004, 2005; Huang et al., 2006). We show that the proposed model reproduces the complex traffic phenomena that are commonly observed on two-lane highways. Following the convention in the literature (Daganzo, 1997; Daganzo et al., 1997; Holland and Woods, 1997; Laval and Daganzo, 2003, 2006; Munjal and Pipe, 1971; Munjal et al., 1971; Michalopoulos et al., 1984; Tang and Huang, 2004, 2005; Huang et al., 2006), the conservation equations for two-lane traffic flow are written as follows.