Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 136451, 13 pages

http://dx.doi.org/10.1155/2015/136451

## An Improved Macro Model of Traffic Flow with the Consideration of Ramps and Numerical Tests

College of Automation, Northwestern Polytechnical University, No. 127 Youyi Road (West), Beilin, Xi’an, Shaanxi 710072, China

Received 8 April 2015; Revised 11 June 2015; Accepted 14 June 2015

Academic Editor: Xiaosong Hu

Copyright © 2015 Zhongke Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present an improved macro model for traffic flow based on the existing models. The equilibrium point equation of the model is obtained. The stop-and-go traffic phenomenon is described in phase plane and the relationship between traffic jams and system instability is clearly shown in the phase plane diagrams. Using the improved model, some traffic phenomena on a highway with ramps are found in this paper. The numerical simulation is carried out to investigate various nonlinear traffic phenomena with a single ramp generated by different initial densities and vehicle generation rates. According to the actual road sections of Xi’an-Baoji highways, the situations of morning peak with several ramps are also analyzed. All these results are consistent with real traffic, which shows that the improved model is reasonable.

#### 1. Introduction

In recent years, traffic jams has become more and more serious. They do not only cause a large number of costs but they also have a negative impact on the environment and energy sustainability. Therefore, researchers have made many efforts to develop transportation electrification to alleviate the impact. Hu et al. [1, 2] analyzed the energy efficiencies of a series plug-in hybrid electric bus with different energy management strategies and battery sizes and comparatively examined three different electrochemical energy storage systems for a hybrid bus powertrain. Recently, they [3] also investigated the optimal component sizing and power management of a fuel cell/battery hybrid bus. Sun et al. [4] presented a traffic data-enabled predictive energy management framework for plug-in hybrid electric vehicles. Furthermore, they [5] studied the velocity predictors for predictive energy management in hybrid electric vehicles. However, many other physicists and engineers have tried to develop traffic models with the aim of optimizing traffic flow. During the past decades, lots of traffic models have been constructed to replicate the formation mechanism and inherent law of the traffic phenomena. In microscopic view, the traffic flow system was regarded as a complex self-driven many-particle system composed of a large number of vehicles. The microscopic traffic flow models investigated the dynamical behavior of a single vehicle and the interactions between the vehicles, so they can also describe the traffic phenomena of the whole system. Among them, the car-following model is a favorable type of traffic models describing the driver’s following behavior in view of the stimulus from its preceding vehicle. On the macroscopic view, physicists paid close attention on the collective behavior of traffic. Due to the analogy of vehicle stream with gas stream or fluid stream, large numbers of the gas kinetic models or fluid-dynamic models have been developed to approximately describe the traffic phenomena. Based on these models, researchers can use a lot of system simulation methods to analyze the traffic phenomena.

The traffic flow on a highway with the ramp has been studied for decades through observation and modeling. Lee et al. [6] studied the presence of the external vehicle flux through ramps and found a new kind of traffic phenomenon, called “recurring humps” (RH). In this state, the density and the flow oscillated periodically and the oscillations concentrated around the ramp. Gupta and Katiyar [7] studied the phase transition on a highway in a modified anisotropic continuum model with an on-ramp. Huang [8] observed an interesting phase: jam-max.-free when the on-ramp is placed before the off-ramp. He also demonstrated that the bulk properties on the roadway are totally controlled by the ramp flow through the boundaries. Tang et al. [9] indicated that ramps often have different effects on the main road traffic during the morning rush period and the evening rush period and that the effects are related to the initial status of the main road traffic flow.

However, these models cannot completely describe the various complex phenomena resulted by different input and output conditions on ramps. In particular, the phenomena of fixed vehicle generation rate but increasing initial homogeneous density with a single ramp, the situation of morning peak, and the congested traffic stream with several ramps are rarely studied in the past. In this paper, we present an improved macro model for traffic flow to analyze these phenomena on a highway with a single ramp and multiple ramps. Moreover, we introduce a completely different method to describe traffic phenomena in the phase plane diagrams from a stability perspective. The variable substitution is adopted in the models and the traffic congestion corresponds to the unstable system in phase plane. So the traffic flow problems can be converted into the system stability problems.

The remainder of the paper is organized as follows. In Section 2, we present an improved continuum model based on the existing traffic flow model. In Section 3, we deduce the equilibrium point equation of the model. In Section 4, we analyze the well-known stop-and-go waves using the phase plane diagrams based on the improved model and compare them with the traditional temporal evolution of vehicle density. In Section 5, we use the improved model to describe various nonlinear phenomena on a highway with a single ramp. In Section 6, the traffic flow on a highway with multiple ramps is also studied and the actual traffic phenomena of Xi’an-Baoji highways are discussed. We conclude the paper in Section 7.

#### 2. Models

The macroscopic traffic flow models consider vehicles as interacting particles and consider traffic flow as a one-dimensional compressible flow of these particles. The study of macroscopic traffic flow models began with the LWR model proposed by Lighthill and Whitham [10] and Richards [11]. To overcome the shortage of the LWR model, Payne [12] developed a higher order model by using a dynamic equation for the mean velocity. Hereafter, many researchers presented a great number of models based on Payne’s model [13–15]. However, these models fail to describe the property that the characteristic speeds are always less than or equal to the macroscopic flow speed. Later, Zhang [16] proposed a macroscopic traffic flow model which overcomes the backward travel problem. Gupta and Katiyar [17] also developed an anisotropic continuum model which is referred to as GK model. Although these models can describe many complex traffic phenomena, they cannot be used to directly explore the effects of ramps since they do not consider this factor. So far, some theoretical models have been developed to study the effects of ramps [6–9]. However, these models cannot completely describe the various complex phenomena resulted by different input and output conditions on ramps. In particular, the phenomena of fixed vehicle generation rate but increasing initial homogeneous density with a single ramp, the situation of morning peak, and the congested traffic stream with several ramps are rarely studied in the past. In this paper, we present an improved macro model for traffic flow on a highway with ramps based on the GK model as follows: where is the density; is the velocity; and represent space and time, respectively; is the driver’s reaction time; is a nonnegative dimensionless parameter; is the optimal velocity function and has the following form [18]:Consider , is the free-flow speed, is the maximum or jam density, and is the traffic sound speed given byConsider is the flow generation rate. For simplicity, we here adopt the definition of flow generation rate in Jiang et al. [19]; that iswhere is the region of the ramp, is the length of the ramp, and is the total ramp flow. We here define as follows: where is the input flow of the on-ramp and is the output flow of the off-ramp.

Furthermore, we employ a simple transformation as follows:

Substituting the variables into (1), we have a new traffic flow model as follows:

Similarly, substituting the variables into (2), the equilibrium velocity is as follows:

According to the variable substitution , we can see that as long as the traffic becomes congested and the vehicles velocity goes to zero, the state variable will approach infinity. Likewise, from the variable substitution, , we can see that if the vehicle density becomes saturated, the state variable will approach infinity in the same way. So we can use the phase plane diagrams about the variable or to describe clearly the relationship between traffic jams and system instability. As long as the traffic has a very small density fluctuation, the value of and will change sharply. Moreover, as long as there is traffic jam formation, the value of and will approach infinity. The more the value of and increases, the greater the fluctuation of the vehicle density is and the more unstable the traffic system is. On the contrary, the system becomes more stable, so the problem of traffic flow could be converted into that of system stability. We can describe all kinds of nonlinear traffic phenomena with the phase plane diagrams and determine whether there will be traffic congestion or other abnormal phenomena from a global stability point. It may be possible to apply some mathematical tools such as branch and bound to the nonlinear stability analysis of traffic system. We can find the equilibrium solutions and some bifurcations of the new model to regulate the stability of traffic system in the future work.

#### 3. The Equilibrium Point Equation Analysis

When the traffic system reaches equilibrium state, the density and velocity of the whole road will not change with time. Moreover, when the traffic system reaches some special equilibrium points, the density and velocity of the whole road will not change with time and displacement at the same time. In order to find these equilibrium solutions of the new model, the equilibrium point equation of the system is analyzed firstly.

When system (7) does not change with time, we have

It is assumed that the input flow of the on-ramp is equal to the output flow of the off-ramp. So we have . By substituting (9) into (7), the equilibrium points satisfy the formula as follows:

Equation (10a) can be rewritten as

The right hand side of (11) can be written as

So,

By integrating (11) at both ends, we have is a nonzero constant and (14) can be rewritten as

By substituting (15) and (11) into (10b), we obtain

In summary, the system equilibrium points satisfy the following equations:

The meaning of every parameter is the same as above. If the initial values of and are given by the solution of (17), the density and velocity of the whole road will not change with time. At the same time, we can see from (15) that the product of density and velocity is equal to .

Furthermore, when system (17) also does not change with displacement, we obtain the equilibrium points equation as follows:

Next, we analyze the solution of (18b). cannot be zero according to . If , then is equal to 0. In this case, is trivial equilibrium point and has no practical significance. So we only need to investigate the following equation:

Equation (19) can be written as

At the same time, (18a) can be written as . So we can see from (20) that .

Therefore, we may conclude that if the value of initial density is set as a random constant in the reasonable range of traffic flow and the initial velocity is given as the equilibrium velocity which is corresponding to the initial density, the density and velocity of the whole road will not change with time and displacement. These conclusions are also consistent with the phenomena observed in realistic traffic flow.

#### 4. The Stop-and-Go Traffic Phenomena on the Phase Plane

The stop-and-go traffic phenomena are international well-known nonlinear phenomena. Traditional researches on it mainly focused on using the figures of temporal development of density through the original traffic flow models. The new model mentioned above can also describe it through the phase plane diagrams from a system stability perspective. The comparisons and discussions between the two methods by numerical experiments were given as follows. Here we assume that the input flow of the on-ramp is equal to the output flow of the off-ramp.

The stop-and-go phenomena can be observed in the amplification of a small disturbance. In this section, we simulate the stop-and-go phenomena with respect to an amplified localized perturbation in an initial homogeneous condition. The following initial variation of the average density is used as in [20]:where is the initial vehicle density, veh/m is the amplitude of localized perturbation, and km is the length of road section under consideration. The dynamic approximate boundary condition was given by

For computational purpose, the space domain was divided into equal intervals of length of m and time interval was chosen as 1 s. The related parameters of our model were as follows:

The critical density values of the GK model corresponding to the parameters above were veh/m and veh/m, which can easily be found out by the stability condition [17]. The traffic flow will be unstable between these critical densities. The small disturbance in these initial homogeneous conditions will be amplified, and the stop-and-go phenomena will occur.

Traditionally, people used the temporal evolution of vehicle density or velocity to describe the stop-and-go waves, such as in [17, 21–24]. In particular, [17] employed the GK model to analyze it. The variation range of vehicle density is 0–0.25 veh/m and velocity is 0–30 m/s. So there are rather limited changes in the diagrams about these variables. When the traffic becomes congested, the vehicle density and velocity both tend to a specific value. We cannot see significant changes from the traditional temporal development of density or velocity. However, through our variable substitutions, the state variable and both tend to infinity. As long as the traffic has a small fluctuation, the value of or will change sharply. Moreover, as long as there is traffic jam formation, the value of or will approach infinity. Using the new model by such variable substitution, we can describe clearly the relationship between traffic jams and system instability in the phase plane. The numerical solution of and can be obtained by applying the finite difference method on the new model. Then we analyze the stop-and-go phenomena with four phase plane diagrams. The coordinate systems of them are , , , and , respectively. Through the four graphs the variation of density or velocity with time or sections can be investigated more clearly. Thus we can completely convert the fluctuations of traffic flow into the stability analysis charts.

Figure 1 shows the unstable traffic situation with small perturbations divergence when the initial density was set to veh/m. Figure 1(a) is the temporal evolution of vehicle density. Since the value of initial density we set was in the unstable range, the amplitude of the initial small perturbations grows in time, leading to traffic instability. A complex localized structure consisting of two or more clusters forms. This situation corresponds to stop-and-go traffic.