Mathematical Problems in Engineering

Volume 2015, Article ID 702695, 9 pages

http://dx.doi.org/10.1155/2015/702695

## Improved Generalized Force Model considering the Comfortable Driving Behavior

MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing 100044, China

Received 17 November 2014; Revised 12 March 2015; Accepted 15 March 2015

Academic Editor: Sebastian Anita

Copyright © 2015 De-Jie Xu 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

This paper presents an improved generalized force model (IGFM) that considers the driver’s comfortable driving behavior. Through theoretical analysis, we propose the calculation methods of comfortable driving distance and velocity. Then the stability condition of the model is obtained by the linear stability analysis. The problems of the unrealistic acceleration of the leading car existing in the previous models were solved. Furthermore, the simulation results show that IGFM can predict correct delay time of car motion and kinematic wave speed at jam density, and it can exactly describe the driver’s behavior under an urgent case, where no collision occurs. The dynamic properties of IGFM also indicate that stability has improved compared to the generalized force model.

#### 1. Literature Review and Introduction

In recent several decades, lots of traffic models have been proposed by scholars to describe the complicated traffic phenomena. In essence, traffic models can be divided into two categories: macroscopic and microscopic models [1]. The former was proposed on the basis of hydrodynamics regarding the traffic flow as a flow of continuous medium, and the latter may focus more on each individual car, which mainly includes cellular automaton models [2–4] and car-following models [5–11]. The original car-following model was proposed by Reuschel and Pipes [12, 13], and its basic assumption is that the driver can adjust the vehicular speed according to the velocity difference of two neighboring cars. Because the velocity difference is just considered in it, the case that only one car exits in the system is neglected, which causes that the model cannot describe the situation correctly which just had one car; besides, when the successive two vehicles travel at the same speed, the headway distance may be very close to each other; this does not conform to the reality. In order to overcome these shortages, Newell [14] presented another model, whose idea is that the velocity depends on the headway distance. So the velocity is expressed as a function of the headway.

In 1995, Bando et al. [15] developed the optimal velocity model (OVM), which has similar idea to the Newell’s model, but it improves the optimal function and can be interpreted simply compared with the previous models, so it attracts many researchers’ interests. The dynamic equation of OVM is as follows:where is the velocity of th car at time , is the sensitivity coefficient, indicates the space headway, and denotes the optimal velocity.

Helbing and Tilch [16] calibrated the parameters of OVM applied to the field data, and they found out that the value of acceleration is unrealistically high. Therefore, they introduced the velocity difference term into the OVM and developed the generalized force model (GFM):where is the sensitivity coefficient, is the acceleration time, and is the optimal velocity function, which readswhere is the maximal desired velocity, is the range of the acceleration interaction. Considerwhere is the minimal headway distance and is the safe time headway.

In order to avoid the vehicle collision, the velocity difference term is added into the GFM and it is only effective when the velocity of the following vehicle is larger than that of the leading vehicle; that is, if , . The sensitivity coefficient of velocity difference term is and we call it acceleration term coefficient, which is given bywhere denotes the braking time and can be interpreted as range of the braking interaction.

However, Jiang et al. [17] pointed out that kinematic wave speed at jam density and delay time of car motion of GFM are poor because of ignoring the effects of positive velocity differences. So, through considering both positive and negative velocity differences, they proposed a full velocity difference model (FVDM) that takes the sensitivity coefficient as a constant.

Then Gong et al. [18, 19] found that FVDM would lead to a problem that the following vehicle may not brake even if the distance to the leading vehicle is extremely close, and they proposed an asymmetric full velocity difference model (AFVDM) that takes the asymmetric characteristic into account. Yu et al. [20] proposed a full velocity difference and acceleration model (FVDAM) by synthetically taking into account headway, velocity difference, and acceleration of the leading car; the following car in FVDAM can accelerate more quickly than that in FVDM.

For the above mentioned models, they have made a great progress in car-following theory. However, they have a problem that the maximum acceleration of leader starting from a standing state is unrealistically high. Synthesizing all the above models, we consider the variation of acceleration under the case that the leading car is initially at rest and unobstructed; then it starts up. The velocity difference is null in all the above models because the space distance is greater than the range of interaction, so the acceleration of the leading car is

From (6), the maximal acceleration of the leading car is and it only related to the sensitivity coefficient of different models (cf. Table 1) if is fixed. Figure 1 shows the acceleration variation of the leading car. We can see that the maximal accelerations are 12.46 m/s^{2} in OVM and 6.96 m/s^{2} in GFM. The optimal velocity functions in FVDM and AFVDM are different from that in GFM, as shown in Figure 2, so their maximal accelerations are 6.01 m/s^{2}. However, according to [16], the empirical acceleration is usually not greater than 4 m/s^{2}. Thus, the accelerations of OVM, GFM, FVDM, and AFVDM are unrealistically high.