Discrete Dynamics in Nature and Society

Volume 2015, Article ID 420581, 11 pages

http://dx.doi.org/10.1155/2015/420581

## Modeling Mixed Bicycle Traffic Flow: A Comparative Study on the Cellular Automata Approach

College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China

Received 10 December 2014; Revised 29 April 2015; Accepted 30 April 2015

Academic Editor: Tetsuji Tokihiro

Copyright © 2015 Dan Zhou 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

Simulation, as a powerful tool for evaluating transportation systems, has been widely used in transportation planning, management, and operations. Most of the simulation models are focused on motorized vehicles, and the modeling of nonmotorized vehicles is ignored. The cellular automata (CA) model is a very important simulation approach and is widely used for motorized vehicle traffic. The Nagel-Schreckenberg (NS) CA model and the multivalue CA (M-CA) model are two categories of CA model that have been used in previous studies on bicycle traffic flow. This paper improves on these two CA models and also compares their characteristics. It introduces a two-lane NS CA model and M-CA model for both regular bicycles (RBs) and electric bicycles (EBs). In the research for this paper, many cases, featuring different values for the slowing down probability, lane-changing probability, and proportion of EBs, were simulated, while the fundamental diagrams and capacities of the proposed models were analyzed and compared between the two models. Field data were collected for the evaluation of the two models. The results show that the M-CA model exhibits more stable performance than the two-lane NS model and provides results that are closer to real bicycle traffic.

#### 1. Introduction

Traffic flow theories are generally divided into two branches: macroscopic and microscopic theories [1]. The macroscopic traffic flow models are based on fluid dynamics and are mostly used to elucidate the relationships between density, volume, and speed (also called the fundamental diagram) in various traffic conditions. The microscopic traffic models, on the other hand, describe the interaction between individual vehicles. The microscopic traffic models generally include car-following models and cellular automata (CA) models. The car-following model is the most important model, describing the detailed movements of vehicles proceeding close together in a single lane. There have been many car-following models produced in the literature over the past 60 years, such as stimulus-response models, safety distance models, action point models, fuzzy-logic-based models, and optimal velocity models [2–5]. For a broader review, refer to Brackstone and McDonald [6] and Chowdhury et al. [7]. Recently, CA models have emerged as an efficient tool for simulating highway traffic flow because of their easy concept, simple rule, and speed in conducting numerical investigations. The rule-184 model, proposed by Wolfram [8], was the first CA model to be widely used for traffic flow. Nagel and Schreckenberg [9] presented the well-known NS CA model, which is an extension of the rule-184 model allowing the maximal speed of vehicles to be more than one cell/s. The NS model and the many improved versions of it reproduce some basic and complicated phenomena such as stop and go, metastable states, capacity drop phenomena (which means the capacity of road experiences a large drop under critical density conditions), and synchronized flow in real traffic conditions.

Most of the aforementioned microscopic traffic models have been developed only for motorized vehicles. Few of them have been used for modeling non-motorized vehicles such as bicycles, tricycles, electric bicycles, and motorcycles because of the complicated characteristics of such vehicles movements. With the increasing usage of bicycles, some researchers have begun focusing on modeling the operation of bicycle facilities. Jiang et al. [10] introduced two different multivalue CA (M-CA) models in order to model bicycle flow. Their simulation results showed that, once the randomization effect is considered, the multiple states in deterministic M-CA models disappear and unique flow-density relations exist. They found the transition from free flow to congested flow to be smooth in one model but of second order in the other. Lan and Chang [11] developed inhomogeneous CA models to elucidate the interacting movements of cars and motorcycles in mixed traffic contexts. The car and motorcycle were represented by nonidentical particle sizes, respectively, occupying 6 × 2 and 2 × 1 cell units, each of size 1.25 × 1.25 meters. The CA models were validated by a set of field-observed data and the relationships between flow, cell occupancy (a proxy of density), and speeds under different traffic mixtures and road (lane) widths were elaborated. A M-CA model for mixed bicycle flow was proposed by Jia et al. [12]. Two types of bicycles, with different maximum speeds (1 cell/s and 2 cells/s), were considered in the system. Different results were analyzed and investigated under both deterministic and stochastic regimes. Li et al. [13] presented a multivalue cellular model for mixed nonmotorized traffic flow composed of bicycles and tricycles. A bicycle was assumed to occupy one unit of cell space and a tricycle two units of cell space. The simulation results showed the multiple state effect of mixed traffic flow. Gould and Karner [14] proposed a two-lane inhomogeneous CA simulation model, an improved version of the NS model combining a lane-changing rule, for bicycle traffic, and collected field data from three UC Davis bike paths for comparison with a simulation model. Yang et al. [15] proposed an extended multivalue CA model that permitted the bicycles to move at faster speeds. The simulation results showed that the mixed nonmotorized traffic capacity increased with an increase in the electric bicycle ratio. Zhang et al. [16] used an improved three-lane NS model to analyze the speed-density characteristics of mixed bicycle flow. The simulation results of the CA model were effectively consistent with the actual survey data when the density was lower than 0.225 bic/m^{2}.

Summarizing the above, none of the aforementioned car-following models have been devoted to mixed traffic with regular bicycles (RBs) and electric bicycles (EBs), but CA models have been widely used for modeling nonmotorized traffic. The modeling of mixed bicycle traffic using CA models can be divided into two branches: NS CA model and M-CA model approaches. The update rules of the NS CA model for bicycle flow are the same as for motorized vehicles, with only the cell size and bicycle speeds being different. As pointed out by Jiang et al. [10] and Jia et al. [12], the M-CA model is more suitable than the NS model for modeling bicycle traffic flow. Because the update rules of the M-CA model do not include direct car-following and lane-changing behavior, it may be appropriate for modeling the nonlane-based behavior of bicycle traffic. The NS CA model and M-CA model have been used for modeling bicycle traffic and mixed bicycle traffic with RBs and EBs. However, there is no evidence in the existing literature as to which model is better for modeling mixed traffic flow, nor as to the differences between these two models. Therefore, a comparison of the NS CA model and the M-CA model in terms of their ability to model mixed bicycle traffic is required so that CA models can be improved efficiently.

This paper attempts to develop two CA models to describe the behaviors of mixed bicycle traffic with RBs and EBs on a separated bicycle path and to compare the characteristics of the NS CA model and the M-CA model. The remaining parts are organized as follows. Section 2 introduces the development of NS and M-CA rules. Section 3 presents the simulation results of these two CA models. Section 4 further discusses differences in the simulation results. Finally, the conclusions and ideas for future studies are addressed.

#### 2. CA Models

##### 2.1. Definition of Cell Size and Bicycle Speed

The main differences encountered when modeling bicycle traffic as opposed to motorized vehicle traffic using a CA model are the cell size and the speed. Mixed bicycle traffic with RBs and EBs on separated bicycle paths is ubiquitous in many Asian countries, such as China, Vietnam, Indonesia, and Malaysia. Because of the different operating speeds of RBs and EBs, mixed traffic produces complicated behavior and characteristics that are likely to lead to safety and efficiency problems. Modeling mixed bicycle traffic is very important for the planning, operation, and management of bicycle facilities. Based on the behavior of cyclists, CA models are the best option for modeling bicycle traffic. The size of cell space and the update rules are two significant aspects of CA models.

Bicycles are shorter and narrower than motorized vehicles. Based on field surveys, the length of most RBs and EBs is 1.7–1.9 m. Meanwhile, bicycle lanes are set at 1 meter wide in both China and the USA [17, 18]. Therefore, the size of a RB or an EB is assumed rectangular, with length 2 m and width 1 m, as is widely used in other CA models [14–16]. The other parameter for modeling bicycle traffic is speed. According to the literature, the reported free flow speed of EBs is larger than that of RBs. Accordingly, in this paper speeds of 2 cells/s (4 m/s or 14.4 km/h) and 3 cells/s (6 m/s or 21.6 km/h) were chosen for RBs and EBs, respectively.

##### 2.2. NS CA Model

The NS CA model used in this paper was proposed by Nagel and Schreckenberg [9]. This model is very widely used in modeling highway traffic and bicycle traffic. The NS CA model includes a car-following rule and a lane-changing rule. The car-following rule is based on four steps, and the lane-changing rule is based on the work of Rickert et al. [19]. Different vehicle behavior rules would lead to different simulation results. With an increase in the number of lanes, the lane-changing logic would become more complicated and make modeling more difficult. Therefore, in this paper only a two-lane bicycle path is simulated and used in the comparison. In the time interval from to , the four basic rules of the NS model evolve according to the following steps.

*Step 1 (longitudinal acceleration). *Considerwhere is the speed of the th bicycle at updating time . is the maximum speed of the th bicycle. This corresponds to the cyclists’ realistic free flow speed.

*Step 2 (longitudinal deceleration). *Considerwhere is the distance between the th bicycle and the bicycle in front of it, at updating time . This step ensures that the bicycle stays safe with no collisions.

*Step 3 (random slowing down). *Considerwhere is a uniformly distributed random number between 0 and 1 and is the random slowing down probability of the th bicycle. The random slowing down effect, which captures one cyclist’s braking maneuver due to a random event (e.g., accident, road, or weather related factors), is one of the most significant parameters of the CA model. This step incorporates the idea of random effects on bicycles that may cause them to slow down.

*Step 4 (motion). *Considerwhere is the position of the th bicycle at time .

The lane-changing logic is shown below. Before the acceleration step, both lanes are examined to evaluate lane-changing opportunities. The following conditions are checked for each bicycle and must be true in order for it to change lanes.(1)The speed of the bicycle currently in th position is larger than or equal to the cell distance to the next bicycle. This condition ensures that this bicycle will need to slow down at the next update:(2)The distance to the next bicycle in the lane adjacent to the lane of the th bicycle () is larger than the distance to the next bicycle in its current lane (). This condition ensures that a benefit is derived from changing lanes:(3)The distance to backward bicycle in the lane adjacent to that of the currently th bicycle () is large enough. This condition ensures that, looking backwards, the closest bicycle in the adjacent lane is sufficiently far away:(4)A uniformly distributed random number between 0 and 1 is less than the probability of a lane change ():

and can be calculated as follows:where , , and are the position, speed, and maximum speed of the nearest following bicycle in the lane adjacent to that of the th bicycle.

The new speed for the bicycle currently in the th position after lane-changing is calculated as follows: where is the speed of this bicycle after the lane-changing.

The motion of the lane-changing bicycle iswhere is the position of the bicycle after the lane-changing.

##### 2.3. M-CA Model

A family of M-CA models has recently been proposed by Nishinari and Takahashi [20–22]. The basic version of the family is obtained from an ultradiscretization of Burgers’ equation. Therefore, it is also called the Burgers CA (BCA). Previously, BCA models were proposed for highway traffic. Recent attempts have included BCA models purported to represent bicycle flow [12, 13] adapted for the unobvious car-following and lane-changing behavior in bicycle traffic. In order to make a comparison with the NS CA model, the M-CA model for mixed bicycle flow is improved upon in this paper.

The numbers of RBs and EBs in location at time are and , respectively. As shown in Section 2.1, RBs with a maximum speed of 2 cells/s and EBs with a maximum speed of 3 cells/s are considered in the simulation systems. Therefore, the updating procedures are changed as follows:(1)all bicycles in location move to their next location if the location is not fully occupied, and EBs have priority over RBs;(2)all bicycles that moved in procedure (1) can move to location if their next location is not fully occupied after procedure (1), and EBs again have priority over RBs;(3)only EBs moved in procedure (2) can move to location if their next location is not fully occupied after procedure (2).

The numbers of RBs and EBs that move one location on from location at time in procedure (1) are and , respectively. The numbers of RBs and EBs that move two locations on from location at time are and , respectively. represents the number of EBs that move three locations on from location at time . is defined as the lane number of the simulation bicycle path. The randomization effect on the RBs is introduced as follows: decreases by 1 with probability if . The randomization effect on the EBs is as follows: decreases by 1 with probability if . The updating rules are as follows.

*Step 1. *Calculation of , , and () is as follows:

*Step 2. *Calculation of , , and is as follows:If , thenIn (12) and (15), and are calculated first because the EBs have priority over the RBs.

*Step 3. *Calculation of is as follows:If , then

*Step 4. *Update , , and :where is a uniformly distributed random number between 0 and 1.

#### 3. Simulation Results

For the comparison of the NS CA model against the M-CA model, the simulation parameters in both models should be set to the same values. In the simulations, a two-lane bicycle path () was selected with length cells (equal to 1000 m). In the initial conditions, RBs and EBs are randomly distributed on the road using the same random number for both models. The default values of the random slowing down probability (), the probability of a lane change (), and the proportion of EBs () are 0.2, 0.8, and 0.5, respectively, for the NS CA model (as in previous studies [14]). The default values of the random slowing down probability of RBs (), the random slowing down probability of EBs (), and the proportion of EBs are 0.4, 0.4, and 0.5, respectively, for the M-CA model. In the M-CA model, the slowing down probability is the probability that the number of bicycles () decreases, which means that one bicycle decreases its speed. In this paper, the simulation is based on two lanes (). Therefore, the maximum value of is 2. If , no bicycle slows down, and the slowing down probability of any bicycle is zero. If , only one bicycle slows down, with probability . If , this means only one bicycle may slow down with probability ; therefore, the total slowing down probability of bicycles is . By summing the above three cases, we assume these three cases have the same percentage. Therefore, the mean of the three cases’ slowing down probabilities is . In order to compare the two models, we used a default value for the random slowing down probability for the M-CA model of half that for the NS CA model.

Periodic conditions that are as close as possible to the actual conditions are used so that the bicycles ride on a circuit. The instantaneous positions and speeds for all particles are updated in parallel, per second. The flow, speed, and density of the mixed bicycle traffic flow can be calculated after a given amount of time (20000 simulation steps) [15], and the averages over the last 5000 steps are used for the calculation in order to decrease the random effect.

##### 3.1. Results of the NS CA Model

In order to show the different characteristics of the NS CA model under different model parameters, speed-density and flow-density plots (the fundamental diagram of bicycle traffic flow) were created so that the results could be analyzed. Example plots are shown in Figures 1, 2, and 3. When , it is a deterministic case, while is a stochastic case. From Figure 1, it can be seen that, with an increase in the slowing down probability , the fundamental diagrams drop quickly, which means that the capacity of the bicycle lane drops quickly with an increase in . When the slowing down probability is equal to one, the stopped RBs will lead bicycle traffic flow to jam, and the speed and flow will both be zero.