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An improved multi-value cellular automata model for heterogeneous bicycle traffic flow
Physics Letters A 379 (2015) 2409–2416
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Physics Letters A www.elsevier.com/locate/pla
An improved multi-value cellular automata model for heterogeneous bicycle traffic flow Sheng Jin a , Xiaobo Qu b , Cheng Xu c,d , Dongfang Ma e,∗ , Dianhai Wang a a
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310058 China Griffith School of Engineering, Griffith University, Gold Coast, 4222 Australia c Department of Transportation Management Engineering, Zhejiang Police College, Hangzhou, 310053 China d College of Transportation, Jilin University, Changchun, 130022 China e Ocean College, Zhejiang University, Hangzhou, 310058 China b
a r t i c l e
i n f o
Article history: Received 7 January 2015 Received in revised form 6 July 2015 Accepted 23 July 2015 Available online 29 July 2015 Communicated by F. Porcelli Keywords: Regular bicycle Electric bicycle Multi-value cellular automata model Fundamental diagram Capacity
a b s t r a c t This letter develops an improved multi-value cellular automata model for heterogeneous bicycle traffic flow taking the higher maximum speed of electric bicycles into consideration. The update rules of both regular and electric bicycles are improved, with maximum speeds of two and three cells per second respectively. Numerical simulation results for deterministic and stochastic cases are obtained. The fundamental diagrams and multiple states effects under different model parameters are analyzed and discussed. Field observations were made to calibrate the slowdown probabilities. The results imply that the improved extended Burgers cellular automata (IEBCA) model is more consistent with the field observations than previous models and greatly enhances the realism of the bicycle traffic model. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The non-motorized traffic trip (e.g. on a regular bicycle (RB) or electric bicycle (EB)) is one of the main trip modes in developing countries, especially in Southeast Asian countries such as China, India, and Vietnam. In recent years, due to their low cost, convenience, and energy efficiency, EBs have quickly become one of the dominant non-motorized travel modes in China [1]. Heterogeneous bicycle traffic flow containing a mixture of EBs and slower-moving RBs is and will continue to be a very common feature of bicycle paths. With the increase in bicycle traffic, the need to realistically model the movement and interactions of heterogeneous bicycle traffic is rapidly gaining importance in the planning, design, management, and operations of bicycle facilities. In response to this need, several approaches to modeling bicycle movements and interactions have been developed [2]. Many traffic flow models (such as the car-following model, lane-changing model, cellular automata (CA) model, and gas dynamics model) have been proposed for motorized vehicles [3–6]. However, the behavior of bicycle traffic is non-lane-based and more complicated, making it more difficult to model. Based on the
*
Corresponding author. Tel.: +86 571 88208704; fax: +86 571 88208685. E-mail address: [email protected] (D. Ma).
http://dx.doi.org/10.1016/j.physleta.2015.07.031 0375-9601/© 2015 Elsevier B.V. All rights reserved.
previous studies, almost all of the bicycle flow microscopic models are based on the concept of CA, and the related studies are very limited. We have summarized the modeling approaches that depict the state of the art in bicycle traffic modeling and have found that overall microscopic models of bicycle traffic can be divided into the modeling of bicycle operations and the modeling of interactions between bicycles and other motorized vehicles [7]. In respect of modeling bicycle traffic, the Nagel–Schreckenberg (NS) model [8], the best-known CA model, is widely used in modeling bicycle flow. The NS model, and the many improvements on it, reproduce some basic and complicated phenomena such as stop and go, metastable states, capacity drop, and synchronized flow in real traffic conditions. Gould and Karner [9] proposed a two-lane inhomogeneous CA simulation model, an improved version of the NS model combined with a lane-changing rule. Field data under uncongested conditions, from Davis, California, were used for calibration. Zhang et al. [10] also used a three-lane NS model and an improved lane-changing rule for analyzing the speed–density characteristics of mixed bicycle flow. Zhao et al. [11] used the CA method to model the characteristics of bicycle passing events in mixed bicycle traffic on separated bicycle paths. An improved three-lane NS model and lane-changing rule were proposed. The dynamic floor field and CA model were introduced to investigate the characteristics of bicycle flow by Yang et al. [12], and a new concept called the lane-changing cost was proposed to study the
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effect of driving styles. Jiang et al. [13] proposed a new three-lane NS model to simulate the flow characteristics, overtaking maneuver patterns and segregation effect in heterogeneous traffic composed of RBs and EBs. Due to its advantage of not considering lane-changing rules, the extended Burgers CA (EBCA) model [14] is very well-suited to modeling bicycle traffic flow. Jiang et al. [15] were the first to introduce two different multi-value CA models (MCA) considering stochastic randomization for modeling bicycle flow. Jia et al. [16] considered the fact that bicycles do not all have the same maximum speed due to differences in the characteristics of cyclists. Therefore, they included two types of bicycles (fast and slow) with different maximum speeds (1 cell/s and 2 cells/s) in the EBCA model. Li et al. [17] presented a MCA 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. In respect of modeling interactions between bicycles and other motorized vehicles, conflict rules between them have been modeled and simulated. Vasic and Ruskin [18] addressed this with a novel technique, based on one-dimensional CA components, for modeling a network infrastructure and its occupancy by vehicles. Using this modeling approach, the simulation combined car and bicycle traffic for two elemental scenarios. The characteristics of mixed traffic flow at the intersection were more complicated than on roadways. Based on motorized-vehicle-bicycle interference characteristics, and the coupling between the vehicle CA model and the bicycle CA model, Zhang et al. [19] presented a different kind of CA model (NS-BCA) to analyze the mixed traffic flow at intersections. Luo et al. [20] proposed a new CA model to simulate car and bicycle heterogeneous traffic on an urban road. The proposed model captured the complex interactions between the two types of vehicles. Ding et al. [21] proposed an improved CA model to simulate mixed traffic flow composed of motor vehicles and bicycles near bus stops. The typical NS CA model for motorized traffic flow and the BCA model for bicycle flow were combined to simulate the mixed traffic flow. Table 1 provides a summary of bicycle traffic flow modeling using CA models. It shows the simulation scenarios, vehicle types, model types, cell size, and maximum speeds of the reported models. From the table, it can be clearly seen that the NS and EBCA models have been widely used in the modeling of heterogeneous bicycle flow. For the NS model, the cell length and the maximum speeds of different bicycles were selected based on real bicycle data (speeds were RBs 4 m/s and EBs 6 m/s), while all of the proposed EBCA models for bicycle simulation used 1 and 2 cells/s as the maximum speeds of RBs and EBs, which are significantly different from real bicycle speed data. However, based on different numbers of bicycle lanes and simulation scenarios, the lanechanging rules of the NS models are significantly different from each other. With an increase in the number of bicycle lanes, these rules become more complex, making them impossible to calibrate, validate, and evaluate. As can be seen from the above, the EBCA models are more suitable for modeling bicycle traffic flow and investigating the characteristics of heterogeneous bicycle flow than the NS models. However, the assumption that RBs have a maximum speed of 1 cell/s and EBs a maximum of 2 cells/s is not realistic. Therefore, this paper expands the maximum speed and update rules for bicycle traffic and proposes an improved EBCA (IEBCA) model for modeling heterogeneous bicycle traffic. Both simulation analysis and field data calibration are presented so as to compare the models. The remaining parts of the paper are organized as follows. Section 2 introduces the IEBCA model. Section 3 presents the simulation results of the proposed model. Section 4 uses field bicycle data for the calibration and validation of the proposed model. Finally, the conclusions and possibilities for future study are addressed.
2. The IEBCA model 2.1. Definition of bicycles’ maximum speeds The size and speed differences between RBs and EBs will inevitably lead to more complicated characteristics and a higher risk of traffic collisions. Accurately describing the size and speed of heterogeneous bicycles is the foundation of modeling bicycle traffic. Based on field surveys [2,22], the typical lengths of RBs and EBs in China are not significantly different from one another, at around 1.7–1.9 m. Adding in a safe distance between two successive bicycles, the length of a bicycle cell is set to 2 m, which has also been widely used in most of the previous bicycle CA models [9–12]. Based on the criteria in the US and China, it is recommended that the standard width of a bicycle lane should be 1–1.2 m [2,10]. Therefore, we use 2 × 1 m as the cell size of a bicycle in this paper. The maximum speeds of different bicycles are the other important parameters that will affect the free flow speed and capacity of heterogeneous bicycle traffic flow. According to the results of field investigations in China [2,22] and reported studies [9,10,12,13], the average speeds of RBs and EBs in free flow are round 14–16 km/h and 20–22 km/h, respectively. Accordingly, in this paper, 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 as the maximum speeds for RBs and EBs, respectively (assuming that the update time step corresponds to 1 second). 2.2. IEBCA model Due to its easy-to-follow concept, simple rules, and speed for numerical investigations, the CA model is an efficient tool for simulating traffic flow [23]. Nishinari and Takahashi [14] were the first to propose a multi-value CA model. The initial multi-value CA model comes from the ultradiscretization of the Burgers equation, and is therefore called the Burgers cellular automata (BCA) model. Its evolution equation is
U j (t + 1) = U j (t ) + min U j −1 (t ), L − U j (t )
− min U j (t ), L − U j +1 (t )
(1)
where U j (t ) represents the number of vehicles in cell j at time t. Then, Nishinari and Takahashi extended the maximum speed of BCA to 2, and presented the extended BCA (EBCA) models [24,25]. As mentioned earlier, the EBs’ maximum speed is considered to be 3 cells/s in this paper. Therefore, the updating rules are more complex than in the EBCA and are extended as follows: (1) Assume that the numbers of RBs, EBs, and all bicycles at location j at time t are U rj (t ), U ej (t ), and U j (t ), respectively. (2) All bicycles at location j move to the next location j + 1 if the next location is not fully occupied, and the EBs have priority over the RBs. (3) All bicycles that have moved based on procedure (2) can move to location j + 2 if location j + 2 is not fully occupied after procedure (2), and the EBs have priority over the RBs. (4) Only the EBs that have moved in procedure (3) can move to location j + 3 if location j + 3 is not fully occupied after procedure (3). Because of the higher speed, stability, and flexibility of EBs, we assume that the EBs have priority to pass the RBs in the proposed model. We assume that the numbers of RBs and EBs that move one cell from location j at time t in procedure (2) are brj (t ) and
bej (t ), respectively; the numbers of RBs and EBs that move two
cells from location j at time t are c rj (t ) and c ej (t ), respectively; the total number of bicycles that move one and two cells from location
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Table 1 Overview of bicycle traffic flow modeling using cellular automata. Reference
Year
Simulation scenario
Vehicle types
Model type
Cell length
Max speed (Cells/s)
Gould and Karner [9]
2009
Separated bicycle path
NS
2.1 m
Zhang et al. [10]
2013
Separated bicycle path
NS
2.0 m
Zhao et al. [11]
2013
Separated bicycle path
Improved NS
2.0 m
Yang et al. [12] Jiang et al. [13]
2013 2014
Separated bicycle path Separated bicycle path
NS and dynamic floor field Improved NS
2.0 m 1.0 m
Jiang et al. [15]
2004
Separated bicycle path
EBCA
1 cell
Jia et al. [16]
2007
Separated bicycle path
EBCA
1 cell
Li et al. [17]
2008
Separated bicycle path
EBCA
1 cell
Vasic and Ruskin [18]
2012
Non-separated roadway and intersection
NS model and Interaction rule
3.75 m
Zhang et al. [19]
2014
Intersection
2015
Non-separated roadway
Ding et al. [21]
2015
Non-separated roadway
NS EBCA NS Improved NS NS EBCA
3.5 m
Luo et al. [20]
RB EB RB EB RB EB RB RB EB RB EB RB EB RB EB Car RB Car RB Car RB Bus/car RB
2 3 2 3 4 5 4 4 6 1 2 1 2 1 2 6 2 5 1 20 6 3/4 2
j at time t are b j (t ) and c j (t ), respectively; the number of EBs that move three cells from location j at time t is d j (t ). In real traffic conditions, bicycles are considered to be constantly subject to perturbation by noise such as pedestrians, parking vehicles, and steep path slopes. Therefore, stochastic randomization is introduced into the IEBCA [16]. The randomization effect on RBs is as follows: c rj (t + 1) decreases by 1 with probability p r if c rj (t + 1) > 0; the randomization effect on EBs is that d j (t + 1)
decreases by 1 with probability p e if dej (t + 1) > 0. The updating rules of the proposed IEBCA model are improved as follows: Step 1: calculating brj (t + 1), bej (t + 1), and b j (t + 1) ( j = 1, 2, 3, . . . , K ):
bej (t + 1) = min U ej (t ), L − U j +1 (t ) brj (t
+ 1) = min
U rj (t ), L
−U
e j +1 (t ) − b j (t
(2)
+ 1)
b j (t + 1) = brj (t + 1) + bej (t + 1)
(3) (4)
where L is the number of bicycle lanes, and K is the total number of cells in the simulation system. Step 2: calculating c rj (t + 1), c ej (t + 1), and c j (t + 1):
− b j +1 (t + 1) + b j +2 (t + 1)
(5)
c rj (t + 1) = min brj (t + 1), L − U j +2 (t ) − b j +1 (t + 1)
+ b j +2 (t + 1) − c ej (t + 1)
(6)
When the random number is less than a preset value p r and the number of RBs that move two cells from location j at time t is more than zero (c rj (t + 1) > 0), the number of moving RBs is minus 1. If rand( ) < p r , then
c rj (t + 1) = max c rj (t + 1) − 1, 0 c j (t + 1)
= crj (t
+ 1) + c ej (t
+ 1)
(7) (8)
4/2 cell 1 cell
where rand() is a uniformly distributed random number between 0 and 1. In equations (2) and (5), bej (t + 1) and c ej (t + 1) are calculated first because the EBs have priority over the RBs due to their higher speed. Step 3: calculating d j (t + 1):
d j (t + 1) = min c ej (t + 1), L − U j +3 (t ) − b j +2 (t + 1)
+ b j +3 (t + 1) − c j +1 (t + 1) + c j +2 (t + 1)
(9)
When the random number is less than a preset value p e and the number of EBs that move three cells from location j at time t is more than zero (d j (t + 1) > 0), the number of moving EBs is minus 1. If rand() < p e , then
d j (t + 1) = max d j (t + 1) − 1, 0 Step 4: updating
U rj (t
+ 1),
U ej (t
(10)
+ 1), and U j (t + 1):
U ej (t + 1) = U ej (t ) − bej (t + 1) + bej−1 (t + 1) − c ej−1 (t + 1) U rj (t
+ 1) =
U j (t + 1) =
c ej (t + 1) = min bej (t + 1), L − U j +2 (t )
1.0 m
+ c ej−2 (t + 1) − d j −2 (t + 1) + d j −3 (t + 1)
(11)
U rj (t ) − brj (t + 1) + brj −1 (t + 1) − crj −1 (t U rj (t + 1) + U ej (t + 1)
(12)
+ 1) + crj −2 (t + 1)
(13)
3. Simulation results For the simulation analysis in this study, a three-lane bicycle path was selected with length K = 100 cells (equal to 200 m). In the initial conditions, RBs and EBs are randomly distributed on the bicycle path. The percentage of EBs is r, which is between 0 and 1. Periodic conditions, which are the closest to the real traffic conditions, were used in this paper. Accordingly, the instantaneous positions and numbers of bicycles for all cells are updated in parallel per second. The flow, speed, and density of the heterogeneous bicycle traffic flow are calculated after 20 000 simulation steps, and the average value of the last 5000 steps is used for the calculation in order to decrease the random effect. The density and flow of the bicycle path are calculated by
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Fig. 1. The fundamental diagrams in the stochastic cases where (a) p r = 0 and (b) p e = 0.
Fig. 2. The fundamental diagrams in the stochastic cases where (a) r = 0.2 and (b) r = 0.5.
ρ (t ) = Q (t ) =
K 1
KL
U j (t )
(14)
j =1
K 1 b j −1 (t ) + c j −2 (t ) + d j −3 (t ) KL
(15)
j =1
where ρ (t ) and Q (t ) are the density and flow per lane of the simulated bicycle path system at time t, respectively. In the following subsections, some deterministic and stochastic cases under different model parameters will be simulated and discussed.
in the free flow region. This is due to the fact that, when considering the slowdown probability of RBs with a slower free flow speed, a steady moving bottleneck will not exist and the quicker EBs will overtake the slower ones. The fundamental diagrams with slowdown probability parameters p e > 0 and p r > 0 are shown in Fig. 2. It can be seen that the multiple states effect disappears in both the free flow and the congested regions. The results for the multiple states are similar to those in previous research, and detailed analysis can be found in Refs. [14–16,24–26]. 3.2. Effect of proportion of EBs
3.1. Effect of slowdown probability In the IEBCA model, the RBs and EBs are assumed to have probabilities p r and p e of reducing their speed because of stochastic interference in the natural traffic situation. Fig. 1 shows the fundamental diagrams of heterogeneous bicycle traffic under the effect of a slowdown probability of p r = 0 or p e = 0. Considering the case where p r = 0 and p e > 0 (Fig. 1a), there are two branches and the multiple states effect occurs in the free flow region. However, the multiple states effect does not occur in the congested region. Fig. 1b considers the case where p e = 0 and p r > 0. Here, the multiple states effect can be seen in the congested region but not
Fig. 3 shows the fundamental diagrams of bicycle traffic flow under different percentages of EB, Fig. 3a showing the deterministic case and Fig. 3b the stochastic case. With an increase in the EB percentage, the capacity of the bicycle path increases and the critical density decreases. This is due to the fact that the EBs have a higher free flow speed. It can also be seen that the multiple states effect occurs in both the free flow and the congested regions in the deterministic case with p r = 0 and p e = 0 (Fig. 3a), and the multiple states effect disappears in Fig. 3b. When r = 0, the IEBCA model is a special case of the EBCA model. Therefore, the phase transition from the free flow to the congested state is second-order.
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Fig. 3. The fundamental diagrams under different proportions of EBs, where (a) shows the deterministic case with p r = 0 and p e = 0 and (b) the stochastic case with p r = 0.3 and p e = 0.3.
Fig. 4. The fundamental diagrams under different numbers of lanes, where (a) shows the deterministic case with p r = 0, p e = 0, and r = 0.5 and (b) the stochastic case with p r = 0.3, p e = 0.3, and r = 0.5.
3.3. Effect of number of bicycle lanes In this subsection, the effect of the number of bicycle lanes on heterogeneous bicycle traffic flow is investigated. The number of bicycle lanes varies widely according to different road conditions. With an increase in the number of bicycle lanes, passing becomes easier. From Fig. 4a, it can be seen that the multiple states effect exists in both the free flow and the congested regions. Meanwhile, the maximum flow increases with the increase in bicycle lanes. This is because when there are more bicycle lanes, the EBs have more opportunities to pass the RBs. Similar results are shown in Fig. 4b where the randomization effect of RBs and EBs is considered. When the number of lanes equals 1, the capacity of the bicycle path is determined by the RBs, and the quicker EBs cannot pass the slower ones. Therefore, the fundamental diagram for L = 1 is lower than those for L greater than one. 3.4. Capacity Because the IEBCA model uses higher speeds, the free flow speed and capacity should be larger than in the previous EBCA model [16]. In this subsection, the fundamental diagrams of the
proposed IEBCA model and the EBCA model are compared. Fig. 5 shows the fundamental diagrams in the stochastic case. It can be seen that the multiple states effect disappears in both the free flow and the congested regions, even if p r and p e are larger than zero. The differences in capacity and critical density between the two models are very large (as shown in Fig. 5a). This could be due to the different maximum speeds of RBs and EBs. In the case of r = 0 for the IEBCA model and r = 1 for the EBCA model, which means that only bicycles with a maximum speed of 2 cells/s exist in the simulation systems, the curves from the two models coincide. 4. Calibration and validation 4.1. Data collection Field data collection is very important for calibrating the parameters of the proposed IEBCA model and evaluating its performance. Previous CA-based bicycle models have not been calibrated, leading to a lack of evaluation and application. In this paper, three survey sites in Hangzhou, China were selected for calibration and validation. All of the bicycle paths have less than a 3% gradient and are separated from the motorized vehicle lanes by physical barri-
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Fig. 5. The fundamental diagrams in the stochastic case where p r = 0.3 and p e = 0.3. The circles and squares correspond to the proposed IEBCA model and the EBCA model, respectively.
Fig. 6. Photos of the camera’s view of the three sites. Table 2 Statistical description of field survey sites. No.
Survey site
Bicycle path width (m)
Number of lanes
% of EBs
% of male cyclists
% of young cyclists
1 2 3
Jiaogong Road Hushu Road Tianmushan Road
2.27 2.93 3.97
2 3 4
68.6% 70.3% 73.9%
63.8% 59.7% 80.0%
80.3% 62.2% 69.8%
ers. Hence, bicycle traffic flow is not interrupted by pedestrians or motorized vehicles. One camera was set up on the side of the bicycle path to take videos to collect data on the bicycles’ operating behavior (see Fig. 6). Cameras were carefully placed so that the cyclists would be unaware they were being observed. The survey days were sunny, there were no crashes, and weather conditions were good. Both free flow and congested flow of a mixture of RBs and EBs were observed during the survey periods, and used to compare the fundamental diagrams of the field observations against those based on simulated data. Using video-processing technology, we were able to identify the moments when bicycles crossed the marked white lines. Then, the traffic flow parameters of the heterogeneous bicycles (such as flow, speed, and density) could be calculated automatically. The bicycle types, and genders and ages of the cyclists, were coded manually. Table 2 presents the statistical description of the field survey sites. The path widths of the selected survey sites are 2.27, 2.93, and 3.97 m, respectively, which can be thought of as approximately 2-, 3-, and 4-lane bicycle paths. The percentages of EBs at the three sites were around 70%. The field bicycle data (flow, speed, and density) were aggregated every 25 seconds.
4.2. Results For the simulations, the numbers of lanes and percentages of EBs were chosen to match the observed results from the three survey sites. The effects of the percentages of male and young cyclists on the fundamental diagrams can be considered as the parameters of the slowdown probability. In order to determine the slowdown probabilities of RBs and EBs, we used different combinations of slowdown probability values for the simulations. The mean absolute percentage error (MAPE) was used as the performance index to evaluate the proposed model in comparison with the field observations [27–29]:
MAPE =
× 100%
M 1 Qˆ ( j ) − Q ( j )
M
j =1
Q ( j)
(16)
where Qˆ ( j ) = the simulated bicycle flow during the jth interval; Q ( j ) = the observed bicycle flow during the jth interval; M = the number of intervals. Fig. 7 shows the fundamental diagrams for the field observations and the simulated results, where (a) shows the two-lane case, (b) the three-lane case, and (c) the four-lane case. It can be seen (Fig. 7 left) that the MAPEs between the field observations and the
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Fig. 7. MAPEs between observations and simulated results under different slowdown probabilities, and simulated fundamental diagrams in which the gray circles represent field observations and the blue solid lines represent simulated results for the minimum MAPE.
simulated fundamental diagrams vary widely under different slowdown probabilities for RBs and EBs. We used the minimum MAPE to determine the calibrated slowdown probabilities for RBs and EBs. The calibrated parameters and MAPEs are shown in Table 3. Fig. 7 (right) shows the observations and calibrated fundamental diagrams. The results show that the IEBCA model gives reasonable results and is suitable for modeling heterogeneous bicycle traffic flow.
5. Conclusions With the wider usage of green and convenient RBs and EBs in recent years, the modeling and simulation of heterogeneous bicycle traffic flow is becoming increasingly important for bicycle path planning, management, and operation. In order to overcome the weaknesses of previous MCA models for bicycle, this paper has proposed an improved MCA model that introduces maximum
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References
Table 3 Calibrated slowdown probabilities and MAPEs. Survey site
Number of lanes
pr
pe
MAPEs
Jiaogong Road Hushu Road Tianmushan Road
2 3 4
0.90 0.85 0.70
0.65 0.70 0.50
24.55% 21.61% 16.38%
speeds of two and three cells/s for RBs and EBs, respectively. New update rules have been proposed for the improved MCA model. The numerical simulation results and fundamental diagrams for bicycle traffic have been analyzed and discussed. Three parameters, the slowdown probability, the percentage of EBs, and the number of bicycle lanes, have been analyzed in both the deterministic and stochastic cases. The results are similar to those of previous research. Field bicycle observations collected from three survey sites were used for the calibration and validation of the simulated fundamental diagrams. The results show that the proposed model matches the field bicycle data better than previous models.
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Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 51338008, 51278454, 51208462, and 61304191), the Fundamental Research Funds for the Central Universities (2014QNA4018), the Projects in the National Science & Technology Pillar Program (2014BAG03B05), and the Key Science and Technology Innovation Team of Zhejiang Province (2013TD09).
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Physics Letters A www.elsevier.com/locate/pla
An improved multi-value cellular automata model for heterogeneous bicycle traffic flow Sheng Jin a , Xiaobo Qu b , Cheng Xu c,d , Dongfang Ma e,∗ , Dianhai Wang a a
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310058 China Griffith School of Engineering, Griffith University, Gold Coast, 4222 Australia c Department of Transportation Management Engineering, Zhejiang Police College, Hangzhou, 310053 China d College of Transportation, Jilin University, Changchun, 130022 China e Ocean College, Zhejiang University, Hangzhou, 310058 China b
a r t i c l e
i n f o
Article history: Received 7 January 2015 Received in revised form 6 July 2015 Accepted 23 July 2015 Available online 29 July 2015 Communicated by F. Porcelli Keywords: Regular bicycle Electric bicycle Multi-value cellular automata model Fundamental diagram Capacity
a b s t r a c t This letter develops an improved multi-value cellular automata model for heterogeneous bicycle traffic flow taking the higher maximum speed of electric bicycles into consideration. The update rules of both regular and electric bicycles are improved, with maximum speeds of two and three cells per second respectively. Numerical simulation results for deterministic and stochastic cases are obtained. The fundamental diagrams and multiple states effects under different model parameters are analyzed and discussed. Field observations were made to calibrate the slowdown probabilities. The results imply that the improved extended Burgers cellular automata (IEBCA) model is more consistent with the field observations than previous models and greatly enhances the realism of the bicycle traffic model. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The non-motorized traffic trip (e.g. on a regular bicycle (RB) or electric bicycle (EB)) is one of the main trip modes in developing countries, especially in Southeast Asian countries such as China, India, and Vietnam. In recent years, due to their low cost, convenience, and energy efficiency, EBs have quickly become one of the dominant non-motorized travel modes in China [1]. Heterogeneous bicycle traffic flow containing a mixture of EBs and slower-moving RBs is and will continue to be a very common feature of bicycle paths. With the increase in bicycle traffic, the need to realistically model the movement and interactions of heterogeneous bicycle traffic is rapidly gaining importance in the planning, design, management, and operations of bicycle facilities. In response to this need, several approaches to modeling bicycle movements and interactions have been developed [2]. Many traffic flow models (such as the car-following model, lane-changing model, cellular automata (CA) model, and gas dynamics model) have been proposed for motorized vehicles [3–6]. However, the behavior of bicycle traffic is non-lane-based and more complicated, making it more difficult to model. Based on the
*
Corresponding author. Tel.: +86 571 88208704; fax: +86 571 88208685. E-mail address: [email protected] (D. Ma).
http://dx.doi.org/10.1016/j.physleta.2015.07.031 0375-9601/© 2015 Elsevier B.V. All rights reserved.
previous studies, almost all of the bicycle flow microscopic models are based on the concept of CA, and the related studies are very limited. We have summarized the modeling approaches that depict the state of the art in bicycle traffic modeling and have found that overall microscopic models of bicycle traffic can be divided into the modeling of bicycle operations and the modeling of interactions between bicycles and other motorized vehicles [7]. In respect of modeling bicycle traffic, the Nagel–Schreckenberg (NS) model [8], the best-known CA model, is widely used in modeling bicycle flow. The NS model, and the many improvements on it, reproduce some basic and complicated phenomena such as stop and go, metastable states, capacity drop, and synchronized flow in real traffic conditions. Gould and Karner [9] proposed a two-lane inhomogeneous CA simulation model, an improved version of the NS model combined with a lane-changing rule. Field data under uncongested conditions, from Davis, California, were used for calibration. Zhang et al. [10] also used a three-lane NS model and an improved lane-changing rule for analyzing the speed–density characteristics of mixed bicycle flow. Zhao et al. [11] used the CA method to model the characteristics of bicycle passing events in mixed bicycle traffic on separated bicycle paths. An improved three-lane NS model and lane-changing rule were proposed. The dynamic floor field and CA model were introduced to investigate the characteristics of bicycle flow by Yang et al. [12], and a new concept called the lane-changing cost was proposed to study the
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effect of driving styles. Jiang et al. [13] proposed a new three-lane NS model to simulate the flow characteristics, overtaking maneuver patterns and segregation effect in heterogeneous traffic composed of RBs and EBs. Due to its advantage of not considering lane-changing rules, the extended Burgers CA (EBCA) model [14] is very well-suited to modeling bicycle traffic flow. Jiang et al. [15] were the first to introduce two different multi-value CA models (MCA) considering stochastic randomization for modeling bicycle flow. Jia et al. [16] considered the fact that bicycles do not all have the same maximum speed due to differences in the characteristics of cyclists. Therefore, they included two types of bicycles (fast and slow) with different maximum speeds (1 cell/s and 2 cells/s) in the EBCA model. Li et al. [17] presented a MCA 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. In respect of modeling interactions between bicycles and other motorized vehicles, conflict rules between them have been modeled and simulated. Vasic and Ruskin [18] addressed this with a novel technique, based on one-dimensional CA components, for modeling a network infrastructure and its occupancy by vehicles. Using this modeling approach, the simulation combined car and bicycle traffic for two elemental scenarios. The characteristics of mixed traffic flow at the intersection were more complicated than on roadways. Based on motorized-vehicle-bicycle interference characteristics, and the coupling between the vehicle CA model and the bicycle CA model, Zhang et al. [19] presented a different kind of CA model (NS-BCA) to analyze the mixed traffic flow at intersections. Luo et al. [20] proposed a new CA model to simulate car and bicycle heterogeneous traffic on an urban road. The proposed model captured the complex interactions between the two types of vehicles. Ding et al. [21] proposed an improved CA model to simulate mixed traffic flow composed of motor vehicles and bicycles near bus stops. The typical NS CA model for motorized traffic flow and the BCA model for bicycle flow were combined to simulate the mixed traffic flow. Table 1 provides a summary of bicycle traffic flow modeling using CA models. It shows the simulation scenarios, vehicle types, model types, cell size, and maximum speeds of the reported models. From the table, it can be clearly seen that the NS and EBCA models have been widely used in the modeling of heterogeneous bicycle flow. For the NS model, the cell length and the maximum speeds of different bicycles were selected based on real bicycle data (speeds were RBs 4 m/s and EBs 6 m/s), while all of the proposed EBCA models for bicycle simulation used 1 and 2 cells/s as the maximum speeds of RBs and EBs, which are significantly different from real bicycle speed data. However, based on different numbers of bicycle lanes and simulation scenarios, the lanechanging rules of the NS models are significantly different from each other. With an increase in the number of bicycle lanes, these rules become more complex, making them impossible to calibrate, validate, and evaluate. As can be seen from the above, the EBCA models are more suitable for modeling bicycle traffic flow and investigating the characteristics of heterogeneous bicycle flow than the NS models. However, the assumption that RBs have a maximum speed of 1 cell/s and EBs a maximum of 2 cells/s is not realistic. Therefore, this paper expands the maximum speed and update rules for bicycle traffic and proposes an improved EBCA (IEBCA) model for modeling heterogeneous bicycle traffic. Both simulation analysis and field data calibration are presented so as to compare the models. The remaining parts of the paper are organized as follows. Section 2 introduces the IEBCA model. Section 3 presents the simulation results of the proposed model. Section 4 uses field bicycle data for the calibration and validation of the proposed model. Finally, the conclusions and possibilities for future study are addressed.
2. The IEBCA model 2.1. Definition of bicycles’ maximum speeds The size and speed differences between RBs and EBs will inevitably lead to more complicated characteristics and a higher risk of traffic collisions. Accurately describing the size and speed of heterogeneous bicycles is the foundation of modeling bicycle traffic. Based on field surveys [2,22], the typical lengths of RBs and EBs in China are not significantly different from one another, at around 1.7–1.9 m. Adding in a safe distance between two successive bicycles, the length of a bicycle cell is set to 2 m, which has also been widely used in most of the previous bicycle CA models [9–12]. Based on the criteria in the US and China, it is recommended that the standard width of a bicycle lane should be 1–1.2 m [2,10]. Therefore, we use 2 × 1 m as the cell size of a bicycle in this paper. The maximum speeds of different bicycles are the other important parameters that will affect the free flow speed and capacity of heterogeneous bicycle traffic flow. According to the results of field investigations in China [2,22] and reported studies [9,10,12,13], the average speeds of RBs and EBs in free flow are round 14–16 km/h and 20–22 km/h, respectively. Accordingly, in this paper, 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 as the maximum speeds for RBs and EBs, respectively (assuming that the update time step corresponds to 1 second). 2.2. IEBCA model Due to its easy-to-follow concept, simple rules, and speed for numerical investigations, the CA model is an efficient tool for simulating traffic flow [23]. Nishinari and Takahashi [14] were the first to propose a multi-value CA model. The initial multi-value CA model comes from the ultradiscretization of the Burgers equation, and is therefore called the Burgers cellular automata (BCA) model. Its evolution equation is
U j (t + 1) = U j (t ) + min U j −1 (t ), L − U j (t )
− min U j (t ), L − U j +1 (t )
(1)
where U j (t ) represents the number of vehicles in cell j at time t. Then, Nishinari and Takahashi extended the maximum speed of BCA to 2, and presented the extended BCA (EBCA) models [24,25]. As mentioned earlier, the EBs’ maximum speed is considered to be 3 cells/s in this paper. Therefore, the updating rules are more complex than in the EBCA and are extended as follows: (1) Assume that the numbers of RBs, EBs, and all bicycles at location j at time t are U rj (t ), U ej (t ), and U j (t ), respectively. (2) All bicycles at location j move to the next location j + 1 if the next location is not fully occupied, and the EBs have priority over the RBs. (3) All bicycles that have moved based on procedure (2) can move to location j + 2 if location j + 2 is not fully occupied after procedure (2), and the EBs have priority over the RBs. (4) Only the EBs that have moved in procedure (3) can move to location j + 3 if location j + 3 is not fully occupied after procedure (3). Because of the higher speed, stability, and flexibility of EBs, we assume that the EBs have priority to pass the RBs in the proposed model. We assume that the numbers of RBs and EBs that move one cell from location j at time t in procedure (2) are brj (t ) and
bej (t ), respectively; the numbers of RBs and EBs that move two
cells from location j at time t are c rj (t ) and c ej (t ), respectively; the total number of bicycles that move one and two cells from location
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Table 1 Overview of bicycle traffic flow modeling using cellular automata. Reference
Year
Simulation scenario
Vehicle types
Model type
Cell length
Max speed (Cells/s)
Gould and Karner [9]
2009
Separated bicycle path
NS
2.1 m
Zhang et al. [10]
2013
Separated bicycle path
NS
2.0 m
Zhao et al. [11]
2013
Separated bicycle path
Improved NS
2.0 m
Yang et al. [12] Jiang et al. [13]
2013 2014
Separated bicycle path Separated bicycle path
NS and dynamic floor field Improved NS
2.0 m 1.0 m
Jiang et al. [15]
2004
Separated bicycle path
EBCA
1 cell
Jia et al. [16]
2007
Separated bicycle path
EBCA
1 cell
Li et al. [17]
2008
Separated bicycle path
EBCA
1 cell
Vasic and Ruskin [18]
2012
Non-separated roadway and intersection
NS model and Interaction rule
3.75 m
Zhang et al. [19]
2014
Intersection
2015
Non-separated roadway
Ding et al. [21]
2015
Non-separated roadway
NS EBCA NS Improved NS NS EBCA
3.5 m
Luo et al. [20]
RB EB RB EB RB EB RB RB EB RB EB RB EB RB EB Car RB Car RB Car RB Bus/car RB
2 3 2 3 4 5 4 4 6 1 2 1 2 1 2 6 2 5 1 20 6 3/4 2
j at time t are b j (t ) and c j (t ), respectively; the number of EBs that move three cells from location j at time t is d j (t ). In real traffic conditions, bicycles are considered to be constantly subject to perturbation by noise such as pedestrians, parking vehicles, and steep path slopes. Therefore, stochastic randomization is introduced into the IEBCA [16]. The randomization effect on RBs is as follows: c rj (t + 1) decreases by 1 with probability p r if c rj (t + 1) > 0; the randomization effect on EBs is that d j (t + 1)
decreases by 1 with probability p e if dej (t + 1) > 0. The updating rules of the proposed IEBCA model are improved as follows: Step 1: calculating brj (t + 1), bej (t + 1), and b j (t + 1) ( j = 1, 2, 3, . . . , K ):
bej (t + 1) = min U ej (t ), L − U j +1 (t ) brj (t
+ 1) = min
U rj (t ), L
−U
e j +1 (t ) − b j (t
(2)
+ 1)
b j (t + 1) = brj (t + 1) + bej (t + 1)
(3) (4)
where L is the number of bicycle lanes, and K is the total number of cells in the simulation system. Step 2: calculating c rj (t + 1), c ej (t + 1), and c j (t + 1):
− b j +1 (t + 1) + b j +2 (t + 1)
(5)
c rj (t + 1) = min brj (t + 1), L − U j +2 (t ) − b j +1 (t + 1)
+ b j +2 (t + 1) − c ej (t + 1)
(6)
When the random number is less than a preset value p r and the number of RBs that move two cells from location j at time t is more than zero (c rj (t + 1) > 0), the number of moving RBs is minus 1. If rand( ) < p r , then
c rj (t + 1) = max c rj (t + 1) − 1, 0 c j (t + 1)
= crj (t
+ 1) + c ej (t
+ 1)
(7) (8)
4/2 cell 1 cell
where rand() is a uniformly distributed random number between 0 and 1. In equations (2) and (5), bej (t + 1) and c ej (t + 1) are calculated first because the EBs have priority over the RBs due to their higher speed. Step 3: calculating d j (t + 1):
d j (t + 1) = min c ej (t + 1), L − U j +3 (t ) − b j +2 (t + 1)
+ b j +3 (t + 1) − c j +1 (t + 1) + c j +2 (t + 1)
(9)
When the random number is less than a preset value p e and the number of EBs that move three cells from location j at time t is more than zero (d j (t + 1) > 0), the number of moving EBs is minus 1. If rand() < p e , then
d j (t + 1) = max d j (t + 1) − 1, 0 Step 4: updating
U rj (t
+ 1),
U ej (t
(10)
+ 1), and U j (t + 1):
U ej (t + 1) = U ej (t ) − bej (t + 1) + bej−1 (t + 1) − c ej−1 (t + 1) U rj (t
+ 1) =
U j (t + 1) =
c ej (t + 1) = min bej (t + 1), L − U j +2 (t )
1.0 m
+ c ej−2 (t + 1) − d j −2 (t + 1) + d j −3 (t + 1)
(11)
U rj (t ) − brj (t + 1) + brj −1 (t + 1) − crj −1 (t U rj (t + 1) + U ej (t + 1)
(12)
+ 1) + crj −2 (t + 1)
(13)
3. Simulation results For the simulation analysis in this study, a three-lane bicycle path was selected with length K = 100 cells (equal to 200 m). In the initial conditions, RBs and EBs are randomly distributed on the bicycle path. The percentage of EBs is r, which is between 0 and 1. Periodic conditions, which are the closest to the real traffic conditions, were used in this paper. Accordingly, the instantaneous positions and numbers of bicycles for all cells are updated in parallel per second. The flow, speed, and density of the heterogeneous bicycle traffic flow are calculated after 20 000 simulation steps, and the average value of the last 5000 steps is used for the calculation in order to decrease the random effect. The density and flow of the bicycle path are calculated by
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Fig. 1. The fundamental diagrams in the stochastic cases where (a) p r = 0 and (b) p e = 0.
Fig. 2. The fundamental diagrams in the stochastic cases where (a) r = 0.2 and (b) r = 0.5.
ρ (t ) = Q (t ) =
K 1
KL
U j (t )
(14)
j =1
K 1 b j −1 (t ) + c j −2 (t ) + d j −3 (t ) KL
(15)
j =1
where ρ (t ) and Q (t ) are the density and flow per lane of the simulated bicycle path system at time t, respectively. In the following subsections, some deterministic and stochastic cases under different model parameters will be simulated and discussed.
in the free flow region. This is due to the fact that, when considering the slowdown probability of RBs with a slower free flow speed, a steady moving bottleneck will not exist and the quicker EBs will overtake the slower ones. The fundamental diagrams with slowdown probability parameters p e > 0 and p r > 0 are shown in Fig. 2. It can be seen that the multiple states effect disappears in both the free flow and the congested regions. The results for the multiple states are similar to those in previous research, and detailed analysis can be found in Refs. [14–16,24–26]. 3.2. Effect of proportion of EBs
3.1. Effect of slowdown probability In the IEBCA model, the RBs and EBs are assumed to have probabilities p r and p e of reducing their speed because of stochastic interference in the natural traffic situation. Fig. 1 shows the fundamental diagrams of heterogeneous bicycle traffic under the effect of a slowdown probability of p r = 0 or p e = 0. Considering the case where p r = 0 and p e > 0 (Fig. 1a), there are two branches and the multiple states effect occurs in the free flow region. However, the multiple states effect does not occur in the congested region. Fig. 1b considers the case where p e = 0 and p r > 0. Here, the multiple states effect can be seen in the congested region but not
Fig. 3 shows the fundamental diagrams of bicycle traffic flow under different percentages of EB, Fig. 3a showing the deterministic case and Fig. 3b the stochastic case. With an increase in the EB percentage, the capacity of the bicycle path increases and the critical density decreases. This is due to the fact that the EBs have a higher free flow speed. It can also be seen that the multiple states effect occurs in both the free flow and the congested regions in the deterministic case with p r = 0 and p e = 0 (Fig. 3a), and the multiple states effect disappears in Fig. 3b. When r = 0, the IEBCA model is a special case of the EBCA model. Therefore, the phase transition from the free flow to the congested state is second-order.
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Fig. 3. The fundamental diagrams under different proportions of EBs, where (a) shows the deterministic case with p r = 0 and p e = 0 and (b) the stochastic case with p r = 0.3 and p e = 0.3.
Fig. 4. The fundamental diagrams under different numbers of lanes, where (a) shows the deterministic case with p r = 0, p e = 0, and r = 0.5 and (b) the stochastic case with p r = 0.3, p e = 0.3, and r = 0.5.
3.3. Effect of number of bicycle lanes In this subsection, the effect of the number of bicycle lanes on heterogeneous bicycle traffic flow is investigated. The number of bicycle lanes varies widely according to different road conditions. With an increase in the number of bicycle lanes, passing becomes easier. From Fig. 4a, it can be seen that the multiple states effect exists in both the free flow and the congested regions. Meanwhile, the maximum flow increases with the increase in bicycle lanes. This is because when there are more bicycle lanes, the EBs have more opportunities to pass the RBs. Similar results are shown in Fig. 4b where the randomization effect of RBs and EBs is considered. When the number of lanes equals 1, the capacity of the bicycle path is determined by the RBs, and the quicker EBs cannot pass the slower ones. Therefore, the fundamental diagram for L = 1 is lower than those for L greater than one. 3.4. Capacity Because the IEBCA model uses higher speeds, the free flow speed and capacity should be larger than in the previous EBCA model [16]. In this subsection, the fundamental diagrams of the
proposed IEBCA model and the EBCA model are compared. Fig. 5 shows the fundamental diagrams in the stochastic case. It can be seen that the multiple states effect disappears in both the free flow and the congested regions, even if p r and p e are larger than zero. The differences in capacity and critical density between the two models are very large (as shown in Fig. 5a). This could be due to the different maximum speeds of RBs and EBs. In the case of r = 0 for the IEBCA model and r = 1 for the EBCA model, which means that only bicycles with a maximum speed of 2 cells/s exist in the simulation systems, the curves from the two models coincide. 4. Calibration and validation 4.1. Data collection Field data collection is very important for calibrating the parameters of the proposed IEBCA model and evaluating its performance. Previous CA-based bicycle models have not been calibrated, leading to a lack of evaluation and application. In this paper, three survey sites in Hangzhou, China were selected for calibration and validation. All of the bicycle paths have less than a 3% gradient and are separated from the motorized vehicle lanes by physical barri-
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Fig. 5. The fundamental diagrams in the stochastic case where p r = 0.3 and p e = 0.3. The circles and squares correspond to the proposed IEBCA model and the EBCA model, respectively.
Fig. 6. Photos of the camera’s view of the three sites. Table 2 Statistical description of field survey sites. No.
Survey site
Bicycle path width (m)
Number of lanes
% of EBs
% of male cyclists
% of young cyclists
1 2 3
Jiaogong Road Hushu Road Tianmushan Road
2.27 2.93 3.97
2 3 4
68.6% 70.3% 73.9%
63.8% 59.7% 80.0%
80.3% 62.2% 69.8%
ers. Hence, bicycle traffic flow is not interrupted by pedestrians or motorized vehicles. One camera was set up on the side of the bicycle path to take videos to collect data on the bicycles’ operating behavior (see Fig. 6). Cameras were carefully placed so that the cyclists would be unaware they were being observed. The survey days were sunny, there were no crashes, and weather conditions were good. Both free flow and congested flow of a mixture of RBs and EBs were observed during the survey periods, and used to compare the fundamental diagrams of the field observations against those based on simulated data. Using video-processing technology, we were able to identify the moments when bicycles crossed the marked white lines. Then, the traffic flow parameters of the heterogeneous bicycles (such as flow, speed, and density) could be calculated automatically. The bicycle types, and genders and ages of the cyclists, were coded manually. Table 2 presents the statistical description of the field survey sites. The path widths of the selected survey sites are 2.27, 2.93, and 3.97 m, respectively, which can be thought of as approximately 2-, 3-, and 4-lane bicycle paths. The percentages of EBs at the three sites were around 70%. The field bicycle data (flow, speed, and density) were aggregated every 25 seconds.
4.2. Results For the simulations, the numbers of lanes and percentages of EBs were chosen to match the observed results from the three survey sites. The effects of the percentages of male and young cyclists on the fundamental diagrams can be considered as the parameters of the slowdown probability. In order to determine the slowdown probabilities of RBs and EBs, we used different combinations of slowdown probability values for the simulations. The mean absolute percentage error (MAPE) was used as the performance index to evaluate the proposed model in comparison with the field observations [27–29]:
MAPE =
× 100%
M 1 Qˆ ( j ) − Q ( j )
M
j =1
Q ( j)
(16)
where Qˆ ( j ) = the simulated bicycle flow during the jth interval; Q ( j ) = the observed bicycle flow during the jth interval; M = the number of intervals. Fig. 7 shows the fundamental diagrams for the field observations and the simulated results, where (a) shows the two-lane case, (b) the three-lane case, and (c) the four-lane case. It can be seen (Fig. 7 left) that the MAPEs between the field observations and the
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Fig. 7. MAPEs between observations and simulated results under different slowdown probabilities, and simulated fundamental diagrams in which the gray circles represent field observations and the blue solid lines represent simulated results for the minimum MAPE.
simulated fundamental diagrams vary widely under different slowdown probabilities for RBs and EBs. We used the minimum MAPE to determine the calibrated slowdown probabilities for RBs and EBs. The calibrated parameters and MAPEs are shown in Table 3. Fig. 7 (right) shows the observations and calibrated fundamental diagrams. The results show that the IEBCA model gives reasonable results and is suitable for modeling heterogeneous bicycle traffic flow.
5. Conclusions With the wider usage of green and convenient RBs and EBs in recent years, the modeling and simulation of heterogeneous bicycle traffic flow is becoming increasingly important for bicycle path planning, management, and operation. In order to overcome the weaknesses of previous MCA models for bicycle, this paper has proposed an improved MCA model that introduces maximum
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References
Table 3 Calibrated slowdown probabilities and MAPEs. Survey site
Number of lanes
pr
pe
MAPEs
Jiaogong Road Hushu Road Tianmushan Road
2 3 4
0.90 0.85 0.70
0.65 0.70 0.50
24.55% 21.61% 16.38%
speeds of two and three cells/s for RBs and EBs, respectively. New update rules have been proposed for the improved MCA model. The numerical simulation results and fundamental diagrams for bicycle traffic have been analyzed and discussed. Three parameters, the slowdown probability, the percentage of EBs, and the number of bicycle lanes, have been analyzed in both the deterministic and stochastic cases. The results are similar to those of previous research. Field bicycle observations collected from three survey sites were used for the calibration and validation of the simulated fundamental diagrams. The results show that the proposed model matches the field bicycle data better than previous models.
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Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 51338008, 51278454, 51208462, and 61304191), the Fundamental Research Funds for the Central Universities (2014QNA4018), the Projects in the National Science & Technology Pillar Program (2014BAG03B05), and the Key Science and Technology Innovation Team of Zhejiang Province (2013TD09).
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