Fundamental Diagram of Bicycle Traffic Based on Logistic Model

Cheng Xu

1, 2

, Xin Wang

and Xiaonan Yu

Department of Traffic Management Engineering, Zhejiang Police College, Hangzhou 310053 China

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

Department of International School, Zhejiang Police College, Hangzhou 310053 China

xucheng@zjjcxy.cn, wangxin@zjjcxy.cn, yxnpolice1474@outlook.com

Keywords: Fundamental diagram, Bicycle traffic, Electric bicycle, Logistic model.

Abstract: Bicycle has become one of the important commuting travel ways to residents who are living in developing

countries including China. Therefore, it’s of great theoretical interest and industrial significance to study on

the fundamental diagram of bicycle traffic flow. According to the operating characteristic, this paper proposes

adopting logistic model to modeling the fundamental bicycle traffic flow and the method of model parameters

calibration based on the maximum likelihood algorithm. This model was verified by the measured data from

three sections of Hangzhou city and estimated the capacity was 2243 bikes/h/m. The fitting precision of this

model was significantly improved compared to the traditional model. Based on the study mentioned above, it

will provide data supporting to the application of fundamental diagram in the bicycle traffic flow field.

1 INTRODUCTION

Fundamental diagram (FD) is the relationship

between density and volume of traffic, and is the

important foundation for traffic state identification,

capacity estimation, and determination the phase

regime of traffic flow. FD is also an important feature

which distinguishes traffic flow from other fluids.

Therefore, FD is of significance for both motorized

vehicle traffic flow and non-motorized traffic flow.

Greenshields firstly presented a linear speed-density

relationship and parabolic density-volume

relationship using the measured field data (B. D.

Greenshields, 1935). After around 80 years’

development of FDs, there are numerous studies of

FDs. The famous FD models includes Greenberg

model (H. Greenberg, 1959), Underwood model (R.

T. Underwood, 1961), Newell model (G. F. Newell,

1961), Pipes model (L. A. Pipes, 1976), and Qu

models (X. Qu, S. Wang, J. Zhang, 2015; X. Qu, J.

Zhang, 2017).

Most of the FD models are single-regime models

which could fit the empirical data samples well for

motorized vehicle traffic flow. However, most of the

FD models were presented for motorized vehicle

traffic flow. There was almost no FD model proposed

for non-motorized vehicles. In recent years, bicycle

traffic (including electric bicycles) has been become

one of the main trip modes in developing countries,

especially in Southeast Asian countries such as

China, Vietnam, and Indian. The electric bicycle

(E-bike) has quickly become one of the main

non-motorized travel modes in China (S. Jin, 2015;

C. Xu, 2016; S. Jin, 2015; M. Zhou, 2017). With the

rapid increase of bicycles, it is very important to

study the traffic flow characteristics of bicycles.

Traditional motorized vehicle-oriented FD models

cannot adapt to the operating characteristics of

bicycle traffic flow. Therefore, the purpose of this

paper is to use logistic model for modeling the FD of

bicycle traffic flow, and proposed a FD model for

estimating the capacity of bicycle flow.

2 MODEL

2.1 Logistic Model of Bicycle

Fundamental Diagram

Logistic curve is a model with many equations and

wildly used in population ecology field. Wang et al.

(H. Wang, J. Li, Q. Chen, 2011) presented a model

that describes traffic flow speed-density relation.

Compared the imitative effect on logistic models of

three or four parameters and five parameters, he used

Xu, C., Wang, X. and Yu, X.

Fundamental Diagram of Bicycle Trafﬁc Based on Logistic Model.

In 3rd Inter national Conference on Electromechanical Control Technology and Transportation (ICECTT 2018), pages 187-190

ISBN: 978-989-758-312-4

187

five parameters to model the motor vehicle flow.

Concrete equation as follows:





()

1exp

vk v













(1)

where, v is the speed of bicycle flow, k is the density

of bicycle flow, v

is the free flow speed, v

is the

speed when traffic is on the condition of stop and go,

is the density from free flow to congested flow, θ

is the parameter which determining the slope of the

logistic curve, and θ

is the parameter which

determining the symmetry of the logistic curve.

Based on the basic relationship of volume

-speed-density model,

qkv



. (2)

The relationship of density-volume is presented

as follow:





1exp

qkv





















(3)

where, q is the volume of bicycle flow.

Considering the effect of E-bikes on bicycle flow,

the volume of E-bikes should be converted to bicycle

volume using the bicycle equivalent unit for the

E-bike,

beb

qq q





(4)

where, q

is the volume of bicycles, q

is the

volume of E-bikes, and β is the bicycle equivalent

unit for the E-bike which can be set as 0.66 (S. Jin,

2015).

Based on this model, we can establish the

fundamental diagram of traffic flow under the

operation of hybrid bicycle. Applying this model can

fit the operation characteristic of bicycle better.

2.2 Parameter Calibration

For Equation (4), the formula is nonlinear, and the

traditional method of least squares cannot be used in

parameter calibration directly. Therefore, a nonlinear

least squares fitting method is proposed for

calibrating the model parameters. For equation

q=f(k), given f() and observation vector q, the main

steps of the LM algorithm are as follows:

Step 1: Set initial value k

and stop control

constant ε, and calculate

()qfk





Step 2: Calculate the Jacobi Matrix J

, and

iiii

NJJ I





, then establish the equation

ii i i







Step 3: Solve the above equation for



(1) If

()

ii i

qfk









, then

1iii







else if







, stop iteration, output; else









, return to Step 2.

(2) If

()

ii i

qfk









, then

1ii









solve the function again and get



, return to Step 1.

Using these three steps, we can complete the

calibration and obtain the parameters that minimize

the model error.

3 RESULTS

3.1 Date Collection

Field data collection is of significance for FD model

estimation and calibration. In this paper, three bicycle

path sections in Hangzhou, China, were selected for

data collection. Every bicycle path was one-way, and

located in the middle of road links, far away from

intersections, which will minimize the effects of

traffic signals and pedestrians on bicycle traffic.

Camera was set up on the side of the bicycle path to

take videos. The filed data was conducted on

weekdays during peak and non-peak hours. Using

video-processing technology, the volume and speed

can be easily recorded automatically.

Table 1 provides the width of bicycle path,

percentage of E-bike and percentage of male cyclist

from these three surveyed sections. Traffic flow data

from these three sections will use to fit and capacity

estimate on fundamental diagram of bicycle traffic

flow as below.

Table 1: Description of field survey data

Section

name

Width of

bicycle path

Percentage

of E-bike

Percentage of

male cyclist

Jiaogong

Road

2.4 m 65.5% 62.7%

Hushu

Road

3.0m 57.9% 67.3%

Wener

Road

3.5 m 59.8% 66.3%

Overall - 61.0% 65.5%

3.2 Results of Estimation

In order to analyze the advantage of logistic model

further, this paper will analyze it by comparing to

Greenshields model, Greenberg model and

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188

Underwood model individual. The above three model

equations as follow:

(1 )

qkv kk

(5)

log

qkv kk

(6)

exp( )

qkv kk

(7)

Calibrating the above four models’ parameter by

applying the maximum likelihood algorithm and

draw the relationship diagram between model

estimation and field observed data. Table1-3 presents

the fundamental diagram which was drawn by field

observed data from three sections and the curve

which is fitting by the four models. The gray circles

are field observed data. The black, red and green

solid lines represent the Greenshields model,

Greenberg model and Underwood model

respectively. The blue dotted line represents the

outcomes of logistic model fitting. The logistic model

has higher fitting precision to these three different

widths of paths from the tables.

Figure 1: FD models of bicycle traffic on Jiaogong Road

Figure 2: FD models of bicycle traffic on Hushu Road

Figure 3: FD models of bicycle traffic on Wener Road

In order to analyze the fitting precision of model

further, Table 2 gives fitting errors and estimated

bicycle capacity of different models. Paper presents

the average error of logistic model which gets higher

fitting precision than others is around 15%.

Meanwhile, the average estimated capacity of these

three sections is about 2243 bikes/h/m, which is more

correspond to actual fact.

Table 2: Results of density-volume relationships

Models

Jiaogong

Road

Hushu

Road

Wener

Road

Average

Capacity

Greenshields 2340 2476 2432 2412

Greenberg 2530 2432 2412 2473

Underwood 2101 2098 2023 2062

Logistic

model

2312 2322 2203 2243

Fitting error

Greenshields 19.41% 17.62% 18.34% 18.33%

Greenberg 34.12% 31.32% 28.23% 30.21%

Underwood 19.32% 17.23% 19.12% 18.54%

Logistic

model

15.32% 14.43% 16.12% 15.12%

4 CONCLUSIONS

The study on the fundamental diagram of traffic flow

focus on the vehicle flow, and few of research

addressed the non-motor vehicle. From the

perspective of establishing the fundamental diagram

of non-motor vehicle traffic flow which was base on

logistic model, this paper verified the feasibility of

fundamental diagram which was applied to the

Fundamental Diagram of Bicycle Trafﬁc Based on Logistic Model

189

non-motor vehicle traffic flow. Following

conclusions were obtained in this paper:

(1) The speed-density relation model of bicycle

traffic flow corresponds to the fundamental diagram,

but the data points are more discrete compared to the

vehicle data.

(2) The outcome of fitting error based on the

logistic model is significantly rising compared to the

classic model.

(3) The capacity and block density of bicycle path

which were calibrated based on the logistic model

correspond more to actual fact.

ACKNOWLEDGEMENT

This work was partly supported by the Zhejiang

Provincial Natural Science Foundation (No.

LQ17E080001), the China Postdoctoral Science

Foundation, and the Key Research and Development

Program of Zhejiang (No. 2018C01007).

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