modeling and analysis of the car-truck heterogeneous

Yang, Jin, Ran, Pu and Yang 1

Modeling and Analysis of the Car-truck Heterogeneous Traffic Flow Based on Intelligent 1 Driver Model 2 3 Da Yang, Ph.D. Candidate 4 Department of Civil and Environmental Engineering, 5 University of Wisconsin-Madison, 6 1415 Engineering Drive, Madison, WI 53706, USA 7 Phone: 1-608-334-4281 8 E-mail: [email protected] 9 10 Jing Jin, Ph.D., Postdoctoral Fellow 11 Department of Civil, Architectural, and Environmental Engineering, 12 The University of Texas-Austin, 13 1616 Guadalupe St., Ste. 4.202, Austin, TX 78701, USA 14 Phone: 1-512-232-3124 15 E-mail: [email protected] 16 17 Bin Ran, Ph.D., Professor 18 Department of Civil & Environmental Engineering, 19 University of Wisconsin-Madison, 20 1415 Engineering Drive, Madison, WI 53706, USA 21 Phone: 1-608-262-0052 Fax: 1-608-262-5199 22 Email: [email protected] 23 and 24 School of Transportation, Southeast University, 25 No.2 Si Pai Lou, Nanjing 210096, China 26 27 Yun Pu, Ph.D., Professor 28 School of Transportation and Logistics, 29 No.111 erhuanlubeiyiduan, Chengdu, 610031, China 30 Email: [email protected] 31 32 Fei Yang, Ph.D., Associate Professor 33 School of Transportation and Logistics, 34 No.111 erhuanlubeiyiduan, Chengdu, 610031, China 35 Email: [email protected] 36 37 Corresponding Author: Da Yang 38

39 Submitted for Presentation and Publication 40 To the 92nd Transportation Research Board Meeting 41 Submission Date: Sep. 15th, 2012 42

43 5449 Words + 1 Table + 6 Figures = 7199 Words 44

TRB 2013 Annual Meeting Paper revised from original submittal.


ABSTRACT 1

The traffic flow heterogeneity caused by the different car-following dynamics among the different types 2

of vehicles has drawn increasing attention recently. This paper explores the characteristics of the four 3

types of car-truck car-following combinations, car-following-car (CC), car-following-truck (CT), truck-4

following-car (TC) and truck-following-truck (TT), and their impacts on traffic flow stability. A 5

heterogeneous traffic flow model based on Intelligent Driver Model (IDM) is proposed and calibrated 6

using the Next Generation Simulation (NGSIM) vehicle trajectory data. Based on the calibrated model, 7

the characteristics of the car-truck heterogeneous traffic flow are evaluated using the linear stability 8

analysis, fundamental diagrams, and shock wave characteristics. The linear stability analysis identifies 9

two critical factors that can influence the stability of the car-truck heterogeneous traffic flow: the 10

stability functions and the proportions of the four types of car-truck combination. Cars and trucks can 11

both stabilize and destabilize the traffic flow depending on the combination type and the equilibrium 12

velocity. Fundamental diagrams of car-truck heterogeneous flow are found to be determined by the 13

distance headways and proportions of the four types of combination. Moreover, the fundamental 14

diagrams of different car-truck combinations converge to several clusters with the same proportion 15

difference between the CC and TT combinations. The slowing-down effect of trucks on shock wave 16

speed in the car-truck heterogeneous traffic flow is also observed in the simulation. 17

18

KEYWORDS: CAR AND TRUCK, HETEROGENEOUS TRAFFIC FLOW, INTELLIGENT 19

DRIVER MODEL, CAR-FOLLOWING COMBINATIONS 20



INTRODUCTION 1

Heterogeneity is a key characteristic of real-world traffic flow. Although traffic flow theories and 2

models are usually developed first for the homogeneous traffic flow, most of them can be easily 3

converted into their heterogeneous forms. However, the difficulty is the lack of sufficient data to 4

calibrate and explore the heterogeneous models. With the development of the data collection technology 5

in recent years, such as the video-based method and the GPS-based method, traffic information, 6

especially individual vehicle dynamics, can be obtained in greater detail. Many researchers have been 7

using these new data sources to investigate the characteristics of car-truck heterogeneous traffic flow in 8

the past few years. Early researchers focused on investigating the differences between the car and truck 9

driving behavior. Huddart and Lafont (1), McDonald et al. (2) and Sayer et al. (3) compared the 10

headway differences between these two cases: car-following-car and car-following-truck, but their 11

studies did not reach conclusive results regarding with which case had the larger headway. Peeta et al. (4, 12

5) analyzed interactions of cars and trucks in multiple lanes. Highway Capacity Manual presented that 13

trucks occupied more space, had poorer operating capabilities, and created lager gaps than cars in most of 14

situations (6). However, these studies (1-5) did not recognize that the car-following behavior also depends 15

on the following vehicle type. Ye’s study (7) first explores the impact of the following vehicle type (car 16

or truck) on traffic flow. He concluded that the four types of car-truck car-following combination should 17

be taken into account in the study of the car-truck heterogeneous traffic flow: the car-following-car (CC), 18

the car-following-truck (CT), the truck-following-car (TC), and the truck-following-truck (TT). Sarvi (8) 19

also studied the driving behavior of these three car-following combinations, car-following-car, truck-20

following-car and car-following-truck. Aghabayk et al. (9) further studied the distance headway, time 21

headway, reaction time and car-following threshold variations among the four types of combination. One 22

major limitation of those early studies on car-truck traffic flow characteristics is the lack of modeling of 23

the dynamic traffic flow characteristics. Mason and Woods (10) developed the homogeneous Optimal 24

Velocity (OV) car-following model into a heterogeneous traffic flow model to describe the interaction 25

between cars and trucks. The derived OV heterogeneous car-following model is as follows, 26

nnnnnn vxxU

dt

txd 12

2 )( (1) 27

where xn(t) and vn(t) respectively denote the location and velocity of the vehicle n at time t, xn-1(t) 28

denotes the location of the vehicle n-1 (the preceding vehicle of vehicle n) at time t, n denotes the 29

sensitivity parameter of the vehicle n, nnn xxU 1 denotes the optimal velocity function the vehicle n 30

wishes to take, and it is the function of the headway of vehicle n. 31

The heterogeneous Optimal Velocity model is used to conduct the comparison of the car-32

following dynamics between homogeneous and heterogeneous vehicles and it is found that trucks may 33



dampen congestion waves. Because of the complication in their multi-species formulations, it is difficult 1

to analyze individually how different car-truck following combinations can affect traffic flow. In this 2

study, based on Intelligent Driver Model (IDM), we developed a heterogeneous model with the different 3

sub-models for each car-truck following combination. To calibrate the new model, the model parameters 4

for each car-truck car-following combination are calibrated separately using the car-following data 5

extracted from the I-80 NGSIM (Next Generation Simulation) data for each combination. Based on the 6

calibrated model, we study the three characteristics of the car-truck heterogeneous traffic flow, the linear 7

stability, fundamental diagrams, and shock wave. 8

9

METHODOLOGY 10

An IDM-based Car-following Model for the Car-truck Heterogeneous Traffic Flow 11

Treiber et al. proposed Intelligent Driver Model in 2000 (11) for the homogeneous traffic flow. This 12

model is a widely explored car-following model (12-14), and its formulation is as follows, 13

ab

tvtvtv

V

tvsstvtvS

ltx

tvtvS

V

tva

dt

txd

nnn

nnn

n

nnnn

2

)()()(

)())(),((

)(

))(),(()(1

)(

10

2

2

2

(2) 14

where a is the maximum acceleration, V is the desired velocity, is the acceleration exponent, S() is the 15

desired minimum gap, s0 and s1 are the jam distances, is the safe time headway, b is the desired 16

deceleration, l is the leading vehicle length, and tvtvtv nnn 1 is the removal rate of the vehicle 17

n to its preceding vehicle n-1. 18

We develop the homogeneous IDM to its heterogeneous form by giving subscripts to the model 19

parameters. The proposed heterogeneous IDM formulates the four different car-truck car-following 20

combinations as follows, 21

nn

nnnn

n

nnnnnn

nn

nnn

n

nn

n

ba

tvtvtv

V

tvsstvtvS

ltx

tvtvS

V

tva

dt

txdn

2

)()()(

)())(),((

)(

))(),(()(1

)(

10

2

2

2

(3)

22

where all the parameters an, n, Vn, 0ns , 1

ns , n and bn have four alternatives. Taking an as an example, an 23

can be acc, act, atc and att. The leading vehicle length l has two alternatives, lc and lt. 24

In the homogeneous traffic flow, all vehicles at the equilibrium state have zero acceleration, the 25

same distance headway and the same velocity; while at the equilibrium state, all vehicles in the 26



heterogeneous traffic flow still have zero acceleration and the same velocity, but their distance headways 1

vary for different vehicles which can be described using the following equations: 2

vvn , 0nv , and nn hh (4) 3

where v is the equilibrium velocity which is the same for all vehicles, nh is the corresponding 4

equilibrium headway of the vehicle n, and nh varies among the vehicles. Substituting (4) into the 5

heterogeneous IDM (3) yields, 6

vvvssvs

lh

vs

v

v

IDM

010

2

0

/)0,(ˆ

0)0,(ˆ

1 (5) 7

Rewriting the above equation yields, 8

lvv

vvvssh

0

010

/1

/ (6) 9

In the car-truck heterogeneous flow, h has the four alternatives, cch ,

cth , tch and

tth which 10

correspond to the four types of combinations. It should be noticed that the vehicle length l in h only 11

depends on the leading vehicle type, namely, l = lc in cch and

tch , and l = lt in cth and

tth . 12

Then, the fundamental diagram of the heterogeneous IDM for the equilibrium state can be 13

derived based on Equation (6). Assume a given heterogeneous traffic flow on a single lane contains N 14

vehicles. The length of the entire traffic flow in the equilibrium state is: 15

N

nntotal hL

1

(7) 16

In the car-truck heterogeneous traffic flow, Ltotal has the following formula: 17

tttttctcctctcccctotal hNhNhNhNL (8) 18

where Ncc, Nct, Ntc and Ntt are the numbers of the CC, CT, TC and TT combinations in the car-truck 19

heterogeneous traffic flow. Hence, the density of traffic flow can be calculated as: 20

tttttctcctctcccc hNhNhNhN

Nk (9) 21

Rewriting equation (9) in the following form: 22

tttttctcctctcccc hPhPhPhP

k1

(10) 23

where Pcc, Pct, Ptc and Ptt are the proportions of the CC, CT, TC and TT combinations. Thus, the 24

fundamental diagram of the car-truck heterogeneous traffic flow has the following relationship: 25



tttttctcctctcccc hPhPhPhP

vq

(11) 1

where q is the flow rate of the car-truck heterogeneous traffic flow in space. 2

Equation (11) indicates that the distance headways and proportions of the four types of 3

combination determine the fundamental diagram of the car-truck heterogeneous traffic flow. 4

5

Linear Stability Criterion of the Car-truck Heterogeneous Traffic Flow 6

The linear stability analysis investigates the perturbation propagation characteristic of a vehicle platoon 7

by adding a small perturbation on the first vehicle of the platoon (13, 15). The stability criterion of IDM 8

can be derived by following the general stability criterion summarized by Wilson and Ward (13). The 9

derivation of the detailed stability criteria of IDM requires the partial differentials of the velocity, 10

headway and velocity difference between the leading and following vehicles with respect to the 11

equilibrium state of IDM. 12

vv Slh

aS

V

vaf 2

12

, vv Slh

aSf

2

2, 32

2 lhSafh (12) 13

where ),0,( hvSS . vS , vS and hS are respectively the partial differences of the desired minimum 14

gap function with respect to the vn, vn and xn at the equilibrium state ),0,( hv . Hence, according to 15

the general stability criterion introduced by Wilson and Ward (13), the stability criterion of IDM is as 16

follows, 17

02

2

v

hvvf

fff (13) 18

Substituting equation (12) into (13) yields the stability criterion of IDM, 19

02

2

2222

2

1

02

1

023

2

lh

S

v

va

lh

SS

v

v

lh

SaS

lh

S vv

(14) 20

The derivation of the linear stability criterion of the car-truck heterogeneous traffic is more 21

complicated. Ward (16) presented the general formula of the linear stability criterion of the 22

heterogeneous traffic flow as follows,

23

02/

2

,2,,,,

n njhjvnhnvnvn fffff (15) 24



where vnf , , vnf , , hnf , are the partial differences of the car-following model adopted by the vehicle n 1

with respect to v, v and x at the equilibrium state where vvn , 0 nv and nn hx . 2

However, Equation (15) cannot directly reflect the proportion information of the different 3

combinations, so we rewrite it as the following form using the induction method: 4

02/1

1

2,

22,

2,,,,

Q

i

Q

jij

NPhj

NPhivihivivii

ii ffffffNP , (16) 5

where Q is the number of the combinations in a given car-truck heterogeneous traffic flow, N is the total 6

number of vehicles in the traffic flow, Ni is the total number of the combination type i in the platoon, 7

N

NP i

i is the proportion of the combination type i in total, and 1

1

M

iiP . In the car-truck 8

heterogeneous traffic flow, Q = 4 and i has four alternatives: CC, CT, TC and TT. 9

Equation (16) can be further simplified as follows, 10

02/

12,

2,,,,

Q

i hi

vihivivii

f

ffffP (17) 11

In Equation (17), two parts determine the stability criterion of the car-truck heterogeneous traffic 12

flow: the fraction part 2,

2,,,, 2/

hi

vihivivi

f

ffff providing the stability characteristic of the combination 13

i and the proportion part iP of the combination i. We define the fraction part as the Stability Function 14

(SF) of the combination, 15

2,

2,,,,

,,,

2/),,(

hi

vihivivihivivii f

fffffffSF

(18) 16

where SFi is the stability function of the combination type i. In addition, comparing the stability criterion 17

of IDM (14) with equation (18), it can be found that SF has the same sign as (14), so SF also can be used 18

to judge the stability of the car-following combination i. The stability effects of combination type i 19

denoted as SFi in the car-truck heterogeneous traffic flow can be written as follows, 20

2

2

1

2,

1

2,

3

2

42

62

2

222

4 ii

i

ii

i

ii

ivi

ii

ii

viii

ii

i

ii

iii

lh

S

V

va

lh

SS

V

v

lh

SSa

lh

S

Sa

lhSF

i

(19)

21

Moreover, we define the left part of equation (17) as the stability function F of the car-truck 22

heterogeneous traffic flow. Thus, in the car-truck heterogeneous traffic flow, there is, 23

tttttctcctctcccc SFPSFPSFPSFPF (20) 24



When F < 0, the car-truck heterogeneous traffic flow is stable. Equation (20) indicates that the 1

proportion of the car and the truck is not the only deciding factor of the stability of heterogeneous traffic 2

flow, and the stability function of each car-truck following combination is also significant. Furthermore, 3

on a ring road, the stability function of the car-truck heterogeneous traffic flow and the proportions of 4

four types of combination have the following relationship: 5

tcct

tttcctcc

tttttctcctctccccl

PP

PPPP

SCPSCPSCPSCPF

1 (21) 6

7

Model Calibration and Experimental Design 8

The car-following model calibration is a nonlinear optimization problem. In this study, we use Genetic 9

Algorithm (GA) to solve this nonlinear optimization problem to obtain the optimal parameters. We also 10

adopt the Theil’s U function as the objective function as suggested in several existing studies (17, 18). 11

M

m

simm

M

m

realm

M

m

simm

realm

objective

yM

yM

yyM

F

1

2

1

2

1

2

11

1

(22) 12

where realmy is the real data, sim

my is the simulation result from the model, and m=1, …, M is the number 13

of the data sample. 14

To calibrate car-following models, we use the NGSIM vehicle trajectory data collected on I-80 15

San Francisco, California on April 13, 2005 (19). Since the data includes lane change and large vehicle 16

gaps, we use the following criteria to extract the car-following datasets of the four combinations 17

separately from NGSIM data. 18

a. Each car-following group contains two vehicles (car-following pair). Car-following group 19

with more than three vehicles will be divided into multiple two-vehicle groups. 20

b. Each pair of car-following vehicles is formed and decomposed based on two spacing 21

thresholds, the engaging threshold DE = 130 ft (39.62 m) and the disengaging threshold DD 22

= 150 ft (45.72 m). These values are determined based on critical density at the experimental 23

site. Using two thresholds instead of one threshold can avoid unnecessary frequent grouping 24

and ungroup of car-following vehicle pairs caused by small fluctuation of spacing value 25

around a single threshold. 26

c. If a vehicle group decomposes due to spacing increase, lane changing or reaching the end of 27

the segment, it will not be exported as a valid sample. 28



The final datasets include the 477712, 25844, 16471, and 10105 trajectory points for the CC, CT, 1

TC, and TT combinations respectively. The calibrated car-following models are also inspected using 2

several error indexes including Mean Error (ME), Mean Absolute Error (MAE) and Mean Absolute 3

Relative Error (MARE) as follows, 4

M

yy

ME

M

m

simm

realm

1

)(

(23) 5

M

yy

MAE

M

m

simm

realm

1 (24) 6

M

yyy

MARE

M

m

realm

simm

realm

1

)(

(25) 7

where m indicates the mth sample, M is the total number of samples, realmy is the mth real data sample (y 8

can be the acceleration, velocity or position), and simmy is the mth simulation data sample. 9

To investigate the stability and characteristics of the car-truck heterogeneous traffic flow, a ring-10

road numerical simulation is conducted by placing 100 vehicles on a single-lane, flat, and ring road, with 11

the first vehicle following the end vehicle. In the simulations, the initial state of the traffic flow is 12

equilibrium state, add a small perturbation on the first vehicle, and then observe the perturbation 13

development in the platoon. Many studies adopted this simulation method to explore the fundamental 14

diagram (20, 21), shock wave (21, 22), linear stability (13, 23), and other properties (24) of traffic flow. 15

16

RESULTS ANALYSIS 17

Preliminary Data Analysis 18

Some preliminary analysis is conducted to obtain the general characteristics of the different car-truck 19

car-following combination types. First, the gap versus velocity relationship for each combination type is 20

explored. The CC combination has the smallest average gap, and the TT combination has the largest 21

average gap for the same velocity. When v = 17 m/s, the CT combination has the same gap with the TC 22

combination. The CT combination has a larger gap than the TC combination before v = 17 m/s, while the 23

TC combination has a larger gap after v = 17 m/s. Second, the desired velocities of cars and trucks are 24

also studied. The statistical results show that the truck has the maximum velocity 18 m/s, and the car has 25

the maximum velocity 30 m/s. The maximum velocity of the CT combination is around 18m/s which is 26

restricted by the maximum velocity of the leading truck. The maximum velocity found for the CT 27

combination is around 23 m/s with the combined impacts of the slow-moving follower truck and fast-28



moving leader car. The CC and TT combinations respectively had the maximum velocities of 30 m/s and 1

16 m/s. Third, the significant difference in response time for the different vehicle types suggested by (25) 2

can be measured using the method introduced by Aghabayk et al. (9). The average response time of the 3

truck drivers is found to be 2.1 s, and the average response time of the car drivers is around 1.3 s. 4

Calibration and Evaluation Results 5

Table 1 lists the calibration results of the heterogeneous IDM. The calibrated desired velocities of the 6

four combinations are similar to the preliminary statistical results. The TT combination has the largest 7

jam spaces ( 0ns and 1

ns ); while the CC combination has the smallest jam spaces. The order of the safe 8

time headway n is CC < CT < TC < TT. Cars have larger deceleration bn than trucks. The acceleration 9

exponents are same for all four types of combination. The values of the error indexes of acceleration, 10

velocity, and location for each combination type can also be found in Table 1. The acceleration 11

simulation errors (MARE) for the four combinations are all less than 15%, and the velocity and location 12

simulation errors (MARE) are all less than 10%. Therefore, the calibration results can reflect the car-13

following behavior characteristics of the four combinations in the I-80 NGSIM vehicle trajectory data. 14

TABLE 1 Calibration and evaluation results of the four combinations 15

Variables CC CT TC TT

Calibration Results

an (m/s2) 1.01 1.03 0.78 0.74

Vn (m/s) 27 19.3 20.6 17.7

0ns (m) 0.85 1.35 1.11 1.53

1ns (m) 0.19 0.27 0.12 0.36

n (s) 1.2 1.4 1.8 2.0

bn (m/s2) 2.26 2.12 1.70 1.61

n 4 4 4 4

Acceleration Evaluation

ME 0.84 0.28 -0.03 0.13

MAE 1.17 1.67 1.24 1.16

MARE 0.11 0.13 0.12 0.1

Velocity Evaluation

ME -0.25 0.15 -0.23 -0.04

MAE 1.31 1.62 1.2 0.75

MARE 0.08 0.09 0.07 0.05

Location Evaluation

ME 2.44 3.00 -1.93 1.23

MAE 9.71 9.6 5.75 5.61

MARE 0.05 0.06 0.03 0.04


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this case, th

traffic flow i

-0.83, and SF

he TC and T

e perturbatio

(the car-truc

ends after th

No.71 (the CC

ffic flow is

portions of th

esponding SF

bility functio

2

nd

he

0,

ur

ks

T

he

is

Ftt

T

on

ck

he

C

a

he

SF

on


Yang

valu11

Equa12

FIGU13

zero 14

area 15

Whe16

the 17

How18

four 19

func20

12

13

FIG15

16

Fun16

We e27

equa28

CC, 29

critic30

fund31

show32

besid33

FIGU34

inter35

incre36

heter37

g, Jin, Ran, P

es of the car

ation (21), th

URE 3 (a) ar

stability fun

above the li

en the propor

traffic flow

wever, when

combination

ction values a

GURE 3 Sc

ndamental D

explore the f

ation (11). W

CT, TC and

cal factor to

damental diag

wn in FIGUR

de the legen

URE 4 (b) th

rsection and

ease with th

rogeneous tra

Pu and Yang

r-truck hetero

he proportion

re the stabilit

nction values

ine and the s

rtion of the C

is always u

v 1 m/s a

ns are stable.

are always les

(a)

catter plots o

Diagram A

fundamental

When examin

TT, we find

determine th

grams with

RE 4 (a). FI

nds are the p

he curves hav

increases af

he increasing

affic flow mo

ogeneous tra

ns of CC and

ty function v

s. The neutra

stable area un

CC combinat

unstable reg

as shown in

. Thus, no ne

ss than zero.

of the stabili

The case v*

Analysis

diagram cha

ing the fund

d that the diff

he shape and

similar Pd v

GURE 4 (b)

roportions o

ve an interse

fter the inter

g of Pd, wh

oves towards

affic in the sp

d TT can dete

alues, and th

al stability li

nder the line

tion is more

gardless of t

FIGURE 3

eutral stabilit

ity function

= 1 m/s. (b)

aracteristic of

amental diag

ference betw

d location of

values locate

) illustrates t

of the four co

ction, and th

rsection. Fur

hich reveals

s the high flo

pace of the C

ermine the pr

he line is the

ne splits the

e. (0, 0.22) a

than 0.5 or t

the proportio

(b), the four

ty line exists

of the car-tr

) The case v*

f the car-truc

gram curves

ween the CC a

f the overall

e close to ea

the cluster fo

ombinations

he flow decre

rthermore, th

s that the m

ow and high d

CC and CT p

roportions of

neutral stabi

quadrant in

and (0.5, 0) a

the CT comb

ons of the o

r SF are all n

s in this case

(

ruck heterog

* = 6 m/s.

ck heterogene

for all propo

and TT comb

fundamental

ach other on

for the case P

CC, CT, TC

ases with the

he flow rate

most unstabl

density area w

proportions (

f CT and TC

lity line conn

nto two areas

are the two c

bination is m

other three c

negative, wh

since the ov

(b)

geneous traf

eous traffic f

ortion combin

binations Pd

l diagram. It

n the flow-de

Pd = 0.2 and

C and TT re

e increase of

and critical

le point of

with the incr

13

(According t

C). The dots i

necting all th

s, the unstabl

critical point

ore than 0.22

combination

hich means a

verall stabilit

ffic flow. (a)

flow based o

nations of th

= Pcc - Ptt is

t is found tha

ensity plot a

d the number

espectively. I

Pcc before th

l density bot

the car-truc

easing of Pd.

3

to

in

he

le

s.

2,

s.

all

ty

on

he

a

at

as

rs

In

he

th

ck

.


Yang

2 3

4

FIG6

7

Sho7

The 16

diffe17

prop18

view19

boun20

traff21

truck22

flow23

shoc24

17

18

FIG20

21

g, Jin, Ran, P

GURE 4 Fu

ck Waves A

propagation

erences betw

pagations in t

wpoint. The a

ndaries betw

fic flow is th

k heterogene

w because the

ck wave trave

x

GURE 5 Pro

Pu and Yang

(a)

undamental d

diagr

Analysis

speed of the

ween cars and

the pure car t

arrow indicat

ween two diff

he same durin

eous traffic fl

e truck needs

els faster in th

(a)

opagation of

flow

)

diagrams of

ams. (b) Fun

e shock wav

d trucks. FIG

traffic flow a

tes the propa

ferent states

ng propagati

flow and the

s more time t

he pure car tr

)

f the shock w

w. (b) The ca

f the car-tru

ndamental d

ve is not unif

GURE 5 (a)

and the car-tr

agation trajec

of traffic flo

ion. Howeve

average slop

to react to th

raffic flow th

t

wave from th

ar-truck het

ck heterogen

diagram clus

form through

and (b) resp

ruck heterog

ctory of a per

ow. The slop

er, the slope

pe decreases

he speed chan

han in the car

x

he trajectory

terogeneous

(b)

neous traffic

ster at Pd =0

h the platoon

pectively dem

eneous traffi

rturbation wh

pe of the sh

changes wh

in the car-tr

nge of its pre

r-truck hetero

(b)

y viewpoint.

traffic flow.

c flow. (a) F

0.2.

n due to the r

monstrate the

ic flow from

hich gives th

hock wave in

en going thr

ruck heterog

eceding vehi

ogeneous tra

t

. (a) The pur

.

14

undamental

response tim

e shock wav

the trajector

he shock wav

n the pure ca

rough the car

eneous traffi

icle. Thus, th

affic flow.

re car traffic

4

l

me

ve

ry

ve

ar

r-

ic

he

c


Yang

8

prop9

traff10

and 11

wave12

heter13

traff14

9

10

FIG12

13

13

CON14

The 23

pape24

car-f25

car-f26

and 27

car-f28

heter29

char30

shoc31

28

using29

of th30

Furth31

espe32

g, Jin, Ran, P

FIGURE

pagation in bo

fic flow, and

30 trucks. G

e takes aroun

rogeneous fl

fic flow can s

GURE 6 Sim

NCLUSION

traffic flow

er, the focus

following-tru

following mo

use the NGS

following mo

rogeneous tr

acteristics of

ck wave.

We deriv

g the ring-ro

he four comb

hermore, the

ecially with th

Pu and Yang

E 6 illustrate

oth pure car

the traffic flo

iving the sam

nd 250 s to p

low, it takes

slow down th

(

mulation of t

(

N

mixed by ca

is the analys

uck (CT), tru

odels. We ex

SIM data to c

odels can ref

raffic flow.

f the car-truc

ve the linear

ad simulation

inations, are

e individual

he increasing

es the shock

and mixed c

ow in FIGUR

me perturbati

propagate to

around 400

he shock wav

a)

the propaga

(b) The car-t

ars and trucks

sis of the imp

ck-following

xtend the orig

calibrate mod

flect the car-

Based on

ck heterogene

stability crit

ns. Two criti

found to det

l stability fu

g of equilibri

k wave propa

car-truck flow

RE 6 (b) is th

ion to the tw

the last veh

s. The simu

ve.

ation speed o

truck hetero

s is the typic

pacts of the f

g-car (TC) an

ginal Intellig

del paramete

-following dy

the calibrat

eous traffic f

erion of the c

ical factors, t

termine the s

unction vari

ium velocity,

agation obse

w. The traffic

he car-truck h

wo types of tra

icle in the pu

ulations illus

of the shock

ogeneous tra

cal heterogen

four types of

nd truck-follo

gent Driver M

ers. The eval

ynamics of t

ted car-follo

flow: the line

car-truck het

the individua

stability of th

es with the

, TC become

erved in a s

c flow in FIG

heterogeneou

affic flow at

ure car traffi

strate that the

(

wave. (a) Th

affic flow.

neous traffic

f combination

owing-truck (

Model (IDM)

luation result

the four com

owing mode

ear stability, f

terogeneous t

al stability fun

he car-truck h

e increasing

es less stable

simulation of

GURE 6 (a) i

us traffic mix

the same pla

ic flow; whil

e car-truck h

b)

he pure car

flow in real

n, car-follow

(TT), using h

) to a hetero

ts show that

mbinations in

el, we explo

fundamental

traffic flow a

nction and th

heterogeneou

of equilibri

than CT. In

15

f perturbatio

is the pure ca

xed by 70 car

ace, the shoc

le in car-truc

heterogeneou

traffic flow.

traffic. In thi

wing-car (CC

heterogeneou

ogeneous form

the calibrate

n the car-truc

ore the thre

diagrams an

and validate

he proportion

us traffic flow

ium velocity

addition, car

5

on

ar

rs

ck

ck

us

.

is

C),

us

m

ed

ck

ee

nd

it

ns

w.

y;

rs



and trucks can both stabilize and destabilize the car-truck heterogeneous traffic flow and their effects 1

depend on their roles (leading vehicle or following vehicle) in car-following and the equilibrium velocity 2

of the car-truck heterogeneous traffic flow. This conclusion is more sophisticated than the previous 3

studies (10, 26) in which the truck or car only has one single effect (stabilize or destabilize) on the traffic 4

flow. We also investigate the impact of the proportions of the four combination types by simulation. The 5

results show that the proportions of the four combinations may not have significant influence on the 6

stability of traffic flow with low equilibrium velocity (e.g. on an extremely congested road); however, in 7

most cases, the neutral stability lines can be found for the car-truck heterogeneous traffic flow. 8

Fundamental diagrams of the car-truck heterogeneous traffic flow are determined by the 9

distance headways and the proportions of the four types of combination. The simulations further reveal 10

that the fundamental diagrams with the same proportion difference between the CC and TT 11

combinations cluster together. The flow rate and the critical density increase with the increasing of the 12

proportion difference between the CC and TT. Finally, the shock wave analysis indicates that trucks can 13

slow down the propagation speed of the shock wave due to their longer response times. 14

15

ACKNOWLEDGMENT 16

This work was supported by the National Natural Science Foundation of China (Grant No. 17

51278429 and Grant No.50908195) and the National High Technology Research and Development (863) 18

Program of China (Grant No. 2011AA110403). 19

20

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