Defining Highway Node Acceptance Capacity (HNAC): Theoretical Analysis and Data Simulation

Abstract

A new concept of Highway Node Acceptance Capacity (HNAC) is proposed in this paper inspired by a field data observation. To understand HNAC in microscopic view, boundary condition of successful merging is found using car-following behaviours and lane-changing rules, which could also explain traffic oscillations. In macroscopic view, linear positive relationship between HNAC and background traffic volume is obtained based on moving bottleneck. To determine the explicit form of the relationship, data simulation considering car-following behaviours and traffic flow theory is used. In the results, the synchronization phenomenon of oscillation in on-ramp (with respect to main road) and intersected road is found. The explicit equation of HNAC is determined based on standard deviation and correlation coefficient analysis, and also proved to be accurate with model validation, which is helpful in studies related to propagation mechanism of traffic emergencies on highway network.

Full text

Research Article
Defining Highway Node Acceptance Capacity (HNAC):
Theoretical Analysis and Data Simulation
Xingliang Liu , Jinliang Xu , Yaping Dong, Han Ru, and Zhihao Duan
Highway School, Chang’an University, Xi’an, China
Correspondence should be addressed to Jinliang Xu; xujinliang@chd.edu.cn
Received 30 August 2019; Revised 1 November 2019; Accepted 28 November 2019; Published 15 January 2020
Academic Editor: Alain Lambert
Copyright © 2020 Xingliang Liu et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new concept of Highway Node Acceptance Capacity (HNAC) is proposed in this paper inspired by a eld data observation. To
understand HNAC in microscopic view, boundary condition of successful merging is found using car-following behaviours and
lane-changing rules, which could also explain trac oscillations. In macroscopic view, linear positive relationship between HNAC
and background trac volume is obtained based on moving bottleneck. To determine the explicit form of the relationship, data
simulation considering car-following behaviours and trac ow theory is used. In the results, the synchronization phenomenon
of oscillation in on-ramp (with respect to main road) and intersected road is found. e explicit equation of HNAC is determined
based on standard deviation and correlation coecient analysis, and also proved to be accurate with model validation, which is
helpful in studies related to propagation mechanism of trac emergencies on highway network.
1. Introduction
In our previous work [1], trac ow eld data was collected on
City Ring Road and Lianhuo Expressway, Xi’an. During the
process of data collection, an interesting phenomenon was
observed on Baqiao Interchange (K450 + 100 of Lianhuo
Expressway, G30). Taking northbound trac merging from City
Ring Road to Lianhuo Expressway as an example (Figure 1),
trac condition of this highway node varies between morning
and aernoon peak hours. In Figure 1(a), trac condition at
8:30 a.m. was depicted. e trac ow in downstream Lianhuo
Expressway had trac volume of 1830 veh/h ∗ l and density of
26 veh/km ∗ l, respectively. ese two values in upstream City
Ring Road were 465 veh/h ∗ l and 11 veh/km ∗ l, and merging
volume of on-ramp was 330 veh/h ∗ l. It could be seen, all sections
mentioned above were in free ow. is is mainly because during
morning peak hour, people go into the city, which means going
south. During aernoon peak hour, people go out northbound,
which leads to a dierent situation. In Figure 1(b), trac
condition in downstream Lianhuo Expressway remained the
same (volume of 1830 veh/h ∗ l and density of 26 veh/km ∗ l), but
congestion was caused in upstream City Ring Road (volume of
272 veh/h∗l and density of 91 veh/km ∗ l) with the increase of
merging trac volume (665 veh/h ∗ l) in on-ramp.
Combining the example given above, some terms are
dened here. e rst term is “main road”, dened as road
section with trac merging in of a highway interchange (like
Lianhuo Expressway in the example). e second term is
“intersected road”, dened as road section with trac
diversion (like City Ring Road in the example). e third
one is “highway node”, which is a part of an interchange,
consists of main road, intersected road and the on-ramp
(with respect to main road) connecting them. Inspired by
the phenomenon observed above, a problem could be
presented here. If there was an oscillation downstream the
main road, whether a specic upper limit exists, when
exceeded by merging trac volume, trac ow of intersected
road will be aected. is is crucial in research on propagation
mechanism of trac emergencies in highway network, which
is very helpful to large scale evacuation and rescue. In 1995,
Daganzo [2] provided a quite detailed study on trac
merging problem using cell transmission model. is is a
classic research followed by many scholars, and there are still
some works remaining to be done, to further improve this
eld. e method in his work mainly focused on the
topological form of the junction and the logic between
adjacent nodes, scilicet a partial mathematical method.
erefore, in consideration of practical application, some
Hindawi
Journal of Advanced Transportation
Volume 2020, Article ID 8939621, 16 pages
https://doi.org/10.1155/2020/8939621

Journal of Advanced Transportation2
practical eorts could be done, including microscopic
relationship among vehicles and trac ow’s physical
characteristics in merging area, e.g.
us, to further ameliorate Daganzo’s classic work, concept
of Highway Node Acceptance Capacity (HNAC) is provided
here. HNAC means the upper limit of trac volume merging
into the main road from the intersected road through the
on-ramp. If this value was exceeded by merging volume, trac
ow in intersected road will be aected, showing obvious oscil-
lations and further forming congestion. is parameter shares
the same unit with trac volume, scilicet
veh/h⋅l
. In micro-
scopic view, study of HNAC is directly related to a specic
vehicle’s driving behaviour when merging into target lane in
main road. Mentioning problems related to microscopic driv-
ing behaviour, car-following models and lane-changing rules
become crucial because they are basic theories in this eld,
and they will be studied in Section 3. In macroscopic view,
essence of studying HNAC is calculating the volume of suc-
cessfully merging vehicles. Typically, a successful merging
behaviour depends on enough large spacing between the lead-
ing vehicle and the follower in target lane, and the volume of
large spacing is related to HNAC. In real trac stream, spacing
between adjacent vehicles is not homogeneous. e existence
of slow vehicles will cause moving bottlenecks and reform the
trac stream. erefore, in macroscopic study of HNAC, the-
ories of moving bottleneck should be taken into consideration,
and these contents will be studied in Section 4. Furthermore,
literature review will be stated in Section 2. e process of data
simulation is given in Section 5, and Section 6 provides the
results, including the synchronization of oscillation in on-ramp
and intersected road, standard deviation and correlation coef-
cient analysis, and the explicit equation of HNAC. Section
7 provides conclusions and some discussions.
2. Literature Review
As depicted in Section 1, HNAC was dened as the upper limit
of trac volume merging into main road through on-ramp.
If this value was exceeded by merging volume, trac ow in
intersected road will be aected. is concept was rstly
mentioned in Kerner’s classical three phase trac theory work
in 2004 [3], by stating that “in free trac of merging area, there
might be random highway capacities which depend on the
ow rate of on-ramp”. is statement is related to the
phenomenon observed in Section 1. However, unfortunately,
in Kerner’s abovementioned and following works [4–6], clear
denition of HNAC and explicit formulas were not provided,
either in other followers’ works [7, 8].
is paper began with the denition of HNAC in Section 1.
As depicted in previous part, car-following model is rstly
considered. e car-following model developed by Newell [9]
showed some signicant importance and practicality in related
works. is model clearly depicts the relationship between the
leading vehicle’s and the follower’s moving condition. It implies
the follower will change its velocity and spacing depends on
the leader’s driving condition with a small spatio-temporal
hysteresis. Besides, compared to traditional car-following
models, the dynamic process of acceleration and deceleration
are considered as instantaneously completed. erefore, this
model showed some advantages in explicit trac ow mac-
roscopic modeling compared to those traditional ones [10–
15]. Moreover, Newell’s model showed the same accuracy and
succinct formation with a dierent logic more suitable to
modern trac. Following theoretical and empirical studies of
classic car-following model, relaxation phenomenon was dis-
covered, showing that driver will accept shorter spacing and
adjust it to more comfortable value in process of lane-changing
or merging behaviour [16–20]. is concept and phenomenon
North
Li
anhuo expressway
K450 + 100
Q
O
O
k
k
v
City ri
ng expressway
Trac
condition of
8:30 a.m.
(a) (b)
F 1:e phenomenon observed on Baqiao interchange. (e red arrow depicts moving directions of the vehicles, green rectangles depict
free ow in certain sections, red rectangle depicts congested ow, and orange rectangle depicts the increase of trac volume in on-ramp). (a)
Trac condition in Baqiao interchange at 8:30 a.m. (b) Trac condition in Baqiao interchange at 16:00 p.m.

Journal of Advanced Transportation 3
could be used as an assumption in microscopic modelling.
Trac oscillations and stop-and-go waves are common phe-
nomena caused by lane-changing behaviours [21–23], which
could be theoretically explained using car-following models.
erefore, this could be used as inspection standard in this
paper. In Laval’s related study [24], a parsimonious theory
explaining the appearance and transformation of trac oscil-
lation was provided based on Newell’s model, and, timid and
aggressive driving behaviors were concluded, which could be
used in macroscopic modelling of this paper. Set aside the
theoretical researches, empirical data were also used in model
formulation. In Chen’s work [25], a behavioral car-following
model was developed based on empirical trajectory data using
NGSIM dataset, which revealed the dynamic behaviour prole
of drivers experiencing trac oscillations. e data charac-
teristics in this work provided some references in data simu-
lation. In recent works, the researches of car-following model
were basically related to trac control strategy and automated
vehicle [26–28]. Among them, the work by Han provided a
novel breakdown probability model based on extending
Newell’s model, and a trac control method to obtain uniform
spacing was developed considering low passing rate of
connected automated vehicle technology.
Based on the concept of HNAC depicted in Section 1,
lane-changing rule and model are problems which could not
be ignored or avoided in microscopic modelling in this work.
e driver’s decision to change lanes derived from his or her
answers to three questions, whether it is possible, necessary,
and desirable to change lanes [29]. In his work, structure of
the driver’s decision process before changing lanes was
modelled. It has been regarded as a procedural and hierarchy
basis in lane-changing models. Following lane-changing rules,
the method of cellular automata was used in model formation
[30, 31]. In anterior work, eects of various rules of lane-
changing on characters of trac ow were studied. e results
showed that most ecient rules would be those allow fast
vehicles to travel as fast as possible without sacricing the total
throughput. In the latter work, a general scheme of lane-
changing rules was proposed based on summary of dierent
approaches. In this work, Wagner’s gap rules [32] were
developed and realistic lane-changing rules were obtained. In
the researches of lane-changing model, data simulation was
frequently used. e classic study by Hidas [33] should be
noticed. e detailed and inspirational lane-changing and
merging algorithms were presented in this work, indicating
that forced and cooperative lane changing are essential, which
could produce realistic volume-velocity relationships during
congested conditions. In 2006, a statement that lane-changing
behaviours were strictly related to moving bottleneck and
trac volume reduction was proved by Laval [34]. In this
work, the mechanism mentioned above was explained by a
model that tracks lane-changing vehicles precisely. Besides,
two phenomena previously thought to be unrelated were
combined by this simplied parameters model, passing rate
drop of bottleneck at the beginning of congestion, and the
relationship between moving bottleneck velocity and its
capacity. Safety criteria should be regarded as a boundary
condition in lane-changing models. In Kesting’s work [35], a
general lane-changing model was proposed for discretionary
and mandatory lane-changing behaviours. e essence of
safety criteria was explored in his paper, in other words, the
merging vehicle should keep safe distance with the leader and
the follower in targeted lane.
As depicted in previous part, the theories of moving
bottleneck could provide a reasonable explanation of vehicle
distribution characters in real trac ow. is phenomenon
was systematically studied by Gazis [36], and the widely used
model was developed by Newell [9]. In their work, the
denition of moving bottleneck was given, and the model of
passing rate, queue behaviour, moving queue growth rate were
also provided. ese results were widely used and developed
in related researches [28, 37–39].
3. Microscopic Modelling of HNAC
In microscopic perspective, research on HNAC could be
related to specic situation stated as below. Imagine that, a
vehicle traveling from on-ramp at a stable velocity, having the
demand to enter the main road. However, the spacing between
the leading vehicle and following vehicle in target lane does
not satisfy the microscopic lane changing condition.
erefore, the vehicle ready to enter the main road has to
decelerate and wait for a proper chance. is situation might
lead to congestion in on-ramp and further aects the trac
condition in intersected road. is situation could be
interpreted as: the volume sent by on-ramp exceeds the
HNAC of the main road.
From the simplied description in previous context, the
microscopic modelling should consist of two important parts,
car-following behaviour and lane-changing condition.
3.1. Car-Following Behavior. For analytical tractability,
vehicles are assumed to follow the classic car-following model
by Newell [9]. is model provides two crucial contents, the
relationship between the leading vehicle and following vehicle’s
trajectory in time and space, and, the relationship between a
vehicle’s spacing and velocity. is model formed the basic
laws in car-following research areas, adopted and developed
by many researchers [18–20, 24, 25, 28].
In Newell’s model, highway is treated as a homogeneous
carrier, and spacing is linear related to travel velocity. When
the leading vehicle changed its operation velocity, the follower
would change its speed according to the leader’s, but there is
a hysteresis that exists in time and space. is relationship
could be depicted in Figure 2.
As depicted in le side of Figure 2(a), the trajectories of
leading vehicle and the follower’s are represented by
n−1()
and
n()
respectively. At a time point, the leader accelerated
from
vn−1
to
v
n−1
(
vn−1 <
v

n−1
). However, the follower did not
change its velocity immediately. Instead, aer a time period
𝜏n
, when the spacing of the follower increased from
𝑆n
to
𝑆󸀠
n
,
the follower also accelerated to
v󸀠
n
. e parameters
𝜏n
and
𝛿n
related to hysteresis phenomenon are independent from
vn
,
they depend on the driver himself. e trajectory in car-fol-
lowing model and the relationship between spacing and veloc-
ity depicted in Figure 2 could be expressed as below.
(1)
n=
n+vnn,

Journal of Advanced Transportation4
lane-changing vehicle enters the target lane, it should keep a
“safe distance” with the leading vehicle and the following one.
It is known that the “safe distance”, referred to as spacing, is
related to operation velocity of the vehicle. erefore, equation
(3) should be cooperated in microscopic modelling.
To simplify the modelling process, assuming that merging
vehicle completes its lane changing process in a short period
without acceleration or deceleration, soon aer it enters the
target lane, the vehicle will adjust its velocity according to the
leader’s driving condition, and the follower will also adjust its
velocity according to the lane-changing vehicle’s driving con-
dition. is process is depicted in Figure 3.
At time point
𝑡0
, a vehicle traveling from on-ramp with
velocity
vin
shows up at any position beside the target lane. In
target lane, the follower at position
𝑥n
is traveling at velocity
vn,
and the leader at position
𝑥n−1
is traveling at velocity
vn−1
.
At
𝑡0
, the distance between each vehicle are
𝐿n
and
𝐿n−1
. At
time point
0+ Δ
, the vehicle has just nished its lane-
changing process. e distance between each vehicle are
𝐿󸀠
n
and
𝐿
n−1
. In this process,
𝐿
n−1
and
𝐿󸀠
n
could be calculated using
Equations (4) and (5).
(4)

n−1 =
n−1 − Δ(
v
in −
v
n−1),
(5)
󸀠
n=
n− Δ(
v
n−
v
in).
In the two equations shown above,
𝛿n
could be dened as the
spacing in totally congested trac, and
𝜏n
could be dened as
the time consumption by which the kinematic-wave travels a
distance of
𝛿n
[37]. As depicted in Figure 2(b), there is a lim-
itation
vn∈ [0,
v
free]
, which means equation (1) only estab-
lished in congested trac ow (the right side of volume-density
curve). To a steady trac ow, all of the vehicles travel at
nearly the same velocity
v
, then we get:
It should be noticed, in real restricted trac ow, the spacing
varied due to the personality of the drivers. Aggressive drivers
tend to choose smaller spacing (
n,
n
) and vice versa, timid
drivers tend to have loose spacing. It should be noted, car-
following model is used in restricted ow, in which the fol-
lower should change its running velocity and spacing according
to the leader’s. In traditional trac ow theory, there is a
boundary concentration
kb
between restricted ow and free
ow [40]. In free ow condition, the driver operates in free
ow velocity
vfree
, and maintains the spacing larger than
n,free =
n+vfreen
owning to the low concentration. When
the concentration equals to
𝑘b
, the trac ow is about to enter
the restricted condition. At this moment, the driver in free
ow gets its smallest spacing equal to
𝑆n,free
.
3.2. Microscopic Modelling Based on Lane-Changing Rules. As
described in previous part, the microscopic modelling of
HNAC could be transferred to a problem related to lane-
changing rules. To be precise, to nd a boundary spacing
between two adjacent vehicles in targeted lane of the main
road for the vehicle from on-ramp at a specic velocity to
enter.
Lane-changing rules are rst developed by P. G. Gipps
in 1985 [29], with the perfections of the followers
[21, 30, 31, 33, 35]. It is commonly accepted that aer the
(2)
n+
n=
n−1()−
n.
(3)
=+
v
,
=
1


∑
=1
k,=1


∑
=1
k
.
Distance
Time
(a)
t
xn–1(t)
x
vn
vfree
Velocity
(b)
Sn
Sn
τn
τn
δn
δn
vn
vn
v
xn(t)
Spacing
Sn
vn
Sn
Sn
F 2:Schematic diagram of Newell’s car-following model. (a) e trajectories of leading vehicle and the follower’s. (b) e relationship
between spacing and velocity in Newell’s model.
xnLn
vn
vin
vn–1
Ln–1 xn–1 t0
t0+∆t
F 3:Schematic diagram of Lane-changing process.

Journal of Advanced Transportation 5
merging vehicle to enter the main road at
vin
, then this spacing
could be called an “eective spacing” (ES). e essence of
HNAC macroscopic modelling is to calculate the volume of
ES passed the entrance in a time unit, and determine the rela-
tionship between ES number (ESN) and trac volume
𝑄
in
main road.
4.1. HNAC Macroscopic Modelling in Congested Flow. As
depicted in previous part, congested ow refers to the right
side of volume-density curve. In this situation, the velocity of
each vehicle in trac ow is assumed to be the same,
v
. en
𝐿limit
could be reformed as below.
From equation (9),
limit =
in +
(v)>
(v)
, which means
there is no chance for merging vehicle to enter the main road,
and this is obviously unrealistic. e truth is, in congested
ow, even the velocity is treated as equal, the spacing diers
from each other due to the variance of
𝛿
and
𝜏
. To aggressive
drivers, they tend to choose smaller
𝛿
and
𝜏
, as for timid ones,
they tend to choose larger values [24], which could be
expressed as below.
erefore, assuming a vehicle platoon’s volume equals to
𝑛
.
Among them, the ratio of ESN derived from timid driver is
assumed as
𝑝
, then the length of the vehicle platoon could be
calculated using equations (12). e time consumption and
trac volume of passing a specic entrance could be calcu-
lated by equations (13) and (14).
en, combining equations (12)–(14), the ESN could be
expressed as below,
which means
ESN
is linear positive correlated to trac vol-
ume. But, it should be claried that the model given above is
an ideal situation. Here provides an example to show the real-
ity in some degree, shown in Figure 5. A platoon consists of
four vehicles is going to pass an entrance located at
=0
, and
they are all operated by timid drivers with velocity
v
. e initial
spacing
tim =
limit
. At the time point
=
0
, a merging vehicle
from on-ramp begins entering the target lane, and adjusts its
velocity from a lower value to
v
. In order to keep the spacing
𝑆tim
, the three followers have to decelerate and lead to an oscil-
(9)

limit
=2 − 
vn−1
−
vn
+
vin
+
vn

=
in ++
v
=
in +
(
v
).
(10)
agg =
agg +
aggv<
limit,
(11)
tim =
tim +
tim
v
>
limit.
(12)
=(−1
)(
v
)=(−1
)(+
v
),
(13)
=
v,
(14)
=

=
nv
.
(15)
ESN =
(−1)=
(
QT
−1)= QL
v−1,
With the lane-changing rules in the 2nd paragraph of this
section, and cooperating equation (3), the boundary spacing
𝐿limit
between two adjacent vehicles in targeted lane could be
obtained.
e modelling of HNAC is strongly related to the boundary
spacing
𝐿limit
. Imaging that, the vehicle traveling from the
on-ramp is ready to merge into the target lane in main road.
However, the spacing of the trac ow in target lane of the
main road is smaller than
𝐿limit
, meaning there is no chance
for this vehicle to merge in. erefore, this vehicle from
on-ramp has to slow down or even stop to wait a proper
chance. is is the situation for one single merging vehicle,
just like an unsuccessful interpolation of two gearwheels. In
real trac condition, merging trac from the on-ramp is con-
tinuous, indicating that if the volume of spacing satised
𝐿limit
in main road trac ow is not enough for the merging trac
volume from on-ramp to consume, a congestion might form
on the on-ramp and further aect the intersected road.
It should be noticed that the merging process varies
depending on dierent relationships among
vn
,
vn−1
, and
vin
.
To further study, six situations are established and the merging
process in each situation is shown in Figure 4.
In all of the 6 situations, the merging vehicle doesn’t
need to change its velocity only in situation (2). When
vn−1 >vin >vn
, the distances among three vehicles will get
larger, and this is the most idealized condition. From situation
(1) and (4), when
vin >vn
, whatever the relationship between
vn
and
vn−1
, the driver has to change the velocity according to
the leader, which means the merging vehicle will join the
downstream platoon and shows no aection on upstream traf-
c. From situation (3), (5), and (6), when
vin <vn
, it is impor-
tant to notice that an oscillation shows up with a transmission
speed equal to
/
, which means the merging vehicle forms a
moving bottleneck that aects the upstream trac. Besides, it
also proved that the lane-changing behaviour will lead to some
trac oscillations, that is, if intersected road is aected when
the transferred volume exceeds HNAC, it is important to dis-
tinguish the aection of oscillations and from the on-ramp,
scilicet HNAC.
4. Macroscopic Modelling of HNAC
In this part, the range of macroscopic study should be rstly
dened. Microscopic study mentioned in Section 3 depicted
a research scope concentrate on single vehicle’s moving con-
dition. Compared to concept of microscopic study, macro-
scopic modeling concentrates on trac ow in a certain
highway node, especially the trac ow in main road. In
Section 3,
𝛿
and
𝜏
were adopted as average values, which
means
𝐿limit
only depends on velocity. If a spacing in target
lane is larger than
𝐿limit
, which means that it’s eective for a
(6)

n−1 ≥
in =+⋅
v
in,
(7)
󸀠
n≥
n=+⋅
v
n,
(8)
limit =2 − Δ(
v
n−1 −
v
n)+(
v
in +
v
n).

Journal of Advanced Transportation6
4.2. HNAC Macroscopic Modelling in Free Flow. e vehicle
distribution character in free ow is dierent from congested
ow. Due to the existence of moving bottleneck, it is
unreasonable to regard every vehicle’s velocity as free ow
speed
vfree
. In the part without moving bottleneck, each vehicle
travels with velocity
vfree
. Taking the description in last part
lation. e actual spacing passing the entrance were
𝑆1
,
𝑆2
, and
𝑆3
. From the gure depicted this process given below,
it is easy to conclude that
1=
3=
limit
,
2<
limit
. erefore,
the actual
ESN
turns from three to two. at is to say, in real
trac ow, the actual
ESN
will be lower than the value given
by equation (15).
t
t
tt
t
t
δ
δ
δ
τ
(1) vin > vn–1 > vn(2) vn–1 > vin > vn
(3) vn–1 > vn > vin
(5) vn
> vin
> vn–1 (6) vn
> vn–1
> vin
(4) vin > vn > vn–1
xx
x
n–1 xn–1
vn–1
vn–1
vn–1
vn–1
vn–1
vn
vnvn
vn
Oscillation
Oscillation
Oscillation
vin
vin vin
vin
Sn
Sn
L′n–1
Ln–1
Ln–1
Ln
vn–1 Ln–1
vin
vn
Ln
vn
t0t0 + ∆t
t0 + ∆tt0 + ∆t
t0 + ∆t
t0 + ∆t
t0 + ∆t
L′n–1
L′n
Ln
L′n
L′n–1
L′n
vin
xn
xx
x
n–1
xn–1
xn
xx
x
n–1 xn–1
xnxn
xn
xn
t0
t0t0
t0
t0
Ln–1
τ
Ln
Ln–1
L′n–1
L′n
Ln
Ln–1
L′n–1
L′nτL′n
Sn
Ln
L′n–1
F 4:Schematic diagram of the Merging process in dierent situations.

Journal of Advanced Transportation 7
In moving bottleneck region,
vb<vfree
, then we have:
For simplication, discrepancy among the drivers is ignored.
erefore, it is reasonable to assume that ES only exists in the
part of free driving. In the part of moving bottleneck, every
spacing does not satisfy the merging condition. Assuming
that the length of a stable moving bottleneck is
𝐿b
, the
distance between the leading vehicle and the entrance is
𝐿free
. In a time period
𝑇
, the last vehicle in moving bottleneck
passed the entrance, and the background trac ow param-
eters counted by an observer on the entrance is average vol-
ume
𝑄
, average speed
v
, average concentration
𝑘
. e
parameters in free ow part and bottleneck part are repre-
sented by
(
v
free,free)
and
(
v
b,b)
, respectively. en
ESN
could be calculated by equation (20).
In equation (20),
𝐿free𝑘free
represents the vehicle number in
free ow part, and
Tqr
stands for vehicle number escaping
from queue region and it entered the target lane. It is known,
b+free =(
QT
/)
, combined with equation (18), we have:
By equation (21), we can draw the conclusion that
ESN
is lin-
ear positive correlated to trac volume and linear inversely
proportional to trac concentration.
5. Data Simulation
According to results obtained from Section 4.1 and 4.2, HNAC
(ESN) is theoretically linear correlated to the background
trac volume of the main road. Besides, from analysis in the
last part of Section 4.1 and equation (21), it is known that the
real ESN in free ow is lower than the value calculated by
equation (15), and the equality relationship is only obtained
in congested trac ow. Moreover, in equation (21), ESN
could not be directly obtained because the values of parameters
𝐿free
,
𝑇
,
𝑞
, and
vb
are undetermined.
(19)
(
v
b)<(
v
free)<
limit =
in +
free.
(20)
ESN =
freefree +Tqr− 2.
(21)
ESN
=
freefree +QT

⋅Cq
(
vfree
−
vb
)
v
free
v
b
.
of Section 3.1, the distance between adjacent vehicles
𝐿n
and
𝐿limit
could be expressed as below.
In the part of a stable moving bottleneck, the trac condition
becomes complex. Moving bottleneck is caused by slow mov-
ing vehicles in free trac ow. Free running vehicles have the
demand to pass the slow one (moving bottleneck), and a queue
would form behind the bottleneck until it reached a stable
length. e trac characteristics in moving bottleneck could
be expressed by Figure 6 [36].
In queue region, it could be seen that the travel velocity
and concentration in all of the lanes are nearly the same,
expressed as
vb
and
𝑘b
. While in downstream, operation speed
basically equals to
vfree
. However, concentration
𝑘blo
in blocked
lane is smaller than
𝑘free
in upstream region, concentration
𝑘un
in unblocked lane is smaller than
𝑘blo
. is could be interpreted
as the vehicles escaped from queue region didnot spread in
each lane in average.
Moreover, when getting back to the calculation of HNAC,
the trac volume escaped from queue region and entered the
target lane should be taken into consideration. When the
capacity of downstream road is known as
𝐶
, the portion of
volume enters the target lane is
𝑞
, then the volume referred in
previous
𝑞r
could be calculated as below [38].
(16)
n>
free =+vfree,
(17)
limit =
in ++vfree =
in +
free.
(18)

r=Cq
(
1−
v
b
v
free ).
x
x
1
x
2
x
3
x
4
S1
t0
Sin
Stim
S2S3
t
Stim
τtim
δtim
Oscillation
F 5:Schematic diagram of the real trac ESN.
Speed Concentration
Upstream region
kfree
vfree vun
vblo
x
x
kb
vb
kblo
kun
Queue region Downstream region
F 6:Schematic diagram of the trac characteristics in moving
bottleneck.

Journal of Advanced Transportation8
last part of Section 3.2, the oscillation derived from exceeding
HNAC shows synchronization with the trac condition in
on-ramp. erefore, in Figure 7, a detector is also deployed to
observe the trac condition in on-ramp. To nd HNAC, the
merging trac volume also varies, shown in Table 2.
6. Results
6.1. e Synchronization of Oscillation in On-Ramp and
Intersected Road. To make the simulation feasible, number of
simulation groups should be relatively limited. erefore, 11
large groups were divided according to heavy vehicle mixing
ratio
𝑟
, seen in Table 2,
=5
% was settled as step distance.
In each large group, 10 sub-groups were divided according
to concentration in main road, based on step distance of
Δmain =4
. So, the concentration in main road varied from
relatively free ow to congested ow, representing the most
common trac conditions in daily life. T aking the data from
four groups when
=10
% as examples, which numbered 10–2–1
(
vmain = 54.9km/h,main = 34veh/km ∗l,main = 1868 veh/
h∗l,in = 100 veh/h∗l
), 10–2–3 (
vmain = 54.9km/h
,
main = 34
veh/km ∗l
,
main = 1868veh/h∗l
,
in = 300veh/h∗l
),
10–2–7 (
vmain = 54.9km/h
,
main = 34veh/km ∗l
,
main = 1868
veh/h∗l
,
in = 700veh/h∗l
), 10–2–10 (
vmain = 54.9km/h
,
main = 34veh/km ∗l
,
main = 1868veh/h∗l
,
in = 1000
veh/h∗l
), respectively. e synchronization of oscillation
in on-ramp and intersected road could be easily observed in
Figure 8.
e gures depicted in Figure 8 come from a simulation
group consisting of 18 simulation tests, in which the merging
trac volume
𝑄in
varies from 100
veh/h∗l
to 1800
veh/h∗l
(
Δin = 100veh/h∗l
). In Figure 8(a),
in = 100veh/h∗l
,
which is a relatively very low volume, caused no oscillation in
both the intersected road and on-ramp. In Figure 8(b),
in = 300veh/h∗l
, higher than previous volume, caused
some oscillations in intersected road. It should be noticed that
there is no oscillation that appears in on-ramp, considering
the analysis in the last part of Section 3.2, this is due to the
lane-changing behaviours in intersected road. In Figure 8(c),
in = 700veh/h∗l
, there is also no oscillation appears in
on-ramp. But the frequency of oscillation in the intersected
road is obviously larger than that in Figure 8(b), which means
the increasing number of lane-changing behaviours lead to
severe velocity dispersion in the intersected road. In
Figure 8(d),
in = 1000veh/h∗l
, when
∈[0s,1500s]
, the
trac condition in intersected road and on-ramp showed the
To determine
𝐿free
,
𝑇
,
𝑞
, and
vb
based on theoretical anal-
ysis is unrealistic and unworthy because they are assumed to
be random. erefore, to obtain the explicit form of equation
(21), method of data simulation is adopted in this paper. In
the process of data simulation, trac model in our previous
work [1, 40] is adopted, shown in equation (22), depicts the
relationship among trac volume
𝑉
, concentration
𝑘
and
heavy vehicle mixing ration
𝑟
.
In this paper, the simulation platform VISSIM is adopted,
which provides a high level of details and exibility in highway
design, vehicle performance and driver’s behaviours.
According to the user manual, three detailed aspects should
be noticed in the process of building HNAC (ESN) simulation
model [41]: vehicle movement at highway merging area, veloc-
ity adjusting area in highway on-ramp, and car-following
behaviours.
In aspect of vehicle movement in highway merging area,
routes of the merging vehicles from on-ramp should extend
beyond the whole weaving area, which ensures that vehicles
from highway on-ramp successfully complete their merging
movement. In consideration of second aspect, a velocity
adjusting area should be dened in simulation model to pro-
vide a space for the drivers to decelerate or accelerate to
vin
when approaching the entrance to identify if a suitable gap
(ES) was available in target lane. e length of this area
adopted in simulation model is 30 m setup 10 m upstream of
the entrance according to previous eld data collection. Taking
car-following behaviors, Wiedemann 99 model is adopted
since it is suitable for interurban trac, and the microscopic
parameters related to driving behaviours are adopted in Table
1 [42], based on the large amount eld data collected on West
3rd Ring Expressway, Beijing. Simulation model derived from
the contents above is shown in Figure 7.
As depicted in previous contents, the purpose of data
simulation is to obtain the explicit form of HNAC (ESN).
Considering the concept of HNAC in Section 1, the mani-
festation of exceeding HNAC is an oscillation in intersected
road conducted through on-ramp from the main road.
erefore, a detector should be deployed upstream from the
ramp in intersected road to observe the trac condition
aected by exceeding HNAC, shown in the below part of
Figure 7. In order to catch the oscillation, the background
trac input in intersected road should possess a relatively
lower robustness, which is the demarcation point of free ow
and congested ow (
vinter = 81.8km/h
,
inter = 33veh/km ∗l
,
inter = 2699veh/h∗l
) [1].
In the upper part of Figure 7, a detector is deployed down-
stream at the terminal of the ramp to supervise the trac
condition of the main road. Besides, as depicted in Section
4.2, HNAC (ESN) is linearly related to trac volume in main
road. erefore, the background trac input in the main road
should vary in trac volume and heavy vehicle mixing ratio.
e specic values of
𝑘main
,
vmain
, and
𝑄main
are shown in
Table 2 (the concentration varies regularly, the volume and
speed varies according to equation (22). As depicted in the
(22)
=
(,)=v
f
1 + exp  − t/−0.23 ∗  + 1.243
0.0025m∗−0.1538
.
T 1:Microscopic parameters related to driving behaviors.
Microscopic parameters simulation model Value
P1 Minimum headway (m) 1.75
P2 Maximum deceleration (m/s2) −4.4
P3 Reduction factor of desired safety distance (m) 0.67
P4 Maximum look ahead distance (m) 246.49
P5 Average standstill distance (m) 1.11
P6 Additive part of desired safety distance 1.81
P7 Multiple part of desired safety distance 3.05

Journal of Advanced Transportation 9
background trac condition. In all of the 110 simulation groups
in this study, the fuzzy boundary in every group could be found.
However, to determine the explicit equation of HNAC, some
further studies are still needed.
6.2. Standard Deviation and Correlation Coefficient Analysis.
From the analysis in Section 6.1, it could be known that with
the increasing of merging trac volume, velocity dispersion
in intersected road also increased gradually. When exceeding
the fuzzy boundary (HNAC), velocity dispersion in on-ramp
shows a steep increase and the synchronization also appears.
In mathematical language, standard deviation (Std) of the
trac velocity in intersected road is increasing with merg-
ing trac volume. In a moment around the steep increase
point, Std from the intersected road will equal to that from
same characters in Figures 8(b) and 8(c). When coming to
∈[1500s, 3500s]
, it is important to notice that velocity col-
lapse and velocity synchronization appeared in both the inter-
sected road and on-ramp, meaning that the merging trac
volume exceeded HNAC, leading oscillations conducting to
intersected road through on-ramp.
In all of the 18 simulation tests in the group of
vmain = 54.9km/h,
main = 34veh/km ∗l
,
main = 1868veh/h∗l
, a fuzzy bound-
ary
in = 1000veh/h∗l
is found. When the merging trac vol-
ume is less than
1000veh/h∗l
, trac condition in the intersected
road is only aected by lane-changing behaviors. When the merging
trac volume is larger than
1000veh/h∗l
, trac condition in the
intersected road would be aected by the exceeding HNAC. is
result corresponds to the theoretical analysis in Section 3.2,
moreover, providing a fuzzy value of HNAC in a specic
Main
road
Main road vehicle
routeConnector
Adjusting area
Ramp vehicle route
Detector on ramp
Intersected
road
Priority rules Connector
Vehicle moving
direction
Detector on main road
Main road vehicle
route
Detector on
intersected road Route decision point Ramp vehicle route
F 7:VISSIM simulation model.
T 2:Background trac input on main road and the merging trac volume.
Heavy vehicle mixing
ratio
𝑟
(%)
Concentration on main
road
𝑘main
(veh/km ∗ l)
Concentration step
Δmain
(veh/km ∗ l)
Merging trac volume
𝑄in
(veh/h ∗ l)
Merging trac volume
step
Δin
(veh/h ∗ l)
033–69 4 100–50%
𝑄main
100
531–67 4 100–50%
𝑄main
100
10 30–66 4 100–50%
𝑄main
100
15 29–65 4 100–50%
𝑄main
100
20 28–64 4 100–50%
Qmain
100
25 26–62 4 100–50%
𝑄main
100
30 25–61 4 100–50%
𝑄main
100
35 23–59 4 100–50%
𝑄main
100
40 22–58 4 100–50%
𝑄main
100
45 21–57 4 100–50%
𝑄main
100
50 20–56 4 100–50%
𝑄main
100

Journal of Advanced Transportation10
vmain =28.3km/h
,
main =1414veh/h∗l
,

in
∈[100,1400]
veh/h∗l
are chosen as presentations, shown in Figure 9.
In these gures, trends of Std in intersected road trac
velocity and on-ramp velocity are depicted in upper parts.
SCCs between intersected road trac velocity and on-ramp
velocity are depicted in lower parts. From the gures shown
above, discontinuity point of Std and SCC implies a same fuzzy
boundary (HNAC) of the merging trac volume. In le side
of this point, both Std and SCC stayed in a very low stable
value. In right side of this point, they stayed in a relatively high
stable value. It could be seen in upper parts that there exists a
point where the Std of the intersected road trac velocity
equals to that in on-ramp. is could provide us the accurate
value of the fuzzy boundary (HNAC). Moreover, with the
value of
𝑄main
decreased, which means the trac condition of
the main road became more severe, the fuzzy boundary
(HNAC) also went down, corresponding to the theoretical
result in Section 4.2.
6.3. e Explicit Equation of HNAC. From Figure 9, the point
where Std of intersected road trac velocity equals to that in
on-ramp refers to the ccurate value of the fuzzy boundary
(HNAC). In that, 10 HNAC values could be obtained in
dierent trac conditions of the main road, when heavy
vehicle mixing ratio equals to 10%. erefore, 110 HNAC
values in trac stream of dierent heavy vehicle mixing ratios
on-ramp, caused by synchronization. Moreover, the synchro-
nization could be expressed by correlation coecient. When
merging trac volume is less than the boundary (HNAC),
the coecient remains in a relatively low level, and when the
merging trac volume is larger than HNAC, the coecient
remains in a high level.
In this part, Std
𝜎
is calculated by equation (23), and
Spearman correlation coecient (
𝜌
) (SCC) is adopted because
it is suitable to randomly distributed data set, calculated by
equation (24).
Taking the data test groups of
=10
% as examples,
main ∈[30,66](veh/km ∗1
)
,
𝑄main
is calculated by equation
(22). e total number of simulation groups of
=10
% is 10,
and information concluded from each simulation group is basi-
cally of same kind. erefore, there is no need to list all the
simulation results. Std and SCC in group
vmain =54.9km/h
,
main = 1868veh/h∗l
,
in ∈[100,1800]veh/h∗l
and group
(23)

=
√∑(
−
)
2
,
(24)
= ∑
=1
(
i
−)(
i
− )
√∑

=1(
i
− )
2
∑

=1(
i
− )
2
.
100
90
80
70
60
50
40
30
20
10
00 500 1000 1500
Time (s)
2000 2500 3000
3500
Velocity (km/h)
Intersected road
Main road
Ramp
(a)
2500
100
90
80
70
60
50
40
30
20
10
00 500 1000 1500
Time (s)
Lane-changing oscillation
2000 3000
3500
Velocity (km/h)
Intersected road
Main road
Ramp
(b)
2500
100
90
80
70
60
50
40
30
20
10
00 500 1000 1500
Time (s)
2000 3000
3500
Velocity (km/h)
Intersected road
Main road
Ramp
Lane-changing oscillation
(c)
2500
100
90
80
70
60
50
40
30
20
10
00 500 1000 1500
Time (s)
2000 3000
3500
Velocity (km/h)
Intersected road
Main road
Ramp
Lane-changing oscillation
Synchronized oscillation
(d)
F 8:e data examples analysis. (a)
main = 1868veh/h∗l
,
in = 100veh/h∗l
, (b)
main = 1868veh/h∗l
,
in = 300veh/h∗l
,
(c)
main = 1868veh/h∗l
,
in = 700veh/h∗l
, (d)
main = 1868veh/h∗l
,
in = 1000veh/h∗l
.

Journal of Advanced Transportation 11
Data group 1 average = –0.0010
0 200 400 600 800 1000 1200 1400 1600
1800
Volume travel to mainlane (veh/hr
∗
1)
Data group 2 average = 0.5993
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.1
Spearman corelation
Spearman data group1
Spearman data group2
Std. of intersected road
Std. of ramp
30
25
20
15
10
5
0
Std.
Average of lower value = 1.4182
0 200 400 600 800 1000 1200 1400 1600 1800
Volume travel to mainlane (veh/hr
∗
1)
Stable value = 15.3658
Average of larger value = 25.5360
(a)
F 9:Continued.

Journal of Advanced Transportation12
values of
Adj −𝑅2
have proved the accuracy of linear tting.
erefore, the explicit equation of HNAC is shown below.
6.4. Model Validation. Following the explicit equation of HNAC
obtained in Section 6.3, model validation is provided in this
part. As depicted in Section 1, HNAC is dened as the upper
limit of trac volume merging into main road through the on-
ramp. If this value was exceeded by merging volume, the trac
ow of intersected road will be aected. To validate the model,
trac dynamic should be supervised in a certain highway node,
including real time variance of trac volume and velocity in
main road, intersected road and on-ramp. When the oscillation
(28)
()=0.68 +0.2,
(29)
()=−690 +93,
(30)
HNAC =(main, ) =0.68main ⋅+0.2main −690 +93.
Table 2 and dierent main road’s trac conditions could be
obtained. e theoretical conclusion in Section 4.2 that HNAC
is linear positive correlated to trac volume of the main road
𝑄main
has been proved by simulation data. With our previous
work [1], the trac volume is also linearly related to heavy
vehicle mixing ratio
𝑟
. us, the assumption of the explicit
equation of HNAC could be reasonably presented as below.
Using the 110 HNAC values obtained in Section 6.2 to linear
t equation (25), the result is shown in Figure 10. e values
of
()
and
ω()
in trac conditions with dierent
𝑟
values are
also obtained, provided in Table 3.
From the linear tting results given above, the explicit
equation of relationship (26) and (27) could be obtained. e
(25)
HNAC =main,  =()main +(),
(26)
()=1+2,
(27)
()=

1
r
+

2.
0 200 400 600 800 1000 1200
1400
Data group 2 average = 0.6904
Data group 1 average = 0.0267
Volume travel to mainlane (veh/hr ∗ 1)
Spearman data group1
Spearman data group1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.1
Spearman corelation
0 200 400 600 800 1000 1200 1400
Average of lower value = 1.2826
Stable value = 10.9291
Average of larger value = 15.8492
Volume travel to mainlane (veh/hr
∗
1)
18
16
14
12
10
8
6
4
2
0
Std.
Std. of ramp
Std. of intersected road
(b)
F 9:Std and SCC when
=10
%, taking two simulation groups for presentation.

Journal of Advanced Transportation 13
method. Another is to determine the boundary condition of
merging to targeted lane under the dimension of space, which
could be used as an intermediate variable in macroscopic mod-
elling. It should be declared that the microscopic merging pro-
cess is simplied, by bypassing the process of deceleration or
acceleration of the merging vehicle. Compared to related
works, this consideration largely reduced the complexity of
microscopic modelling based on lane-changing rules.
Moreover, the microscopic explain of trac oscillations, the
result of macroscopic model in Section 4.2, and data simulation
also proved its rationality. e macroscopic model of HNAC
is built using the boundary condition mentioned above.
Macroscopic models were built in congested ow and free ow
based on dierent assumptions respectively, providing the
function of HNAC and also explained the production of oscil-
lations. And this is an advantage which made it stay closer to
actual trac situations. Besides, it provided a thought of con-
necting microscopic problem and macroscopic problem.
e data simulation method is chosen instead of eld data
because it is dicult to collect enough real road data with
dierent heavy vehicle mixing ratios and merging trac vol-
ume in one specic highway node. erefore, the simulation
model is built based on rigid car-following behaviors organi-
zation and also controlled by macroscopic trac ow model,
to guarantee not only the typical trac ow phenomenon such
as oscillations and trac waves, but also the comprehensive
background trac condition needed in this paper. However,
it should be noticed, driving characters were simplied and
idealized in theoretical models, and for instance, vehicle accel-
eration and deceleration processes were neglected. is might
cause some dierences. From Figure 8 in Section 6.1, though
synchronization phenomenon could be signicantly observed,
lines depicting velocity in on-ramp and intersected road were
not closely matched, and volatility of velocity curves were
obvious. is phenomenon was mainly caused by the gap
between theoretical models and real trac conditions repre-
sented by simulation. Furthermore, from Figure 10 and Table
mentioned in Section 6.1 happens, merging volume should be
recorded and compared to theoretical HNAC obtained from
equation (30), to verify the accuracy of the model.
In order to prove the accuracy of HNAC model introduced
in this article, Maoerliu interchange of Xi’an Ring Expressway
(G3001) was chosen as the supervision objective. Taking
northbound section as the main road, and westbound as the
intersected road, detectors were deployed as shown in
Figure 11. Besides, all detectors have avoided the merging and
diversion areas, to reduce the impact of weaving trac. Trac
volume and velocity of the three detection points were super-
vised during 2019.09.15–2019.09.26.
In eld detection, the oscillation mentioned in previous could
only be caught in peak hours, and 28 oscillations in intersected
road caused by exceeding HNAC were observed. From equation
(30), it is known that HNAC varied with trac volume and heavy
vehicle mixing ratio
()
in main road. In selected part of G3001,
𝑟
remains in about 12–14%, taking 13% as average, the HNAC
distribution curve could be obtained. Moreover, merging volume
and trac volume of 28 oscillations were collected. Comparing
the theoretical values with the supervised values, the accuracy of
the model could be observed, shown in Figure 12.
From Figure 12, it can be seen, 28 supervised HNACs
spread around the theoretical curve obtained from equation
(30), basically in range of
[1100,1650]veh/h∗l
. e distribu-
tion of 28 supervised HNACs is random and homogeneous,
and the average relative error of supervised HNACs is 14.6%,
which is an acceptable level, proving the accuracy of the model
introduced in this article. Moreover, owning to the lack of
related studies, it is unfortunately that the model in this paper
could not be compared to other similar ones.
7. Discussion and Conclusions
e purpose of building microscopic model is not complex.
e rst one is to analyze the cause of HNAC in microscopic
800
700
600
500
400
300
200
100
600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800
Traffic volume of main lane (veh/hr ∗ 1)
HNAC = f (Qmain, r) = φ(r) Qmain + ω(r)
Heavy vehicle mixing ratio 0%
Heavy vehicle mixing ratio 5%
Heavy vehicle mixing ratio 10%
Heavy vehicle mixing ratio 15%
Heavy vehicle mixing ratio 20%
Heavy vehicle mixing ratio 25%
Heavy vehicle mixing ratio 30%
Heavy vehicle mixing ratio 35%
Heavy vehicle mixing ratio 40%
Heavy vehicle mixing ratio 45%
Heavy vehicle mixing ratio 50%
HNAC (veh/hr ∗ 1)
F 10:Linear t of equation (25).

Journal of Advanced Transportation14
Unfortunately, accsurate number or proportion of timid and
aggressive drivers could not be obtained. erefore, to make
HNAC more accurate and practical, large amount of real trac
data are needed, including detailed information of drivers in
real road, geometric conditions, roadside facilities and trac
rules.
e denition of HNAC provided in Section 1 has implied
that this concept lies in the macroscopic aspect, which is more
related to a specic highway node than a road section. is
concept derived from a simple observation towards the trac
condition of a specic highway node, and it is very likely to
be noticed by other researchers. However, directly related
works were not found, owning to the contribution of car-fol-
lowing models and moving bottleneck theory, which provide
sucient and ecient methods to solve many important exist
-
ing trac problems, lane-changing behaviours, trac oscilla-
tions and breakdown, stop-and-go waves, relaxation
phenomenon, etc. But, moving the consideration to large scale
macroscopic problems, especially the propagation mechanism
of trac emergencies on highway network, which is very help-
ful to large scale evacuation and rescue, the systematic research
of HNAC becomes important. In policy and planning aspects,
when an emergency happens downstream a highway node,
the trac condition in road section could be obtained through
kinetic wave models. In this situation, if the trac volume in
on-ramp exceeded HNAC of the main road, congestion will
form in on-ramp and further aects the trac ow in inter-
sected road, meaning that the emergency eect will spread
into the intersected road. Based on the judgement mentioned
above, the approximate range of emergency eect in road net-
work could be obtained, providing a base line to trac man-
agement departments in dealing trac emergency.
Data Availability
Data in this article are available only with permission of cor-
responding author.
Conflicts of Interest
e authors declare that they have no conicts of interest.
3 provided in Section 6.3, accuracy obtained from simulation
data were signicantly higher than that from the eld test,
depicted in Figure 12. is might be caused by ignorance of
timid and aggressive drivers mentioned in Section 4. When
timid drivers took the majority part of on-ramp trac ow,
they might lose some opportunities of merging into the trac
ow in main road, and HNAC of this situation might be
smaller than theoretical value. Points below theoretical curve
in Figure 12 stand for the situation mentioned above, and
points above the curve represent the opposite situation.
T 3:e values of
()
and
()
.
𝑟
(%)
()
()
Adj −
2
0% 0.2155 89.82 0.9641
5% 0.2475 57.93 0.9674
10% 0.2747 23.35 0.9711
15% 0.3004 −9.10 0.9736
20% 0.3213 −36.68 0.9737
25% 0.3517 −77.48 0.9683
30% 0.3850 −108.80 0.9697
35% 0.4372 −162.40 0.9558
40% 0.4652 −181.20 0.9511
45% 0.5131 −215.70 0.9473
50% 0.5692 −252.00 0.9443
Detection point 1 (main road)
Detection point 2 (on-ramp)
Detection point 3 (intersected road)
Maoerliu interchange of G3001 expressway
Traffic moving direction of main road
Traffic moving direction of on-ramp
Traffic moving direction of intersected ro
ad
F 11:Maoerliu interchange of G3001 and the detectors deployment.
550
500
450
400
350
300
HNAC (veh/h ∗ 1)
250
200
150
600 800 1000 1200 1400 1600 1800
Traffic volume in main road (veh/h ∗ 1)
Theoretical HNAC when r = 13%
Supervised HNAC when oscillation happens
F 12:Comparison between theoretical HNAC and supervised
values.

Journal of Advanced Transportation 15
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