A microscopic investigation into capacity drop: impacts of a bunded acceleration and
reaction time
Yuan, Kai; Knoop, Victor; Hoogendoorn, Serge
Publication date 2016
Published in
Proceedings of the 95th Annual Meeting of the Transportation Research Board
Citation (APA)
Yuan, K., Knoop, V., & Hoogendoorn, S. (2016). A microscopic investigation into capacity drop: impacts of a bunded acceleration and reaction time. In Proceedings of the 95th Annual Meeting of the Transportation Research Board: Washington, United States (pp. 1-21)
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TIME
3 4
Kai Yuan, PhD candidate 5
TRAIL research school 6
Department of Transport and Planning 7
Faculty of Civil Engineering and Geosciences 8
Delft University of Technology 9
Stevinweg 1, P.O. Box 5048, 2600 GA Delft, The Netherlands 10 Phone: +31 15 278 1384 11 Email: k.yuan@tudelft.nl 12 13
Victor L. Knoop, Assistant Professor 14
Department of Transport and Planning 15
Faculty of Civil Engineering and Geosciences 16
Delft University of Technology 17
Stevinweg 1, P.O. Box 5048, 2600 GA Delft, The Netherlands 18 Phone: +31 15 278 8413 19 Email: v.l.knoop@tudelft.nl 20 21
Serge P. Hoogendoorn, Professor 22
Department of Transport and Planning 23
Faculty of Civil Engineering and Geosciences 24
Delft University of Technology 25
Stevinweg 1, P.O. Box 5048, 2600 GA Delft, The Netherlands 26 Phone: +31 15 278 5475 27 Email: s.p.hoogendoorn@tudelft.nl 28 29 30 July 2015 31 32 33 34 35 36 37 Word count 38 nr of words in abstract 201 39
nr of words in manuscript (including abstract and references) 5545 40
nr of figures & tables 7 41
total 7496 42
43 44
ABSTRACT 1
2
The capacity drop indicates that the queue discharge rate is lower than the free-flow 3
capacity. Studies show that queue discharge rates vary under different traffic conditions. 4
Empirical data show that the queue discharge rate increases as the speed in congestion 5
increases. Insights into the underlying behavioral mechanisms that result in such variable 6
queue discharge rates can help minimize traffic delays and eliminate congestion. 7
However, to the best of the authors’ knowledge, few efforts have been devoted to testing 8
impacts of traffic behaviors on the queue discharge rate. This paper tries to fill this gap. 9
We investigate to what extent the acceleration spread and reaction time can influence the 10
queue discharge rate. It is found that the (inter-driver) acceleration spread does not reduce 11
the queue discharge rates as much as found empirically. Modelling reaction time might 12
be more important than modeling acceleration for capacity drop in car-following models. 13
A speed-dependent reaction time mechanism for giving variable queue discharge rates is 14
proposed. That is, decreasing reaction time as the speed in congestion increases can give 15
the same queue discharge rate as found empirically. This research suggests that 16
motivating drivers to speed up earlier could increase the queue discharge rate and thereby 17
minimize delays. 18
1. INTRODUCTION 1
2
Road congestion can be categorized into two classes: standing queues with heads fixed at 3
a bottleneck and stop-and-go waves with queue fronts moving upstream. The bottleneck 4
is a fixed point where the congestion head is located. Once congestion sets in, the flow 5
out of congestion is the queue discharge rate. This flow is generally lower than the free-6
flow capacity, i.e., the maximum flow. This phenomenon is called the capacity drop. 7
8
The magnitude of the capacity drop is not constant. Empirical data show that the queue 9
discharge rate vary considerably at the same location [1, 2]. This is shown to correlate 10
well with congestion states [3, 4]. Yuan et al. [3] reveal a linear relation between the 11
speed in congestion and the queue discharge rate (see Figure 1). The specific relation is 12
based on empirical data collected on freeway A4 and A12 in the Netherlands. Road 13
design and control measures can contribute to varying queue discharge rates [5, 6]. These 14
findings show that there might be promising strategies that can increase the queue 15
discharge rate to reduce delays. However, to determine effective approaches, an insight is 16
needed into the underlying behavioral mechanisms that cause the capacity drop. 17
Therefore, this paper tries to investigate the impacts of driver behavior on the queue 18
discharge rate. 19
20
More specifically, this paper studies the impacts of acceleration and reaction time on the 21
queue discharge rate. The acceleration can give the capacity drop with inter-driver 22
acceleration spread. Inter-driver acceleration spread (or in short: acceleration spread) 23
means that vehicles do not have the same acceleration. As a result, voids will be created 24
between a low-acceleration vehicle and its high-acceleration predecessor. The reaction 25
time indicates how long a following vehicle needs to take to react to the change of its 26
leader’s driving behavior. Voids can also be created if the follower’s reaction time is 27
longer than Newell’s reaction time (see section 3.3). In this paper, we call such long 28
reaction time the extended reaction time. To what extent the inter-driver acceleration 29
spread and the extended reaction time contribute to the capacity drop is unknown. Hence, 30
we here study the impacts of the acceleration spread and the extended reaction time on 31
the queue discharge rate. 32
33
This paper develops analytical models to investigate the independent impact of 34
accelerations and reaction time. Furthermore, we design numerical experiments for two 35
objectives. First, the experiment is used to validate the analytical model to ensure the 36
approximation in the model is accurate enough. Second, we use the experiment to see the 37
combination effects of acceleration spread and reaction time on the queue discharge rate. 38
The empirical relation revealed in [3] is the reference used in our analyses, see Figure 1. 39
40
Our study excludes several factors that may influence the queue discharge rate. Firstly, 41
drivers’ perspectives, i.e., whether drivers are aggressive or timid, are excluded. 42
Secondly, lane changing is not considered in this paper. As argued in [7], if we simulate a 43
stop-and-go wave moving on a homogeneous road section, lane changing frequency 44
should be very low in an acceleration mode. 45
The outline of the paper is as follows: we start with a literature review in section 2. Then 1
section 3 presents the analytical investigation on the capacity drop. In section 4, we use 2
simulations to validate the analytical model (section 4.2) and investigate the combination 3
of acceleration and reaction time (section 4.3), followed by discussions and conclusions 4 in section 5. 5 6 : 29 5000 ; Linear y x Empirical data in A4 Empirical data in A12 Rainy day (8.8 mm, A12) Linear fitting Quadratic fitting 0 10 20 30 40 50 60 70 80 4000 4500 5000 5500 6000 6500 7000 7500 Speed in congestion (km/h) Queue dis ch arg e ra te (veh /h) 7 8 Figure 1 Relation between queue discharge rate and the speed in congestion [3]. 9 10 2. LITERATURE REVIEW 11 12
A wide range of capacity drop values have been observed, which are reviewed in section 13
2.1. The wide range of the capacity drop values could be due to various queue discharge 14
rates which correlate well with different congested states. The research objective of this 15
paper is to investigate the relation between driving behavior and the queue discharge rate. 16
Hence, section 2.2 reviews previous traffic behavioral mechanism of the capacity drop. 17
18
2.1 Empirical features of the capacity drop 19
20
The capacity drop was reported for the first time in 1991, with a drop of 6% [8] and 3% 21
[9]. In the past decades, the capacity drop have been studied more often, with values of 22
the drop ranging between 3% and 18% [6]. In [10] the capacity drop ranges from 8% to 23
10%. In [5] the capacity falls by 15% at an on-ramp bottleneck. Chung et al. [1] show a 24
range of capacity drop from 3% to 18% with data collected at three active bottlenecks, 25
which shows a drop from 8% to 18% at the same location. Cassidy and Rudjanakanoknad 26
[11] observe capacity drop between 8.3% and 14.7%. 27
28
We argue that the wide range of capacity drop values in literature correlates well with the 29
congestion state. Yuan et al. [3] show a positive correlation between the queue discharge 30
rate and the speed in congestion with empirical data collected on freeways in the 31
Netherlands. Oh and Yeo [4] find that the queue discharge rate is related to the severity 32
of congestion by analyzing microscopic trajectory data. Hence, the research question is: 33
what is the mechanism behind the dependency of discharge rate on the congested states? 34
Answering this question might help to better understand the microscopic mechanism of 35
the capacity drop. 36
2.2 Overview of assumptions on mechanisms of the capacity drop 1
2
Many studies have been reporting the capacity drop in the past decades. Table 1 3
summarizes most of the existing most popular assumptions on the traffic behavioral 4
mechanism of capacity drop. Generally, we can divide them into three categories: 5
bounded acceleration capability, inter-driver/vehicle spread, and intra-driver spread. 6
7
Bounded acceleration capability means vehicles cannot accelerate instantaneously. 8
Consequently lane change manoeuvers can create voids in the traffic stream. The limited 9
acceleration causes that the lane changing vehicle cannot catch up with its new 10
predecessor [12-14]. Coifman and Kim [15] show that lane changing in the far 11
downstream of the congestion can result in the capacity drop, too. Insertions result in 12
shockwaves in the new lane and the divergences in the old lane create voids which cannot 13
be filled in duo to the bounded acceleration capability. So an aggregated flow detected in 14
the downstream of queue could be lower than the capacity. 15
16
Inter-driver/vehicle spread indicates the spread of drivers and vehicles. Papageorgiou et 17
al. [7] state that the capacity drop is due to the acceleration difference between two 18
successive vehicles. Voids can be created between a low-acceleration vehicle and its 19
high-acceleration predecessor. Chen et al. [16] try to explain the capacity drop in a view 20
of drivers’ perspectives. Wong and Wong [17] reproduce the capacity drop when 21
modeling the multi-class traffic flow. 22 23 Table 1 Possible mechanisms of the capacity drop 24 25
Basic mechanisms Assumptions on mechanisms: References: a) Bounded
acceleration capability
Lane changing Laval and Daganzo [12]
Leclercq et al. [13] Leclercq et al. [14] Coifman and Kim [15] Coifman et al. [18] b)
Inter-driver/vehicle spread
Drivers’ perspective Chen et al. [16] Acceleration variance Papageorgiou et al. [7]
Multi-class vehicles Wong and Wong [17]
c) Intra-driver spread
Variance-driven time headways Treiber et al. [19] Multiphase car-following theory Zhang and Kim [20] Asymmetric driving behavior theory Yeo [21]
Activation level Tampère [22]
26
The third popular explanation, intra-driver spread, assumes driver behaviors vary 27
depending on traffic conditions. Treiber et al. [19] assume drivers would choose a longer 28
time headway in congestion than that in free flow. The preferred time headway in 29
congestion increases as density increases. This assumption, also called variance-driven 30
time headways, is based on an empirical observation of an increasing time gaps between 31
one vehicle’s front bumper and the rear bumper of the preceding vehicle after a 32
considerable queuing time in [23]. Zhang and Kim [20] propose a multi-phase car-33
following traffic flow theory to reproduce the capacity drop. They highlight that the 1
capacity drop is a result of driver behavior spread across three phases, i.e., acceleration, 2
deceleration and coasting. Yeo [21] validates the acceleration and deceleration curves to 3
further develop the asymmetric microscopic traffic flow theory based on empirical 4
trajectory data, explaining the capacity drop as a difference of the maximum flow 5
between the acceleration and the deceleration curve in density-flow fundamental 6
diagram. The asymmetric driver behavior theory is also applied in [4] to understand the 7
impacts of stop-and-go waves on the capacity drop. Tampère [22] assumes drivers’ 8
behavior depends on a temporary, traffic condition dependent variable “activation level”. 9
The low activation level used to accounted for a loss of motivation. They reproduce the 10
capacity drop as a result of low activation level in case studies. 11
12
In this paper, we focus on studying the impacts of acceleration spread and reaction time 13
on the queue discharge rate and its correlation with the congestion state. 14
15
3. ANALYTICAL INVESTIGATION 16
17
This section analytically investigates to what extent the acceleration spread (3.1) and 18
reaction time extension (section 3.2) can independently account for the capacity drop. In 19
each of section 3.1 and section 3.2, we firstly present a numerical expression of the queue 20
discharge rate, followed by analysis of the model properties. 21
22
Using mathematical derivations show that including acceleration spread in car-following 23
models does not give sufficient capacity drop compared to empirical observations, and 24
that intra-driver reaction time extension mechanism can model similar queue discharge 25
rates as reality. For practical purpose, these conclusions indicate that pushing slowly 26
driving vehicles to speed up earlier, rather than managing vehicular acceleration, might 27
be an approach for minimizing capacity drops and delays. 28
29
3.1 Capacity drop due to accelerations spread 30
31
In this section we derivate analytical formula for the capacity drop in section 3.1.1 and 32
find the acceleration spread does not give sufficient queue discharge rate reduction 33
compared to empirical observations in section 3.1.2. 34
35
3.1.1 Analytical expressions of queue discharge rates 36
37
Let us consider a stop-and-go wave moving upstream on a homogeneous road section 38
shown as the grey block in Figure 2. Bold lines are vehicular trajectories. The traffic in 39
the scenario is described by a triangular fundamental diagram with positive wave speed 40
w , free-flow speed v and capacity C . The critical density and maximum jam density f 41
are given by cri and jam, respectively. There are n vehicles in total in the queue in a 42
single lane, obeying the first-in-first-out (FIFO) rule. Each vehicle is numbered 43
1, 2,...,
i i n , increasing from the head of the queue
i 1
to the tail
in
. The speed 44and density in the queue are v and q q, respectively. When all vehicles reach the free-45
flow speed after leaving the queue, the free-flow spacing and time headway between 1
vehicle i and i is given by 1 s and i h , respectively. The minimum free-flow spacing i 2
min
s for all vehicles should be
cri
1
(or the minimum time headway min
1 h
C
), indicating 3
no capacity drop at all. Each vehicle i is described by two constants, its desired 4
acceleration aidesire and acceleration a . In principle, every vehicle accelerates with its i 5
desired acceleration. However, s is at the low end bounded by i smin. Therefore, if 6
desire
i i
a a will result in sismin , we set desire
i i
a a to ensure sismin , Note that 7
desire 1 1
a a . Desired accelerations fall within the interval
amin,amax
. The reaction time of 8vehicle i is denoted as t . All vehicles have reached free-flow speed at r x where The 1 9
sum of free-flow time headways from the second vehicle to the last vehicle is denoted as 10
H. 11
12
If all vehicles follows continuous Newell car-following model [24], constructed by 13
shifting its predecessor’s trajectory by spacing
jam 1 s and time jam 1 s t w w , see 14
Figure 2a, there is no capacity drop, 1 1 ( 1) d i n n q C H n h . 15 16
It is impossible that all vehicles have the same acceleration. We assume the desired 17
acceleration follows an uniform distribution bounded by amin and amax , i.e., 18
desire
min, max
a U a a . We exclude the impact of reaction time extensions by setting 19
0s
ex
t
. When the desired acceleration of vehicle i is higher than its leader’s 20
acceleration, settingai ai1 ensures the follower can neither overtake nor be too close 21
sismin
to its leader. Otherwise,desire i i a a . In summary, 22
desire 1 1 desire 1 desire , min , , otherwise i i i i i i i a a a a a a a (1) 23 24A void is created between two successive vehicles if the follower’s desired acceleration is 25
lower than the predecessor’s acceleration. In Figure 2c, a dashed line is the Newell 26
trajectory of vehicle i . Note the void between the Newell trajectory and the trajectory of 27
vehicle i. The void means the free-flow spacing is extended by siextension: 28
2 2 1 extension 1 1 1 1 1 2 2 j f i i i j f i i i i v v a a s v v a a a a (2) 29Now let us consider n vehicles within a stop-and-go wave as in Figure 2d. The queue 30
discharge rate q is expressed as: d 31 1 d n q H (3) 32
congestion t s
s
i s w vehic le i vehi cle 1 i f v 1 t t2 H f v w vehi cle n veh icle 1 w 1 x 1 2(a) Two vehicles, no capacity drop (b) n vehicles, no capacity drop
3 4 congestion L o ca ti o n vehi cle i vehi cle 1 i vehi cle 1 i w vf w 1 t extension i s f v C 2 t t3 t4 t5 L o c at io n ve hic le 1 vehi cle n cri H H extension 1,n
s
w 1 x 5 6 7 8 9 congestion w sshiftvj ex t t 1 t t2 t3 t4 vehic le i vehi cle 1 i extension i s f v C veh icle 1 vehi cle n n1tex n1 / crivf H j v n1t vex f w 1 x 10 11 12 13 14 Figure 2 Measurements of queue discharge rates 15(c) Spacing extensions due to acceleration variability
(d) Queue discharge rates measurements with acceleration variability.
(e) Spacing extensions due to reaction time extensions
(f) Queue discharge rates measurements with reaction time extensions
We firstly assume the first vehicle has the same acceleration as the last vehicle, see the 1
dashed line in Figure 2d. There is no capacity drop, Hcri n 1 C
. Next, we relax such 2
assumption by setting a1a1desireU a[ min,amax]. An extenstion of spacing s1,nextention that 3
denotes the free-flow spacing between the first hypothezied trajectory and the first 4
vehicle’s trajectory can be estimated in Equation (2). Hence, we have: 5
extention 2 1, cri 1 1 1 1 1 2 n j f f f n s n H H v v v C v a a (4) 6We are interested in the average headway, but since the acceleration of the first and the 7
last vehicle are stochastic, we compute the expected value of H: 8
2 1 2 1 1 1 1 1 2 1 1 1 1 2 j f f n j f f n n E H v v E C v a a n v v E E C v a a (5) 9Since a1a1desireU a[ min,amax], the expected value of
1 1 a is: 10 max min 1 max min ln 1 a a E a a a (6) 11 Now we need 1 n E a
. Equation (1) indicates that the last vehicle always has the slowest 12
acceleration mong all vehicles: 13
desire
desire desire desire
1 1 1
min , min , ,..., ,
n n n n m n m n n
a a a a a a a (7) 14
We choose m from set
1,n 1
. Let
desire desire
(1) ,..., ( )na a denote the corresponding order 15
statistics of the random sample
desire desire
1 ,..., na a so that a(1)desirea(2)desire a( )desiren . So 16
Equation (7) means desire (1)
n
a a . Hence, the probability density function of a equals to n 17
the probability density function of the smallest order statistic a(1)desire. According to the 18
order statistic [25], the probability distribution function f of A a(1)desire is: 19
desire
desire
1
desire
1 n
A
f a n F a f a (8) 20
desire
F a and f a
desire
are the cumulative distribution function and probability21
distribution function of the desired acceleration: 22
desire
desire min desire
min max max min , for , a a F a a a a a a (9) 23
desire
desire
min max max min 1 , for , f a a a a a a (10) 24Hence, incorporating Equation (9) and (10) into Equation (11) gives the probability 1 density function f of N a : n 2
1 max min max max min max min, for , n n N n n a a n f a a a a a a a a (11) 3
We estimate the second-order approximation of E g a
n
with the Delta method. 4Setting Function g x
as the inverse of x , i.e., g x
1 x , we can have 5
1 n n E E g a a . Thus, 6
1
2
2 n n n n E g a g E a g E a E a (12) 7
E an and 2
an are the expected value and the standard spread of a , respectively. n 8They can be deduced from Equation (11): 9
max min 1 n a a n E a n (13) 10
2 2max min max min max
2 max mi 2 2 1 2 n 1 1 2 n a n a a a n a a n a n n a n n (14) 11 Because
3 max min 1 2 n n g E a a a n , combining Equation (12)~(14) gets: 12
2 2max min max m ma in ma x min 3 2 max min 2 max min x 1 2 2 1 1 1 1 n n E g a a a n a a a n a a n n n n a a n n a a n n (15) 13
Incoporating Equation (6) and Equation (15) into Equation (4), we get the expected 14 value of H: 15
2 max 2 min max min max min 2 3 2 max min 3 2 max mi 2 2max min max min max
n ln 1 1 2 2 1 2 1 1 2 2 1 j f j f f f j f f a v v n a a v v n a n E H C v a a a n a a n n n v a a n v v a n v a n n a a n (16) 16 17
This gives get the expected value of the queue discharge rate: 18
1 d n E q E H (17) 19 20 213.1.2 Analysis of model properties 1
We set a triangular fundamental diagram with w 18km/h, v f 114km/h, C 6840 2
veh/h, cri60 veh/km and jam 440 km/h. This fundamental diagram indicates a 3
similar traffic situlation as that in [3]. Different bounds for accelerations are reported: for 4
instance 0.5m/s2 - 3m/s2 [13], or 1.5m/s2 - 2m/s2 [26]. We combine these and set the 5
limits for desired accelerations from 0.5m/s2 to 2m/s2. We will limit the range further. 6
7
Consider a stop-and-go wave that propagates at speed w for 10 minutes. Variational 8
theory [27] gives the number of vehicles in the queue jam 1320 60 w n veh. This 9
section analyses the queue discharge rate for this queue. 10
11
As shown in Equation (16), E H is a function of
amin, amax and n . The sensitivity of 12the queue discharge rate to the average desired accelerations, standard spread of desired 13
accelerations and number of vehicles are evaluated with Equation (16) and (17), 14
presented in Figure 3. 15
16
Figure 3a presents a relation between the speed in congestion v and the queue discharge j 17
rate q when setting d E a
desire
as 0.75m/s2, 1.25m/s2, 1.75m/s2 respectively and 18
2 2 0.5 12 desire a . We obtain so by setting the pair
amin,amax
to (0.5m/s2,1m/s2), 19(1m/s2,1.5m/s2) and (1.5m/s2, 2m/s2). We see that the faster the average desired 20
acceleration, the higher the queue discharge rate. 21
22
Figure 3b presents the relation between v and j q when d
2 desire a equals to 2 0.5 12 , 23 2 0.9 12 and 2 1.5 12 , setting
2 1.25m/s desireE a . That is, the pair
amin,amax
are chosen to 24(1m/s2,1.5m/s2), (0.8m/s2,1.7m/s2) and (0.5m/s2,2m/s2) respectively. It indicates that the 25
larger the spread, the lower the queue discharge rate. 26
27
If we fix amin 0.5m/s2 and decrease amax from 2m/s2 to 1m/s2, then both of
desire
E a 28
and 2
desire
a
decreases. Figure 3c shows that the decrease of amax increases the queue 29
discharge rates. Since the decrease of E a
desire
and 2
adesire
will decrease and 30
increase the queue discharge rate respectively, the increase of queue discharge rates in 31
Figure 3c indicates that 2
adesire
has more influences on the queue discharge rate than32
desire
E a .
33 34
Figure 3d shows the sensitivity to n with amin 0.5m/s2 and
2 max 2m/s
a . The more 1
vehicles, the higher queue discharge rates. It is not a surprise because the follower’s 2
acceleration is always limited by its leader’s acceleration, that makes the acceleration 3
spread decrease as the vehicle number increases. Since n 1320 means the congestion 4
only propagates for 10min, the queue discharge rate can be even higher when setting a 5
longer time of congestion propagation. 6 7 0 20 40 60 80 100 120 6650 6700 6750 6800 6850 Speed in congestion (km/h) Qu eu e d isch ar ge r at e ( v eh /h ) <2(adesire) = 0:5 2 12 <2(adesire) = 0:92 12 <2(adesire) = 1:52 12
2 2 desire 0.5 12 a
2 2 desire 0.9 12 a
2 2 desire 1.5 12 a 8 9 10 11 12 13 0 20 40 60 80 100 120 6550 6600 6650 6700 6750 6800 Speed in congestion (km/h) Qu eu e d is cha rg e rat e (v eh /h) <2(adesire) = 1:5 2 12; E (a desire ) = 1:25 <2(adesire) = 12 12; E (a desire) = 1 <2(adesire) = 0:5 2 12; E (a desire ) = 0:75
2
2 desire 1.5 desire , 1.25 12 a E a
2
2 desire 1 desire , 1 12 a E a
2
2 desire 0.5 desire , 0.75 12 a E a 14 15 16 17 18 19 20 21Figure 3 Sensitivity of queue discharge rates when capacity drop is due to the
22 acceleration spread. 23 24 0 20 40 60 80 100 120 6740 6760 6780 6800 6820 6840 Speed in congestion (km/h) Q ue ue d is cha rg e r at e ( v eh/ h ) E (adesire) = 0:75 E (adesire) = 1:25 E (adesire) = 1:75 0 20 40 60 80 100 120 6500 6600 6700 6800 6900 Speed in congestion (km/h) Q ueu e discharg e rat e ( v eh/h) n= 660 n= 1320 n= 1980 desire ( ) 0.75 E a desire ( ) 1.25 E a desire ( ) 1.75 E a 660 n 1320 n 1980 n
a) Sensitivity of the analytical model to the mathematic expectation of desired accelerations.
b) Sensitivity of the analytical model to the standard deviations of desired accelerations.
c) Comparisions of impacts on queue discharge rates between the mathematical expectation and the standard deviations of desired accelerations.
d) Sensitivity of the analytical model
Setting amin 0.5m/s2,
2 max 2m/s
a and n 660veh gives a considerable influence of 1
the acceleration spread on queue discharge rates, shown as the line with circles in Figure 2
3d. However, the contribution of acceleration spread to the queue discharge rate 3
reduction is still marginal. In Figure 3d when v j 0km/h, the minimum queue discharge 4
rate (6522veh/h) is still much higher than the empirical value (5000veh/h) shown in 5
Figure 1. 6
7
Note that the hypothesis about the uniform desired acceleration distribution has already 8
maximized the 2
desire
a
. In reality, the desired acceleration could follow some 9
distribution with peaks (such as shown in [28]) which will have smaller 2
adesire
.
10
Therefore, we can conclude that the acceleration spread is not a dominant factor for 11
capacity drop. 12
13
3.2 Capacity drop due to reaction time extension 14
15
This section shows that the reaction time extension can considerably influence the queue 16
discharge rate. A negative relation between the reaction time and the speed in congestion 17
could result in a similar queue discharge rates as empirical findings. We give analytical 18
expressions of queue discharge rates and the sensitivity analyses in section 3.2.1 and 19
3.2.2, respectively. 20
21
3.2.1 Analytical expressions of queue discharge rates 22
If all vehicles have the same acceleration while the reaction time of each driver is larger 23
than t , the queue discharge rate will be lower than the capacity. We consider only the 24
cases when the reaction time is longer. Therefore, we can define 25
r ex
t t t (18) 26
t
is considered as a fixed reaction time (related to the fundamental diagram) and tex as 27
a reaction time extention. As shown in Figure 2e, two bold solid lines are trajectories of 28
two successive vehicles accelerating from speed v up to free speed j v . The follower’s f 29
reaction time is extended by tex from t . The dashed line is the follower’s trajectory 30
when tex . The follower’s trajectory can be considered as a shifted trajectory from 0 31
the dashed line in time (by tex) and space (by sshift) . Hence,
32 j shift ex s v t (19) 33 extension i shift f ex s s v t (20) 34 So we can have: 35
cri cri 1 extension 1 i i f j ex s s v v t (21) 36Consider n vehicles accelerating from a queue with the same acceleration (see Figure 2f. 37
the spacing between the first and last vehicle is: 38
1 1, 1 cri cri 1 1 1 1 n extension n i f j ex i n s n s n v v t
(22) 1 Hence, 2
1, cri 1 1 j f ex n f f f n v v t s n H v v v (23) 3So the queue discharge rate equals to: 4
cri cri 1 1 f d j f ex v n q H v v t (24) 5 63.2.2 Analysis of model properties 7 8 0 20 40 60 80 100 120 5000 5500 6000 6500 7000 Speed in congestion (km/h) Q u eue dischar ge r ate ( ve h/h ) capacity empi rica l rel atio n in Fig ure 1 " tex= 0:2 " tex= 0:15 " tex= 0:1 " tex= 0:05 0.2 ex t 0.15 ex t 0.1 ex t 0.05 ex t 9 10 11 12 13 14 Figure 4 Sensitivity of queue discharge rates to reaction time extensions 15 16
The independent impact of the reaction time extension is evalued with Equation (24), see 17
Figure 4a. We examine the relation between the speed in cognestion and the queue 18
discharge rate, setting reaction time extension tex to 0.05s, 0.1s, 0.15s and 0.2s. Figure 19
4a firstly indicates that reaction time extension tex can give a positive relation between 20
the speed in congestion and the queue discharge rate. As the reaction time extension 21
increases even slightly, the queue discharge rate will decrease considerably. When 22
0s
ex
t
, the queue discharge rate equals to the capacity. Secondly, a dynamic reaction 23
time extension can model the empirical observation. The bold line in Figure 4a is the 24
empirical relation revealed in [3] (see Figure 1). The intersections between the bold line 25
and the other lines indicates that to give empirical observations we may need to decrease 26
the reation time extension as the speed in congestion increases. When the vehicular speed 27
in queue reached around 63km/h, there is no capacity drop. That is, the reaction time 28 0 20 40 60 80 100 120 4500 5000 5500 6000 6500 7000 7500 Speed in congestion (km/h) Q ueu e discharg e rat e ( v eh/h) reac tion tim e de crea ses Modelled relation
Empirical relation in Figure 1
a) Sensitivity of the analytical model
to the reaction time extension
b) An intra-driver reaction time extension mechanism for giving empirical observations
extension might be zero. We use vmaxj to indicate the lowest speed in congestion leading 1
to no capacity drop. Hence, we set 2 max max 0, j ex j v t v (25) 3
is a parameter indicating the reaction time extension when the speed in congestion is 0 4
km/h. Varying with , we find a good relationship if we set 0.195s. The modelled 5
realtion with Equation (25) is shown as dark triangulars in Figure 4b. The bold line is the 6
empirical realtion as in Figure 1. The modelled relation can fit the empirical relation quite 7
well, see Figure 4b. 8 9
4. Numerical experiments
10 11In this section, we use numerical experiments to firstly validate the analytical model 12
presented in section 3.1.1. The estimation of queue discharge rate is an approximation. 13
So we need to check whether the approximation is accurate enough. This validation step 14
aims to make our conclusions solid. 15
16
Secondly, we present the combination effects of bounded acceleration spread and the 17
reaction time extensions. A positive reaction time extension can allow a following 18
vehicle to have a faster-than-predecessor acceleration. So the acceleration of the last 19
vehicle in the queue will not follow Equation (8) any more, i.e., a does not have to be n 20
the slowest acceleraion among all vehicles in the queue. The distribution of the last 21
vehicle’s acceleration is difficult to deduce, so we decide to use numerical experiments to 22
see the combination effects of bounded acceleration spread and the reaction time 23
extensions. 24
25
Thirdly, we try to see how to give a same relation between the speed in congestion and 26
the queue discharge rate as empirical observations, considering combination effects of the 27
acceleration spread and the reaction time extension. 28
29
The simulation results in this section correlate quite well with our analytical findings in 30
section 3. No matter whether the reaction time is included or not, the acceleration spread 31
does not contribute sufficiently to the capacity drop. No matter whether the acceleration 32
spread is considered, a negative relation between the reaction time and the speed in 33
congestion can give similar queue discharge rates as empirical observations. 34
35
4.1 Simulation model used 36
37
Figure 5 shows trajectories of two vehicles accelerating from congestion. Vehicle i is 1 38
the leader of vehicle i . Let us set an acceleration difference a . The free-flow spacing 39
between Vehicle i and i will be 1 1
cri
if ai ai1 .So if a aiai1 , the free-a
flow spacing between two vehicles will be smaller than the critical spacing 1
cri
. In
1
Figure 5 we use a dashed line to present a trajectory of vehicle i according to Newell’s 2
model.. Finally the trajectory of vehicle i will ovaerlap with the dashed line. Vehicle i 3
reached the free-flow speed earlier than the dashed trajectory by . The whole t2 4
acceleration process of vehicle i last . Hence, we can have: t1 5
1 1 2 f j i ex v v a t t t (26) 6
1
1 f j i v v a a (27) t 7
2 2 2 2 1 1 3 2 2 f j f j j ex i i f v v v v v t a a a v t (28) 8Equation (26) and (29) describe the acceleration process of the dashed trajectory and 9
vehicle i , respectively. Equation (28) means finally vehicle i will overlap the dashed 10
trajectory when ai aidesireai1 . a 11 12 Time Lo ca ti o n
without reaction time extension with reaction time extension
vehi cle 1 i veh icle i f v w t t1 t2 13 14 Figure 5 Measurement of accelerations when reaction time is extended. 15 16
Combination of Equation (26) - (28) can give: 17 2 1 1 1 2 , for 2 2 i ex f j i ex f j i ex a t a v v a t v v a t (29) 18
Equation (29) shows that the following vehicle can catch up with its predecessor with 19
1
i i
a a when a vf vj 2ai1tex. If ai1 a aidesire, then aiai1 , the free-a 20
flow spacing between vehicle i and i will be critical spacing. If 1 ai1 a aidesire, then 21
desire 1
i i i
a a a , the free-flow spacing between two successive vehicles are: a 22
2 1 1 1 1 2 j f i i i f j ex cri v v a a s v v t (30) 1When vf vj2ai1tex, i.e., the reaction time is too long, and it is impossible for the
2
follower to catch up with the leader. In the this case, the follower’s acceleration will not 3
be limited by its predecessor, i.e., desire
i i
a a . The free-flow spacing between two vehicles 4
will be larger than the critical spacing, calculated according to Equation (31). In 5 summary: 6
desire
1 min , i i i a a a a (31) 7
desir 1 2 1 1 e 1 , for 1 1 1 , ot 2 and herwise 2 i i cri i j f i i f j ex cr x i f j i e v v a a a s v v a a v v t a t (32) 8Finally, in the numerical experiment we calculate the queue discharge flow as: 9
, 2,..., f d i v q i n E s (33) 10Equation (32) and (33) are general expressions for estimating queue discharge rates in the 11
three experiments, that is for the validation of analytical models, the examination of 12
combination effects and the reproduction of empirical observations respectively. 13
14
Since in section 3.1, we found the independent impact of acceleration on the queue 15
discharge rate is marginal. We hypothesize that when considering reaction time 16
extensions, the acceleration spread cannot contribute to queue discharge rate reduction 17
greatly, either. The consequence of the hypothesis is that to obtain the empirically 18
observed queue discharge rate (Figure 1), it is more important to model the impact of the 19
reaction time extension than that of the acceleration spread. Hence, we still use Equation 20
(25) to give the queue discharge rate. 21
22
4.2 Simulation set-up 23
24
For validations of the analytical model in section 3.2.1, we let tex0s. For examining 25
combination effects of the acceleration spread and the reaction time extension, we set two 26
scenarios, i.e., tex0.1s and tex 0.2s . Finally, in the third experiment we give 27
0.18s
.
28 29
At the beginning of the experiment, we set all vehicles’ desired acceleration and reaction 30
time extension. With Equation (31) - (33), we can directly have the final queue discharge 31
rate. The notations of models and the set-up of fundamental diagram are the same as 32
those in section 3. To draw the relation between the speed in congestion and the queue 33
discharge rate, in each scenario set-up we run one simulation with newly distributed 34
desired accelerations for each speed in congestion. We run the simulation for 1000 times 1
to get the expected value and standard spread of queue discharge rates. we set 2
660veh
n , amin 0.5m/s2 and amax 2m/s2. 3
4
4.3 Validations of analytical models 5
6
We approximate the mean queue discharge rate by approximating the expected value of 7
the time-headway in section 3.1. So we need to check whether the approximations are 8
accurate enough to draw conclusions on the independent impacts of accelerations. The 9
comparison between the numerical experiment result and the analytical result is shown in 10
Figure 6a. In Figure 6a, we use error bars and plus signs to indicate the standard spread 11
and the expected value of queue discharge rates respectively for experiment results. 12
Circles show the analytical approximations of queue discharge rates from section 3.1.1. 13
14
We find that the analytical approximations of queue discharge rates fit the numerical 15
experiment results well. Secondly, the queue discharge rate spread increases as the speed 16
in congestion decreases. The fluctuation of queue discharge rate might be a related to the 17
order of desired accelerations. But the spread is not high. All in all, the analysis of the 18
independent impacts of accelerations on the queue discharge rate in section 3.1 is correct. 19
20
4.4 Combination effects of the accelerations spread and reaction time extension 21
22
The combination effects of the acceleration spread and the reaction time extension is 23
examined in numerical experiments, shown in Figure 6b. The experiment results, i.e., 24
mean and standard spreads of queue discharge rates, are shown as plus signs and error 25
bars in Figure 6b, respectively. As a reference, we use circles to indicate the mean queue 26
discharge rate with the independent impact of reaction time, which is the same as shown 27
in Figure 4a. 28
29
In Figure 6b, the acceleration spread hardly contribute to the queue discharge rate 30
reduction. The maximum reduction in experiments is 180 vehicles (around 3% reduction) 31
when tex0.1s and v j 0km/h . Meanwhile, increasing tex from 0.1s to 0.2s 32
decreases the queue discharge rate considerably. When v j 0km/h, the queue discharge 33
rate decreases around 13% (with acceleration spread) and 14% (without acceleration 34
spread). It also means a slight decrease of reaction time can contribute a considerable 35
increase of queue discharge rates. 36
37
Because the acceleration spread can only reduce the queue discharge rate slightly, we use 38
Equation (25) to model mechanism of capacity drop to give queue discharge rates. The 39
experiment results are in Figure 6c. As reaction time decreases when congestion gets 40
lighter, queue discharge rates can fit empirical observations well. 41
0 20 40 60 80 100 120 4500 5000 5500 6000 6500 7000 Speed in congestion (km/h) Qu eu e d isc ha rg e r at e (v eh /h ) combination effects
independent impacts of reaction time 0.1 ex t 0.2 ex t 1 2 3 4 5 6 7 0 20 40 60 80 100 120 4500 5000 5500 6000 6500 7000 7500 Speed in congestion (km/h) Qu eu e disc h arg e rat e (v eh /h )
numerical enperiment results empirical observations reac tion tim e decr ease s 8 9 10 11 12 13 Figure 6 Results in experiments 14 15 6 CONCLUSIONS 16 17
This paper reveals the impacts of bounded accelerations and reaction time on the queue 18
discharge rate. Firstly, we find the impact of inter-driver acceleration spread on the queue 19
discharge rate is rather small. No matter whether the reaction time is considered or not, 20
the acceleration spread can hardly decrease the queue discharge rate. Secondly, a speed-21
dependent reaction time extension mechanism, that is the reaction time decreases as the 22
speed in congestion increases, yields a similar relation between the speed in congestion 23
and the queue discharge rate as found in empirical observations. 24 25 0 20 40 60 80 100 120 6400 6500 6600 6700 6800 6900 Speed in congestion (km/h) Q ueu e d is ch ar g e r at e (v eh/h)
numerical mean and deviations analytical approximations
a) Validation of the analytical model for the inter-driver acceleration spread
b) Combination effects of inter-driver acceleration spread and reaction time extension on queue discharge rates
c) Combination effects on queue discharge rates with intra-driver
Therefore, we conclude that including the acceleration spread when modelling the 1
capacity drop within car-following models is not essential, but including reaction time 2
variations is. Also, this paper gives reasons to believe that a control approach motivating 3
drivers accelerate earlier might be able to considerably benefit maximizing queue 4 discharge rates. 5 6 ACKNOWLEDGEMENTS 7 8
This research is financially supported by China Scholarship Council (CSC) and the NWO 9
grant "There is plenty of room in the other lane". 10
11
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