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An analysis of four methodologies for estimating highway capacity
from ITS data
Zhao Li
•
Rilett Laurence
Received: 12 December 2014 / Revised: 14 April 2015 / Accepted: 16 April 2015 / Published online: 21 May 2015
Ó The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract With the recent advent of Intelligent Transporta-
tion Systems (ITS), and their associated data collection and
archiving capabilities, there is now a rich data source for
transportation professionals to develop capacity values for
their own jurisdictions. Unfortunately, there is no consensus
on the best approach for estimating capacity from ITS data.
The motivation of this paper is to compare and contrast four of
the most popular capacity estimation techniques in terms of
(1) data requirements, (2) modeling effort required, (3) esti-
mated parameter values, (4) theoretical background, and (5)
statistical differences across time and over geographically
dispersed locations. Specifically, the first method is the
maximum observed value, the second is a standard funda-
mental diagram curve fitting approach using the popular Van
Aerde model, the third method uses the breakdown identifi-
cation approach, and the fourth method is the survival prob-
ability based on product limit method. These four approaches
were tested on two test beds: one is located in San Diego,
California, U.S., and has data from 112 work days; the other is
located in Shanghai, China, and consists of 81 work days. It
was found that, irrespective of the estimation methodology
and the definition of capacity, the estimated capacity can vary
considerably over time. The second finding was that, as ex-
pected, the different approaches yielded different capacity
results. These estimated capacities varied by as much as 26 %
at the San Diego test site and by 34 % at the Shanghai test site.
It was also found that each of the methodologies has
advantages and disadvantages, and the best method will be the
function of the available data, the application, and the goals of
the modeler. Consequently, it is critical for users of automatic
capacity estimation techniques, which utilize ITS data, to
understand the underlying assumptions of each of the different
approaches.
Keywords Capacity estimation method Van Aerde
model Breakdown identification PLM
1 Introduction
The Highway Capacity Manual (HCM) has been updated
regularly (1965, 1985, 2000, and 2010) since it was first pub-
lished in 1950 and its underlying theory has remained consis-
tent [1–5]. While the HCM provides a uniform methodology
for estimating the capacity for any highway in the U.S., many
jurisdictions, both within the U.S. and outside the U.S., would
prefer to use capacity values reflective of their own local
conditions. With the recent advent of Intelligent Transporta-
tion Systems (ITS), and their associated data collection and
archiving capabilities, there is now a rich data source for
transportation professionals to develop capacity values for
their own jurisdictions. Unfortunately, there is no consensus on
the best approach for estimating capacity from ITS data.
While this study is concerned with estimating capacity,
the problem is a subset of a much broader issue—how to
identify the fundamental speed–flow–density relationship
for a given facility. If the form of the underlying speed–
flow–density fundamental diagram for a given facility is
known, the capacity may be readil y obtained given the
appropriate empirical data. Needless to say the assumptions
underlying the speed–flow–density function will affect the
resulting capacity estimate.
Z. Li (&)
University of Nebraska Lincoln, 2200 Vine St, 330P Whittier
Research Center, Lincoln 68503, USA
e-mail: li@huskers.unl.edu
R. Laurence
University of Nebraska Lincoln, 2200 Vine St, 262D Whittier
Research Center, Lincoln 68503, USA
123
J. Mod. Transport. (2015) 23(2):107–118
DOI 10.1007/s40534-015-0074-2
This paper first estimates capacity concept using the (1)
maximum method, (2) Van Aerde model, (3) breakdown
identification, and (4) product limit method from ITS data
collected in San Diego, California, U.S. and Shanghai,
China. The values are then compared across time to ex-
amine the variability of capacity estimates as a function of
location and methodology. In addition, the paper compares
these capacity values with those obtained from the HCM.
The paper concludes with a description of the advantages
and disadvantages of each approach.
2 Literature review
2.1 Related work
The HCM 2010 [4] defines the capacity of a facility as ‘‘the
maximum sustainable hourly flow rate at which persons or
vehicles reasonably can be expected to traverse a point or a
uniform section of a lane or roadway during a given time
period under prevailing roadway, environmental, traffic,
and control conditions’’. The HCM capacity is a function of
the free-flow speed. For example, if free-flow speed equals
110 km/h, the capacity is 2,400 pcphpl; if free-flow speed
equals 80 km/h, the capacity is 2,200 pcphpl. This manual
has been adopted in many jurisdictions around the world,
mainly because it provides a single, deterministic value
that represents average conditions. The HCM is based on
speed–flow–density fundamental diagrams that were de-
veloped using empirical flow rate data collected across the
U.S. for similar facilities. The flow rate at the apex of the
speed–flow curve is regarded as the maximum throughput
of the facility and thus is treated as the capacity [6].
While the use of the HCM is straightforward, identifying
capacity from empirical observations is not. Consider the
data shown in Fig. 1 which is from a detector site on the
westbound I405 in San Diego California. Note that this site
will be described in detail later in this paper. Intuitively, a
number of curve fitting algorithms could be used for esti-
mating the speed–flow relationship in Fig. 1. The purple
curve was developed using a standard generic curve fitting
model, and it may be seen that the capacity for this ex-
ample is 2,110 veh/h/ln. However, it may be seen that there
are observed flows that are greater than this estimated ca-
pacity. A number of authors [7– 9 ] have shown that even
under ‘‘constant’’ conditions, both the maximum traffic
flow and the capacity can vary over time and space. Other
authors [10] have argued that the actual maximum ob-
served flow rate of the roadway is the best measure of
capacity because it: (1) is closer to the definition of ca-
pacity listed in the HCM and (2) does not assume a prior
speed–flow–density relationship. Referring to the example
in Fig. 1, the highest observed flow rate is 2,200 veh/h/ln,
and this would represent the capacity according to the
maximum capacity definition. The focus on this paper is to
examine four popular methods of estimating capacity for
ITS data similar to that shown in Fig. 1.
Minderhoud et al. [10] compared several capacity esti-
mation methods including the headway method, bimodal
distribution method, selected maxima method, the funda-
mental diagram method, and the on-line procedure estima-
tion method. The comparison was done based on theoretical
characteristics and no field data was used. The authors
pointed out that there has been no comprehensive study of
the validity and accuracy of these methods and, at the time
the paper was written, this was an open research question.
Similar work was conducted by Geistefeldt and Brilon [11]
who compared stochastic methods for estimating capacity
using empirical data. They found that the estimates varied
over time, and that this should be accounted for in the es-
timation approach. In recent years a number of authors have
developed new methods for estimating highway capacity
using stochastic approaches [12, 13]. In general, these new
approaches are based on existing popular methods such as
model fitting, stochastic distribution, and breakdown-relat-
ed methods that are examined in this paper. The accuracy of
these new methods has not been compared empirically.
This paper compares both deterministic and stochastic
methods using data from the same test beds. Sp ecifically,
four of the most popular methods are first introduced:
Maximum method and Van Aerde method (deterministic)
and breakdown method and PLM method (stochastic).
Capacity values over time were estimated for each test site.
The distribution of estimated capacity values were then
compared statistically. Lastly, the four capacity estimation
techniques were compared in terms of (1) data require-
ments, (2) modeling effort requi red, (3) estimated pa-
rameter values, (4) theoretical background, and (5)
statistical differences across time and over geographically
dispersed locations.
0
20
40
60
80
100
120
140
0 500 1000 1500 2000 2500
Speed (km/h)
Flow rate (veh/h/ln)
Max (fitted) = 2,110
Max (observed) = 2,200
Fig. 1 Flow–speed diagram of westbound I405, San Diego, April
3rd, 2013 (5-min aggregation period)
108 Z. Li, R. Laurence
123
J. Mod. Transport. (2015) 23(2):107–118
2.2 Capacity estimation approaches for a single day
This paper will examine four of the mos t popular capacity
estimation techniques for ITS data. A number of authors
feel that capacity can change as a function of time and
space, all else being equal, and therefore a particular focus
of this paper will be on the techniques that examine the
stochastic nature of capacity. Four of the most widely used
methods (e.g., maximum method, fundamental diagram
curve fitting using the Van Aerde model, breakdown
identification method, and product limit method) were
chosen for capacity estimation . A brief overview of the
calibration and estimation procedures of the first three ca-
pacity methods are provided in this section.
2.2.1 Maximum capacity methodology
Arguably the easiest way to estimate capacity at a given
location is to obtain the maximum flow rate from observed
data measured over a given time period (e.g., a day), as
shown in Eq. (1).
C
i
¼ max f
i;d
8 d ¼ 1; 2; ...; N; ð1Þ
where, C
i
denotes the maximum flow rate (e.g., capacity)
over a given time period d for location i; d denotes the time
interval (e.g., 5 min); f
i,d
denotes the observed flow rate
during time interval d at location i; and N denotes the
number of time periods considered, e.g., N = 288 for
d = 5 min.
When applying this method, the time interv al d will, by
definition, affect the resulting capacity value C
i
. All else
being equal as the time interval d increases, the capacity
value C
i
will decrease. For this reason, it is critical to define
the time interval when presenting capacity values and to
never compare capacity values that were developed from
different time durations. Typical time durations range from
1–60 min. The HCM uses a 15-min time interval when
defining capacity.
2.2.2 Van Aerde capacity methodology
The first and most famous deterministic speed–flow–den-
sity model was developed by Greenshields [14] and is
based on the assumption of a linear speed–density rela-
tionship. Van Aerde [15] proposed a four-parameter model
that provides more degrees of freedom to capture the range
of behavior across different regimes and facility types.
Note that this approach may still be considered a single
regime traffic flow model. The Van Aerde model, which
requires four input parameters, was designed specifically to
be calibrated using empirical ITS data. While the original
applications used field inductance loop data, any detector
data, such as that from radar and video detectors, can be
used. The functional form of Van Aerde model is shown in
Eq. (2).
C
i
¼
u
i
c
1
þ
c
2
u
f ;i
u
i
þ c
3
u
i
; ð2Þ
where, C
i
denotes the estimated capacity for location i; u
i
denotes the space mean speed (km/h) for location i; u
f,i
denotes the free flow speed (km/h) for location i; and, c
1
,
c
2
, and c
3
denote the headway constant coefficients.
The model parameters are calculated using Eqs. (3–6 ).
m ¼
2u
c
u
f
ðu
f
u
c
Þ
2
; ð3Þ
c
1
¼ mc
2
; ð4Þ
c
2
¼
1
k
j
m þ
1
u
f
2
; ð5Þ
c
3
¼
c
1
þ
u
c
q
c
c
2
u
f
u
c
u
c
; ð6Þ
where, u
c
denotes the speed at capacity (km/h); q
c
denotes
the flow at capacity (veh/h); and k
j
denotes the jam density
(veh/km).
Once the model is calibrated, the capacity is identified as
the maximum flow defined by the calibrated speed–flow
curve. Because of its simplicity, this approach is relatively
easy to program.
2.2.3 Breakdown capacity methodology
While widely used around the world, the general approaches
described above (e.g., HCM, Van Aerde, etc.) have been cri-
ticized because they do not consider the stochastic nature of
congestion and thus may be unsatisfactory for traffic op-
erations applications [16, 17]. It has been argued that the point
of traffic breakdown might be more appropriate for estimating
capacity. Elefteriadou et al. [7, 18] showed that the traffic
breakdown does not necessarily occur at the same volume
level over different days and therefore that capacity should not
be treated as a deterministic value. It should be noted that
while the newly updated HCM 2010 acknowledges that ca-
pacity is stochastic, it does not identify a methodology to
estimate the stochastic component of capacity.
In g eneral, a breakdown is defined when the speed de-
crease between two consecutive time intervals exceeds a
pre-specified threshold, and this lower speed is sustained for
a predefined length of time [18]. A review of the literature
indicates that the breakdown speed threshold and the con-
gested time duration are location dependent. For example,
Lorenz and Elefteriadou [19] defined breakdown as occur-
ring when the average speed of all lanes drops below 90 km/
h for a period of at least 5 min. Brilon et al. [17 ] used 70 km/
h for their studies in Germany. Zhang et al. [15] studied
An analysis of four methodologies for estimating highway capacity from ITS data 109
123
J. Mod. Transport. (2015) 23(2):107–118
several freeway sites in the US and concluded that the ‘‘low
speed’’ condition has to be sustained for at least 15 min to be
recognized as a ‘‘true’’ breakdown instance.
Based on Elefteriadou [18], the underlying logic for
identifying roadway capacity is shown by the flow chart in
Fig. 2. The breakdown time duration in this case is 3
consecutive time intervals or 15 min.
As discussed above, the speed threshold and the break-
down durati on time are location dependent, and conse-
quently, it is critical that the user picks appropriate values.
Intuitively, if the thresholds are too ‘‘liberal,’’ many
‘‘breakdowns’’ will be identified, and if the thresholds are too
‘‘conservative’’, no breakdown will be identified. Unfortu-
nately, there is no theoretical approach for identifying these
values and engineering judgment is often used. A downside
to this approach is that two researchers using the sam e data
may choose different thr eshold values, which could result in
different capacity estimates.
Banks et al. [19] also proposed a multi-regime traffic
flow model based on the concept of breakdown. The au-
thors proposed a two-capacity model consisting of pre-
queue flow (PQF) and queue discharge flow (QDF). They
observed that flow immediately downstream of the bottle-
necks decreased by a small amount at the breakdown point
(i.e., PQF [ QDF), which is termed as ‘‘capacity drop’’
[20, 21]. Similarly, Cassidy and Hall et al. [22, 23] claimed
that there was an approximately 10 % reduction in max-
imum flow rates after the onset of congestion. Such a flow
breakdown appeared to be triggered by speed instability.
In summary, the above three capacity estimation methods
(Sects. 2.2.1–2.2.3) are widely used to estimate capacity at a
given location for a single day. In general, the first two
methodologies (e.g., maximum flow and curve fitting) are
used in planning applications, and the breakdown method is
used in traffic operation applications. What is important to
note is that the three methods can lead to different capacity
values. As an example, consider Fig. 3, which shows a speed–
flow diagram for westbound I405 in San Diego, California,
where the green dots represent the observed flow rates over
5 min time durations for 288 periods (e.g., one complete day).
The largest observed flow is 2,289 veh/h/ln and is indicated as
Method 1 on the graph. The purple line shows the line of best
fit using the Van Aerde approach. This results in a capacity
estimate of 1,774 veh/h/ln and is indicated as Van Aerde
Capacity in Fig. 3. Using the definition of breakdown by
Elefteriadou [18], the breakdown flow rate of 2,223 veh/h/ln
can be identified. This is indicated as Breakdown Capacity in
Fig. 3. This breakdown capacity point delimits the congested
and uncongested parts of the observed traffic flows.
In addition, each of these approaches can be repeated
across days. This would then yield a histogram of estimated
capacity values and the user could use some measure of
central tendency (e.g., mean or median) to identify the
capacity. The following sections describe one approach for
identifying capacity based on the assumption that capacity
is intrinsically stochastic.
2.3 Capacity estimation method for multiple days:
product limit method
Based on daily observations of traffic data collected over
several months, Brilon et al. [17] argued that capacity is
Weibull-distributed with a nearly constant shape pa-
rameter. Based on this observation, the authors developed
the product limit method (PLM) for estimating the capacity
distribution function from empirica l data.
The PLM is related to (1) the breakdown flow rate, and
(2) those flow rates that do not result in a breakdown oc-
currence. There might be certain intervals that exceed the
threshold, and these are regarded as censored data (i.e., data
in intervals that do not provide breakdown information). In a
similar manner, traffic breakdown is regarded as a failure
Fig. 2 Flow chart of breakdown capacity estimation
110 Z. Li, R. Laurence
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J. Mod. Transport. (2015) 23(2):107–118
event and it is analogous for estimating the capacity as the
lifetime in a lifetime data analysis [24]. The statistics of this
lifetime analysis can be then used to estimate the parameters
of the distribution function, which includes the censored
data. The survival function, described by the non-parametric
PLM [25], is shown in Eq. (7).
F
c
ðqÞ¼1
Y
i:q
1
q
k
i
d
i
k
i
; ð7Þ
where F
c
(q) denotes the distribution functi on of capacity c;
q denotes the traffic flow rate; q
i
denotes the traffic flow
rate in interval i; k
i
denotes the number of intervals with a
traffic volume of q C q
i
; and i denotes the 5 min intervals
belonging to the lifetime T (i.e., a day).
The following five steps are used to define the capacity:
(1) Identify breakdowns using flow chart in Fig. 2. Then,
delete all the data in the interval that a breakdown
happens. Order the 5-min interval flow and speed
censored data over time.
(2) Group the data in 15 veh/h incremental of flow levels.
Count the total number of intervals y
i
and the
breakdown intervals d
i
that fall into each group i;
(3) Calculate the survival probability p
ci
¼ 1
d
i
y
i
for
each 15 veh/h increment of flow levels.
(4) Calculate the overall bottleneck capacity by con-
tinuously production: Sðq
j
Þ¼
Q
j
i¼1
p
ci
.
(5) Plot the survival probability curve. Choose a survival
probability (e.g., 95 %), and estimate the capacity
from the curve.
3 Sites and data
3.1 The study sites
The two study sites used in this paper were chosen because
they contained known ‘‘bottlenecks’’ where breakdown
phenomena would more readily be observed [17]. The first
is the westbound section of Interstate 405 (I405) in San
Diego, California, as shown in Fig. 4a. It may be seen that
there are four lanes and an off-ramp upstream at the detector
location. The posted speed limit is 70 mi/h (110 km/h). The
area is primarily urban and is oft en congested. Ap-
proximately 6 months of dual inductive loop data, lasting
from March 3, 2013 through Sep. 27, 2013, were collected
from the Caltrans Performance Measurement System
(PeMS) website [26] at loop detector ID = 1217573. The
30-second flow and speed by-lane data were aggregated into
5-min intervals over the entire 24 h period (e.g., 288 5 min
periods per day). A total of 112 weekdays were observed.
The second site is an eastbound section of the Inner-
Ring Expressway located at the Wuning crossroa d in
Shanghai, China, as shown in Fig. 4b. There are two main
lanes, and the posted speed limit is 80 km/h. The loop
detector (ID = NHWN_40) is located approximately
102 m downstream from the Merge End, shown in Fig. 4b.
The area is prima rily urban, and the site experiences sig-
nificant congestion. Approximately 5 months of dual in-
ductive loop data, from June 1, 2010 through Octobe r 29,
2010, were collected. The 20-second flow and speed by-
lane data were aggregated into 5-min intervals over the
entire 24 h period (i.e., 288 5-min periods per day). A total
of 81 weekdays were observed.
Note the two test beds were chosen because both sys-
tems have been studied extensively and have been
calibrated on a regular basis [27–30]. In addition, prior to
the analysis, the data were analyzed to identify detector
malfunctions (e.g., missing data or abnormal data) or ex-
traordinary events (e.g., congestion time and occupancy
were order-of-magnitude larger than average). No outlier
events were identified.
3.2 Preliminary data analysis
Figure 5a shows the speed and flow, aggregated to a 5 min
average, as a function of time of day for the San Diego test
site on April 17, 2013. It may be seen that the traffic flows
fairly smoothly as evidenced by the relatively high speed
experienced for the majority of the day. The exception is
during the period from 14:45 to 19:05 where there is a
considerable decrease in both flow rate and speed, which is
an indication of congested conditions. After this period, the
speed recovers to pre-congested conditions. At ap-
proximately 14:45, there is a 22 % speed drop, and this is
the breakdown time identified using the logic in Fig. 2.It
should be noted that this pattern is typical for the weekdays
in the test dataset.
Figure 5b shows the flow and speed as a function of
time at the Shanghai site on July 1, 2010. This day may be
considered typical of weekdays for the data set. It may be
0
10
20
30
40
50
60
70
80
90
100
110
120
0 500 1000 1500 2000 2500
Speed (km/h)
Flow rate (veh/h/ln)
2. Van Aerde capacity
3. Breakdown
capacity
1. Max capacity
congested
Uncongested
Fig. 3 Three capacity estimation methods (data in 5-min aggrega-
tion, westbound I405, San Diego)
An analysis of four methodologies for estimating highway capacity from ITS data 111
123
J. Mod. Transport. (2015) 23(2):107–118
seen that congestion lasts for approximately 16 h from 7:00
AM until approximately 11:00 PM. It can be seen in
Fig. 5b that the speed gradually decreases over time
starting at approximately 7:00 AM. This can be contrasted
with Fig. 5a where the speed decrease is much more
abrupt. However, the breakdown identification approach
shown in Fig. 2 can be used for both situations.
As discussed in Sect. 2.2.3, in order to find the break-
down point, both a threshold speed and breakdown dura-
tion need to be identified. In this study, the threshold values
were based on a study of the 5-min average speed–flow
data at the study sites. It was decided that 95 and 50 km/h
were appropriate threshold values for the San Diego site
[28] and Shanghai site [29], respectively. This paper
identified a breakdown when (1) the speed decreased below
the corresponding threshold, and (2) the lower speed was
sustained for three consecutive 5-min intervals.
4 Capacity estimation analyses
This section first applies the maximum method, Van Aerde
method, and breakdown method to estimate capacity for
each day in both data sets. The resulting six capacity dis-
tributions (e.g., for both sites and all three methods) are
plotted. Subsequently, the capacity over all the study days
is estimated using the PLM approach for both test sites and
compared with the measures of central tendency for the
capacity distributions obtained by the three methods.
4.1 Capacity estimation and comparison
A preliminary analysis of the San Diego indicated the
speed–flow relationships could be federated into two dis-
tinct categories. Representative days for these two cate-
gories are April 17, 2013 and June 4, 2013, and the speed–
flow diagrams for these days are shown Fig. 6a and b, re-
spectively. Note that in these figures the data is aggregated at
5 min and (1) the ‘‘Ma x Capacity’’ is the largest five minute
Fig. 4 The layout of the study sites. a Fairview Rd off-ramp, I405, San Diego site. b Wuning Rd on-ramp, inner-ring expressway, Shanghai site
(b)
(a)
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00
Speed (km/h)
Flow rate (veh/h/ln)
Time
Flow
Speed
Congestion
24:00
Breakdown
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36
Speed (km/h)
Flow rate (veh/h/ln)
Time
Flow
Speed
24:00
Congestion
Breakdown
Fig. 5 Typical traffic flow and speed profiles at the two sites. a San
Diego site, April 17th, 2013. b Shanghai site, July, 1st, 2010
112 Z. Li, R. Laurence
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J. Mod. Transport. (2015) 23(2):107–118
flow rate observed for that particular day, (2) the ‘‘Van
Aerde Capacity’’ is fitted by Van Aerde model from the
empirical speed-flow relationship, and (3) the ‘‘Breakdown
Capacity’’ is iden tified by the breakdown occurrence that is
immediately prior to the start of speed drop, as shown in
Fig. 2. In Fig. 6a, the capacities estimated using the max-
imum capacity, Van Aerde capacity, and Breakd own ca-
pacity methods are 2,109, 2,111 and 2,019 veh/h/ln,
respectively. It should be noted that the differences among
the three values are within 5 % of each value.
In contrast, consider the empirical data plotted in Fig. 6b
which shows capacities estimated on June 4, 2013 for the San
Diego site. In this case, there is a discontinuity between the
uncongested and congested traffic regimes. This is also known
as a capacity drop as discussed in Sect. 2.2.3. From the data in
Fig. 6b, the capacities estimated using the maximum capacity,
Van Aerde capacity, and Breakdown capacity methods are
2,360, 1,756 and 2,240 veh/h/ln, respectively. It should be
noted that the differences among these three values are within
26 % of each estimated capacity.
Note that the speed–flow pattern shown in Fig. 6ais
typical for the San Diego site in that for 82 % of the days
the differences of the three capacities are within 5 % of
each other. Specifically, the capacity drop phenomena,
identified in Fig. 6b, were observed only on nine out of the
112 days. During these 9 days the smallest and largest
differences in capacity among the three techniques were
9.7 % and 26 %, respectively.
In addition , for all 112 days at the San Diego test site the
average difference between the breakdown point and the
maximum capacity is 5 %. Consequently, on a typical day
there is little change in average speed as the flow rate
approaches its maximum value. Based on this fact, it could
be argued that for this location breakdown capacity is a
good approximation for maximum capacity. However, this
is not always the case as will be demonst rated in the
Shanghai test site analyses.
Figure 7 illustrates the relationship between speed and
flow rate at the Shanghai test site on July 3, 2010. It may be
seen that the speed gradually decreases as flow rate increases,
which is in marked contrast to what was observed at the San
Diego site. The pattern shown in Fig. 7 was typical for all
81 days studied. The breakdown point, identified using the
logic in Fig. 2, occurs at a considerably lower flow rate than
the maximum flow rate. This is referred to ‘‘early onset
breakdown’’ by Sun et al. [29, 30]. The estimated capac ities
for maximum method, Van Aerde method, and breakdown
method are 2,086, 2,076, and 1,866 veh/h/ln, respectively.
The differences of the three values are within 11 % of each
value, which is much higher than observed at the San Diego
site (Fig. 6a). In addition, for all 81 days, the smallest and
largest differences in capacity among the three tec hniques
were 14 % and 34 %, respectively.
4.2 Overall capacity at each site
Histograms and boxplots of the estimated capacities at the
San Diego test site for the Maximum Capacity, the Van
Aerde Capacity, and the Breakdown Capacity methods are
given in Fig. 8a, b, and c, respectively. It may be seen in
Fig. 8a that the capacities for the maximum capacity
method range from 1,760 to 2,504 veh/h/ln with a mean of
2,188 veh/h/ln. There is considerable spread in the data as
evidenced by the standard deviation of 180 veh/h/ln. It can
be seen in the boxplots that the distribution is skewed left
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 500 1000 1500 2000 2500
Speed (km/h)
Flow rate (veh/h/ln)
Fitting Curve
Speed Flow Data
Speed threshold
Van Aerde
capacity
Breakdown capacity
(a)
Max
capacity
Van Aerde
fitting curve
Speed flow data
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 500 1000 1500 2000 2500
Speed (km/h)
Flow rate (veh/h/ln)
Fitting Curve
Speed Flow Data
Breakdown capacity
Van Aerde
capacity
Speed threshold
(b)
Max
capacity
Van Aerde
fitted curve
Speed flow data
Fig. 6 Capacity estimates at San Diego site. a April 17th 2013. b June 4th 2013
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 500 1000 1500 2000 2500
Speed (km/h)
Flow rate (veh/h/ln)
Fitting Curve
Speed Flow Data
Speed threshold
Breakdown capacity
Max
capacity
Van Aerde capacity
Van Aerde fitted curve
Speed flow data
Fig. 7 Comparison of the two capacity estimates at Shanghai site,
July 3rd 2010
An analysis of four methodologies for estimating highway capacity from ITS data 113
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with a kurtosis value of -1.1, which indicates that more
than half of the observed daily maximum capacities are
higher than the mean value.
The histogram and boxplot of the capacities from the
Van Aerde method are shown in Fig. 8b. It may be seen
that the daily capac ity has a range of 1,732–2,580 veh/h/ln,
a mean of 2,082 veh/h/ln, and a standard deviation of 213
veh/h/ln. The Van Aerde capacity also has a bi-modal
distribution and is also skewed left with a kurtosis value of
-1.2. It was hypothesized that this occurred because of the
9 days in which a capacity drop was observed.
The histogram and boxplot of the daily capacities from
the breakdown method are shown in Fig. 8c. In contrast to
the first two graphs, this distribution is unimodal and
approximately uniform, though the Kolmogorov–Smirnov
test did not show significant under uniform distribution (see
Table 1). The mean value is 2,126 veh/h/ln, with a median
of 2,096 veh/h/ln, and both of these metrics are good
indicators of the central tendency.
Histograms and boxplots of the estimated capacities at
the Shangha i test site for the maximum capacity, the Van
Aerde capacity, and the breakdown capacity methods are
given in Fig. 9a, b, and c, respectively. It may be seen in
Fig. 9a that the capacities for the maximum capacity method
range from 1,923 to 2,178 veh/h/ln with a mean of 2,049
veh/h/ln. There is considerably less spread in the estimated
capacities, as compared to the San Diego test site, as evi-
denced by the standard deviation of 56 veh/h/ln. The Van
Aerde capacity has a similar distribution as seen in Fig. 9b.
The daily capacity values estimated using this method range
from 1,986 to 2,178 veh/h/ln with a similar mean of 2,066
veh/h/ln and a standard deviation of 47 veh/h/ln.
It may be seen in Fig. 9c that the capacities for the
Breakdown Capacity method range from 1,434 to 2,128
veh/h/ln. The mean of this distribution was 1,789 veh/h/ln,
which is 14 % and 13 % lower than that of the maximum
method and Van Aerde method, respectively. In addition,
there is considerably more spread in the estimated ca-
pacities, as compared to the maximum capacity and Van
Aerde capacity method, as evidenced by the standard de-
viation of 146 veh/h/ln. It may be seen that the distribution
is unimodal and is approximately Gaussian shaped.
The stochastic nature of the estimated capacities for
each methodology is captured in Figs. 8 and 9. While a
visual inspection can indicate the general form (e.g., uni-
form, Gaussian, etc.) for each method, it is unclear which
distribution, if any, best fits the estimated values. Four test
distributions, the Normal/Gaussian, Lognomal, Weibull,
and Uniform, were tested using a one-sample Kolmogorov-
Simirnov test. The testing was conducted at the 5 % level
of significance, and the results are shown in Table 1.
The last column in Table 1 shows the ‘‘best’’ distribu-
tion based on the p value. It can be seen at the San Diego
site none of the generic distributions was found to provide
a statistically significant fit for the maximum and Van
Aerde methods. This is not unexpected given the bi-modal
nature of these latter distributions and the fact that the
tested distributions are unimodal.
It can be seen from Figs. 8 and 9, and Table 1 that the
capacities vary widely depending on test site and capacity
estimation technique. For example, the capacity estimated
by maximum method at San Diego is similar to a bimodal
distribution. In contrast, the capacities estimated at
Shanghai test site (Fig. 9) were normally distributed with a
comparatively tight range.
Based on the PLM described in Sect. 2.3, the capacity
value in this paper is taken to be the 5th percentile value of
Fig. 8 Histogram and box plots of estimated capacity by maximum
(a), Van Aerde (b), and breakdown (c) Methods at San Diego Test
Site
114 Z. Li, R. Laurence
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J. Mod. Transport. (2015) 23(2):107–118
the cumulative distribution function breakdown [12]. This
means there is 95 % probability that a bottleneck will have
dispersed when the flow rate is greater than the PLM ca-
pacity as shown in Fig. 10. The PLM-based capacity is
identified as 2,217 and 1,733 veh/h/ln at San Diego site and
Shanghai site, respectively, as show n in Fig. 10.
The PLM capacity is compared with the capacity esti-
mated from the maximum method, Van Aerde method, and
breakdown method as shown in Table 2. At the San Diego
site, the 25th and 75th percentile of capacity range esti-
mated by all the three methods includes the PLM capacity.
In contrast, at the Shanghai site, only the breakdown
method is within the 25th and 75th of capacity range, while
the PLM capacity does not fit in the ranges for either the
maximum flow method or Van Aerde method.
The Wilcoxon signed-rank test (nonparametric test) is
used to test the hypothesis that there are no significant
differences between the PLM capacity and capacities es-
timated by the three methods. The tests were conducted
with a-value equal to 0.05. In other words, a p value larger
than 0.05 will mean the alternative hypothesis is rejected.
As seen in Table 2, it can conclude that at the San Diego
site, the capacity estimated by the PLM method is statis-
tically the same as the capacity values obtained by the
maximum method and the Van Aerde method. In contrast,
it is concluded that the capacity estimated by the PLM
method is statistically different than the capacity estimated
by the breakdown method.
At the Shanghai site, the PLM capacity is statistically
the same as the breakdown capac ity as evidenced by the
fact that the p value of 0.09 is greater than 0.05. The 25th–
75th percentile ranges for the maximum and Van Aerde
methods, however, do not include the PLM capacity. This
would be expected because of the natu re of the PLM ap-
proach which first identifies a breakdown and then uses
only censored data to estimate capacity.
As a point of reference, HCM capaciti es for San Diego
and Shanghai, in terms of units, are also shown in Table 2.
Table 1 One-sample Kolmogorov–Smirnov Test (at 5 % level of significance)
Normal Lognormal Weibull Uniform Distribution type
San Diego
Maximum method 0.014 \0.001 0.018 0.034 None
Van Aerde method 0.003 \0.001 \0.001 \0.001 None
Breakdown method 0.016 0.010 0.076 0.029 Weibull
Shanghai
Maximum method 0.200 0.150 \0.001 0.005 Normal/lognormal
Van Aerde method 0.007 \0.001 0.155 0.006 Weibull
Breakdown method 0.028 0.059 0.036 \0.001 Lognormal
H
0
the data is from the to-be-tested distribution. Small p values reject H
0
Italics values signifies at 5% sigificant level
Fig. 9 Histogram and box plots of estimated capacity at Shanghai by
maximum, Van Aerde and breakdown method, respectively
An analysis of four methodologies for estimating highway capacity from ITS data 115
123
J. Mod. Transport. (2015) 23(2):107–118
These were based on their free-flow speeds, and it shoul d
be noted that heavy vehicle factor was not accounted.
Comparing the capacities obtained by the PLM method-
ology and the HCM methodology, it may be seen that the
values are much closer at the San Diego site. In contrast,
the capacity values estimated by the PLM and HCM ap-
proaches are considerably different at the Shanghai site.
However, as seen in Table 2, in comparison to the HCM
capacity, the maximum method and Van Aerde model
provide capacity estimates that are 7 % closer.
5 Concluding remarks
A wide variety of authors have developed methodo logies
for estimating capacity from empirical data sets that are
collected automatically as part of an ITS. These include (1)
simply observing the maximum flow rate, (2) using basic
curve fitting techniques based on simple assumptions re-
lated to the fundamental diagram, (3) calculating capacity
estimates based on breakdown phenomena, and (4) ap-
proaches for estimating average daily capacity based on the
stochastic nature of capacity and multiple days of data.
Note that a comprehensive literature review is beyond the
scope of this paper. Instead, the goal was to compare and
contrast four of the most popular capacity estimation
techniques in terms of (1) data requirements, (2) modeling
effort required, (3) estimated parame ter values, (4) theo-
retical background, and (5) statistical differences across
time and over geographically dispersed locations.
In summary, there were three major conclusions arising
from these analyses. The first is that irrespective of the
estimation methodology and the definition of capacity, the
estimated capac ity varies over time. For this case study,
three methods (e.g., maximum method, Van Aer de model,
and breakdown method) were used to estimate the capacity
at the San Diego test site and the Shanghai test site over
multiple days. It was found that at the San Diego site, the
differences of the estimated capacity were within 5 %
among 103 of the 112 days (e.g., 82 % of the time). For
nine of the 112 days a ‘‘capacity drop’’ was identified, and
on these days, the smallest and largest difference s in esti-
mated capacity were 9.7 % and 26 %, respectively. In
contrast, at the Shanghai site the capacity estimates for the
three methodologies were between 14 % and 34 % for all
81 days. There was not capacity drop observed at the
Shanghai site. While the two tes t sites were located on
major thoroughfares in two large metropolitan areas, the
estimated daily capacity values were markedly different. It
is hypothesized that this occurred because of differences in:
(1) vehicle types/capabilities, (2) vehicle distributions, and
(3) driving behavior.
It was also found that the estimated capacity at the
Shanghai site was markedly different than the equivalent
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1000 1500 2000 2500
Survival probability
Capacity (veh/h/ln)
2217 veh/h/ln
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1000 1500 2000 2500
Survival probability
Capacity (veh/h/ln)
1733 veh/h/ln
(b) Shanghai site
(a) San Diego site
Fig. 10 Capacity estimate by PLM at each site
Table 2 Estimated capacity distribution statistics (unit: veh/h/ln)
Site Method Mean Median [25th, 75th] percentile PLM capacity Nonparametric
test (a = 0.05)
HCM capacity
San Diego Maximum 2,188 2,264 [1,976, 2,360] 2,217 0.7925 2,400
Van Aerde 2,188 2,260 [1,968, 2,364] 0.1641
Breakdown 2,078 2,092 [1,892, 2,231] 0.0162
Shanghai Maximum 2,049 2,035 [2,006, 2,097] 1,733 \0.0001 2,200
Van Aerde 2,066 2,058 [2,028, 2,100] \0.0001
Breakdown 1,789 1,782 [1,686, 1,860] 0.0910
Unit in HCM: passenger car per hour per lane
Italic values signifies at 5% sigificant level
116 Z. Li, R. Laurence
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J. Mod. Transport. (2015) 23(2):107–118
HCM values. Using the HCM methodology the est imated
capacity would be 2,400 and 2,200 pcphpl for fre e flow
speeds of 110 and 80 km/h, respectively. In contrast, the
estimated capacity at the Shanghai test side ranged from
1,686 to 2,100 veh/h/ln. It is hypothesized that the differ-
ence in capacity may come from the heavy vehicle factor
which was ignored in this study and the fact that the HCM
is based on 15 min aggregate flows, while this study uti-
lized 5 min aggregate flows.
Regardless, it is advantageous to understand the distri-
bution of capacities over time when selecting the ‘‘true’’
capacity at a site. Simply picking a day at random will, in
all likelihood, lead to an erroneous result unless the traffic
flow rates are homogeneous over time.
The second finding was that, not surpris ingly, the esti-
mated capacities were different for each of the techniques.
In general, the maximum method focused on the maximum
value of the traffic flow rates over a day, and therefore it
tends to define the upper bound of the estimated capacity
range. The Van Aerde method estimated the theoretical
capacity value by regressing all the flow rate data over a
single day. It was found that the differences of the esti-
mated capacity values are within 5 % over all days. Ca-
pacities obtained by the breakdown method vary largely
depending on the traffic situation at a particular site.
Usually, this approach identifies a lower capacity than the
other techniques, and this may serve as a lower bound of
the estimated capacity range. The PLM estimation values
typically are within the ranges identified b y the first three
methods. This is not surprising because the PLM estima-
tion explicitly accounts for day by day variability.
The third finding was that each of the methodologies has
advantages and disadvantages, and the best method will be
function of the available data, the application, and the goals of
the modeler. It was found that the maximum method is the
easiest method, from a computational perspective, for esti-
mating capacity from empirical data. However, there was very
little variation in capacity estimates over time using this
method, and the method cannot be used to identify major
changes in flow rate over short periods of time. It was found
that the Van Aerde model is not tied directly to breakdown
events and thus can be used to obtain deterministic capacity
over the study period. This methodology is based on the traffic
flow theory and does not require the user to identify the status
of traffic flow on freeway a priori. It is also easy to automate
and is particularly useful for uniform traffic flow such as the
San Diego site. The breakdown identification method ac-
counted for the stochastic nature of capacity, which many
authors believe that it leads to more credible results. The au-
thors argued that it is necessary to identify the breakdown
conditions, otherwise the modeler is unsure of whether a
higher flow rate could be observed. Similar to the previous two
models, the breakdown method is easily adapted to different
types of freeways. Although the determination of an appro-
priate breakdown capacity is not straightforward for many
applications, it is critical to identify breakdown points, and in
these situations, such as at the Shanghai site, breakdown-re-
lated approaches are very useful.
In future work, several aspects that would increase the
accuracy of capacity estimation could be addressed in-
cluding (1) a larger sample size with more days and mul-
tiple locations, (2) an analysis of the effect of vehicle type
including percentage of heavy vehicle s, (3) an analysis of
the traffic flow distribution across lanes, and (4) an analysis
of the traffic flow distribution in terms of the effect of
weather. Lastly, it would be interesting to relate individual
driver behavior characteristics under distinctive traffic si-
tuations to the capacity estimation.
Acknowledgments Special thanks are due to Dr. Jian Sun of the
Tongji University in China and the PeMS in California for their as-
sistance in providing data. The analysis and findings related to this
data are strictly those of the authors and not necessarily those of the
people and organizations who provided the underlying data.
Open Access This article is distributed under the terms of the Crea-
tive Commons Attribution 4.0 International License (http://creative-
commons.org/licenses/by/4.0/), which permits unrestricted use,
distribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a link
to the Creative Commons license, and indicate if changes were made.
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