Dynamic Traffic Flow Model for Travel Time Estimation

Abstract

Travel time, as a fundamental measurement for intelligent transportation systems, is becoming increasingly important. Because of the wide deployment of fixed-point detectors on freeways, if travel time can be accurately estimated from point detector data, the indirect estimation method is cost-effective and widely applicable. This paper presents a modified dynamic traffic flow model for accurately estimating the travel time of freeway links under transition and congestion conditions with fixed-point detector data. The modified estimation model is based on a thorough analysis of the dynamic traffic flow model. The applications and the limitations of the model are analyzed for theory, equation derivation, and modifications. Through a simulation study and real traffic data, the (modified) dynamic models are compared according to performance measurements. A comparison of the estimated results and measurement errors shows the accuracy of the modified dynamic model for estimating the travel times of freeway links under transition and congestion traffic conditions.

Full text

70
Transportation Research Record: Journal of the Transportation Research Board,
No. 2526, Transportation Research Board, Washington, D.C., 2015, pp. 70–78.
DOI: 10.3141/2526-08
Travel time, as a fundamental measurement for intelligent transpor-
tation systems, is becoming increasingly important. Because of the wide
deployment of fixed-point detectors on freeways, if travel time can be
accurately estimated from point detector data, the indirect estimation
method is cost-effective and widely applicable. This paper presents a
modified dynamic traffic flow model for accurately estimating the travel
time of freeway links under transition and congestion conditions with
fixed-point detector data. The modified estimation model is based on
a thorough analysis of the dynamic traffic flow model. The applica-
tions and the limitations of the model are analyzed for theory, equa-
tion derivation, and modifications. Through a simulation study and real
traffic data, the (modified) dynamic models are compared according to
performance measurements. A comparison of the estimated results and
measurement errors shows the accuracy of the modified dynamic model
for estimating the travel times of freeway links under transition and
congestion traffic conditions.
With the dramatic growth of urban traffic congestion and the accom-
panying growth in the need for accurate traffic information, intelli-
gent transportation systems (ITS), which involve the application of
advanced electrical engineering technologies and methods, have taken
on an important role in managing traffic flow and providing real-time
traffic information. Travel time, as a fundamental measurement, is a
basic input for ITS.
Various methods have been developed in recent decades to
acquire travel time data. Some methods sample travel time by using
test vehicles, license plate matching, electronic distance measuring
instruments, video imaging, and probe vehicles, such as automatic
vehicle identification and automatic vehicle location. However, these
methods are not widely deployed because of costs and privacy
issues (1, 2).
The predominant detection technology applied on freeways uses
fixed-point detectors, that is, the inductive loop detector. Fixed-point
detection records continuous traffic condition data for the entire
traffic stream. Single loop detectors can provide traffic flow and
occupancy data but cannot directly measure vehicle speed or travel
time. Dual loop detectors can provide spot speed measurements in
addition to traffic flow and occupancy data but likewise cannot
provide direct travel time measurements. A method that can accu-
rately estimate link travel time from point measurements would be a
cost-effective and widely applicable travel time estimation tool (3).
Nam and Drew developed a dynamic traffic flow model for estimat-
ing freeway travel time from flow measurements taken at upstream
and downstream stations (4). The model was based on the charac-
teristics of a stochastic vehicle counting process and the principle
of the conservation of vehicles. In the model, the initial traffic on
the link is required, the density is calculated from the cumulative
flow measurements, cumulative arrival–departure flow diagrams
are used to estimate travel time, and the vehicles considered as targets
for the travel time estimation are different for both the free-flow and
congestion conditions, which provides two separate models with
different diagrams in different traffic situations.
In some previous studies (5–7), it was pointed out that the accuracy
of the determined flow rates and calculated densities will affect the
model. This model requires the number of vehicles that initially exist
on the link. Moreover, the model uses an inductive approach along
with geometric interpretations of cumulative arrival–departure flow
diagrams (8), so it is not appropriate for low-volume conditions
because of the smoothing of the cumulative flow curve. Vanajakshi
et al. modified the Nam–Drew model by combining the two separate
models into one that uses the same targeted vehicles (2, 3). To decrease
the sensitivity of the density and flow parameters, Vanajakshi et al.
suggested use of occupancy values to estimate density (3). For low
traffic flow conditions, the speed extrapolation model can be used to
estimate the travel time (3).
In addition to the mentioned limitations, the Nam–Drew model
assumes constant downstream flow rates during consecutive time
intervals. However, in the field situation, the flow changes with time,
and the assumption is not applicable. This paper improves the model
by using the cumulative flow diagram. The theory, equation deriva-
tion, and modification of the model are described. To test the accuracy,
a simulation study and field data are used to compare the dynamic
traffic flow models.
DYNAMIC TRAFFIC FLOW MODEL
The theoretical concept of the dynamic traffic flow model, also called
the Nam–Drew model, is based on the stochastic process and con-
servation of vehicles. The equation of the model can be derived from
the flow–density–speed relationship and the diagram of the cumulative
flows of the two stations, as the following procedures indicate.
Stochastic Process
In the stochastic process, the two stations’ cumulative flows Q(xu, tn),
Q(xd, tn) by time tn (n ∈ [1, n]), defined as the number of vehicles
Dynamic Traffic Flow Model
for Travel Time Estimation
Ting Yi and Billy M. Williams
T. Yi, Transportation Planning and Traffic Engineering, CDM Smith, 8140 Walnut
Hill Lane, Suite 1000, Dallas, TX 75231. B. M. Williams, Department of Civil,
Construction, and Environmental Engineering, North Carolina State University,
Campus Box 7908, Raleigh, NC 27695-7908. Corresponding author: T. Yi, yit@
cdmsmith.com.

Yi and Williams 71
passing the fixed stations located at the upstream location xu and the
downstream location xd of the freeway link without on- or off-ramps,
are treated as random variables because they have nonnegative and
nondecreasing characteristics.
Q(x
u
, t
n
)Q(x
d
, t
n
)
xuxd
Accordingly, the number of vehicles on the freeway link at time tn
is Q(xu, tn) − Q(xd, tn) and Q(xu, tn) − Q(xd, tn) ≥ 0. The vehicles passing
the upstream and downstream stations during the time interval (tn−1, tn]
are Q(xu, tn) − Q(xu, tn−1) and Q(xd, tn) − Q(xd, tn−1).
Conservation of Vehicles
The values q(xu, tn) and q(xd, tn) are the flow rates at the upstream
and the downstream at the time interval (tn−1, tn], k(tn) is the density
over the freeway link at the time interval (tn−1, tn], Δt is the time
interval fixed length, that is, Δt = tn − tn−1, n ∈ [1, n], and Δx is the
length of the freeway link, that is, Δx = xd − xu. Then, during the time
interval (tn−1, tn], the vehicles passing the upstream and downstream
locations are q(xu, tn) ∗ Δt and q(xd, tn) p Δt. The vehicles traveling
over the freeway link at time tn are k(tn) p Δx.
According to the stochastic analysis in the preceding section, the
equations can be derived as
Qx
tQxt qx tt
un un un
p
()()()
−=∆
−
,,
,(
1)
1
Qx
tQxt qx tt
dn dn dn
p
()()()
−=∆
−
,,
,(
2)
1
Qx
tQxt kt x
un dn np
()()
()
−=∆
,, (3)
Subtracting Equation 2 from Equation 1 gives
Qx tQxt Qx tQxt
qx ttqx ttqx ttqx
tt
Qx tQxt Qx tQxt
un un dn dn
un dn un dn
un dn un dn
pp
pp
[][]
[]
[]
()()()()
() () () ()
()() ()()
−−−
=∆−∆⇒∆
−∆
=−−−
−−
−−
,, ,,
,,
,,
,, ,, (4)
11
11
Inserting Equation 3 into Equation 4 gives
qx
ttqx ttkt xktx
un dn nn
pp
() ()
() ()
∆− ∆= ∆− ∆
−
,, (5)
1
where q(xu, tn) p Δt − q(xd, tn) p Δt is the difference between the
number of vehicles entering the link flow and the number departing
it during the time interval (tn−1, tn], and k(tn)Δx − k(tn−1)Δx represents
the change in the number of vehicles traveling on the freeway link
during the same time interval.
Therefore, according to Equation 5, it can be verified that the
difference between the number of vehicles entering the link flow and
the number departing it during the time interval corresponds to the
change in the number of vehicles traveling on the freeway link during
the same time interval. This is the principle of the conservation of
vehicles.
Flow–Density–Speed Relationship
From Equations 1 and 2, the relationship between cumulative flow
and flow rate is expressed as
Qx
tQxt qx
tt
Qx
tQxt qx
tt
un un un
dn dn dn
p
p
()()()
()()()
−=∆
−=∆
−
−
,, ,
,, ,
1
1
Qx tqxt tQxt
qx ttqx ttQx t
qx ttqx tt qx
tt
tqxt
un un un
un un un
un un u
ui
i
n
∑
()() ()
() () ()
() () ()
()
⇒= ∆+
=∆+∆+
=∆+∆++ ∆
=∆
−
−−
−
=
,, ,
,, ,
,,
... ,
,(
6)
1
12
11
1
p
pp
pp p
Qx ttqx t
dn di
i
n
∑
() ()
⇒=∆
=
,, (7)
1
From Equation 3, the relationship between density and cumulative
flow is
kt Qx tQxt
x
n
un dn
()()
()
−
∆
,, (8)
The initial conditions are
Qx
tQxt nt kt nt
x
ud
() ()
() ()
()
−=−=
∆
,0,,
,(
9)
0000
0
where n(t0) is the number of vehicles traveling on the freeway link
at time t0.
Let n(tn) be the number of vehicles traveling on the freeway link
at time tn:
nt Qx tQxt
nundn
()()
()
=−,, (10)
Let m(tn) be the number of vehicles entering and leaving the link
during the time interval (tn−1, tn], which can be derived as
mt Qx tQxt nt
Qx tQxt Qx tQxt
Qx tQxt
ndndnn
dn dn un dn
dn un
[]
()()
()()()()
()()
() ()
=− −
=− −−
=−
−−
−−−
−
,,
,, ,,
,, (11)
11
111
1
Let uf be the free-flow speed of the vehicles over the link. Under
some appropriate Δt,
tx
uf
∆>
∆
the model assumes that m(tn) is the performance measurement
necessary to determine the traffic flow conditions:
• Normal flow, m(tn) > 0, and
• Congestion flow, m(tn) ≤ 0.
For the different traffic conditions, two separate equations from
the cumulative flow diagrams are used to estimate travel time; these
equations are described in the next section.

72 Transportation Research Record 2526
Travel Time Estimation in Normal Flow Condition
Figure 1 is a diagram of the cumulative flows at the upstream and
the downstream under normal flow conditions, m(tn) > 0, Q(xd, tn) >
Q(xu, tn−1). The normal flow model was modified by Vanajakshi such
that the travel time can be estimated from all vehicles that enter the
link during the time interval (2, 3).
For this normal flow model, the following factors are assumed:
the known initial condition, the smooth cumulative flow at the time
interval [constant q(xu, tn), q(xd, tn) in the time (tn−1, tn)], and the same
downstream flow rates at consecutive intervals, q(xd, tn) = q(xd, tn+1).
During the time interval (tn−1, tn), the total travel time T(tn) of all the
vehicles n(tn) entering the link is shown as the shaded area in Figure 1
and is calculated as follows:
Tt
tt ttnt
nnnn
[]
()()
() ()
=′′ −+
′′′ −
−
1
2
(12)
1
where t′′ is the expected departure time from the link of the first
vehicle that enters the link at the time interval (tn−1, tn] and t′′′ is the
expected departure time from the link of the last vehicle that enters
the link at the time interval (tn−1, tn]
The total travel time is represented as
Tt
kt x
qx t
kt x
qx tnt xktkt
qx tnt
n
n
dn
n
dn
n
nn
dn
n
()() ()
()
() ()
()
() ()
()
=∆+∆




=∆+
−
+
−
1
2, ,2,
(13)
1
1
1
The average travel time tt(tn) is the total travel time T(tn) divided
by n(tn):
tTt
nt
xktkt
qx t
n
n
n
nn
dn
()
()
()
()
() ()
==
∆+
−
tt
2, (14)
1
Travel Time Estimation
in Congested Flow Condition
Figure 2 is a diagram of the cumulative flow at the upstream and the
downstream under congestion flow conditions, m(tn) ≤ 0, Q(xd, tn)
≤ Q(xu, tn−1).
The assumptions in the diagram include the known initial condition,
the smooth cumulative flow curves per time interval, and the down-
stream flow rate of the interval that is the same as that of the next
time interval, q(xd, tn) = q(xd, tn+1).
Because m(tn) ≤ 0, no vehicles enter the link during the time interval
and exit the link during the same interval; thus, the travel time is based
on all the vehicles n(tn) that enter the link during the time interval
(tn−1, tn]. The total travel time T(tn) for the vehicles n(tn) is shown in
Figure 2 as the shaded area.
Thus, the total travel time is
Tt tt ttnt
nnnn
[]
()()
() ()
=′′ −+
′′′ −
−
1
2
(15)
1
Figure 2 shows that
tt Qx tQxt
qx t
kt x
qx t
n
un dn
dn
n
dn
()()
() ()
()
′′ −=
−
=
∆
−−−−
,,
,, (16)
1
111
tt
Qx tQxt
qx t
kt x
qx t
n
un dn
dn
n
dn
()()
() ()
()
′′′ −=
−
=
∆
++
,,
,, (17)
11
Inserting Equations 16 and 17 into Equation 15, and assuming that
q(xd, tn) = q(xd, tn+1), the total travel time is
Tt kt x
qx t
kt x
qx tnt
kt x
qx t
kt x
qx tnt xktkt
qx tnt
n
n
dn
n
dn
n
n
dn
n
dn
n
nn
dn
n
()()
()() ()
() () () ()
() () () () ()()
=∆+∆





=∆+∆




=∆+
−
+
−−
1
2, ,
1
2, ,2,
(18)
1
1
11
The average travel time tt(tn) is the total travel time T(tn) divided
by n(tn):
tTt
nt
xktkt
qx t
n
n
n
nn
dn
()
()
()
()
() ()
==
∆+
−
tt 2, (19)
1
With all entering vehicles considered during the same interval
in the normal flow, the modified dynamic traffic flow model by
Vanajakshi combines the two separate Nam–Drew models into one,
which appears to be the most appropriate model for estimating the
travel time for transition flow.
t′′′
Q (xu, tn)
q (xd, tn)
Q (xu, t)
Q (xd, t)
q (xd, tn+1)t n (tn)
m (tn)
Q (xd, tn)
Q (xu, tn–1)
Q (xd, tn–1)
tn+1 t′′ t′tn
Total Travel Time
Ti
me
Number of Vehicles
Traveling at Time tn
Cumulative Flow
FIGURE 1 Schematic representation of total travel time during
interval (tn21, tn) under normal flow conditions.
Q(xu, tn–1)
Q(xu, t)
Q(xd, t)
n(tn)
Q(xu, tn)
q(xd, tn)
Q(xd, tn)q(xd, tn+1)
Q(xd, tn–1)
tn+1
tn–1 tnt′′ t′′′
Cumulative Vehicle Counts
Time
Total Travel Time
FIGURE 2 Schematic representation of total travel time during
interval (tn21, tn) under congested flow conditions.

Yi and Williams 73
MODIFIED DYNAMIC TRAFFIC FLOW MODEL
Both the dynamic traffic flow models assume that the flow rates of the
downstream during consecutive time intervals are the same, which
is not always applicable in the field. In reality, the flow rates vary
under normal and transition flow conditions. Given the consecutive
interval flow rate, a modification to the models is proposed in the next
analysis.
In the dynamic model, the distinction between normal and con-
gested flow, which is based on the vehicle that enters and exits the
link during the same interval, is not necessarily appropriate for all
conditions. For example, if the time interval is long enough, under
congestion situations, some vehicles that enter the link can exit,
while in the dynamic model this occurrence is considered to be the
normal flow condition. In the modified model, the number of vehicles
that enter and exit during the same interval is not used to determine
the traffic conditions; rather, the different cumulative flow diagrams
and the modified models are based on the number of vehicles.
Use of Downstream Data of Subsequent
Time Interval to Modify Model
Figure 1 shows the cumulative flow at two detection locations under
varying traffic flow conditions, given the downstream flow rate of
the subsequent time interval q(xd, tn+1) and q(xd, tn+1) ≠ q(xd, tn),
whereby m(tn) > 0 ⇒ Q(xd, tn) > Q(xu, tn−1) and n(tn) are considered
in estimating the travel time.
Given q(xd, tn+1) and q(xd, tn+1) ≠ q(xd, tn), the total travel time T(tn)
of all vehicles n(tn) entering the link during the time interval (tn−1, tn)
is indicated by the shaded area in Figure 1 and is calculated as
Tt
tt ttmt
tt ttnt mt
nnnn
nnnn
[]
[]
[]
()()
()()
() ()
() ()
=′′ −+−′
+−
′+′′′ −−
−
1
2
1
2
(20)
1
where t′ is the entering time into the link of the last vehicle that exits
the link during interval (tn−1, tn].
Also, the following equations can be derived from Figure 1:
Then total travel time is
Tt
xkt
qx t
kt
qx tmt
xkt
qx t
kt
qx tnt mt
xkt
qx t
kt
qx tmt
xkt
qx t
kt
qx tnt
n
n
dn
n
un
n
n
un
n
dn
nn
n
dn
n
dn
n
n
un
n
dn
n
[]
()()
()()
()()
()()
() () () ()
() () ()
()
() () ()
() () ()
=∆+





+∆+




−
=∆−





+∆+





−
+
−
+
+
2, ,
2, ,
2, ,
2, ,
1
1
1
1
1
The average travel time tt(tn) is the total travel time T(tn) divided
by n(tn):
tTt
nt
x
qx t
kt kt
x
t
kt
qx t
kt
qx t
n
n
nun
nn
n
dn
n
dn
[][]
() ()()
() ()
()
() ()
() ()
==
∆
+
+∆
∆−















−
+
−
tt
2,
,,
(21)
1
2
1
1
2
Figure 2 shows the cumulative flow at two detection locations
whereby m(tn) ≤ 0 ⇒ Q(xd, tn) ≤ Q(xu, tn−1) and n(tn) is considered in
estimating the travel time.
Given q(xd, tn+1) and q(xd, tn+1) ≠ q(xd, tn), the total travel time T(tn)
of all the vehicles n(tn) entering the link during the time interval
(tn−1, tn) is indicated as the shaded area in Figure 2 and is calculated as
Tt tt ttnt
tt tttnt
nnnn
nnn
[]
[]
()()
()()
() ()
()
=′′ −+
′′′ −
=∆+′′ −+
′′′ −
−
1
2
1
2(22)
1
Then the total travel time is
Tt tkt xqxt t
qx t
kt x
qx tnt
tkt xktxqx tt
qx tnt
n
ndn
dn
n
dn
n
nndn
dn
n
()
() ()
()
()
()
() ()
()
() () ()
=∆+∆− ∆+∆





=∆+∆+ ∆− ∆





−
++
−
+
1
2
,
,,
1
2
,
,(23)
1
11
1
1
The average travel time tt(tn) is the total travel time T(tn) divided
by n(tn):
tTt
nt
tktxkt xqxt t
qx t
n
n
n
nndn
dn
()
()
()
()
()
() ()
==
∆
+
∆+ ∆− ∆
−
+
tt 2
,
2, (24)
1
1
This modified model includes two cumulative flow diagrams,
which are based on the vehicles n(tn) that enter the link and can exit
it during the same interval, and two equations, which are derived to
estimate the average travel time of all the vehicles that enter the link
during the interval. The model avoids the distinction between traf-
fic flow conditions based on n(tn). Moreover, given the subsequent
interval flow rate, the modification considers the varying flow rates
to improve the accuracy of the travel time estimation.
Use of Updating Downstream Data
to Modify Model
From the diagram of the cumulative flows in Figure 1, besides the
assumptions q(xd, tn+1) ≠ q(xd, tn) and Q(xd, tn) > Q(xu, tn−1), Equa-
tion 21 also is based on the assumption that Q(xd, tn+1) > Q(xu, tn) or
Q(xd, tn+1) ≤ Q(xu, tn) and q(xd, tn+2) = q(xd, tn+1).
Given the updating downstream data, a complete diagram of the
cumulative flows at two detection locations is shown in Figure 3. The
necessary updated data include (a) the time tz that the downstream
cumulative flow is equal to that of upstream at time tn and (b) the
downstream cumulative flow Q(xd, tn+j+1) at time tn+j+1 supposing
tn+j ≤ tz ≤ tn+j+1 j = 1, 2, . . . . The downstream flow rate data q(xd, tn+1),
q(xd, tn+2), . . . , q(xd, tn+j), q(xd, tn+j+1) is used to introduce n(tn) to
estimate the travel time.
The total travel time TT of all vehicles n(tn) that enter the link during
the time interval (tn−1, tn) is indicated in Figure 3 by the shaded area
and is calculated as shown in Equation Box 1.
The average travel time tt is the total travel time TT divided by
n(tn) and shown in Equation Box 2.
For the diagram of the cumulative flow in Figure 2, besides the
assumption q(xd, tn+1) ≠ q(xd, tn) and Q(xd, tn) ≤ Q(xu, tn−1) ≤ Q(xd, tn+1),
Equation 24 is also based on the assumption that Q(xd, tn+1) > Q(xu, tn)
or Q(xd, tn+1) ≤ Q(xu, tn) and q(xd, tn+2) = q(xd, tn+1).

74 Transportation Research Record 2526
Q(xu, t) Q(xd, t)
Q(xd, tn+j+1)
Q(xu, tn)
q(xu, tn)
q(xd, tn)
q(xd, tn+1)
q(xd, tn+j)
q(xd, tn+j+1)
Q(xd, tn+j)
Q(xd, tn+1)
Q(xd, tn)
Q(xu, tn–1)
Q(xd, tn–1)
t
n–1
t
n
t
n+1
t
n+j
t
n+j+1
t
zTime
n(tn)
Cumulative Flow
FIGURE 3 Schematic representation of total travel time during interval (tn21, tn).
Qx tQxt
qx t
Qx tQxt
qx tQx tQxt
Qx tQxt
qx ttQx tQxt
qx tQx tQxt
tQx tQxt
qx ttQx tQxt
qx tQx tQxt
jt
Qx tQxt
qx tjt
Qx tQxt
qx tQx tQxt
jt
Qx tQxt
qx tjt
Qx tQxt
qx tQx tQxt
un dn
dn
un dn
un
dn un
un dn
un
un dn
un
dn dn
un dn
un
un dn
un
dn dn
un dnj
un
un dnj
un
dnjdnj
un dnj
un
un dnj
dnj
un dnj
[]
[]
() ()
()
()
() ()
() ()
[]
[]
[]
()()
()
()()
() ()()
()()
() ()()
() ()()
()()
() ()()
() ()()
() ()
()
()
()
()
()
() ()
=
−+−




−
+−+∆ +−




−
+∆+−+∆+−




−
+
+−∆+ −+∆+−




−
+∆+−+∆+−




−










































−− −
++
++
++
+− +
++
−
++
++
+
TT
1
2
,,
,
,,
,,,
1
2
,,
,
,,
,,,
1
2
,,
,2,,
,,,
......
1
21,,
,
,,
,
,,
1
2
,,
,
,,
,,,
11
1
1
1
12
21
1
1
1
pp
ppp
EQUATION BOX 1
EQUATION BOX 2
nt qx tt
Qx tQxt
qx t
Qx tQxt
qx tQx tQxt
tQx tQxt Qx t
qx tqx tt
tQx tQxt Qx t
qx tqx tt
jt
Qx tQxt Qx t
qx tqx tt
jt
Qx tQxt
qx t
Qx tQxt
qx tQx tQxt
nun
un dn
dn
un dn
un
dn un
un dn dn
un
dn
un dn dn
un
dn
un dnjdnj
un
dnj
un dnj
un
un dnj
dnj
un dnj
[]
()()
()
() ()
() ()
[]
()
()()
()
()()
() ()()
()()()
() ()
()()()
() ()
() ()
()
()
()
() ()
()
== ∆
−+−




−
+∆+−−




∆
+∆+−−




∆
+
+−∆+ −−




∆
+∆+−+−




−










































−− −
++
++
+
+− +
+
++
++
+
tt TT 1
2,
,,
,
,,
,,,
2,,,
,,
32, ,,
,,
......
21 2, ,,
,,
2,,
,
,,
,,,
(25)
11
1
1
1
12
2
1
1
pp

Yi and Williams 75
Given the updating downstream data, a complete diagram of the
cumulative flows at two detection locations is shown in Figure 4.
The necessary updated data are the time tA that the downstream
cumulative flow is equal to that of upstream at time tn−1 and the down-
stream cumulative flow Q(xd, tn+i+1) at time tn+i+1 with tn+i ≤ tA ≤ tn+i+1,
i = 1, 2, . . .; the time tz that the downstream cumulative flow is equal
to that of upstream at time tn; and the downstream cumulative flow
Q(xd, tn+j+1) at time tn+j+1 supposing tn+j ≤ tz ≤ tn+j+1, j = 1, 2, . . . . With
the downstream flow rate data q(xd, tn+1), q(xd, tn+2), . . . , q(xd, tn+i),
q(xd, tn+i+1), . . . , q(xd, tn+j), q(xd, tn+j+1), n(tn) is introduced to estimate
the travel time.
The total travel time TT of all the vehicles n(tn) that enter the link
during the time interval (tn−1, tn) is shown in Figure 4 as the shaded
area and is calculated as shown in Equation Box 3.
The average travel time tt is the total travel time TT divided by
n(tn) and is shown in Equation Box 4.
In all the (modified) dynamic traffic flow models, the assump-
tion of smooth cumulative flow curves is not practical for normal
flow conditions, especially for the low-flow condition, and thus
the dynamic traffic flow models are not appropriate for these
conditions.
SIMULATION STUDY
The study area is the segment upstream of the on-ramp. Detectors
are located upstream and downstream of the studied segment. The
freeway links comprise different link lengths between the detectors
(500, 750, and 1,000 m) and different numbers of lanes (one, two,
and three lanes).
To design the various traffic conditions, such as free-flow, tran-
sition, and congestion, the input traffic flows of the major road and
its on-ramp are changed according to the saturation flow rate of the
freeway links. There are 10 runs for each simulation state, and the
various scenarios are as follows:
• Desired speed distribution, v = 60 to 90 km/h;
• Traffic composition, 90% car and 10% truck;
Q(xd, t)
Q(xu, t)
Q(xd, tn+i+1)
Q(xd, tn+j+1)
Q(xu, tn)
q(xu, tn)
q(xd, tn)
q(xd, tn+i)
q(xd, tn+j)
q(xd, tn+j+1)
q(xd, tn+i+1)
Q(xd, tn+j)
Q(xu, tn–1)
Q(xd, tn)
Q(xd, tn+i)
Q(xd, tn–1)
tn–1 tntn+itn+jtn+j+1
tn+i+1 tz
tA
Time
n(tn)
Cumulative Flow
FIGURE 4 Schematic representation of total travel time during interval (tn21, tn).
it
Qx tQxt
qx tit
Qx tQxt
qx tQx tQxt
it
Qx tQxt
qx tit
Qx tQxt
qx tQx tQxt
jt
Qx tQxt
qx tjt
Qx tQxt
qx tQx tQxt
jt
Qx tQxt
qx tjt
Qx tQxt
qx tQx tQxt
un dni
dni
un dni
un
dniun
un dni
un
un dni
un
dn
id
ni
un dnj
un
un dnj
un
dnjdnj
un dnj
un
un dnj
dnj
un dnj
[]
[]
() ()
()
()
() ()
() ()
[]
[]
() ()()
() () ()()
() ()
()
() ()()
() () ()()
() ()
()
() ()
()
()
()
()
()
() ()
=
+∆+−++∆+ −




−
++∆+ −++∆+ −




−
+
+−∆+ −+∆+−




−
+∆+−+∆+−




−


































−+
++
++ ++ −
++ ++ ++ ++
+− +
++
−
++
++
+
TT
1
21,,
,1,,
,
,,
1
21,,
,2,,
,
,,
......
1
21,,
,
,,
,,,
1
2
,,
,
,,
,,,
1
1
1
11
12
21
1
1
1
p
p
pp
ppp
EQUATION BOX 3

76 Transportation Research Record 2526
• Total simulation time length, 4 h;
• Time interval lengths, 2 min; and
• Data sampling interval of flow measurement, Δ = 1 s.
In the simulation study, because of the design of the on-ramp, which
causes queues to form or disappear from downstream to upstream,
the approximate range and significant difference of the average
space–mean speed at the upstream and the downstream can be used
to determine the traffic conditions. When both space–mean speeds
are in the desired speed range, the traffic can be considered to be in the
free-flow condition. If one of the space–mean speeds is in the desired
speed range and the other is not, the traffic condition is transition.
If neither of the space–mean speeds is in the range, the condition is
congestion.
The performance measures that are based on the estimation errors,
mean absolute error (MAE) and mean absolute percentage error
(MAPE), during varying traffic conditions are calculated.
MAE for a data series is calculated as
TT
n
tt
i
n
∑
=
−
=
MA
E
ˆ
(27)
1
MAPE is calculated as
TT
T
n
tt
t
i
n
∑
=
−
=
MAPE
ˆ
100
(28)
1
p
A comparison of the results of the dynamic traffic flow models is
given in Table 1. The comparison uses data from various freeway
links comprising different link lengths and numbers of lanes during
different time intervals.
In the transition condition, for the travel time estimation obtained
from the 500-m freeway link data, MAPE decreases from between
5.6% and 9.6% to between 3.9% and 6.7% when the modified dynamic
traffic flow model is used. For the 750-m data, MAPE decreases
from between 3.9% and 7.5% to between 2.6% and 5.4%. For the
1,000-m data, MAPE decreases from between 4.4% and 8.8% to
between 2.8% and 4.0%.
In the congestion condition, for the 500-m freeway link data, MAPE
decreases from between 5.9% and 9.4% to between 3.3% and 3.4%.
For the 750-m data, MAPE decreases from between 4.1% and 10.8%
to between 1.6% and 2.0%. For the 1,000-m data, MAPE decreases
from between 6.2% and 11.0% to between 1.5% and 1.6%.
The errors decrease significantly when the modified dynamic
traffic flow model is used. The modified dynamic traffic flow models
can accurately estimate the travel time in transition and congestion
conditions.
FIELD DATA STUDY
The real traffic data are collected from NGSIM. Besides the wide-area
detector data, the data sets include detailed vehicle trajectory data.
The true travel time can be measured from these detailed data, and
thus the NGSIM data can be used to develop and test the proposed
estimation model. The field data selected from the NGSIM data sets
consist of US-101 data, I-80 prototype data, and new I-80 data.
The detection site for US-101 data is 2,100 ft long. This section has
five lanes and an extra sixth lane between the two ramps. Complete
vehicle trajectory data were transcribed at a resolution of 10 frames
per second, and data for 45-min periods, from 7:50 to 8:35 a.m.,
present transitional and congested flow conditions. The study area
is approximately 698 ft (213 m) long and has six lanes.
The prototype I-80 data were collected at the Berkeley Highway
Laboratory site, which is approximately 1,300 ft long. There are
six lanes in this section between the on-ramp and the off-ramp.
The data collection time was approximately 30 mins, from 2:35 to
3:05 p.m. The detection area for the new I-80 data was just at the
merge section of the on-ramp downstream. The merge site is only
approximately 1,230 ft long. A total of 45 min of data were collected
during the afternoon peak hours.
For the three field data sets, the study areas are all weaving seg-
ments between an on-ramp and an off-ramp of the freeway. Weaving
segments require intense lane changing maneuvers, and thus traffic
in a weaving segment is more turbulent than that normally present
on basic freeway segments (9). To improve the accuracy of travel
time estimation, the time window T should be divided into smaller
parts, T1, . . . , Tn. For the US-101 data and prototype I-80 data, each
2-min interval is divided into eight 15-s intervals. For the new I-80
data set, each 2-min interval is divided into four 30-s intervals. The
flow rate q1, . . . , qn can be calculated separately, the travel times
nt qx tt
it
Qx tQxt
qx t
Qx tQxt
qx tQx tQxt
it
Qx tQxt Qx t
qx tqx tt
jt
Qx tQxt Qx t
qx tqx tt
jt
Qx tQxt
qx t
Qx tQxt
qx tQx tQxt
nun
un dni
dni
un dni
un
dniun
un dnidni
un
dni
un dnjdnj
un
dnj
un dnj
un
un dnj
dnj
un dnj
[]
()()
()
() ()
() ()
[]
()
() ()()
() ()()
() ()()
() ()()()
() ()
() ()
()
()
()
() ()
()
== ∆
+∆+−+−




−
++∆+ −−




∆
+
+−∆+ −−




∆
+∆+−+−




−


































−+
++
++ ++ −
++ ++ ++
+− +
+
++
++
+
tt TT 1
2,
21 ,,
,
,,
,,,
23 ,, ,
,,
......
21 ,, ,
,,
2,,
,
,,
,,,
(26)
1
1
1
11
12
2
1
1
p
pp
EQUATION BOX 4

Yi and Williams 77
tt1, . . . , ttn are estimated at the smaller intervals T1, . . . , Tn, and the
average travel time tt at the 2-min time interval can be determined as
qttq
qq
nn
n
=
++
++
tt ... tt
...
11
1
pp
As shown in Table 2, the modified dynamic model produces the
smallest measurement errors, that is, MAE of 0.1 to 1.8 s and MAPE
of 0.6% to 3.9% for the US-101, new I-80, and prototype I-80 data
sets. Therefore, the proposed model can accurately estimate travel
times for different freeway links under transition and congestion
conditions.
CONCLUSIONS
The accuracy of the dynamic traffic flow models by Nam and Drew,
or the modified model by Vanajakshi et al., is limited because of the
assumption of constant downstream flow rates. This study consid-
ered the changes in cumulative downstream flow rates, proposed a
modified dynamic traffic flow model, and improved the modified
dynamic traffic model by using updated downstream data.
For the simulation data, a comparison of the estimated results and
the measurement errors showed the accuracy of the modified dynamic
model for estimating travel times of freeway links under transition
and congestion conditions.
For the real traffic data, from the testing for frequency changes
in the traffic parameters, the determined 2-min interval was divided
into smaller time divisions. A comparison of the measurement errors
of all the estimation methods verified that the modified model is
appropriate for travel time estimation of freeway links.
The study presented a modified dynamic traffic flow model for
estimating the travel times of freeway links under transition and
congestion traffic conditions.
However, the study depends entirely on correct fixed-point data.
Errors in measured spot speeds and occupancy related to detector
inaccuracies would be reflected in any estimates of link travel time.
The flow data would be affected even more than the occupancy and
speed data if detector malfunctions occur. Therefore, future work
is needed to verify and improve the modified model in the case of
TABLE 1 Error Comparison Under Transition and Congestion Condition
Transition Congestion
Modified Traffic Flow Model Modified Traffic Flow Model
Freeway Link Statistic
Dynamic
Traffic Flow
Model
Downstream Data
of Subsequent
Time Interval
Updating
Downstream
Data
Dynamic
Traffic Flow
Model
Downstream Data
of Subsequent
Time Interval
Updating
Downstream
Data
500 m
One lane MAE (s) 4.4 2.8 2.8 12.9 7.4 3.9
MAPE (%) 9.6 6.8 6.7 9.4 6.2 3.4
Two lanes MAE (s) 3.6 1.8 1.8 8.3 5.1 4.6
MAPE (%) 6.4 4.1 4.0 7.2 4.5 4.1
Three lanes MAE (s) 2.1 1.7 1.7 4.8 4.8 2.6
MAPE (%) 5.6 3.9 3.9 5.9 5.8 3.3
750 m
One lane MAE (s) 5.4 3.4 3.3 21.9 9.2 3.2
MAPE (%) 7.5 5.5 5.4 10.8 4.7 2.0
Two lanes MAE (s) 6.9 2.6 1.8 12.5 6.4 4.3
MAPE (%) 7.9 3.7 2.9 7.1 3.7 2.5
Three lanes MAE (s) 2.2 2.0 1.5 4.8 4.5 1.8
MAPE (%) 3.9 3.1 2.6 4.1 3.8 1.6
1,000 m
One lane MAE (s) 9.6 7.2 3.7 29.1 15.5 3.7
MAPE (%) 8.8 6.0 4.0 11.0 5.8 1.6
Two lanes MAE (s) 9.4 7.5 3.5 17.0 9.7 5.2
MAPE (%) 7.6 5.1 2.9 7.1 4.0 2.2
Three lanes MAE (s) 3.4 2.9 2.2 9.6 5.4 3.5
MAPE (%) 4.4 3.4 2.8 6.2 3.5 1.5
TABLE 2 Error Comparison for Field Data
US 101 Data Set Prototype I-80 Data Set New I-80 Data Set
Model MAE (s) MAPE (%) MAE (s) MAPE (%) MAE (s) MAPE (%)
Modified dynamic flow model
Updating downstream data 0.1 0.6 0.5 3.9 1.8 2.5
Downstream data of next time period 0.4 1.5 0.5 3.9 2.4 3.1
Dynamic traffic flow model 1.8 7.2 2.0 13.2 6.4 10.0

78 Transportation Research Record 2526
incomplete or wrong simulation or field fixed-point detector data.
Possible enhancements of fixed-point detectors should be studied to
gain high-quality data sets.
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Travel Time Estimation Method from Point Detector Data for Freeways.
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The Standing Committee on Urban Transportation Data and Information Systems
peer-reviewed this paper.