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3 Uncontrolled intersections
3.1 Introduction
The capacity of an intersection is determined by the shared use of conflict areas
by conflicting traffic streams. The capacity is reached if one of the conflicting
streams does not get enough opportunities to handle its traffic. So, the capacity
depends on the ratio of flows and the time assigned to the various conflicting traffic streams.
Conflict
areas
Figure 3.1 Conflict areas
Remark:
The capacity is not the maximum number of vehicles per hour that can pass the
intersection. By assigning all the time to the main road, the capacity of the main
road determines the maximum number of vehicles that can pass. However, there
will be a growing queue on the minor road.
So, it is better to use the term performance. The performance can be expressed
in several items (average delay, number of stops, queue length, emissions, fuel
consumption, but also the maximum number of vehicles that can pass the intersection based on the volume ratios of the streams in the critical conflict group
(group of mutual conflicting streams).
3.1 Priority intersection
At intersections without traffic lights, priority rules and priority signs determine the
time assigned to various streams and so the performance of the intersection.
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STOP
Qmain
Qmain
STOP
Qside
Qside
a
b
Figure 3.1.1 Priority intersections with yield signs (a) and with stop signs (b)
There are several methods for the calculation of the maximum throughput of uncontrolled intersections:
- the method originally developed by Harders (SVT 22, 1983),
- the CAPCAL method (TV131 method) developed by the Swedish ministry of
transport
- the TRB special report 165.
The input for the calculation is:
- Geometry of the intersection (number of approaches, width of the lanes, number of lanes
- Type of regulation for priority on the intersection
- Traffic flows on each intersection
- Composition of the traffic flow (cars, pedestrians, cyclists etc.)
The capacity of the non-priority road is determined by the gaps in the priority flow
and the gap acceptance:
qside = ∫ f(t) . g(t) dt
(3.1.1)
where:
qside maximum traffic volume departing from the stop line in the minor stream in
vehicles per second
f(t)
probability density function for the distribution of the gaps in the major
stream
g(t)
the number of minor stream vehicles that can enter into a major stream
gap of length t.
If we assume constant values of gap acceptance and an exponential distribution
of gaps, we may assume for g(t):
g(t) = 0 for t < t0
= (t – t0) / h for t ≥ t0
(3.1.2)
t0 = tcritical – h/2
(3.1.3)
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where:
h
average headway between two following vehicles (≈ 2 s)
tcritical minimum gap accepted by drivers on the non-priority road (≈ 4 s)
Following time distribution (1200 pcu/h)
0.30
Probability
0.25
0.20
0.15
0.10
0.05
sec
Gap for
1 vehicle
to cross
7.50
7.00
6.50
6.00
5.50
5.00
4.50
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0.00
Gap for
2 vehicles
to cross
Figure 3.1.2 Probability of gaps
The resulting capacity is
qside = h-1 exp(-qmain.tcritical)
Qside= 3600* qside
where:
qmain
(3.1.4)
major stream volume in vehicles per second
The maximum throughput of an intersection depends on the distribution of flows
at the conflicting streams (ratio of flow at the side road and the priority road)
h=2 s
2250
Tcritical=4 s
Qmain+Qside
2000
1750
1500
50%
40%
30%
20%
10%
0%
1250
Qside/Qmain
Figure 3.1.3 Maximum throughput uncontrolled intersection (see figure 3.1.1)
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3.2. Pedestrian crossing
The waiting times of pedestrians at crossings depend also on gap acceptance. De
Haes and van Zuylen (1982) found out the following expressions from a theoretical analysis and observations of crossing behavior:
P = 1 – exp(-0.00028 Q.tcritical – 0.00026 Q + 0.066)
(3.2.1)
and
D = (0.00058 Q. tcritical + 0.2)2
Where:
tcritical
Q
P
D
(3.2.2)
average value of the accepted gaps [s],
volume on the road to be crossed [veh/h]
fraction of pedestrians that have to stop before they cross
average delay [s].
From observations done by de Haes it appears, that if the approaching vehicle
moves faster, the value of tcritical is less. Apparently it is difficult to estimate the
gaps if speeds increase.
Furthermore, for older people tcritical is longer.
If pedestrians carry luggage, they need longer gaps.
If the first vehicle is a lorry, the value of tcritical is longer than for a passenger car.
The longer pedestrians wait, the shorter the gaps they accept.
Furthermore, also the purpose of crossing has some influence on tcritical.
Table 3.2.1 Example of accepted gap tcitical
depending on purpose of crossing and on waiting time
waiting time < 30 s
waiting time > 30 s
to post office
from post office
tcritical = 5.3 s
tcritical = 5.2 s
tcritical = 4.9 s
tcritical = 5.0 s
The delay of pedestrians can be reduced in some cases by traffic lights, but it is
also possible to have an isle in the middle of the road that cuts the crossing into
two parts. The delays are reduced considerably in all cases.
The acceptance by pedestrians of delays at uncontrolled intersections depends
on the average delay and the probability that pedestrians have to wait longer than
a certain time. A standard that is often used is that the probability that pedestrians
have to wait longer than 30 s should be less than 5%. De Haes and van Zuylen
derived expressions that bring this standard back to volumes and average delay.
The acceptance of delays depends on the presence of traffic light. It appears that
pedestrians are more patient if there are traffic lights and that they accept about
20 s longer waiting times than in situations without traffic control.
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3.3 Roundabout
3.3.1 Introduction
If the capacity of a priority intersection is insufficient, a roundabout can be an alternative. By separating conflict areas, the performance of a roundabout is higher
than the performance of most regular uncontrolled priority intersections; delays
are often less and the acceptance by road users higher. The geometry and the
priority rules determine the capacity of the roundabout.
Figure 3.3.1 new build roundabouts nearby Roelofarendsveen
Due to its strong reduction of severe accidents and its high capacity in comparison with original uncontrolled intersections, roundabouts have become rather
popular in the past decades. The number of roundabouts is growing fast: each
year about 50 to 60 new roundabouts are built in the Netherlands (see for example figure 3.3.1).
Until the mid 80th, roundabouts were hardly used by Dutch road designers. By
changing the priority rules and the design, roundabouts gradually became widely
accepted.
Figure 3.3.2 shows a modern roundabout. In contrast to traditional roundabouts
this type is characterized by the fact that the circulating traffic has priority. As a
consequence, traffic can leave the roundabout unhindered and buffer space on
the roundabout is hardly needed. So, the size of the roundabout decreased.
Another major advantage of the new roundabout concerns the achieved safety.
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Circulatory roadway
1. Central area radius
2. Inscribed circle radius
3. Apron width
4. Number of lanes
5. Width
Entries
6. Width
7. Number of lanes
8. Radius
Exits
9. Width
10. Number of lanes
11. Radius
Splitter island
12. Width
Cyclists
13. Bike lane or bike path
14. Bike path form
15. Yielding signs
Figure 3.3.2 Schematic representation of design elements of the standard type
roundabout
3.3.2 Capacity estimators
Several models estimate the capacity of roundabouts (e.g. the ‘ROTONDEVERKENNER’ (van Arem, 1992, 1993). However this model has some limitations. It
only estimates the capacity of single lane roundabouts and it does not take into
account the influence of bike and pedestrian traffic, which is of particular importance in the Dutch situation.
Models of non-Dutch origin may also be used, such as Bovy’s (1991) model that
was developed in Switzerland. Although such models may have some advantages over their Dutch equivalents, their main disadvantage is that they were developed for non-Dutch traffic. Consequently, their validity for the Netherlands with
substantial bike traffic still has to be shown.
De Leeuw (1997) developed and verified a capacity model for Dutch roundabouts.
His model takes into account the influence of slow traffic explicitly.
The next section discusses the structure of the new model and some background
information of the used sub-models. Thereafter follows a practical method to determine the capacity of one entry of a standardized roundabout.
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3.3.2.1 Structure and background of analytic roundabout model
The general framework of the model is shown graphically in figure 3.3.3.
The model results in capacities of all entries. These entry capacities are required
to calculate performance indicators, such as Volume/Capacity-ratios, delay, and
queue lengths for the given OD-matrix.
The entry capacity depends on the volumes of the conflicting streams, which at
roundabouts are the volumes just upstream of each entry. These volumes are
calculated from the given OD-matrix.
If one or more of the entries is saturated, the calculated volumes in the OD-matrix
are no longer valid as the flows from the saturated entries then are blocked. In
that case both the volumes of the conflicting traffic and the capacities cannot be
calculated directly. Instead, they must be calculated iteratively until the volumes
and the capacities are in conformity.
OD-Matrix
Calculation of the
main conflict capacity
Calculating the
circulating volumes
Calculating
capacities
No
Cyclists have priority
For
each
entry
->>
For
each
entry
No
Yes
Calculation
of influence
of bike traffic
at entry
Calculated volumes
in conformity with capacity
Calculation
of influence
of bike traffic
at exit
Reduce main
conflict capacity
Yes
Capacity of
each entry
Figure 3.3.3 Base framework of a roundabout capacity model
The capacity of each entry depends on conflicts. Drivers at the entry have to yield
to traffic in the circulatory roadway. At roundabouts where cyclists have priority,
entry capacity also depends on bike traffic conflicts. In fact, two bike traffic conflicts can be distinguished: drivers have to yield to cyclists before they can enter
the circulatory roadway, and at some roundabouts (often in urban areas) they
have to yield to cyclists again before they can leave the circulatory roadway (in
the Netherlands in urban areas).
The slow traffic conflict at the entry directly influences entry capacity. The slow
traffic conflict at the exit can only influence entry capacity indirectly when it causes
a sufficiently long queue in the circulatory roadway to block an upstream entry. In
practice, queues that block more than one entry rarely occur.
Entry capacity can be calculated for each of the three conflicts separately. In other
words, three sub-models for entry capacity are identified. Because the main con-
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flict occurs at every roundabout, it is convenient to first calculate the capacity that
results from the main conflict. When slow traffic has right-of-way, this capacity
value should be reduced. The two sub-models for the slow traffic conflicts calculate the extent of the reduction.
3.3.2.2 Calculation of main conflict capacity
The sub-model for the main conflict is based on the method of Bovy (1991). Bovy
developed his model to calculate the capacity of one- and two-lane roundabouts.
Bovy’s method considers separately the influence of the number of lanes of the
entry and the circulatory roadway. Further, the model takes into account the socalled pseudo conflict. The model is given by equation 3.3.1.
Centry = 1/ γ * (1500 – 8/9.(β.Qcirc+α.Qexit) )
xit
Entry capacity
Circulating flow
Exiting flow
Influence of pseudo conflict
Influence of number of lanes circulatory
Influence of number of lanes entry
Qe
Where:
Centry
Qcirc
Qexit
α
β.
γ.
(3.3.1)
Qcirc
Centry
Figure 3.3.4 Conflict streams
The model assumes a maximum entry capacity of 1500 pcu/h. If the amount of
traffic in the circulatory roadway increases, the capacity of the entry decreases.
The influence is less than proportional, which means that traffic uses the roundabout more efficiently for higher volumes of conflicting traffic.
The model uses the coefficients β and γ to calculate the capacities of roundabouts
for different numbers of lanes. Appropriate values for these coefficients, as given
by Bovy (1991), are shown in table 6.2.2. They show that the increase of capacity
when the number of lanes is doubled is less than one hundred percent.
Table 3.3.1 Recommended factors (Bovy 1991)
Single lane roundabout
Two lane roundabout
β
0.9 - 1.0
0.6 - 0.8
γ
1.0
0.6 - 0.7
The essence of the pseudo conflict is that drivers at the entry sometimes wait for
exiting vehicles. They perceive a part of the exiting flow as conflicting. Consequently, the effectively conflicting flow consists of the actually conflicting flow and
a part α of the exiting flow that does not indicate the exiting movement clearly (in
the Netherlands, drivers are supposed to use their direction indicators when leaving the roundabout).
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Bovy determines the coefficient based on the geometric design of the roundabout:
the distance between the ultimate point where the decision to leave the roundabout becomes obvious and the entry.
This distance is measured between the points C and C’ (see figure 3.3.5). If this
distance is large, the pseudo conflict will not occur. The time needed to cover the
distance is larger than the critical gap time of waiting drivers. However, if the distance decreases, the share of drivers with critical gap times larger than the clearance time increases. So does the value of α.
1
0.9
0.8
0.7
high speeds
low flow rates
C’
C
0.6
0.5
0.4
0.3
Low speeds
high flow rates
Centry
0.2
0.1
0
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Distance C-C' [m]
Figure 3.3.5 Relation distance c-c’ and α
The rate by which α increases, mainly depends on the probability density distribution of drivers’ critical gap times. The graph also shows that the value of α depends on the flow rate and the speeds at the roundabout. At higher volumes more
drivers tend to use their direction indicators correctly, which has an alleviating effect on the occurrence of the pseudo conflict. When speeds at the roundabout are
higher, the time needed to cover the distance between C and C’ is less and more
drivers have critical gaps larger than the clearance time. So, the significance of
the pseudo conflict increases.
The sub-model of the main conflict was calibrated for The Netherlands on two
one-lane roundabouts. Table 3.3.2 shows the Dutch parameters of the main conflict model.
Table 3.3.2 Dutch calibration results
Sub-model
Parameter
Main conflict Pseudo conflict (α)
Value
According to
calculation
not changed
Influence number of
lanes circular ( )
One lane
roundabout
1
Influence number of
One lane entry
1
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lanes entry ( )
Slope
Free flow
one lane
9/9
capacity
1500 pcu/h
3.3.2.3 Calculation of influence of slow traffic at exit
For the capacity reduction caused by cyclists crossing the downstream exit, the
method of Marlow & Maycock for crossing pedestrians is used (see Semmens
1985).
This method calculates the probability of blocking the exit by crossing cyclists.
This probability is determined via an intermediate variable, the virtual V/C-ratio, y.
This is determined as a function of:
• Volume of crossing cyclists (Qbike);
• Time required by cyclists to cross the exit (tcross);
• Minimum follow-up time of exiting vehicles (s).
Marlow & Maycock present a table with blocking chances based on this virtual
V/C-ratio. For the new model this table is represented by a function, leading to the
'reduction factor for the exit', fexit
3.3.2.4 Calculation of influence of slow traffic at entry
This calculation consists of two parts. Part one concerns the capacity in case of a
bike lane and part two calculates the capacity increase of a bike track in comparison with a bike lane.
Cycle lane:
The reduction of capacity caused by crossing cyclists is modeled by the formula
of Siegloch, based on gap-acceptance. It presents the probability that the entry is
not blocked by crossing cyclists. The formula assumes random arrivals of slow
traffic. As slow traffic does not influence each other as much as motor vehicles
do, this assumption seems realistic.
Cycle path:
The increase of capacity caused by the extra space between cycle path and
roundabout has been modeled by Brilon (1996). At the entry of a roundabout the
motor vehicles have to yield the conflicting flows in two phases. At first the vehicles are yielding the crossing cyclists, then they can use the space of five meters
between the conflicting flows, and then they are yielding the circulating motor vehicles.
This completes the model for calculation of the capacity of one roundabout entry.
The model is too complex to be calculated by hand. However, the next section
gives a fast calculation method, which can be done by hand, to calculate the ca-
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pacity of one entry of a one-lane roundabout. This method is set up by choosing
some defaults in geometric properties and some fixed cycle flows.
3.3.2.5 Calculation of capacity of one entry of a standardized one-lane roundabout with cycle path
This section gives a fast calculation method by hand, which calculates the capacity of one entry of a one-lane roundabout. This calculation is done according to
the scheme in figure 3.3.3.
The following defaults are used for the geometric properties of a one-lane roundabout in build up areas, according to a publication by CROW (#) about roundabouts: central island radius of 10.5 m; inscribed circle radius of 16.0 m; entry radius of 12 m; exit radius of 15 m; width of entry 4 m; width of exit of 4.5 m; width
of splitter island of 2.5 m; and, finally, cyclists have right of way on a separate cycle path.
The capacity can be calculated in case of 0, 100, 200 and 300 cyclists/h.
Step one: Main conflict
The formula for calculation of the main conflict with the new Dutch parameters is
as follows:
Qq
circ
circ
Centry1 = 1/1*(1500-9/9.(1.Qcirc+α.Qexit))
This reduces to:
Centry1 = 1500-Qcirc-α.Qexit))
Qq
exit
exit
Centy
C
entry
Figure 3.3.6 Roundabout with bike path
The parameter α depends on the distance C-C’ (see figure 3.3.4). For the standardized roundabout this distance C-C’ is equal to 18.5 m and consequently α is
equal to 0.2.
If no cycle facility is present or cyclists don’t have right of way, the correct capacity is already calculated. Otherwise step two should follow.
Step two: Calculation of influence of slow traffic at exit
It is assumed that only the conflict between exiting cars and through going cyclists
of the first downstream exit can cause a queue that influences the capacity of the
entry.
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q
Qexit
exit
qQcyclists
bike
Centry
entry
C
Figure 3.3.7 Determination of fexit
Step three: Calculation of influence of slow traffic at entry
First multiply the exit capacity due to the main conflict (Centry1) by factor fexit.
Now, figure 3.3.8 gives the reduction factor for crossing cyclists at the entry, fentry.
qcyclists
Qbike
C
Centry
entry
Figure 3.3.8 Determination of fentry
Step four: Reduce main conflict capacity
The final capacity of the entry is equal to:
Centry = Centry1.fexit.fentry
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Example
700
Qcirc
qcirc
6 00
qQbike
c yclist
s
30 0
qcyc list
Qbike
s
1 00
3 00
qexit
Qexit
qex it
Qexit
C
entry
Centry
Qbike entry
Qbike exit
Qexit leg
Qexit downstream
Qcirc
Figure 3.3.9 Example
Step 1
Centry1 = 1500 – 600 - 0.2 . 300 = 840
Step 2
Qexit = 700
Qbike = 300
fexit = 0.95
Step 3
Centry2 = 840 . 0.95 = 798
Qbike = 100
fentry = 0.995
Step 4
Centry3 = 798 . 0.995 = 794
100
300
300
700
600
3
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