Unsignalised Intersection Capacity

TRAFFIC ENGINEERING LECTURE

TRAFFIC ENGINEERING CAPACITY OF INTERSECTIONS

INTERSECTIONS

–

UNSIGNALIZED

1. GENERAL A highway intersection is required to control con flicting and merging streams of traf fic so that delay is minimised. This is achieved through choice of geo-metric parameters that control and regulate the vehicle paths through the intersection. These determine priority so that all movements take place with safety. The three main types of junction dealt are:

-

Priority intersections, either simple T-junctions, staggered T-junctions or crossroads

-

Signalised intersections

-

Roundabouts.

The starting point for any junction design is the determination of the volume of traf fic incident on it together with the various turning, merging and con flicting movements involved. The basis for the design will be the flow estimate for some point in the future – the design reference flow (DRF). It is an hourly flow rate. Anything from the highest annual hourly flow to the fiftieth highest hourly flow can be used.

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2. UNSIGNALIZED INTERSECTIONS Unsignalized intersections are the most common intersection type. Although their capacities may be lower than other intersection types, they do play an important part in the control of traffic in a network. A poorly operating unsignalized intersection may affect a signalized network or the operation of an Intelligent Transportation System. Unsignalized intersections give no positive indication or control to the driver. The driver alone must decide when it is safe to enter the intersection. The driver looks for a safe opportunity or "gap" in the traffic to enter the intersection. This technique has been described as gap acceptance. Gaps are measured in time and are equal to headways.

At unsignalized intersections a driver must also respect the priority of other drivers. There may be other vehicles that will have priority over the driver trying to enter the traffic stream and the driver must yield to these drivers. Most of analysis procedures have relied (spoléhají) on gap acceptance theory. The theory describing the operations at an unsignalized intersection and includes four basic elements: 1. Relative priority of traf fic streams 2. Distribution (availability) of gaps in higher priority streams 3. Usefulness of gaps to yielding vehicles (gap acceptance) 4. Queue discharge

The availability of gaps is described by a probability distribution of headways in higher priority streams. A gap acceptance function describes the usefulness of headways as well as queue discharge.

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3. RELATIVE PRIORITY OF TRAFFIC STREAMS There is a hierarchy of priorities – streams of Rank 1 – Rank 4:

Through and right-turning movements on the major road (2, 3, 8, and 9) have absolute priority. This group of movements is called Rank 1. Left turning traf fic from the major road (1, 6) and right turning traf fic from the minor road (7, 12) have Rank 2. They must give priority to Rank 1 movements. Through traf fic on the minor road (5, 11) has Rank 3. It must give priority to all movements on the major road (Ranks 1 and 2). Finally, left turning traf fic from the minor road (4, 10) is subordinate to all other streams, and it has Rank 4. All yielding vehicles, which do not find an acceptable gap at arrival must decelerate and/or stop to wait for the next safe gap. There may, however, also be vehicles or pedestrians of higher rank waiting for an acceptable gap. Accordingly, a vehicle must find an acceptable gap after all waiting higher rank vehicles and pedestrians have passed the con flict area.

NOTE: Right turning stream 3 does not have a con fl ict with minor road streams 4, 5 and 6. It is, however, possible that minor road drivers are not con fident that a major street vehicle will turn right, until they see a clear indication of its intentions. Accordingly, a stream may have an effect on the performance of another stream, even though no con fl ict exists between the two streams.

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3.1

UNCONTROLLED INTERSECTION OF TWO ONE-WAY ROADS

In an uncontrolled intersection of two one-way roads displayed in figure below the south bound traf fic (11) must yield to east-bound traf fic (2). The intersection operates in the same way, if the northern approach has a yield sign. If there is a stop sign, vehicles must stop before passing the sign. An intersection of two one-way roads is the simplest case to analyze, and it is used in the derivation of potential capacity and delay equations.

Uncontrolled intersection of two one-way roads

3.2

T-INTERSECTION

The traf fic streams at a T-intersection are presented in next Figure. There are only three hierarchical levels of priorities (ranks). Consequently, there are less possible con flicts between streams, so that it is easier for a driver to observe the safe gaps in priority streams.

Traffic streams in T-intersection with unified rankings of streams

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3.3

FOUR LEG INTERSECTION

The traf fic steams at a four-leg intersection are presented in below. Streams 1 – 12 are vehicles and streams 13–16 are could be pedestrians crossing the road (they are not in figure).

Traffic streams in 4-leg intersection with unified rankings of streams

The operations are more complicated than in the simple case of previous Figures. All minor road vehicles yield the right-of-way to major road vehicles. Left-turning vehicles in all approaches yield right-of-way to opposing through and ri ght-turning vehicles.

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4. DISTRIBUTION (AVAILABILITY) PRIORITY STREAMS

OF

GAPS

IN

HIGHER

For distribution of gaps in higher priority streams is the best using exponential distribution which is the most common distribution. This distribution is based on the assumption (př edpoklad) that vehicles arrive at random without any dependence on the time the previous vehicle arrived. The distribution can be derived from assuming that the probability of a vehicle arriving in a small time interval (t, t+t) is a constant. It can also be derived from the Poisson distribution which gives the probability of n vehicles arriving in time t, that is: n

P ( n ) = (m * t) where:

-m * t

e

n!

n is number of vehicles e is Euler number m is flow rate in veh/sec.

For n = 0 this equation gives the probability that no vehicle arrives in time t. The headway, h – denoted according to Czech standard as t CO, must be then greater than t and the probability, from Equation below is:

P ( t CO  t) = e where:

-m * t

P(tCOt) is probability of occurrence (výskyt) of gaps tCO (headway) greater than t e is Euler number m is flow rate in veh/sec.

The cumulative probability function of headways is then:

P ( t CO  t) = 1 - e

-m * t

The probability distribution function is then:

f ( t ) = m * e

-m * t

This is the equation for the negative exponential distribution. The parameter m can be estimated from the flow or the reciprocal of the average headway. EXAMPLE: If there were 228 headways observed in half an hour, then the flow is 228/1800 i.e. m = 0,127 veh/sec. The proportion of headways expected to be greater than 5 seconds is then P(h>5) = e –m* t=0,531. The expected number of headways greater than 5 seconds observed in half an hour is then 0,531*228 or 116. If the flow was 1440 veh/h or 0,4 veh/sec then the number of headways less than 0,1 seconds is then m * [P(h>0.1)]*3600 or 56 per hour. This over-estimation of the number of very short headways is considered to be unrealistic and the displaced exponential distribution is often used instead of the negative exponential distribution.

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NOTE: For traffic streams with larger volumes, when interactions between vehicles cannot be neglected, other models are used, which give better results than the simple Poisson process. Some of them actually represent certain generalizations of the simple Poisson process - Displaced Exponential Distribution, Dichotomized Headway Distributions.

5. USEFULNESS OF ACCEPTANCE)

GAPS

TO

YIELDING

VEHICLES

(GAP

The gap acceptance theory commonly used in the analysis of unsignalized intersections is based on the concept of defining the extent (measure of) drivers who will be able to utilize a gap of particular size or duration. EXAMPLE: Will drivers be able to leave the stop line at a minor road if the time between successive vehicles from the left is 10 seconds; and, perhaps how many drivers will be able to depart in this 10 second interval?

The gap acceptance parameters:

-

CRITICAL GAP

The minimum gap that most of drivers in the minor stream are assumed to accept at all similar locations is the critical gap - tg. Thus, the driver's critical gap is the minimum gap that would be acceptable. According to the driver behavior model ( gap acceptance theory usually assumes that drivers are consistent and homogeneous) usually assumed, no driver will enter the intersection unless the gap between vehicles in a higher priority stream (with a lower rank number) is at least equal to the critical gap – tg. For example, if the critical gap was 4 seconds, a driver would require a 4 second gap between Rank 1 stream vehicles before departing. They will require the same 4 seconds at all other times he or she approaches the same intersection and so will all other drivers at that intersection.

-

FOLLOW - UP TIME

The time between the departure of one vehicle from the minor street and the departure of the next vehicle using the same major-street gap, under a condition of continuous queuing on the minor street, is called the follow-up time – tf . Thus, t f is the headway that defines the saturation flow rate for the approach if there were no conflicting vehicles on movements of higher rank.

It has been found that the gap acceptance parameters tg and tf may be affected by the speed of the major stream traffic (Harders, 1976 and Troutbeck, 1988). It also expected that drivers are influenced by the difficulty of the maneuver. The more difficult a maneuver is, the longer are the critical gap and follow-up time parameters. There has also been a suggestion that drivers require a different critical gap when crossing different streams within the one maneuver. For instance a turn movement acrossa number of different streams may require a driver having a different critical gap or time period between vehicles in each stream (Fisk 1989). This is seen as a unnecessary complication given the other variables to be considered.

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NOTE: Note that other researchers have used a different concept for the critical gap and the follow-up time. McDonald and Armitage (1978) and Siegloch (1973) independently described a concept where a lost time is subtracted from each major stream gap and the remaining time is considered 'useable.' This 'useable' time divided by the saturation flow gives an estimate of the absorption capacity of the minor stream. But the effect of this different concept is negligible (bezvýtnamný) – unimportant.

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5.1. BASE VALUES OF T g AND Tf FOR PASSENGER CARS ACCORDING TO CZECH STANDARD

Mean values of critical gap:

Type of traffic stream

Number of traffic stream

tg in dependence on travel speed on major street v85% (km/h)

Left turning from major street

7/1

tg=3,4 + 0,021 * v85%

Right turn from minor street

6/12

tg=2,8 + 0,038 * v85%

Passing through from minor street

5/11

tg=4,4 + 0,036 * v85%

Left turn from minor street

4/10

t =5,2 + 0,022 * v85%

Mean values of follow – up times:

Type of traffic stream

Number of traffic stream

tf in dependence on traffic sign control

Left turning from major street

7/1

2,6

Right turn from minor street

6/12

3,1

3,7

Passing through from minor street

5/11

3,3

3,9

Left turn from minor street

4/10

3,5

4,1

NOTE:

Yield control

Stop

t f t g

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5.2.

BASE CAPACITY ACCORDING TO CZECH STANDARD

The gap acceptance model used in this method computes the base capacity of each minor traffic stream in accordance with Equation (modified from Grabe method by Siegloch, 1973):

G n where:



3600 t f



I H

* e 3600

( t g



t f 2

)

Gn is base capacity of traffic lane of n – th traffic stream (veh/h) IH is ruling traffic flow – flow rate (veh/h) e is Euler number tg is critical gap (s) tf is follow – up time (s)

5.3.

-

CAPACITY OF TRAFFIC STREAMS ACCORDING TO CZECH STANDARD streams of Rank 1:

don’t determine – capacity of streams of Rank 1 equals to the capacity of free flow (about 1800 veh/h) -

streams of Rank 2:

for streams of Rank 2 is valid: Cn = Gn -

streams of Rank 3 and 4:

No analytical solution is known for the derivation of the capacity of Rank 3 movements like the left-turner from the minor street at a T-junction (movement 3) or passing through at a four leg intersection (movements 5, 11). Here, the gap acceptance theory uses the impedance factors p0 as an approximation. p0 for each movement is the probability that no vehicle is queuing at the entry.

This is given with sufficient accuracy by Equation:

p0 , n where:

 1 - a v  1  I n

C n

av is exploitation ratio; In is design traffic flow (veh/h); Cn is capacity

Therefore, for Rank 3 and 4 movements, the basic value Cn for the potential capacity must be reduced to get the real capacity for T – intersection and cross intersection.

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T – intersections:

C 4 where:

 p

0,7

* G 4

C4 is capacity of stream 4; p0,7 is impedance factor of higher priority stream 7 G4 is base capacity of stream 4

Cross intersection:

C 5 C 11 where:

 p  p

x

* G 5

x

* G 11

C5,C11 is capacity of stream 5 or 11; px is impedance factor of higher priority stream 1 and 7

p x

 p

0,1

* p0,7

G5, G11 is base capacity of stream 5 and 11

5.4

RESERVE CAPACITY ACCORDING TO CZECH STANDARD

Re z  C n - I n where:

In is design traffic flow (veh/h); Cn is capacity

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6. QUEUE DISCHARGE – other parameters 6.1 AVERAGE DELAY

Kimber and Hollis (1979) derived the following expression as a measure of the average delay per arriving vehicle over an interval with using reserve capacity as a parameter in the model.

t w D1

1

F = 0

 q0

 D 

1 2

1

1

 G  F )

( F 2

 T  2 (  q ) y  ( y   G =



0

 q0  )  E 

2 * T * y  q 0

   (  q ) E   

 q0

q0

E = 0

y = 1 where:

 E 

(

0



 q 0 ) 0

 q0

q

tw is average delay in time interval T (s) T is length of demanded time interval (s); T=3600 s

 is capacity of lower priority movement in demanded time interval (veh/s); =Cn/3600 q is flow rate of lower priority movement (veh/s); q=I n/3600

0 is capacity after peak interval (veh/s); 0=1600/3600 q0 is flow rate of movement after peak interval (veh/s)

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6.2 QUEUE LENGHT

Estimation of queue length is an important consideration at unsignalized intersections. Theoretical studies and empirical observations have demonstrated that the probability distribution of queue lengths for any minor movement at an unsignalized intersection is a function of the capacity of the movement and the volume of traffic being served during the analysis period. Queue length on approaches to the unsignalized intersection is design to the 95thpercentile considered queue length. This means that for 95% time during peak hour is the queue shorter than considered length. Queue length depends on exploitation ratio av and capacity Cn:

a where:

v

=

I n C n

av is exploitation ratio; In is design traffic flow (veh/h); Cn is capacity

Than the queue length is:

N 95%

 = C n  a v  1  2  3

( 1  a ) 2  3 ,0

8 * a v C n

  

(m)

capacity C (veh/h)

m t h n e l e u e u

av - exploitation ratio

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Unsignalised Intersection Capacity

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