Basics of the capacity calculation for signalised intersections

Theintergreen differencebetween the minimum intergreen times t_ig,min and the actually used intergreen timest_igis calculated as∆t_ig=t_ig,min−t_ig. Intergreen time differences can be assigned to signal groups, leading togreen time extensions∆t_G. These green time extensions can be converted to vehicle volumes and, thus, are a measure of the maximum optimisation potential of intergreen times from a capacity point of view. It has to be noted that this maximum optimisation potential can only be achieved by a controlled vehicle flow or complete information on the behaviour of every single vehicle approaching and crossing the intersection. Deterministic intergreen times would have to be calculated dynamically for every single change of stages. This won’t, of course, be possible in the near future.

Achievable intergreen times

For random driver behaviour safety margins have to be added to the intergreen times, leading to em-pirical intergreen times. These emem-pirically to be derived safety margins are not in the focus of this research. Nevertheless, the outcome of this research will be an estimate of the capacity impacts of different safety margins. The achievable capacity of an intersection, i.e. the capacity achievable with consideration of safety margins, will be somewhere between the effective capacity and the maximum capacity.

to signal groups. Thus, lanes are easier to observe than vehicle streams (as an alternative to sum-ming up volumes on lanes; streams are more difficult to observe, because they may share the same lanes).

C =

n_l

X

i

C_l,i (4)

where C intersection capacity (veh/h)

C_l,i lane capacity (veh/h)

i index indicating approach lane

TRB (2000) proposes the use oflane groups, a concept grouping lanes with similar characteristics and identical signalisation together. This concept is based on the same principals as focussing on single lanes. The concept can also be found in FGSV (2001), even without the mentioning of the term lane group. To determine the capacity of lanes, it can be useful to analyse lane groups first. However, the basic procedure remains the same. The details of the capacity calculation for lane groups can be found in TRB (2000).

3.1.4.2 Lane capacity

The capacity of each lane is determined by the saturation flow and the green time. Saturation flow and green time can be defined in different ways. The most common ones, i. e. the definitions from FGSV (2001) and TRB (2000), and the one used in this study are given in Section 3.1.4.3. The traffic flow rate is the reciprocal of the time headway (Eq. 5), therefore the headway can be used as a subsitute for the saturation flow rate.

q= t_obs

h 3600^s/^h (5)

where q traffic flow rate (veh/observation time) t_obs observation time (commonly1 h) (h)

h time headway (s/veh)

The capacity of approach lanes at signalised intersections in comparison to free flow sections is reduced for three reasons:

• the near-side intersection geometry (i. e. curb radius, grade, lane width etc.) and the traffic situa-tion (i. e. influences through parking, public transport vehicles etc.)

• the right-of-way regulation (i. e. permitted streams)

• the signalisation (i. e. red time, intergreen times, other lost times)

The near-side intersection geometry and the traffic situation can have both an influence on the saturation flow and the effective green time. The right-of-way regulation leads to stops during the green time with the consequent capacity reductions (capacity vs. base capacity). Of primary interest in this research, how-ever, is the effective green time which is influenced by the signal program. The headway, nevertheless, is needed for a quantification of capacity reductions in terms of vehicles per time.

3.1 Introduction 29

3.1.4.3 Saturation flow and effective green time

Since at signalised intersections the flow is interrupted by the signal program (conflicting streams are receiving the right-of-way subsequently), every hour consists of intervals with traffic flow and intervals without traffic flow. The intervals which serve the traffic demand are the green times of the different signal groups. But even during the green times, the traffic flow varies over time. Particularly the first few vehicles have greater headways than the following vehicles.

Two ways exist to derive the lane capacity from saturation flow and green time:

• The approach pursued in this research is based on the definitions in TRB (2000): a saturation headway is determined by observing vehicles in the middle of the green time under saturated con-ditions. The effective green time is then determined to take all capacity reductions with reference to the saturation flow into account. Hence, the effective green time is a ficticious interval which cannot be observed directly on site, but the saturation flow can be measured.

• The alternative is the approach followed in FGSV (2001): no effective green time, but the signalled green time is used, and the saturation flow is adjusted to take capacity reductions into account.

Hence, the green time is known, but the saturation flow is a ficticious value which cannot be observed directly on site.

However, FGSV (2001) does not define an effective saturation flow rate, as is used when following this approach. Moreover, most capacity reductions occur as time differences between thesignalled green time and theeffective green time or between the saturation headway and the individual headway. Thus, the calculation of the capacity is facilitated by using the former approach.

Consequently aneffective green timet_gis defined as the (ficticious) interval during which the traffic flows with the headway under free flow conditions, called the saturation headway h_s. All capacity changes caused by the signal change are determined as time differences with reference to the green time and the saturation headway. By relating the quotient of effective green time and saturation headway to an observation time t_{o bs}, commonly one hour, using thecycle time t_C, the capacity of an approach lane can be calculated as in Eq. 6.

C_l,eff= t_obs·t_g

h_s·t_C = 3600^s/^h·t_g

h_s·t_C (6)

where C_l,eff effective lane capacity (^veh/^h) t_g effective green time (s)

t_C cycle time (s)

h_s saturation headway (s)

The traffic flow has to be determined at a reference location. The most straight forward location is the stop line at the approach. It is fixed for all movements, easy to determine for all lanes, and it is the reference location for the signal program design. The capacity is, thus, derived from the number of vehicles crossing the stop lines of all approach lanes.

Capacity reductions

Capacity reductions occur due to changes in drivers’ following behaviour (increased headways or acceler-ation processes) and due to time differences between the signalled green time and the crossing of the stop line. Both reductions can be expressed in time units (green time difference∆t_gG).

The lane capacity can be therefore determined by measuring the saturation headway and investigate the difference∆t_gGbetween thesignalled green time t_Gand theeffective green timet_g(Eq. 7 and Eq. 8. Neg-ative green time differences, thus, mean capacity reductions. The green time differences are examined in Section 3.2.

t_g=t_G+ ∆t_gG (7)

and thus

C_l,eff= 3600^s/^h

h_s ·t_G+ ∆t_gG

t_C (8)

where C_l,eff lane capacity (veh/observation time) t_g effective green time (s)

t_G signalled green time (s)

∆t_gG green time difference (s)

t_C cycle time (s)

h_s saturation headway (s/veh) 3.1.4.4 Connection between conflict areas and capacity

As has been explained before, the capacity depends directly on the duration of intergreen times. Inter-green times are derived from the occupation of conflict areas. The time needed by vehicles to reach or clear the conflict areas determines the intergreen times. Instead of calculating the capacity at the stop line the capacity could be directly derived from the occupation of conflict areas as has been proposed by M^OSKOWITZand W^EBB(1955). This approach, though, has some major drawbacks:

• Decisive conflict area

Every intersection has numerous conflict areas. Not all of them are decisive. For every change of stages commonly only one conflict area becomes decisive. Decisive means that the time needed by the last clearing vehicle to clear this conflict area minus the time needed by the first entering vehicle to reach it is longer than for all other conflict areas. To calculate the capacity, thus, it has to be determined which conflict area is decisive. This conflict area may only be decisive for one specific change of stages.

• Unoocupied interval of conflict areas

During the stage changes for which a conflict area is not decisive, it may be unoccupied for some interval. This interval has to be known to calculate the capacity of the conflict area. It depends on the intersection layout and the stage settings. Furthermore, during certain stages the conflict area may not be used at all, which also depends on the stage settings and the intersection layout.

• Position of conflict area

The exact position of the conflict area depends on the vehicle trajectories. The exact conflict point determining for a movement sequence may vary.

• Headways at the conflict area

To calculate the capacity of a conflict area, the headways at the conflict area have to be known. For accelerated movements (e.g. entering vehicles), these headways are different from the ones at the stop line. To determine headways inside of the intersection requires an elaborate survey layout.

The random variation of the conflict point has to be considered.

3.1 Introduction 31

The capacity of the conflict areas can be determined if the stage settings, the vehicle trajectories, and the capacity at the stop line are known. It will be shown later in this report, that the capacity at the conflict area is suitable to derive conclusions for the intersection layout, particularly the role of entering behaviour and the entering distances.

3.2 Green time differences