5.4 Model application
5.4.3 Maximum capacity
5.4.3.1 Exemplative quantitative results
Basis for the calculation of the maximum capacity is the conflict tree (cf. Section 3.3.4). For every conflict the speed, distance, and time differences (cf. Section 3.3.2) have to be calculated. The overall intergreen time difference depends on the stage sequence. For variable stage sequences, the probability of the different possible stage sequences would have to be considered. Here – as has been mentioned before – a fixed sequence is assumed to improve clarity. Furthermore, the actual traffic actuated sig-nal program showes only margisig-nal variations, the result is therefore a good estimate for the actual situation.
Intergreen time differences
The intergreen time differences∆tig,m,ifor the individual movement sequences are determined according to Eq. 20 together with Eq. 21 and Eq. 22. The parameters needed are:
• the difference of the crossing times of entering vehicles∆tcr,e,i (cf. Section 5.3.2)
• the difference of the entering time∆te,i (calculated from differing entering distances and entering speeds)
• the difference of the crossing time ∆tcr,i (cf. Section 5.3.3; the same effective crossing times of through vehicles and turning vehicles have been assumed)
• the difference of the clearance time∆tcl,i (calculated from differing clearance distances and clear-ance speeds)
The entering speed has been calculated based on the speed measurements at the example intersec-tion (cf. Secintersec-tion 5.3.4, Table 13). The clearance speed is derived from the measurements from non-coordinated approaches (Table 14). Since it equals the clearance speed in the Guidelines, no difference occurs. Systematic distance differences are taken into account as described in Section 3.3.2.2. The varia-tion error has been estimated from the local situavaria-tion. Only the distances in connecvaria-tion with left turning vehicles are corrected. The variation of vehicle length has been neglected.
The intergreen times and the intergreen time differences for all signal group combinations are given in Table 20. The complete conflict tree of the survey intersection is shown in Ap-pendix B.3.
Stage Signal group tig,s,i ∆tig,g,i Clearing Entering Clearing Entering
(-) (-) (-) (-) (s) (s)
1 2 5 11 0 0.0
1 2 5 12 5 -4.8
1 2 11 12 0 0.0
2 3 11 2 5 -5.8
2 3 11 8 8 -7.3
2 3 12 2 6 -4.9
2 3 12 8 8 -7.1
3 5 2 8 6 -5.5
5 1 8 5 8 -7.5
5 1 8 11 5 -5.8
Table 20:Intergreen times and intergreen time differences for example intersection
Capacity improvement potential
With the cumulated intergreen differences for signal groups∆tig,g,i, the green time extensions∆tG,i can be obtained. An optimisation with the Simplex Algorithm delivers extensions as shown in Table 21. With the saturation headways and the number of lanes, the capacity improvement potential can be calculated according to Eq. 32. The results are listed in the last column of Table 21.
The capacity improvement potential ∆Cmax (based on times calculated to an accuracy of the tenth of a second) amounts to more than 1400veh/h. This is the maximum improvement potential for the survey intersection under the prerequisite of complete information and deterministic driver be-haviour.
5.4 Model application 99
Approach Signal group ∆tG No of lanes hs ∆Cmax,i (s) (-) (s) (veh/h)
N FV 2 6.7 2 1.9 282
E FV 5 0.0 1 1.8
S FV 8 8.2+8.6+5.8=22.6 2 1.8 1004
W FV 11 1.7 1 2.0 34
W FV 12 0.5+4.9=5.4 1 1.9 114
1434
Table 21:Green time extensions and resulting capacity improvements for the example intersection
Consideration of saturation headways at the conflict points
The maximum capacity improvement potential is calculated under neglect of post-encroachment times.
Even under optimal conditions, the capacity could not further be improved with respect to intergreen times. In reality post-encroachment will have to be considered, reducing the capacity (cf. achievable capacity, Section 4.2.5). The post-encroachment time needed for safety reasons is not in the focus of this research. However, when comparing the signal change interval to a continuous stream, it is apparent, that the capacity calculated without consideration of post-encroachment times will lead to higher values (at least at the decisive conflict point) than the capacity of the continuous stream. In a continuous stream the minimum headway is the saturation headway. It is, therefore, of interest to consider the saturation headway at the conflict point.
The saturation headway consists of the time needed for the vehicle to cover its own vehicle length and the net time gap to the following vehicle. The former time is already considered at the conflict area. The post-encroachment time has to equal the mentioned net time gap to fulfill the requirement explained above. If an average vehicle length of five metres is assumed and the clearance speed is set to vcl=10m/s, the saturation headway has to be reduced by half a second to deliver the post-encroachment time.
The post-encroachment time reduces the intergreen time differences and, thus, the green time exten-sions. The capacity improvement potential with consideration of saturation headways at the conflict points can be calculated with the reduced green time extensions. The green time extensions of Table 21 are consequently recalculated with intergreen time differences (cf. Appendix B.3) reduced by the post-encroachment time of tPE=hs,i−0.5 s. The resulting capacity improvement potential (Table 22) is still more than 80 % of the maximum one.
Approach Signal group ∆tG’ ∆Cmax,i’ (s) (veh/h)
N FV 2 6.7-1.4 223
E FV 5 0.0
S FV 8 8.2+8.6+5.8-3·1.3 831
W FV 11 1.7 34
W FV 12 0.5+4.9-1.4 105
1193
Table 22:Green time extensions and resulting capacity improvements for the example intersection (sat-uration headway at conflict points)
5.4.3.2 Analysis of the result Minimum intergreen times
For the example intersection the results show that integreen times could be nearly reduced to naught under the assumptions on which the maximum improvement potential is based (cf. Table 20). The decisive intergreen time differences for the signal group combinations range between 80 % and 120 % of the decisive intergreen times calculated according to the German Guidelines for Traffic Signals (RiLSA) with the average being nearly 100 %. The parameters leading to this extreme potential reduction of intergreen times are highlighted in the following paragraph.
Relative importance of parameters
The relative importance of the different parameters for the improvement potential at the survey inter-section is given in Table 23 and Figure 32. For the dermination of the different parameters refer to Section 3.3 on page 36.
Figure 32:Relative influence of different factors on the overall intergreen time difference
Factor Relative weight
Conflict difference time ∆tc,i 56%
Starting response and entering time ∆tSR+∆te,i 29%
Crossing time ∆tcr,i 11%
Safety margin tsaf 4%
Clearance time ∆tcl,i 0%
Table 23:Relative influence of different factors on the overall intergreen time difference
To illustrate the role of the determining conflict, in Table 24 the determining conflicts for the example intersection are given together with the probability of their occurence. The conflicts with particularly long intergreen times are highlighted. It can be seen, that always either bicycles or turning traffic is involved, both of which have commonly only low probability of occurence. Specifically the bicycle traffic leads to long intergreen times, even if only very few cyclists are present at the particular intersection.
5.4 Model application 101
tig clearing direction clearing vehicle entering direction pi
5 right MV left 0.13
5 through bike left 0.07
10 through bike right 0.01
5 through MV left 0.09
8 through bike right 0.00
8 through bike right 0.00
6 left bike left 0.00
8 left bike through 0.00
6 right MV left 0.04
6 through MV left 0.21
Table 24:Determining conflicts for the example intersection
This, of course, only applies to large intersections, where bicycle traffic leads to longer intergreen times than motorised vehicles.17 The influence of certain movement sequences on the capacity should therefore always be considered together with the likeliness of the underlying movements.
Conclusions for the number of stages
Commonly the number of stages is kept as low as possible to improve the capacity as long as no safety concerns have to be raised. As could be seen, the intergreen times don’t have to be of major importance in this context. If moving starts can be excluded and unfavourable conflicts are prohibited, the intergreen times could be significantly reduced. Taking the effective capacity into account, the capacity will be only insignificantly reduced by the intergreen times for additional stages.
However, the main reason for capacity reductions by additional stages has to be seen in the number of lanes receiving green time simultaneously. Because the total capacity is the sum of the lane capacities and the lane capacities will be reduced on average by additional stages, the capacity will be lower the more stages are used. This holds only true, of course, as long as the capacities of permitted streams is higher than the capacity reductions by additional stages.
For the example intersection, for instance, stages three and five could be reduced to one stage. In this way signal group FV 2 would receive additional green time. The capacity would be higher. However, the capacity for the left turning traffic of FV 8, which would be permitted only in this case, could be too low to serve the demand.
Thus, the capacity impact of additional stages has to be evaluated individually. It depends mainly on the capacity of permitted streams, number of lanes and saturation headways, and com-monly to a minor degree, depending on the intergreen time calculation method, on the intergreen times.
5.4.4 Uncertainty of model results