1. NEW VOLUME DELAY FUNCTION Wacław Jastrzębski
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2. The Overcapacity Problem
Volume>>Capacity
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3. Scott Wilson Ltd – Poland Branch
Problem
Using standard VDF functions, sometimes the forecasted demand results in volumes greater than capacity, whereas the actual capacity may in fact be sufficient .
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4. Scott Wilson Ltd – Poland Branch
Model – representation of human behaviors using the language of mathematic
To travel or not to travel…. ?
To the city center or closer to home ?
By car or by transit ?
Which route?
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5. Scott Wilson Ltd – Poland Branch
Four Step Model
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6. Reality vs. Model Curves
capacity
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7. Mathematical Conditions for VDF Function
- F(x) is a strictly increasing function for the variable between 0 and + (F’(0)>0) - F(0) = T0, where T0 is the free-flow time; - F’(x) existing and is strictly increasing – that means that function is convex – this last condition is not essential but desirable; The calculation time for the new function should not use more CPU time than BPR function,
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8. Behavioral Conditions
Time spent in traffic congestion weights much more for the traveler than the travel time at the acceptable speed;
Within the range of of capacity, the average speed of traffic shows little sensitivity to the volume of traffic. After reaching the capacity level the travel time increases substantially;
Traveler chooses a path based on previous experience
Traveler can adjust the path as new information on traffic situation is acquired.
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9. The Modeling Conditions
The function should „force” the algorithm to seek additional paths in order to minimize the number of links with volume greater then capacity;
The free-flow-speed is the actual average speed as determined through the surveys (regardless of legal limitations such as speed limits).
The function takes into account that traffic lights decrease the average speed;
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10. Various Mathematical Formulas for VDF
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11. Various Mathematical Formulas for VDF
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12. Various Mathematical Formulas for VDF
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13. Scott Wilson Ltd – Poland Branch
Surveys’ Results
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14. Scott Wilson Ltd – Poland Branch
Surveys’ Results t
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15. Scott Wilson Ltd – Poland Branch
Surveys’ Results
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16. Scott Wilson Ltd – Poland Branch
New Function
b – odd integer >1
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17. Mathematical Condition
b – odd integer >1 so b –1 is even
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18. Scott Wilson Ltd – Poland Branch
Continuity V
0<V<C*s
V=C*s
C*s<V<C
V=C
V>C
T’’ - + T’ T To
INCREASE
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19. Scott Wilson Ltd – Poland Branch
New Function
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20. What Does it Mean “free flow speed”?
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21. Scott Wilson Ltd – Poland Branch
Function and Surveys
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22. Scott Wilson Ltd – Poland Branch
EMME Implementation
a fd27 =el1 * ( * ((volau / el2) ^ 9) * volau / el2) +
.2 * (volau .gt. el2) * (volau - el2)
a fd30 =el1 * ( * ((volau / el ) ^ ^ 7) +
.45 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)
a fd31 =el1 * ( * ((volau / el ) ^ ^ 7) +
.44 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)
a fd32 =el1 * ( * ((volau / el2 - .4) ^ ^ 7) +
.3 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)
a fd33 =el1 * ( * ((volau / el ) ^ ^ 5) +
.28 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)
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23. Equilibrium Assignment
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24. Equilibrium Assignment
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25. Scott Wilson Ltd – Poland Branch
How Does It Work?
- capacity 700 pcu/h
- free flow speed 70 km/h
- speed on the capacity limit 20 km/h
- practical capacity 0,65 capacity
- speed on the practical capacity limit ~ 45 km/h
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26. Scott Wilson Ltd – Poland Branch
Various Functions
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27. Results for Various Functions
link
Vatzek
wykładnicza
Overgaard’a
Generalised BPR
Conical
S logit
INRETS
Oslo 1
700
705
709
701
1212
711 2
703
745 3
713
704
656 4
698
699
872
697 5
474 6
689
690
473 7
692
467
iterations 33 41 19 28 48 9 21
overcapacity traffic 22
468 25
vehicle-hours
2522
2453
2593
2452
2312
2449
average time
30,88
30,03
31,75
30,02
28,31
29,98
average speed
19,43
19,98
18,90
21,19
19,99
20,01
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28. Matrix Reduction to Eliminate Overcapacity
Vatzek
expotential
Overgaard’a
BPR
generalised
Conical
S logit
INRETS
Oslo
Trip
matrix
4900
4803
4837
4871
4209
4827
4817
[%]
100,00
98,02
98,71
99,41
85,90
98,51
98,31
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29. Scott Wilson Ltd – Poland Branch
NO FUNCTION IS PERFECT!
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30. Scott Wilson Ltd – Poland Branch
Disadvantages
ASSIGNMENT 2035
~ min
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31. Reason? No alternative paths
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32. Scott Wilson Ltd – Poland Branch
Solution
Check network carefully and add new possible links – even local to add extra capacity
Add extra capacity or additional centroid connector
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