NEW VOLUME
DELAY
FUNCTION
Wacław Jastrzębski
Scott Wilson Ltd – Poland Branch
The Overcapacity Problem
Volume>>Capacity
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 .
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?
PDPi  0,5756  M1960 / 65
Scott Wilson Ltd – Poland Branch
Four Step Model
TRIP GENERATION
TRIP DISTRIBUTION
MODAL SPLIT
TRIP ASSIGNMENT
Scott Wilson Ltd – Poland Branch
Reality vs. Model Curves
capacity
Scott Wilson Ltd – Poland Branch
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,
Scott Wilson Ltd – Poland Branch
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 0.2-0.8 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.
Scott Wilson Ltd – Poland Branch
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;
Scott Wilson Ltd – Poland Branch
Various Mathematical
Formulas for VDF
Irvin, Dodd and von Cube I
Irvin, Dodd and von Cube II
5
a0,0005
4,5
b0,0026
4
T  T0 a *CP a *V  CP  dlaV  CP
3,5
T  T0 a *CP  b *V  CP  dla C  V  CP
T  T0 a *CP  b *C  CP    *(V  C) dlaV  C
czas [min.]
3
2,5
2
1,5
1
0,5
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
potok/przepustowość
Scott Wilson Ltd – Poland Branch
0,9
1
1,1
1,2
Various Mathematical
Formulas for VDF
S-logitowa
S-logitowa
5
4,5
T  T0 
4
(TS  T0 )
V
t *(1 )
1 e
3,5
C
time [min.]
3
t8
2,5
Ts=4,5
2
1,5
1
0,5
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
volume/capacity
Scott Wilson Ltd – Poland Branch
1,5
1,6
1,7
1,8
1,9
2
Various Mathematical
Formulas for VDF
BPR
BPR
5
4,5

 V

T  T0 * 1  a * 

 CC

4
3,5



b

   *V


czas [min.]
3
2,5
2
1,5
1
0,5
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
volume/capacity
Scott Wilson Ltd – Poland Branch
0,9
1
1,1
1,2
Surveys’ Results
Popiełuszki street G 2x3
1129 pcu/h
50
45
40
measured
speed
speed [km/h]
35
average
speed
31,90
std.dev.
30
26,79
std. dev.
25
23,09
20
15
10
5
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
nr pomiaru
Scott Wilson Ltd – Poland Branch
37
39
41
43
45
Surveys’ Results
Puławska streert G 2x3
1430 pcu/h
60
55
50
42,71
45
measured
speed
average
speed
speed [km/h]
40
32,21
35
std.dev.
std. dev.
30
25
25,85
20
15
10
5
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
nr pomiaru
Scott Wilson Ltd – Poland Branch
Surveys’ Results
Punkt pomiarowy Janki - sierpień 1998
TRANSPROJEKT - WARSZAWA
140
Średnia prędkość chwilowa [km/godz]
120
100
L
80
P
60
l
p
40
20
0
0
200
400
600
800
1000
Natężenie ruchu [prz./godz]
Scott Wilson Ltd – Poland Branch
1200
1400
New Function
b


 V

V

T  T0 * 1  a *        b    * 
 C



C





for V  C
b


 V

V

T  T0 * 1  a *        b    *  (V  C ) *  
 C



C





b – odd integer >1
a  R
  R  {0}
   0,1 
   0,1)
Scott Wilson Ltd – Poland Branch
for V  C
Mathematical Condition
b 1


dT
V
1



 T0  a * b *     *  V  C *   


dV
C
C
C




b – odd integer >1 so b –1 is even
a  R
  R  {0}
   0,1 
   0,1)
Scott Wilson Ltd – Poland Branch
Continuity
lim
(V  C ) *   0
V C
V
0
0<V<C*

V=C*
C*<V
<C
V=C
V>C
T’’
-
-
0
+
+
+
T’
+
+
+
+
+
+
T
To
INCREASE
INCREASE
Scott Wilson Ltd – Poland Branch
INCREASE
New Function
PRZYKŁADY FUNKCJI
120
110
100
E2x3(Z)
90
80
prędkość [km/h]
GP2x2 (C)
70
60
G2x3(U)
50
40
Z 1x2 (C)
30
20
10
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
potok/przepustow ość
Scott Wilson Ltd – Poland Branch
1
1,1
1,2
What Does it Mean
“free flow speed”?
70,00
65,00
60,00
55,00
50,00
45,00
Main Arterial
40,00
35,00
Secondary
30,00
Main Arterial
BPR
25,00
20,00
Secondary
BPR
15,00
10,00
5,00
50
00
24
50
24
00
23
50
23
00
22
50
22
00
21
50
21
00
20
50
20
00
19
50
19
00
Scott Wilson Ltd – Poland Branch
18
50
18
00
17
50
17
00
16
50
16
00
15
50
15
00
14
50
14
00
13
50
13
00
12
50
12
00
11
50
11
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
00
10
10
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
0
50
0,00
Function and Surveys
ULICE GŁÓWNE 2x2
120
G2x2(C)
110
G2x2(U)
90
G2x2(P)
80
G2x2(Z)
70
al.Niepodległoś
ci [WBR-98]
60
al.Niepodległoś
ci [IDiM]
50
Popiełuszki
40
Podwale I
30
Podwale II
20
10
0
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
prędkość [km./godz.]
100
natężenie ruchu [p.u./godz.]
Scott Wilson Ltd – Poland Branch
EMME Implementation
a fd27 =el1 * (1 + 1.35 * ((volau / el2) ^ 9) + .65 * volau / el2)
+
.2 * (volau .gt. el2) * (volau - el2)
a fd30 =el1 * (1 + 100 * ((volau / el2 - .44) ^ 7 + .44 ^ 7) +
.45 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)
a fd31 =el1 * (1 + 90 * ((volau / el2 - .43) ^ 7 + .43 ^ 7) +
.44 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)
a fd32 =el1 * (1 + 70 * ((volau / el2 - .4) ^ 7 + .4 ^ 7) +
.3 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)
a fd33 =el1 * (1 + 28 * ((volau / el2 - .42) ^ 5 + .42 ^ 5) +
.28 * volau / el2) + .4 * (volau .gt. el2) * (volau - el2)
Scott Wilson Ltd – Poland Branch
Equilibrium Assignment
Scott Wilson Ltd – Poland Branch
Equilibrium Assignment
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
7*0,05 km
7

 V

V 

7
 1  84* 
 0,425   0,425  0,543*

  700


T  T0*
700





 V  700 *(V  700)*0,25

Scott Wilson Ltd – Poland Branch
Various Functions
80
70
funkcjaautorska
60
Overgaard
50
prędkość [km/godz.]
BPR
zgeneralizow
ana
Conical
40
Slogit
30
INRETS
20
OSLO
Punktystałe
10
0
0
50
100
150
200
250
300
350
400
450
500
550
600
650
natężenieruchu[poj./godz.]
Scott Wilson Ltd – Poland Branch
700
750
800
Results for Various Functions
wykładnicza Generalised
Conical
Overgaard’a
BPR
link
Vatzek
1
700
700
700
700
700
699
700
33
705
703
713
698
699
689
692
41
709
709
704
699
697
690
692
19
0
21
2522
30,88
19,43
2453
30,03
19,98
2
3
4
5
6
7
iterations
overcapacity
traffic
vehicle-hours
average time
average speed
S logit
INRETS
Oslo
701
701
701
700
699
698
700
28
1212
745
656
872
474
473
467
48
701
701
701
700
700
698
699
9
711
711
703
697
699
690
689
21
22
7
468
25
25
2593
31,75
18,90
2452
30,02
19,98
2312
28,31
21,19
2452
30,02
19,99
2449
29,98
20,01
Scott Wilson Ltd – Poland Branch
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
Scott Wilson Ltd – Poland Branch
NO
FUNCTION
IS PERFECT!
Scott Wilson Ltd – Poland Branch
Disadvantages
ASSIGNMENT 2035
~132000 min
Scott Wilson Ltd – Poland Branch
Reason? No alternative paths
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
Scott Wilson Ltd – Poland Branch