Scott Wilson Ltd – Poland Branch NEW VOLUME DELAY FUNCTION Wacław Jastrzębski

Scott Wilson Ltd – Poland Branch

NEW VOLUME DELAY

FUNCTIONWacław Jastrzębski


The Overcapacity Problem

Volume>>Capacity


Problem

Using standard VDF functions, sometimes the forecasted demand results in volumes greater than capacity, whereas the actual capacity may in fact be sufficient .


Model – representation of human behaviors using the language of mathematic

65/60195756,0 MPDPi

• To travel or not to travel…. ?

• To the city center or closer to home ?

• By car or by transit ?

• Which route?


Four Step Model

T R IP A S S IG N M E N T

M O D A L S P L IT

T R IP D IS T R IB U T IO N

T R IP G E N E R A T IO N


Reality vs. Model Curves

capacity


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,


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.


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;


Various Mathematical Formulas for VDF

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2

potok/przepustowość

cza

s [

min

.]

Irvin, Dodd and von Cube I Irvin, Dodd and von Cube II

a0,0005

b0,0026

CVdlaCVCCCTT

CVCdlaCVCTT

CVdlaCVCTT

PP

PPP

PPP

)(***

**

**

0

0

0

babaaa



S-logitowa

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

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 1,5 1,6 1,7 1,8 1,9 2

volume/capacity

tim

e [

min

.]

S-logitowa

t8

Ts=4,5

)1(*

00

1

)(

C

V

S

e

TTTT

t



BPR

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2

volume/capacity

czas

[m

in.]

BPR

VC

VTT

C

**1*0 ab


Surveys’ Results

Popiełuszki street G 2x31129 pcu/h

0

5

10

15

20

25

30

35

40

45

50

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

nr pomiaru

sp

eed

[k

m/h

]

measuredspeed

averagespeed

std.dev.

std. dev.

31,90

26,79

23,09


Surveys’ Results

Puławska streer G 2x31430 pcu/h

0

5

10

15

20

25

30

35

40

45

50

55

60

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

spee

d [

km/h

]

measuredspeed

averagespeed

std.dev.

std. dev.

42,71

32,21

25,85

t


Surveys’ ResultsPunkt pomiarowy Janki - sierpień 1998

0

20

40

60

80

100

120

140

0 200 400 600 800 1000 1200 1400

Natężenie ruchu [prz./godz]

Śre

dn

ia p

ręd

ko

ść c

hw

ilo

wa

[km

/go

dz]

L

P

l

p

TRANSPROJEKT - WARSZAWA


New Function

b – odd integer >1

CVforCVC

V

C

VTT

CVforC

V

C

VTT

a

a

bb

bb

*)(**1*

**1*

0

0

)1,0

1,0

}0{

a

R

R


Mathematical Condition

b – odd integer >1 so b–1 is even

)1,0

1,0

}0{

a

R

R

CCV

CC

VT

dV

dT bab

*1

***1

0


Continuity

0*)(lim

CVCV

V 0 0<V<C*

V=C* C*<V<C

V=C V>C

T’’ - - 0 + + +

T’ + + + + + +

T ToINCREASE INCREASE INCREASE


New Function

PRZYKŁADY FUNKCJI

0

10

20

30

40

50

60

70

80

90

100

110

120

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2

potok/przepustowość

prę

dko

ść [

km/h

]

E2x3(Z)

GP2x2 (C)

G2x3(U)

Z 1x2 (C)


What Does it Mean “free flow speed”?

0,00

5,00

10,00

15,00

20,00

25,00

30,00

35,00

40,00

45,00

50,00

55,00

60,00

65,00

70,00

0 50 100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

900

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

1550

1600

1650

1700

1750

1800

1850

1900

1950

2000

2050

2100

2150

2200

2250

2300

2350

2400

2450

Main Arterial

Secondary

Main ArterialBPR

SecondaryBPR


Function and SurveysULICE GŁÓWNE 2x2

0

10

20

30

40

50

60

70

80

90

100

110

120

0

100

200

300

400

500

600

700

800

900

100

0

110

0

120

013

00

140

0

150

0

160

0

170

018

00

190

0

200

0

210

0

220

0

230

024

00

250

0

260

0

270

028

00

290

030

00

310

0

320

0

330

034

00

350

036

00

370

0

380

0

390

040

00

natężenie ruchu [p.u./godz.]

prę

dko

ść

[k

m./

go

dz.

]

G2x2(C)

G2x2(U)

G2x2(P)

G2x2(Z)

al.Niepodległości [WBR-98]

al.Niepodległości [IDiM]

Popiełuszki

Podwale I

Podwale II


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) +


a fd32 =el1 * (1 + 70 * ((volau / el2 - .4) ^ 7 + .4 ^ 7) +


a fd33 =el1 * (1 + 28 * ((volau / el2 - .42) ^ 5 + .42 ^ 5) +



Equilibrium Assignment


Equilibrium Assignment

0

10

20

30

40

50

60

70

80

0 100 200 300 400 500 600 700 800 900 1000 1100 1200

prę

dk

oś

ć [

km

/go

dz.]

autorska

autorska

BPRzgeneralizowana

BPRzgeneralizowana

S logit

S logit

01200 1100 1000 900 800 300400500600700 100200

v2

v1


How Does It Work?

7*0,05 km

- 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

700)*0,25*(V700V

700

V0,543*0,4250,425

700

V84*1

*TT7

7

0


Various Functions

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800

natężenie ruchu [poj./godz.]

prę

dk

oś

ć [

km

/go

dz.]

funkcja autorska

Overgaard

BPRzgeneralizowana

Conical

S logit

INRETS

OSLO

Punkty stałe


Results for Various Functions

link VatzekwykładniczaOvergaard’a

Generalised BPR

Conical S logit INRETS Oslo

1 700 705 709 701 1212 701 7112 700 703 709 701 745 701 7113 700 713 704 701 656 701 7034 700 698 699 700 872 700 6975 700 699 697 699 474 700 6996 699 689 690 698 473 698 6907 700 692 692 700 467 699 689

iterations 33 41 19 28 48 9 21overcapacity

traffic0 21 22 7 468 25 25

vehicle-hours 2522 2453 2593 2452 2312 2452 2449average time 30,88 30,03 31,75 30,02 28,31 30,02 29,98

average speed 19,43 19,98 18,90 19,98 21,19 19,99 20,01


Matrix Reduction to Eliminate Overcapacity

Vatzekexpotential

Overgaard’aBPR

generalisedConical 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


NO FUNCTION IS PERFECT!


Disadvantages

~132000 minASSIGNMENT 2035


Reason? No alternative paths


Solution

• Check network carefully and add new possible links – even local to add extra capacity

• Add extra capacity or additional centroid connector

Documents

Scott Wilson Ltd – Poland Branch NEW VOLUME DELAY FUNCTION Wacław Jastrzębski