Benjamin Heydecker JD (Puff) Addison Centre for Transport Studies UCL Dynamic Modelling of Road Transport Networks

Benjamin HeydeckerJD (Puff) Addison

Centre for Transport StudiesUCL

Dynamic Modelling of Road Transport Networks

MoN 7: 27 June 2008 Centre for Transport Studies

University College London2

Transport Networks

Dominated by link travel time:1km ~ 100s

Sioux Falls:24 nodes76 links552 OD pairs



Serve individual needs for travel Demand reflects travellers’ experience – response to change Dimensions of choice:

Origin Destination O-D pair Frequency of travel Mode Departure time Route

Transport Networks

Equilibrium analysis Demand-Supply Equilibrium

0

200

400

600

800

400 500 600 700

Cost

Flo

w

Throughput

Demand

C= F(T, p)

T = D(C)



link state xa (t)

link exit time a (t)

link outflow ga[a (t)] .

Dynamic Link Traffic Model

ea(t) ga(t)xa(t)

Link a OutflowInflow

Link inflow ea(s)

x t e t g t



Transport Networks: Features

Conservation of traffic at nodes

c ca a

a B n a A n

g t e t



First-In First-Out: Accumulated flow

Flow propagation

Flows and travel times interlinked

Dynamic Traffic Flows

Time t

Tra

ffic

A

0 s (s)

A = E(t) A = G(t)

ssgse pppp

ss

s t

e s ds g t dt



Traffic Modelling

First In First Out (FIFO):Entry time s , exit time (s)

Flow propagation:Entry flow e(s) , exit flow g(s)

Multi-commodity FIFO: Papageorgiou (1990)

τ τe s g s s

0τ s

τ τp pa a a ae s g s s

xaep



Link characteristics:Free-flow travel time Capacity (Max outflow) Q

Exit time:

Travel Time Models

State xa(t)

Link aFree-flow Capacity Q s s x s Q

35

40

45

50

55

60

Tra

vel tim

e (

s)

0 20 40 60 80 100 120 140

Entry time (s)

Travel time



Accumulate link costs according to time ap(s) of entry

Travel time:

Nested cost operator

Calculation of Costs

p a ap pa p

C s c s s s




Travel time:


Origin-specific costs: ho(s)

Destination-specific costs: fd[p(s)]


p a ap pa p

C s c s s s

Time-Dependent Costs

-50

0

50

100

150

200

0 100 200

Time t

Co

st Origin

Destination




Travel time:


Origin-specific costs: ho(s)

Destination-specific costs: fd[p(s)]

Total cost associated with journey:


p a ap pa p

C s c s s s

p o p p d pC s h s s s f s

Time-Dependent Costs

-50

0

50

100

150

200

0 100 200

Time t

Co

st Origin

Destination



Dynamic equilibrium condition

Path inflow ep(s) , path p , departure time s

Cost Cp(s)

0 ,p p od ode s C s k s p P s



A Variational Inequality (VI) approach

Smith (1979) Dafermos (1980) Variational Inequality

Set of demand feasible assignments: D(s)

Assignment e D(s) is an equilibrium if

Then (set f = e )

Equilibrium assignment solves (solution is 0 )

where

Solve forwards over time s : forward dynamic programming

0T s D s f e C f

Max 0T

D ss

ff e C

Max Tv

D sZ s

f

e f e C

Min v

D sZ

ee



Demand for Travel

Dynamic trip matrix T(s) = {Tod(s)}

Fixed: T(s) is exogenous - estimation?

.



Demand for Travel



.

Dynamic equilibrium inflows

0

1

2

3

4

5

6

0 200 400 600 800 1000

Departure time (seconds)

Inflo

w (

vehi

cles

/s) Demand

Route 1

Route 2

Equilibrium route costs

0

100

200

300

400

0 200 400 600 800 1000

Departure time (seconds)C

ost

(sec

onds

)

Route 1

Route 2



Demand for Travel



Departure time choice:

T(s) varies according to C(s)

- endogenous

Cost of travel is determined uniquely for each o – d pair

*

*

0, ,

0

p odp od

p od

C s Ce s p P od s

C s C



Demand for Travel



Departure time choice:

T(s) varies according to C(s)

- endogenous

Elastic demand:

s

s ds T D C

Demand-Supply Equilibrium

0

200

400

600

800

400 500 600 700

Cost

Flo

w

Throughput

Demand

C= F(T, p)

T = D(C)



Dynamic Traffic Assignment

Route choice in congested road networks Flows vary rapidly by comparison with travel times Travel times and congestion encountered vary

Planning and management: Congestion Capacities Free-flow travel times Tolls …



Analysis of Dynamic Equilibrium Assignment

Wardrop’s user equilibrium (1952) after Beckmann (1956):

To maintain equilibrium:

Necessary condition for equilibrium:

ododp

p Ppskds

dCse 0

sPp

sCsCse

sCsCseod

odpp

odpp

*

*

0

0

od

p pp od

q qq P

g se s T s

g s



Dynamic Equilibrium Assignment with Departure Time Choice

Hendrickson and Kocur: cost of all used combinations is equal

Necessary condition for equilibrium:

Cost of travel is determined uniquely for each o – d pair

*

*

0, ,

0

p odp od

p od

C s Ce s p P od s

C s C

, ,op p p od

d p

1 - h se s = g s p P od s

1 + f s



Logit: Assigned flows ep(s) given by

ep(s) is continuous in path costs Cp(s)

Cp(s) is continuous in state xa(s)

for finite inflows, xa(s) is continuous in time s

ep(s) is continuous in time s

Can use recent costs to estimate assignments

Dynamic Stochastic Equilibrium Assignment

exp

exp

od

r p

p od

r qq P

C se s T s

C s



Example Dynamic Stochastic Assignments

DSUE assignments Costs and Inflows

0

0.5

1

1.5

2

Infl

ow

(v

eh

icle

s/s

)

0 500 1000 1500 2000Entry time (s)

Route 1 Route 2

Stochastic assignments

0

1

2

3

4

Infl

ow

(v

eh

icle

s/s

)

0

500

1000

Co

st

(s

)

0 500 1000 1500 2000

Entry time (s)

Costs and inflows



Equilibrium Network Design: structure

Design p variables

Response variables T(p)

Evaluation

S(C(T, p)) - U(p)

S(C(T, p)): Travellers’ surplus

U(p): Construction costs

Bi-level Structure



Equilibrium Network Design:

Formulation:

Bi-level structure: Costs C depend on

Throughput T Design p

Demands T areconsistent with costs C

CDT

pTFC

pCCTp

,

toSubject

Max,,

US

Demand-Supply Equilibrium

0

200

400

600

800

400 500 600 700

Cost

Flo

w

Throughput

Demand

C= F(T, p)

T = D(C)



Optimality Conditions

No feasible variation p in design improves objective S - U

Using properties of S

Sensitivity analysis for d C / d p

0

ppp d

dU

d

dS

0T

ppp

CCD

d

dU

d

d



Sensitivity of costs C to design p:

Partial sensitivity to origin-destination flows:

Partial sensitivity to design:

FFC

DFI

p

CpTT

11

d

d

d

d

Sensitivity Analysis of Equilibrium

11T CF ET

CCFF pETp 1T



Sensitivity Analysis: Volume of Traffic Er

Cost-throughput:

Start time:

Dependence on values of time f ’(.) and h ’(.)

rr

r

Qfhfh

fhfh

E

C 11100

1100

rr

r

Qfhfh

fh

E

s 11100

110



Dynamic System Optimal Assignment

sTse

sese

dssesc

odPp

p

p

paa

s

aLa

a

od

:toSubject

Mine

Minimise total travel costs (Merchant and Nemhauser, 1978)

Specified demand profile T(s)



Dynamic System Optimal Assignment

Solution by Optimal Control TheoryChow (2007)

ododpppppp PpskssssCse

τ0

Private cost

Direct externality

Costate variables



Comment on Optimal Control Theory solution

ododpppppp PpskssssCse

τ0

Necessary condition

• Hard to solve• Non-convex (non-linear equality constraints)

Curse of dimensionality



Analysis: Recover convexity

Carey (1992):

FIFO as inequality constraints

Convex formulation

Not all traffic need flow – holding back

τ τ 0p pa a a ag s s e s



Illustrative example

o

d1

d2

Qo

Q1

Q2

g1

g2




o

d1

d2

Qo

Q1

Q2

g1

g2

g1+g2 < Q0

hi < Qi

DSO as LP




o

d1

d2

Qo

Q1

Q2

g1

g2

g1+g2 < Q0

hi < Qi

g1

g2

Q2

Q1

Q0

Q0

DSO as LP




o

d1

d2

Qo

Q1

Q2

g1

g2

g1+g2 < Q0

hi < Qi

g1

g2

Q2

Q1

Q0

Q0

Demand

DSO as LP




o

d1

d2

Qo

Q1

Q2

g1

g2

g1+g2 < Q0

hi < Qi

g1

g2

Q2

Q1

Q0

Q0

Demand

Solution region

DSO as LP




o

d1

d2

Qo

Q1

Q2

g1

g2

g1+g2 < Q0

hi < Qi

g1

g2

Q2

Q1

Q0

Q0

Demand

Solution region

DSO as LP

Not proportional to demand



Directions for Further Work

Investigate:

Network effects

Heterogeneous travellers

Pricing

Time-Specific Costs

0

50

100

150

200

-650 -600 -550 -500 -450

Departure time (seconds)C

ost (

seco

nds)

Type 1 Type 2

Type 2Type 1 Type 1

Documents

Benjamin Heydecker JD (Puff) Addison Centre for Transport Studies UCL Dynamic Modelling of Road Transport Networks