Dynamic Modelling of Road Transport Networks
Benjamin Heydecker
JD (Puff) Addison
Centre for Transport Studies
UCL
Transport Networks
Dominated by link travel time:
1km ~ 100s
Sioux Falls:
24 nodes
76 links
552 OD pairs
MoN 7: 27 June 2008
2
Centre for Transport Studies
University College London
Transport Networks
 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
MoN 7: 27 June 2008
Demand-Supply Equilibrium
800
C= F(T, p)
Throughput
Equilibrium
analysis
600
Flow

Demand
400
200
T = D(C)
0
400
500
600
Cost
3
Centre for Transport Studies
University College London
700
Dynamic Link Traffic Model
Link inflow ea(s)
Inflow
ea(t)
Link a
Outflow
ga(t)
xa(t)
 link state xa (t) x  t   e  t   g  t 
 link exit time a (t)
 link outflow ga[a (t)] .
MoN 7: 27 June 2008
4
Centre for Transport Studies
University College London
Transport Networks: Features
Conservation of traffic at nodes

aB n 
MoN 7: 27 June 2008
5
g ac  t  

aA n 
eac  t 
Centre for Transport Studies
University College London
Dynamic Traffic Flows
s

e  s  ds 
s
 s 

Accumulated flow
g  t   dt 
t 
 
Traffic A
First-In First-Out:
A = E(t)
A = G(t)
ep s  g p  p s p s
0
Flow propagation
s
(s) Time t
Flows and travel times interlinked
MoN 7: 27 June 2008
6
Centre for Transport Studies
University College London
Traffic Modelling
First In First Out (FIFO):
Entry time s , exit time (s)
τ s   0
Flow propagation:
Entry flow e(s) , exit flow g(s)
e  s   g  τ  s  τ  s 
Multi-commodity FIFO:
Papageorgiou (1990)
ep
MoN 7: 27 June 2008
eap  s   g ap  τ a  s  τ a  s 
xa
7
Centre for Transport Studies
University College London
Travel Time Models
Link characteristics:
Free-flow travel time

Capacity (Max outflow) Q
Exit time:   s   s    x  s  Q
Free-flow 
Travel time
Capacity Q
State xa(t)
60
Travel time (s)
Link a
55
50
45
40
35
0
20
40
60
80
100
120
140
Entry ti m e (s)
MoN 7: 27 June 2008
8
Centre for Transport Studies
University College London
Calculation of Costs
Accumulate link costs according to time ap(s) of entry
Travel time:
Cp s 
 ca ap  s    p  s   s
a p
Nested cost operator
MoN 7: 27 June 2008
9
Centre for Transport Studies
University College London
Calculation of Costs
Accumulate link costs according to time ap(s) of entry
Travel time:
Cp s 
 ca ap  s    p  s   s
a p
Time-Dependent Costs
200
Nested cost operator
Cost
150
Origin-specific costs: ho(s)
Origin
Destination
50
0
Destination-specific costs: fd[p(s)]
MoN 7: 27 June 2008
100
0
100
-50
Time t
10
Centre for Transport Studies
University College London
200
Calculation of Costs
Accumulate link costs according to time ap(s) of entry
Travel time:
Cp s 
 ca ap  s    p  s   s
a p
Time-Dependent Costs
200
Nested cost operator
Cost
150
Origin-specific costs: ho(s)
100
Origin
Destination
50
0
Destination-specific costs: fd[p(s)]
0
100
200
-50
Time t
Total cost associated with journey: C p  s   ho  s     p  s   s    p  f d   p  s  
MoN 7: 27 June 2008
11
Centre for Transport Studies
University College London
Dynamic equilibrium condition
Path inflow ep(s) , path p , departure time s
 
 
e p  s    0  C p  s    kod  s  p  Pod , s
 
 
Cost Cp(s)
MoN 7: 27 June 2008
12
Centre for Transport Studies
University College London
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
Max  f  e  C  s   0
T
T
(set f = e )
f D s 
Equilibrium assignment solves
 f  e  C  s   0 f  D  s 
Min Zv  e  (solution is 0 )
eD s 
Max  f  e  C  s 
where Zv  e   f
D s 
T
Solve forwards over time s : forward dynamic programming
MoN 7: 27 June 2008
13
Centre for Transport Studies
University College London
Demand for Travel
Dynamic trip matrix T(s) = {Tod(s)}
Fixed: T(s) is exogenous - estimation?
.
MoN 7: 27 June 2008
14
Centre for Transport Studies
University College London
Demand for Travel
Dynamic trip matrix T(s) = {Tod(s)}
Fixed: T(s) is exogenous - estimation?
Dynamic equilibrium inflows
Inflow (vehicles/s)
6
Equilibrium route costs
Demand
Route 1
Route 2
400
Cost (seconds)
5
4
3
2
1
.
300
200
Route 1
Route 2
100
0
0
0
200
400
600
800
0
1000
400
600
800
1000
Departure time (seconds)
Departure time (seconds)
MoN 7: 27 June 2008
200
15
Centre for Transport Studies
University College London
Demand for Travel
Dynamic trip matrix T(s) = {Tod(s)}
Fixed: T(s) is exogenous - estimation?
Departure time choice:
T(s) varies according to C(s)
- endogenous
  0  C  s   C* 
p
od 

ep  s 

*
  0  C p  s   Cod

p  Pod , od , s
Cost of travel is determined uniquely for each o – d pair
MoN 7: 27 June 2008
16
Centre for Transport Studies
University College London
Demand for Travel
Dynamic trip matrix T(s) = {Tod(s)}
Fixed: T(s) is exogenous - estimation?
Demand-Supply Equilibrium
Elastic demand:
 T  s  ds  D C
s
MoN 7: 27 June 2008
800
C= F(T, p)
Throughput
600
Flow
Departure time choice:
T(s) varies according to C(s)
- endogenous
Demand
400
200
T = D(C)
0
400
500
600
Cost
17
Centre for Transport Studies
University College London
700
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
…
MoN 7: 27 June 2008
18
Centre for Transport Studies
University College London
Analysis of Dynamic Equilibrium Assignment
Wardrop’s user equilibrium (1952) after Beckmann (1956):
*
s 
e p s   0  C p s   Cod
 p  Pod s 
*
e p s   0  C p s   Cod s 
To maintain equilibrium:
dC p
e p s  0 
 kod s p  Pod
ds
Necessary condition for equilibrium:
ep  s 
g p   p  s  

qPod
MoN 7: 27 June 2008
19
g q  q  s  
Tod  s 
Centre for Transport Studies
University College London
Dynamic Equilibrium Assignment with
Departure Time Choice
Hendrickson and Kocur: cost of all used combinations is equal
  0  C  s   C*
p
od

ep  s 
*

0

C
s

C



p
od






p  Pod , od , s
Necessary condition for equilibrium:
 1 - h  s  
o
 g p   p  s  p  Pod , od , s
ep  s = 


 1 + f d   p  s  



Cost of travel is determined uniquely for each o – d pair
MoN 7: 27 June 2008
20
Centre for Transport Studies
University College London
Dynamic Stochastic Equilibrium Assignment
Logit: Assigned flows ep(s) given by
ep  s  
exp   r C p  s  
 exp r Cq  s 
Tod  s 
qPod
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
MoN 7: 27 June 2008
21
Centre for Transport Studies
University College London
Example Dynamic Stochastic Assignments
DSUE assignments
Costs and Inflows
Stochastic assignments
Costs and inflow s
2
Route 1
Route 2
1.5
1
1000
3
500
2
1
0.5
0
0
500
1000
1500
Entry time (s)
MoN 7: 27 June 2008
0
0
2000
22
500
1000
Entry time (s)
1500
0
2000
Centre for Transport Studies
University College London
C os t (s)
In fl o w ( v e h i c l e s /s )
In fl o w ( v e h i c l e s /s )
4
Equilibrium Network Design: structure
Bi-level Structure
Design p
variables
Evaluation
S(C(T, p)) - U(p)
Response
variables T(p)
S(C(T, p)): Travellers’ surplus
U(p):
MoN 7: 27 June 2008
23
Construction costs
Centre for Transport Studies
University College London
Equilibrium Network Design:
Formulation:
Max
p , T, C
S C  U p 
Subject to
C  FT, p 
T  DC
Bi-level structure:
Demand-Supply Equilibrium
800
Costs C depend on
 Throughput T
 Design p
600
Demand
400
200

Demands T are
consistent with costs C
MoN 7: 27 June 2008
C= F(T, p)
Throughput
Flow

T = D(C)
0
400
24
500
600
Cost
700
Centre for Transport Studies
University College London
Optimality Conditions
No feasible variation p in design improves objective S - U
æ dS dU ö
çç
÷÷  p  0

è dp dp ø
Using properties of S
æ T
dC dU ö
çç D C
÷÷  p  0

dp dp ø
è
Sensitivity analysis for d C / d p
MoN 7: 27 June 2008
25
Centre for Transport Studies
University College London
Sensitivity Analysis of Equilibrium
Sensitivity of costs C to design p:
1
1
æ
æ dD ö
 ö÷
dC ç
 ç I  Ñ T F ç
÷  Ñ T F  ÷Ñ p F
dp ç
÷
è dC ø




è
ø
Partial sensitivity to origin-destination flows:

Ñ T F  G Ñ E C G
1
T

1
Partial sensitivity to design:
Ñ p F  Ñ T F G T Ñ E C Ñ p C
1
MoN 7: 27 June 2008
26
Centre for Transport Studies
University College London
Sensitivity Analysis: Volume of Traffic Er
Cost-throughput:
Start time:


 ö÷ 1
÷ø Q
C r æ h 0  f  0 h1  f 1
 çç 0
E r è h  f  0  h1  f 1

 


s r0 æ
h1  f 1
 çç 0
E r è h  f  0  h1  f 1

 
r

ö 1
÷÷
ø Qr
Dependence on values of time f ’(.) and h ’(.)
MoN 7: 27 June 2008
27
Centre for Transport Studies
University College London
Dynamic System Optimal Assignment
Minimise total travel costs
Min
(Merchant and Nemhauser, 1978)
Specified demand profile T(s)
e
  c s  e s  ds
s aL
a
a
Subject to :
ea s    eap s 
p
p
e
 s   Tod s 
pPo d
MoN 7: 27 June 2008
28
Centre for Transport Studies
University College London
Dynamic System Optimal Assignment
Solution by Optimal Control Theory
Chow (2007)

e p s    0  C p s   p s    p s    p τ p s 


Private cost


  kod s  p  Pod
 
Costate variables
Direct externality
MoN 7: 27 June 2008
29
Centre for Transport Studies
University College London
Comment on Optimal Control Theory solution
Necessary condition

e p s    0  C p s   p s    p s    p τ p s 




  kod s  p  Pod
 
• Hard to solve
• Non-convex
(non-linear equality constraints)
Curse of dimensionality
MoN 7: 27 June 2008
30
Centre for Transport Studies
University College London
Analysis: Recover convexity
Carey (1992):
FIFO as inequality constraints
g ap  τ a  s  τ a  s   eap  s   0
Convex formulation
Not all traffic need flow – holding back
MoN 7: 27 June 2008
31
Centre for Transport Studies
University College London
Illustrative example
o
Qo
g1
g2
MoN 7: 27 June 2008
Q1
d1
Q2
d2
32
Centre for Transport Studies
University College London
Illustrative example
DSO as LP
o
Qo
g1
g2
MoN 7: 27 June 2008
Q1
d1
g1+g2 < Q0
hi < Qi
Q2
33
d2
Centre for Transport Studies
University College London
Illustrative example
DSO as LP
o
Qo
g1
g2
g2
Q0
Q1
d1
g1+g2 < Q0
hi < Qi
Q2
d2
Q2
MoN 7: 27 June 2008
Q 1 Q0
g1
34
Centre for Transport Studies
University College London
Illustrative example
DSO as LP
Qo
o
g2
Demand
g1
g2
Q0
Q1
d1
g1+g2 < Q0
hi < Qi
Q2
d2
Q2
MoN 7: 27 June 2008
Q 1 Q0
g1
35
Centre for Transport Studies
University College London
Illustrative example
DSO as LP
Qo
o
g2
g1
Demand
g2
Q0
Q2
Q1
d1
g1+g2 < Q0
hi < Qi
Q2
d2
Solution region
MoN 7: 27 June 2008
Q 1 Q0
g1
36
Centre for Transport Studies
University College London
Illustrative example
DSO as LP
Qo
o
g2
g1
Demand
g2
Q0
Q2
Solution region
MoN 7: 27 June 2008
Q 1 Q0
g1
Q1
d1
g1+g2 < Q0
hi < Qi
Q2
d2
Not proportional to demand
37
Centre for Transport Studies
University College London
Directions for Further Work
Investigate:
Network effects

Heterogeneous travellers

Cost (seconds)

Time-Specific Costs
200
150
50
0
-650
38
Type 2
100
Pricing
MoN 7: 27 June 2008
Type 1
Type 1 Type 2
-600
Type 1
-550
-500
Departure time (seconds)
Centre for Transport Studies
University College London
-450