Upload
edith-riley
View
214
Download
0
Embed Size (px)
Citation preview
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London3
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)
MoN 7: 27 June 2008 Centre for Transport Studies
University College London4
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London5
Transport Networks: Features
Conservation of traffic at nodes
c ca a
a B n a A n
g t e t
MoN 7: 27 June 2008 Centre for Transport Studies
University College London6
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London7
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London8
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London9
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London10
Accumulate link costs according to time ap(s) of entry
Travel time:
Nested cost operator
Origin-specific costs: ho(s)
Destination-specific costs: fd[p(s)]
Calculation of Costs
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London11
Accumulate link costs according to time ap(s) of entry
Travel time:
Nested cost operator
Origin-specific costs: ho(s)
Destination-specific costs: fd[p(s)]
Total cost associated with journey:
Calculation of Costs
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London12
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London13
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London14
Demand for Travel
Dynamic trip matrix T(s) = {Tod(s)}
Fixed: T(s) is exogenous - estimation?
.
MoN 7: 27 June 2008 Centre for Transport Studies
University College London15
Demand for Travel
Dynamic trip matrix T(s) = {Tod(s)}
Fixed: T(s) is exogenous - estimation?
.
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London16
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
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London17
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
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)
MoN 7: 27 June 2008 Centre for Transport Studies
University College London18
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 Centre for Transport Studies
University College London19
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London20
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London21
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London22
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London23
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London24
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)
MoN 7: 27 June 2008 Centre for Transport Studies
University College London25
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London26
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London27
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London28
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)
MoN 7: 27 June 2008 Centre for Transport Studies
University College London29
Dynamic System Optimal Assignment
Solution by Optimal Control TheoryChow (2007)
ododpppppp PpskssssCse
τ0
Private cost
Direct externality
Costate variables
MoN 7: 27 June 2008 Centre for Transport Studies
University College London30
Comment on Optimal Control Theory solution
ododpppppp PpskssssCse
τ0
Necessary condition
• Hard to solve• Non-convex (non-linear equality constraints)
Curse of dimensionality
MoN 7: 27 June 2008 Centre for Transport Studies
University College London31
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London32
Illustrative example
o
d1
d2
Qo
Q1
Q2
g1
g2
MoN 7: 27 June 2008 Centre for Transport Studies
University College London33
Illustrative example
o
d1
d2
Qo
Q1
Q2
g1
g2
g1+g2 < Q0
hi < Qi
DSO as LP
MoN 7: 27 June 2008 Centre for Transport Studies
University College London34
Illustrative example
o
d1
d2
Qo
Q1
Q2
g1
g2
g1+g2 < Q0
hi < Qi
g1
g2
Q2
Q1
Q0
Q0
DSO as LP
MoN 7: 27 June 2008 Centre for Transport Studies
University College London35
Illustrative example
o
d1
d2
Qo
Q1
Q2
g1
g2
g1+g2 < Q0
hi < Qi
g1
g2
Q2
Q1
Q0
Q0
Demand
DSO as LP
MoN 7: 27 June 2008 Centre for Transport Studies
University College London36
Illustrative example
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London37
Illustrative example
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
MoN 7: 27 June 2008 Centre for Transport Studies
University College London38
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