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David Watling, Richard Connors, Agachai Sumalee
ITS, University of Leeds
Acknowledgement: DfT “New Horizons”Dynamic Traffic Assignment Workshop, Queen’s University, Belfast
15th September 2004
Encapsulating between day variability in demand in analytical, within-day dynamic, link travel time functions
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Aims
Dynamic modelling of network links subject to variable in-flows comprising:
Within-day variation described by inflow, outflow and travel time profiles
Between-day variation = random variation in these profiles
Thus identify mean travel times under doubly dynamic variation in flows
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UK’s Department for Transport Work
Reliability impacts on travel decisions through generalised cost
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Dynamic Models
Cellular Automata Microsimulation Analytical ‘whole-link’ models
Many shown to fail plausibility tests (FIFO) e.g. = f [x(t)], with x(t) = number cars on link
Carey et al. “improved” whole-link models guarantee FIFO and agree with LWR behaviour.
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Modelling Within-Day Variation:Whole-link model (Carey et al,
2003)))(),(()( tvtuht
travel time for vehicle entering at time t
))(()1()()()( ttvtuhtwht
in-flow at entry time out-flow at exit time
)(τ1
)())(τ(
t
tuttv
Flow conservation (Astarita, 1995)
ttt
dssvdssu
00
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Whole-link Model
)()(
)()1(1)(
1 tuh
tut
Combining gives a first-order differential equation:
1)('FIFO t
No analytic solution for most functions h(.), u(.). Can solve using backward differencing, applied in
forward time (to avoid FIFO violations).
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Flow Capacity Should the link travel-time function h(w) inherently
define max (valid) w and hence capacity, c?
Out-flow can exceed capacity in computation so long as inflow ‘compensates’ such that w=βu(t)+(1-β)v(t+τ(t))< c
Can ensure outflows respect flow capacity by adapting the numerical scheme.
τ0
τ
wc
Scenarios for h(w) with finite capacity c
Desired meaning of capacity requires careful definition of h(w)
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Day-to-day variation
Introduce day-to-day variation of inflow Derive expected travel time profile in terms of mean,
variances, co-variances of day-to-day varying in-flows
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Mean travel time under between-day varying inflows
Travel time at mean inflow
Day-to-day variation
)(,2
1)]([)]([ tHtuEtE
Inflation term for between-day variation. Comprising: Variance-Covariance matrix of inflow variability and Hessian matrix“sensitivity of travel timeto inflows”Not a constant!
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Day-to-day parameterisation
Practically unrestrictive: discretised case with N time slices
Univariate Case
General Case
Vart
ttE2
2 ,
2
1,,
CovtHttE ,
2
1,,
u(t) = u(t, )
each day has different value of (vector)
u(t) = = [θ1, θ2,…, θN]
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Methodology
Monte Carlo simulations of day-to-day inflows drawn from a normal distribution gives many u(t, i)
Whole-link model gives travel time i(t)=(u(t, i)) Calculate mean of all the Monte Carlo days travel
times. This is the experienced mean travel time. Calculate travel time at mean inflow, using whole-link
model with inflow E[u(t,)] Calculate the “Inflation” Term: combination of the
Hessian and Covariance matrix Compare inflation term with ,, ttE
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Numerical Example
BPR-type link travel time function
4
1c
wffwh
ff = 10mins
c = 2000 pcus/hour (‘capacity’)
In-flow profile with random day-to-day peak
240120
12060
600
240
πsin)ε4000(
)ε4000(120
πsin)ε4000(
)(
5
t
t
t
t
t
tU )1000,0(ε 2N
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Solving Carey’s model with = 1, so that = h[u(t)]
No dependence on outflows.
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Std dev of inflows
Travel time calculated for the mean inflow ][uE
Mean travel time over the days (with c.i.s)
Mean inflow over the days uE
)(uE
Numerical difference from plot above
Inflation term by calculation
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Example: =0.1
Asymptotic link travel time function
cw
ffwh
1ff = 10mins
c = 7000 pcus/hour (‘capacity’)
In-flow profile with random day-to-day peak
)20,80( 2N
2
2
2exp
740000),,(
t
tu
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Compare Two Link Travel Time Functions
0 1000 2000 3000 4000 5000 6000 700010
15
20
25
30
35
40
AsympBPR
w
τ=h(w)
7000
1
10w
wh
4
2000110
wwh
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Example: =0.5
Asymptotic link travel time function
cw
ffwh
1ff = 10mins
c = 7000 pcus/hour (‘capacity’)
In-flow profile with random day-to-day peak
240120
12060
600
240sin)4000(
)4000(120
sin)4000(
500)(
5
t
t
t
t
t
tU
)1000,0(ε 2N
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Example: =varying
Asymptotic link travel time function
cw
ffwh
1ff = 10mins
c = 7000 pcus/hour (‘capacity’)
In-flow profile with random day-to-day peak
240120
12060
600
240sin)4000(
)4000(120
sin)4000(
500)(
5
t
t
t
t
t
tU
)1000,0(ε 2N
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Further Work
Analytic derivation of the correction term?
Modify whole-link model to limit outflows Augment with dynamic queuing model?
Conditions for FIFO?
Follow this approach on the links of a network to investigate its reliability under day-to-day varying demand.
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Questions/Comments?