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

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

UK’s Department for Transport Work

Reliability impacts on travel decisions through generalised cost

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.

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

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).

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)

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

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!

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]

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

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

Solving Carey’s model with = 1, so that = h[u(t)]

No dependence on outflows.

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

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

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

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

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

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.

Questions/Comments?