CONTINUOUS PRICE AND FLOW DYNAMICS OFTRADABLE MOBILITY CREDITS

CONTINUOUS PRICE AND FLOW DYNAMICS OFTRADABLE MOBILITY CREDITS

Hongbo YE and Hai YANGThe Hong Kong University of Science and Technology

21/12/2012

Outline

Introduction Continuous price and flow dynamics

Homogeneous value of time (VOT) Fixed demand

Numerical example Conclusion

2

Introduction1.

Background

Travel Demand ManagementRationing

Direct and expedient

May lose effectiveness in the long runPricing

Effective and efficient

Not equal among different income levels, government is not revenue-neutral

4

Tradable Credit Scheme: Yang and Wang (2011) The social planner

initially distributes a certain number of credits to all eligible travelers

set the expiration date of the current credits charges a link-specific number of credits from travelers

using each link allows free trading of the credits among travelers

Effective, efficient, equitable, revenue-neutral Unique equilibrium flow pattern and credit price

Literature Review

In practice, price and flow may fluctuate from day to day. 5

Static case

Day-to-day Dynamics of Traffic Flows Smith (1984) Cascetta (1989) Friesz et al. (1994) Zhang and Nagurney (1996) Watling and Hazelton (2003) Cho and Hwang (2005) Yang and Zhang (2009)

Literature Review

6

Day-to-day Dynamics of Traffic FlowsHorowitz (1984)

Discrete-time dynamic process, stochastic UE, two-link network

Time-varying learning parameter would affect system’s stability through day-to-day evolution

Cantarella and Cascetta (1995) A general framework of day-to-day traffic dynamics

based on path flow’s demand and supply interaction Existence, uniqueness and stability of equilibrium

conditionsWatling (1999)

The stability of a general network taking a specific form in Cascetta and Cantarella (1995)

Bie and Lo (2010) Stability and attraction domain

Literature Review

7

Objective

How the traffic flow and credit price will impact each other and evolve together, considering Travelers’ learning behavior on travel time Travelers’ route choice behavior based on their

perceived path travel cost Price adjustment rule according to the

fluctuation of credit demand and supply

8

Price and Flow Dynamics2.

Notations

Links: A ; OD pairs: W

Routes between OD pair w W : wR , ww WR m

Travel demand between OD pair w W : 0wd , fixed.

1 1, , , ,w w W WD diag d I d I d I

Travelers’ value of time: β

Perceived path travel time t and perceived path travel cost c Link path incidence matrix: Link flow v and path flow f Path credit charge κ

Link travel time function ,at a A t v v

Path travel time : T t v 10

Notations

Time interval: 0,T

Total number of credits initially distributed: 0K kT

Credit trading price on time 0,s T : p s

Path flow on time s : sf

Perceived path travel time on time s : st

Perceived path travel cost on time s : sc

11

Model Assumptions

① Travelers’ learning behavior. Travelers update their perception of path travel times based on their previous perception and new traffic information.

Td

ds D s p s s

s

tt F t κ t

12

Real Traffic InformationPerception>0

Model Assumptions

② Travelers’ route choice. Probabilities for travelers choosing routes depend on the perceived travel time on all the routes.

,

0,1 ,

,r w

r

D D p

F r R

r

w W

F

f F c F t κ

F c c

c

13

Model Assumptions

③ Credit price adjustment. The credit price depends on the expected daily excess credit demand, defined as the difference between the credits consumed on the current day and the average credits per day available during the rest of the period. d

, d

p sQ p s Z s

s

14

T

T 0d

skT z z

TZ s s

s

κ ffκ

,Q p Z satisfies:

i) ,Q p Z is continuous and increasing on both p and Z ;

ii) 0 , 0,Z Q p Z p

Model Assumptions

③ Credit price adjustment.

15

daily credit demand

d,

d

p sQ p s Z s

s

total available credtits

remaing timedaily credit

supply

Continuous Evolution Model

Combine the three assumptions

with initial conditions

T

T

T

,

, ,

p D p

kT up g p Q p D p

T s

u D p

t h t t F t κ t

t κ F t κ

κ F t κ

0 00 0 , 0 0, p p u t t

16

Trajectories of price and flows

If , ph t and ,g pt are smooth, the trajectories of credit

price and network flows are unique on 0,s T . However, when

s T , it cannot be assured that the relationship (credit conservation or feasible condition)

T

0lim d

s

s TkT D z p z z

κ F t κ

always holds, which is a constraint that the total credit consumption could not exceed the total credit supply during the whole time horizon. Furthermore, we also want to know

lim ?s T

p s

CreditSupply

Credit Consumption

17

Existence of the Equilibrium Point

s 0,T ,

T

T

T 0d

,

0

0

s

s D s p s s

kT D z p z zp Q p s D s p s

T s

t t F t κ t

κ F t κκ F t κ

T * * *

* T * *, 0

D p

Q p D p k

t F t κ t 0

κ F t κ

T

T,

D p

p p Q p D p k

t t F t κ

κ F t κ

Brouwer’s fixed point theorem.

Every continuous function from a convex compact subset of a Euclidean space to itself has a fixed point.

18

Existence of the Equilibrium Point

Theorem 1. There exists at least one equilibrium solution

* *, pt , *p R , if the credit scheme (including daily credit

supply k and credit charging scheme , , ,r w wr R w W )

satisfies

, ,min maxw ww r R r w w r R r ww W w W

d k d .

19

Uniqueness of the Equilibrium Point

Theorem 2. Assuming

i) the price adjustment function ,Q p Z is strictly increasing at 0Z , i.e.,

, 0 0,Q p Z Z p

ii) the link travel time function is strictly monotonically increasing, i.e.,

T

1 2 1 2 1 20 v v t v t v v v

iii) the path flow function is monotonically decreasing, i.e.,

T

1 2 1 2 1 20D D c c F c F c c c

then if , ,min maxw ww r R r w w r R r ww W w W

d k d ,

(1) the equilibrium point is unique, *p R ;

(2) *p is strictly increasing when k decreases;

(3) if ,Tmin

ww r R r ww

SUE

Wkd

κ f , where SUEf is the equilibrium flow

pattern without credit scheme, then * 0p .

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Existence and Uniqueness of Equil. Point

Theorem 3. If ,Q p Z satisfies

T

T

T

0, 0

0, 0

p D p kQ p D p k

p D p k

κ F t κκ F t κ

κ F t κ

and the credit scheme (k and ,r w ) satisfies ,minww r R r ww W

k d , then there exists

at least one equilibrium solution.

Theorem 4. With the same assumptions for ,Q p Z in Theorem 3, link travel time

function and path flow function in Theorem 2, if ,minww r R r ww W

k d , then the

equilibrium point is unique, * 0p . Furthermore, *p is strictly increasing when k

decreases in T, ,min

ww r R r wE

w

U

W

Sd κ f .

21

System Stability

T

T

T 0d

s

s D s p s s

kT D z p z zp Q D s p s

T s

t t F t κ t

κ F t κκ F t κ

,T s

T

T

,

,

p D p

p g p Q D p k

t h t t F t κ t

t κ F t κ

time-variant system

time-invariant system

22

System Stability

Theorem 5. (Khalil, 2002) Let *x be an equilibrium point for the

nonlinear system fx x where : mf D R is continuously

differentiable and D is a neighborhood of *x . Let

*

fJ

x x

xx

then *x is asymptotically stable if the real part of all the

eigenvalues of J are negative.

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System Stability

Definition. Suppose *x is an equilibrium point of the autonomous (or time-invariant) system

fx x

The equilibrium point *x is

stable if 0 , 0 , s.t.

*0 0t t x x x ;

unstable if it is not stable; asymptotically stable if it is stable and can be chosen

such that

*0 lim 0 0t

t t

x x x .

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Full paper submitted to the 20th ISTTT

Theorem 6. If , ph t and ,g pt are continuously differentiable in a neighborhood of

* *, pt , and let *J t and *JF be the Jacobian matrix of t and F at the equilibrium point

* *, pt , respectively, then the equilibrium point * *, pt is asymptotically stable if

(a)

* *,

,0

p

p ZQ

p

t

and * *,

,0

p

p Z

Z

Q

t

(b) *TJ t is symmetric and positive definite

(c) F satisfies the following assumptions (i)-(v) and

(i)

0i

i

F

c

c

(ii)

0i

j

F

c

c

, , ,wi j R i j

(iii) 0i

j

F

c

c

, , ,w vi R j R w v (iv) ji

j i

FF

c c

cc

(v) 1 0

w w

ii w

i R i R j

FF j R

c

c

c

(d) there is at least one OD pair owning two paths with different credit charges.

System Stability

price adjustment function

Link travel time function

Logit model can satisfies (c)

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credit charging scheme

Numerical Example3.

Numerical Example

300d 1 4 5 0l l l , 2 2l , 3 1l

Path 1: 1 3O D 1,1 1

Path 2: 2 4O D 2,1 2

Path 3: 2 5 3O D 3,1 3

Link travel time function: 4

1 0.15 100i

i ivt v

.

Route choice probability: the logit function with a unit scaling parameter.

Price adjustment: ,Q p x bx , where 0b is a constant.

Travelers’ value of time is 1 .

Set 450k , then * 4.0776, 3.0913, 4.7507t and * 1.1521p .

O D

1(0)

4(0)

5(0)

2(2)

3(1)

27

Numerical Example (1)

0 0,1.2, 2.0, 2.5

5,15, 200

p

T

Price evolution with different lengths of time horizon and different initial prices

28

Numerical Example (2)

Evolution of perceived travel time with different initial values

29

Numerical Example (3)

Sensitivity of equilibrium points with respect to different credit schemes

30

Numerical Example (3)

Sensitivity of equilibrium points with respect to different credit schemes

31

Numerical Example (4)

Influence of system parameters on price evolution

32

Conclusion4.

Conclusion

A continuous-time model to describe the dynamics of price and perceived travel time under the tradable credit scheme based on fixed demand and homogeneous VOT considering

travelers’ route choice and learning behavior price adjustment process with the variation of credit demand

and supply Some important property of the dynamic model

Existence and uniqueness of the evolution trajectories Existence and uniqueness of the equilibrium point Stability and convergence when time horizon goes infinite

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Conclusion

Numerical example The choice of time horizon of the credit scheme is

critical for the system performance, especially the stability and convergence issues of the system.

When time is long enough, the system with different initial conditions will eventually be stable and convergent.

Choosing a proper credit scheme is also critical.

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THANK YOU!