Continuous price and flow dynamicsof Tradable mobility credits

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

ISTTT2017/07/2013

2

Outline

Introduction Tradable mobility credits Day-to-day flow dynamics

Price and flow dynamics: assumptions & models Fixed demand & homogeneous travelers

Theoretical results Numerical example Conclusion

Introduction1.

4

Typical strategies dealing with traffic congestion

Why a Tradable Credit Scheme

Desirable features Congesti

on pricing

Road space rationing

Short-term and long-term effectiveness /

Equity /

Economic efficiency Governmental revenue-

neutrality

Tradable credit

scheme

5

Each participating agent receives a proportion of credits (on a periodic basis such as a month or a quarter) Equitable

Initial distribution for free Revenue-neutral incentives for mobility and

environmental quality Credit charging scheme Link-specific or cordon-based; distance or time-

based; time-invariant or time-varying

What is a Tradable Credit Scheme

Yang, H., Wang, X.L., 2011. Managing network mobility with tradable credits. Transportation Research Part B 45 (3), 580-594.

6

What is a Tradable Credit Scheme

A policy target in terms of fix-quantity travel credits can be easily achieved. Example: Distance-based credit charge for

achieving control of total veh-km traveled on the network

The equilibrium price of credits is determined by the market through free trading. Market driven Credit: from the higher income groups to the

lower Money: from the wealthy to the less Enhance income distribution or financial transfer

confined only to within the predefined group of travelers

7

Mathematical Model of Traffic Equilibrium under Tradable Travel

Credit SchemesEquivalent model formulation:

subject to:

, , , w

a r w a rw W r R

v f a A

0,min d

av

av f

a

t

, , w

r w wr R

f d w W

a aa A

v K

* *, , 0, ,a a a a r w r w w

a A

t v p f r R w W

*, 0, ,a a a a r w w

a A

t v p r R w W

, 0, ,r w wf r R w W

* 0a aa

v K p

*a a

a

v K

*, 0, ,r w wf r R w W

0p

First-order optimality conditions:

Link travel time: 1 1 18 2t v v , 2 2 216t v v

O-D demand: 1 2 10d ,

Initial credit allocation: 30K , 1 2 3k

Link credit charge: 1 4 , 2 2 .

2

1

1Link 1; 4

User equilibrium and credit market equilibrium conditions:

1 2

1 2

1 2

8 2 4 16 2

10

4 2 30

v p v p

v v

v v

A unique equilibrium solution:

Link flow: * *1 2 5v v

Link travel time: *1 18t , *

2 21t

Unit credit price: * 1.5p

Traffic Equilibrium and Market Equilibrium with Tradable Credits: An

Example

2Link 2; 2

8

Traffic Equilibrium and Market Equilibrium with Tradable Credits: An

Example

For a given credit scheme, a unique equilibrium flow pattern exists; the equilibrium credit price is unique subject only to very mild assumptions.

A properly designed tradable credit scheme can emulate a congestion pricing system and support various desirable traffic flow optima:

Social optimum

Capacity-constrained traffic flow pattern

Pareto-improving and revenue-neutral

9

10

Extensions

Transaction costNie, Y., 2012. Transaction costs and tradable mobility credits. Transportation Research Part B 46 (1), 189-203.

User heterogeneityWang, X., Yang, H., Zhu, D., Li, C., 2012. Tradable travel credits for congestion management with heterogeneous users. Transportation Research Part E 48 (2), 426-437.

Zhu, D., Yang, H., Li, C., Wang, X., 2013. Properties of the multiclass traffic network equilibria under a tradable credit scheme. Transportation Science (revised version under review).

Managing parkingZhang, X., Yang, H., Huang, H.J., 2011. Improving travel efficiency by parking permits distribution and trading. Transportation Research Part B 45 (7), 1018-1034.

11

Extensions

Managing bottleneck congestion and mode choiceNie, Y., Yin, Y., 2013. Managing rush hour travel choices with tradable credit scheme. Transportation Research Part B 50, 1-19.

Tian. L.J., Yang, H., Huang H.J., 2013. Tradable credit schemes for managing bottleneck congestion and modal split with heterogeneous users. Transportation Research Part E 54, 1–13.

Xiao, F., Qian, Z., Zhang, H.M., 2013. Managing bottleneck congestion with tradable credits. Transportation Research Part B (in press).

Implementation issue under limited informationWang, X., Yang, H., 2012. Bisection-based trial-and error implementation of marginal cost pricing and tradable credit scheme. Transportation Research Part B 46 (9), 1085-1096.

Wang, X., Yang, H., Han, D., Liu, W., 2013. Trial-and-error method for optimal tradable credit schemes: The network case. Journal of Advanced Transportation (in press).

12

Extensions

Incorporation of income effectsWu, D., Yin, Y., Lawphongpanich, S., Yang, H., 2012. Design of more equitable congestion pricing and tradable credit schemes for multimodal transportation networks. Transportation Research Part B 46 (9), 1273-1287.

Mixed equilibrium behaviorsHe, F., Yin, Y., Shirmohammadi, N., Nie, Y., 2013. Tradable credit schemes on networks with mixed equilibrium behaviors. Transportation Research Part B (submitted).

Design issueWang, G., Gao, Z., Xu, M., Sun, H., 2013. Models and a relaxation algorithm for continuous network design problem with a tradable credit scheme and equity constraints. Computers and Operations Research (in press)

13

Our Motive

Static Case Given some target flow and price Credit charging and distribution scheme

Could the target be achieved in practice? Flow will change in the network Price will fluctuate in the market

14

Deterministic Process

Stochastic Process Cascetta (1989)

Watling and Hazelton (2003)

Parry and Hazelton (2013)

Day-to-day Traffic Flow Dynamics

Smith (1984)

Friesz et al (1994)

Zhang and Nagurney (1996)

Cho and Hwang (2005)

Yang and Zhang (2009)

He et al (2010)

Smith and Mounce (2011)

Han and Du (2012)

He and Liu (2012)

Continuous-time / Discrete-time

Link-based / Path-based

15

Travelers’ perception on travel time and learning behavior Horowitz (1984) Cantarella and Cascetta (1995) Watling (1999) Bie and Lo (2010)

Day-to-day Traffic Flow Dynamics

Model Description 2.

17

Basic Consideration

How the traffic flow and credit price will impact each other and evolve together.

Travelers’ learning behavior of route choice based on their perceived path travel cost and credit price.

Price adjustment with the fluctuation of credit demand and supply.

18

Notations

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 c and perceived path travel cost C

Time interval: 0,T

Total number of credits initially distributed: 0K kT

Path credit charge κ

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

Perceived path travel time & cost on time t : tc & tC

20

Path Choice

① Travelers’ path choice. Probabilities for travelers choosing paths depend on the perceived travel costs on all the paths.

,

,

T

0 1

,

,

,

, ,

r w

r w

wr R w W

r w

ψ C C

f Dψ C

21

Learning Behavior

② Travelers’ learning behavior. Travelers update their perception according to the revealed traffic condition.

T c Δ c ΔDψ C c

Real travel timePerception>0

22

Price Evolution

③ Credit price evolution. The changing of credit price depends only on the current price and excess credit demand.

Excess credit demand is the difference between the current credit consumption rate and the average credits per unit time available during the rest of the period.

, Zp Q p

T

T 0d

tkT s s

TZ t t

t

κ f

fκ

total credit consumption 0,t

remaing timeaverage credit

supplyCredit demand

Total available credits

,t T

Price Evolution Function

23

Model Assumptions

③ Credit price evolution. The changing of credit price depends only on the current price and excess credit demand. , Zp Q p

,Q p Z satisfies:

i) 0p , ,Q p Z is strictly increasing with Z R ;

ii) If 0p , 0,Q Z is strictly increasing with 0Z ;

iii) ,0 0Q p ， 0p ； 0, 0Q Z ， 0Z .

25

Continuous Evolution Model

Combine the three assumptions

with initial conditions

T

T

T

,

t p

kT up Q p p

T t

u p

c Δ c ΔDψ c κ c

κ Dψ c κ

κ Dψ c κ

0 00 0 , 0 0, p p u c c

T

0d

tu t s p s s κ Dψ c κ

Theoretical Results3.

28

t 0,T ,

T

T

T 0d

0

, 0

t

p

kT s p s sp Q p p

T t

c Δ c ΔDψ c κ c

κ Dψ c κκ Dψ c κ

Existence of the Equilibrium Point

Brouwer’s fixed point theorem

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

Fixed-point Problem

T

T

p

p p p k

c Δ c ΔDψ c κ

κ Dψ c κ

29

Existence of the Equilibrium Point

Theorem 1. If

Tmin lim

pk k p

κ Dψ c κ

then there exists at least one equilibrium point * *, pc

of the evolution model.

mink is the lower bound for the average credit supply k

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Uniqueness of the Equilibrium Point

Theorem 2. Assume the condition in Theorem 1 is satisfied, and furthermore, if

(a) link travel time function c v is strictly monotone w.r.t link flow v

T

1 2 1 2 1 20 v v c v c v v v

(b) negative of path flow demand function Dψ C is monotone w.r.t path cost C

T

1 2 1 2 0 C C Dψ C Dψ C

(c) total credit demand function T p κ Dψ c κ is strictly decreasing with credit price

T1 2 1 2 1 20p p p p p p κ Dψ c κ Dψ c κ

then the equilibrium point is unique. Furthermore, *p is strictly decreasing with k

when T UEmin ,k k κ f , where UEf is the equilibrium flow pattern without the credit

scheme, and * 0p when T UEk κ f .

31

System Stability

T

T

T 0d

,

t

t p

kT s p s sp Q p p

T t

c Δ c ΔDψ c κ c

κ Dψ c κκ Dψ c κ

,T t

T

T,

p

p Q p p k

c Δ c ΔDψ c κ c

κ Dψ c κ

time-variant system

time-invariant system

32

System Stability

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

fx x

The equilibrium point *x is

(1) stable if 0 , 0 , s.t.

*0 0t t x x x ;

(2) 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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System Stability

Theorem 3. (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.

34

Theorem 4. If Q , c and ψ are continuously differentiable in

a neighborhood of the equilibrium point * *, pc , then * *, pc is

asymptotically stable if the following conditions hold:

Price evolution function Q

** ,

0ZQ p

p

,

** ,0

ZQ p

Z

Link travel time function c

*TcΔ J Δ is symmetric and positive definite

Route choice probability ψ

conditions (i)-(v) (The logit model satisfies the conditions)

Credit charging scheme

There is at least one OD pair connected by at least two paths

with different credit charges

System Stability

Numerical Example4.

36

Numerical Example

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

Path 1: 1 3O D 1 1

Path 2: 2 4O D 2 2

Path 3: 2 5 3O D 3 3

Link travel time function 4

1 0.15 100a

a avc v

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

Price adjustment: 0, max 0, 0

bZ pQ p Z bZ p

, 0b constant.

VOT 1 .

Set 450k , then * 4.08, 3.09, 4.75c and * 1.15p .

Evolution process: MATLAB with the function ode45

O D

1(0)

4(0)

5(0)

2(2)

3(1)

37

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

𝑇=5

𝑇=15

𝑇=200

38

Numerical Example (2)

Evolution of perceived travel time

with different initial values

Set

Fix and

,

39

Numerical Example (3)

Sensitivity of equilibrium points

w.r.t. different credit distribution

𝑘≤300

1 2 3

min

1, 2, 3

300, 300d k

mink k

40

Numerical Example (3)

Sensitivity of equilibrium points

w.r.t. different credit distribution

Conclusion5.

43

Conclusion

A continuous-time model to describe the dynamics of price and perceived travel time under the tradable credit scheme fixed demand and homogeneous travelers travelers’ route choice and learning behavior price evolving with the variation of credit demand

and supply Some important property of the dynamic model existence and uniqueness of the equilibrium point stability and convergence when time horizon goes

infinite

THANK YOU !