continuous price and flow dynamics of tradable mobility credits
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Continuous price and flow dynamicsof Tradable mobility creditsHongbo 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
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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.
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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
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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 dav
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
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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 optimumCapacity-constrained traffic flow patternPareto-improving and revenue-neutral
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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.
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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).
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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)
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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
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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-timeLink-based / Path-based
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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.
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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.
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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
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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
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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 ffκ
total credit consumption 0,t
remaing timeaverage credit
supplyCredit demand
Total available credits
,t T
Price Evolution Function
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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 .
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Continuous Evolution Model
Combine the three assumptions
with initial conditions
T
T
T
,
t p
kT up Q p pT 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
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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
T1 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 .
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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
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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 (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.
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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
** ,0
ZQ pp
, ** ,
0ZQ p
Z
Link travel time function c *T
cΔ 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, 0bZ 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)
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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
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Numerical Example (2)
Evolution of perceived travel time with different initial values
Set Fix and ,
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Numerical Example (3)
Sensitivity of equilibrium pointsw.r.t. different credit distribution
𝑘≤300
1 2 3
min
1, 2, 3300, 300d k
mink k
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Numerical Example (3)
Sensitivity of equilibrium pointsw.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 !
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