continuous price and flow dynamics of tradable mobility credits hongbo ye and hai yang the hong kong...
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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
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
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 .
20
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.
23
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 .
24
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)
25
credit charging scheme
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
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
34
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.
35
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