1.225J
1.225J (ESD 205) Transportation Flow Systems
Lecture 7
Freeway Traffic Control:
PrePre-timed, Coordinated Ramp Metering
Prof. Ismail Chabini and Prof. Amedeo R. Odoni
Lecture 7 Outline
‰ Introduction:
• Freeway traffic control
• Ramp metering
‰ Pre-timed, coordinated ramp metering (R11, R13)
• Example
• LP formulation for the example
• General LP formulation
‰ Summary
1.225, 11/20/02
Lecture 7, Page 2
1
Freeway Traffic Control
‰ Lane control:
• Variable speed limit signs
• Keep-lane signs
• Congestion and incident warnings
• Environmental condition warnings, such as fog, ice, rain and snow
‰ Freeway network control:
• Variable Message Signs (VMS) for driving information and/or
guidance
• Individual route guidance
‰ Two types of freeways: Urban freeways and Intercity freeways
‰ Ramp metering is applicable to urban freeways due to:
• Large number of on and off ramps
• Many ramp-to-ramp trips
• Recurrent as well as non-recurrent (usually dynamic) congestion
1.225, 11/20/02
Lecture 7, Page 3
The Ramp Metering Concept
Merging Section
Detection Station
Upstream Flow
fr(t)
Flow after merging ( less than MR or B)
On-Ramp Flow
X(t)
K-Q diagram at merging
q
qmax
u=
(MR or B)
‰ Find X(t) such that:
• X(t) ≤ MR – fr(t)
• This results in a stable flow
immediately upstream and
downstream of the merging section
stable
unstable
kc
1.225, 11/20/02
q
k
kjam
k
Lecture 7, Page 4
2
Ramp Metering: General Aspects
‰ Examples of objectives:
• Preserve freeway capacity
• Maximize total vehicles served
• Reduce duration and extent of recurrent congestion
• React to non-recurrent congestion
‰ Implementation: Traffic lights (one car at a time or traffic cycles)
‰ Main geographical areas of applications:
• Europe
• Asia
• California and Texas
‰ Controllability limitations:
• Amount of controlled ramps
• Minimum ramp volumes
• Ramp length
1.225, 11/20/02
Lecture 7, Page 5
Types of Ramp Metering Methods
‰ Types of ramp metering methods:
• Isolated vs. Coordinated
• Pre-timed (time-of-day) vs. Traffic responsive
‰ Pre-timed (also called fixed-time) ramp metering:
• Does not need real-time measurements
• Calculations are done off-line and are based on historic demands
• Assumes no dynamics, which is valid for sufficiently long roads
and time periods only
‰ Focus of this lecture: Pre-timed, Coordinated ramp metering
1.225, 11/20/02
Lecture 7, Page 6
3
LP Formulation: Notations
Section m
Xi
Xi+1
Section k
Section 1
X1
Xn
X3
X2
‰ Notations
• X j : input volu mes to the freeway system ( j = 1,2,..., n)
• Akj : decimal fraction of vehicles entering at input j which pass
through section k (k = 1,2,..., m; j = 1,2,..., n)
• Bk : capacity of freeway section k ( k = 1,2,..., m )
• D j : hourly demand at input ramp j
1.225, 11/20/02
Lecture 7, Page 7
Pretimed,
Pretimed, Coordinated Ramp Metering: Example
Section 3
X4
X5
Section 2
Section 1
X1
X6
X3
X2
‰ Input data (6 ramps (n=6), 3 sections (m=3))
A jk
1
2
3
k
j
Dj
1.225, 11/20/02
j
1
2
3
4
5
6
1.000 1.000 0.949 0.933 0.842 0.519
1.000 1.000 0.922 0.619
1.000 0.969 0.777
1
600
2
475
3
450
4
500
5
825
k
Bk
1
2
3
5900
6000
6450
6
6800
Lecture 7, Page 8
4
LP Formulation and Solution: Example
‰ Maximize total vehicles served:
max X 1 + X 2 + X 3 + X 4 + X 5 + X 6
s.t.
X 1 + X 2 + 0.949 X 3 + 0.933 X 4 + 0.824 X 5 + 0.519 X 6 ≤ 5900
X3 +
(S 2 )
X 4 + 0.969 X 5 + 0.777 X 6 ≤ 6450
(S 3 )
≤ 600
X1
X2
X3
X4
(S 4 )
≤ 475
(S 5 )
≤ 450
(S 6 )
≤ 500
(S 7 )
≤ 825
X5
X 6 ≤ 6800
X j ≥ 0,
(S1 )
X 4 + 0.922 X 5 + 0.619 X 6 ≤ 6000
(S 8 )
(S 9 )
j = 1, 2,3,4,5,6
‰ An optimal solution: X1 = 447, X 2 = 475, X 3 = 450, X 4 = 367, X 5 = 825, X 6 = 6800
‰ Objective function value = 9364 (maximum flow)
‰ Slack variables: S1 = 0, S2 = 213, S3 = 0, S4 = 153, S5 = 0, S6 = 0, S7 = 133, S8 = 0, S9 = 0
1.225, 11/20/02
Lecture 7, Page 9
General LP Formulation
‰ Problem: max
n
∑X
j =1
n
s.t.
∑A
kj
j
X j ≤ Bk , k = 1,2,..., m (section capacity constraints)
j =1
X j ≤ Dj ,
j = 1,2,..., n (on - ramp demand constraints)
X j ≥ 0,
j = 1,2,..., n (non - negativity constraints)
‰ Solution method:
• The above LP can be solved by the Simplex method
• There are numerous software solvers for LPs (XPRESS-MP,
LINDO, CPLEX, EXCEL)
• The Excel LP Solver may be used for problems of moderate size
only
1.225, 11/20/02
Lecture 7, Page 10
5
Lecture 7 Summary
‰ Introduction:
• Freeway traffic control
• Ramp metering
‰ Pre-timed, coordinated ramp metering
• Example
• LP formulation for the example
• General LP formulation
1.225, 11/20/02
Lecture 7, Page 11
6