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The impact of speed limits on traffic equilibrium and
system performance in networks
Hai Yang
Chair ProfessorDepartment of Civil and Environmental
Engineering, The Hong Kong University of Science and Technology, Clear Water Bay,
Kowloon, Hong Kong, PR China
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OUTLINE
• Introduction
• Impact of speed limit at a macroscopic network level
– Travel time-flow relationship– Traffic equilibrium and system performance
• Comparison of speed limits and road pricing for traffic regulation
• Conclusion
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Reference
Yang, H., Wang, X.L. and Yin, Y.F. (2012) The impact of speed limits on traffic equilibrium and system performance in networks. Transportation Research 46B, No.10, 1295-1307.
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1. INTRODUCTION
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Main Objectives of Speed Limit
• Enhance safety– High speed leads to high risk of crash involvement
(i.e. Solomon 1964, Cirillo 1968, Garber and Graham 1990, Ashenfelter and Greenstone 2002 )
– The severity of injuries is an increasing function of speed (Joksch 1975)
• Reduce fuel consumption– 1974, U.S. 55 mph speed limit on highways
reduce gasoline and diesel fuel consumption during energy crisis
• Reduce vehicle emissions– 2003, southern Switzerland, 80 Km/h on some
motorways
reduce ozone level
– 2003, Rotterdam, 80 Km/h on urban motorways
reduce ozone levels
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• Taylor (2000), Woolley et al. (2002) and Madireddy et al. (2011)
– Microscopic simulation of the network impact of speed limit
– Traffic reallocation is observed and travel time is reported to increase with reduced speed limits but not in a direct proportion to the change of speed limits
Network Impact of Speed Limit
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2. IMPACT OF SPEED LIMIT
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Travel Time – Flow Relationship with a Speed Limit
C
Traffic volume v
t0
Travel time t
v
t
Normal flow
Forced flow
C Traffic volume v
Speed s
v
s
maxs
cs
Assumption 1. For any link a A , the original link travel time function without a speed
limit, a at v is separable, continuous, convex and strictly increasing with its link flow av
Assumption 2. A speed limit s on a road, if imposed, is assumed to be located within the
normal flow regime, namely, maxcs s s .
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Travel Time Function with a Speed Limit
Continuous, convex and increasing
but no longer differentiable and strictly increasing
,
,
t t v v vt v
t v v v
C
Traffic volume v
t0
Travel time t
v
t
: original travel time function
: modified travel time function
t v
t v
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Link-specific Speed Limit Law
T
,
T
T
: set of links
: set of O-D pairs
: set of paths within O-D pair
, , : vector of path flows
, : vector of link flows
, : vector of fixed demand
w
r w w
a
w
A
W
R w W
f f r R w W
v v a A
d d w W
T
max: speed li
, : vec
mit o
tor of lin
n lin
k sp
k ,
eed li
,
mit a
ca a a as a
s
A s
s a A
s s
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User Equilibrium under Speed Limit
* *, ,
* *, ,
, if 0, ,
, if 0, ,
a a a r w r w wa A
a a a r w r w wa A
t v f r R w W
t v f r R w W
* *
0
The UE flow patterns , can be obtained by solving the following
optimization problem:
min d
or equivalently, the following variational inequality problem (V
a
v
v
av
a A
v f
t
* *
IP):
0, a a a a va A
t v v v v
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Uniqueness of UE Solutions
0
is no longer strictly increasing, so d is no longer
strictly convex, and thus the uniqueness of UE link flow can not be readily
established.
av
a a aa At v t
* * *
* *
*
: UE link flow patterns
, , set of links with binding speed limit
, set of rest links with non-binding speed limit
v
a a
c
v v
A v a v v a A
A v
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Appearance of Non-unique Link Flows on Links with Binding Speed Limit
The Parallel Network
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Appearance of Non-unique Link Flows on Links with Binding Speed Limit
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Appearance of Non-unique Link Flows on Links with Binding Speed Limit: The
General NetworkProposition 2. Given the unique link travel time on all links and unique link flows on link set
cA at UE under speed limit s , the set of UE link flows on the complementary link set A is
identical with the set of feasible link flow patterns T,av a A contained in the following
polyhedron: *,, ,
w
ca r a
w W r Rr wf v a A
,, , w
a rW r R
r w aw
f v a A
, a av v a A
, , w
wr R
r w d w Wf
, 0, , wr w r Rf w W
, 0, , r wc
wr R wf W
where wR is the set of equilibrated paths within O-D pair w W defined by
* *, , , ,min , ,
c cw a a a r a a r a a a r a a r w w
a A a Aa A a A
R r t v t t v t r R r R
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Pareto-improvement in total travel time and vehicular emissions with a speed limit law:
A Numerical Example
31
2
4
1
2
3
4 0.2038 exp 0.7962a a a a a a ae v t v l t v
4
0 1 0.15 aa a a
a
vt v t
C
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3. SPEED LIMIT V.S.
PRICING FOR
TRAFFIC REGULATION
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Comparison of Speed Limits and Road Pricing for Traffic Regulation: Speed Limits VS Non-negative
Toll Charge
*Non-negative toll scheme ,a a a a at C t v a A
* *Generalized link cost: , , a a a a a a a ac v t v t C a A
*
max
, , Speed limit:
,
a a a a
ac
a
l c v a As
s a A
* *Link travel time: , ,a a a a at v c v a A
0, , 0,ca aA a a A A a a A
Proposition 3. If a target flow pattern within the normal flow regime can be
sustained as equilibrium by a nonnegative link toll scheme with
* ,a a a a at C t v a A , then we can always find a link-specific speed
limit scheme such that one of its UE link flow patterns is the target flow
pattern. Furthermore, if the UE link flow pattern under the speed limit
scheme is unique, then the target flow pattern is sustainable by the speed limit
scheme.
T* *Target flow pattern: ,av v a A
max, ,ca a as s s a A
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Comparison of speed limits and road pricing for traffic regulation
Speed Limits VS Non-negative Link Tolls
Traffic regulation
mechanism
Individual
travel cost
Total
travel cost
Total
revenue
Speed limit Increase travel time to drive
traffic away *,c v T* *,c v v 0
Road pricing
Increase travel cost by toll to
drive traffic away, but leads
to a reduced travel time
*,c v T* *t v v T *v
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Comparison of speed limits and road pricing for traffic regulation
Speed Limits VS Non-negative Link Tolls
The difference of total network emissions under the speed limit and the
toll charge schemes depends on the target link flow pattern and the link
emission functions.
Emission
Total emission Link without speed
limit
Link with speed
limit
Speed limit * *,a a a ae v t v * *,a a a ae v t v *T
* *,t ve v v
Road pricing * *,a aa ae v t v * *,a aa ae v t v *T
* *,t ve v v
* *a a a at v t v * *
a a a at v t v
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Speed Limits VS Non-negative Link Tolls
1 2 1 2
30.0, VOT=1 min/$
60km 50km, 30
d
l l C C
,
1 2
1
2
T Tue *UE flow under no policy: 12.0,18.0 Target flow: 6.0,24.0v v
Policy Scheme
Individual travel
cost /min
Link travel
time /minTotal
time/min
Link emission
Total emissio
n1 2
1 2
No policy 34.0 34.0 34.0 1020.0 28.2 22.3 741.1
link tolls (18,0)$ 40.0 22.0 40.0 1092.0 39.3 22.1 766.2
speed limit
(90.0,187.5)Km/h
40.0 40.0 40.0 1200.0 26.9 22.1 690.8
0.2038 exp 0.7962 , 1,2a
a a a aa a
le v t v a
t v
1 1 1 2 2 22 10, 16t v v t v v
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Comparison of speed limits and road pricing for minimum traffic emissions
Speed Limits VS Negative Link Tolls (Subsidy)
1 2
Link 2
Link 1
1 2 2 11, 1d d
1000, 1, 2
9i ii
s v iv
9, 1, 2
10i
i i
vt v i
280 50, 1, 2i i ie s s i
PolicyOptimal scheme
Link flowTotal time
Total emissio
n
No policy intervention N/A (1.0,1.0) 2 900
Minimize emission through
toll and subsidy*
(3.32, 3.32)
8.178 341.071
Minimize emission through
speed limit(1.0,1.0) 2.5 100
1 2 4.089
1 2 80s s
* If only non-negative tolls are allowed, then no emission reduction can be achieved.
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Conclusion
• The standard traffic assignment method still applies.
• The uniqueness of link travel times at user equilibrium (UE) remains valid, and the UE flows on links with non-binding speed limit are still unique.
• For other links with binding speed limits, a polyhedron which contains all UE link flows on these links are explicitly given, and the uniqueness of UE link flows can be determined by solving a linear system of equalities and inequalities.
• Although from different perspectives for regulating traffic flows with a different mechanism, a speed limit law can play the same role as a positive toll scheme and overwhelm some negative (rebate) toll schemes under certain conditions for network flow management.