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

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

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.

1. INTRODUCTION

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

• 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

2. IMPACT OF SPEED LIMIT

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 .

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

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

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

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

Appearance of Non-unique Link Flows on Links with Binding Speed Limit

The Parallel Network

Appearance of Non-unique Link Flows on Links with Binding Speed Limit

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

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

3. SPEED LIMIT V.S.

PRICING FOR

TRAFFIC REGULATION

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

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

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

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

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.

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.