Capacity maximising traffic signal control policies

NORTHMOST (12 December 2016)

Capacity-maximising

traffic signal control policies

“If you eliminate the impossible,

whatever remains, however improbable,

must be the truth.”


Capacity-maximising


“If you eliminate the impossible,

whatever remains, however improbable,

must be the truth.”

(Sherlock Holmes,

Sign of the Four, 1890)


Capacity-maximising


Mike Smith, Ronghui Liu, Takamasa Iryo, Tung Le, Hai Vu

The University of York, UK

ITS, University of Leeds, UK

Kobe University, Japan

Swinburne University of Technology, Melbourne, Australia

University of Monash , Melbourne, Australia

RELEVANCE / IMPORTANCE / AIM

TO REDUCE CONGESTION /

POLLUTION IN CITIES

RELEVANCE / IMPORTANCE / AIM

TO REDUCE CONGESTION /

POLLUTION IN CITIES

IN PART AUTOMATICALLY

Traffic Control and Route Choice

Traffic Control Route Choice

Modelling Signal Control and Route

Choice Allsop, Dickson, Gartner, Smith, Van Zuylen,

Meneguzzer, Gentile, Noekel, Taale, Cantarella,

Mounce, Ke Han, Viti, Schlaich, Haupt, Lo, Rinaldi,

Cantelmo, Cascetta, Tung Le, Hai Vu, . . .

●

●

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Previous Work

Origin Destination

Route-swaps

C

s2

s1

Origin Destination

Route-inflow swaps and

Stage green-time swaps

C

s2

s1

Origin Destination

Route flow and stage green-time

swaps motivated by route costs and

“stage pressures”

C

s2

s1

HOPE: Stability V(x) = 4

V(x) = 2

V(x) = 1

V(x) = 3

EQUILIBRIUM:

V(x) = minimum

HOPE: Stability V(x) = 4

V(x) = 2

V(x) = 1

V(x) = 3

EQUILIBRIUM:

V(x) = minimum

●

HOPE: Stability

V(x) = 4

V(x) = 2

V(x) = 1

V(x) = 3

EQUILIBRIUM:

V(x) = minimum

●

●

●

HOPE: Stability

V(x) = 4

V(x) = 2

V(x) = 1

V(x) = 3

EQUILIBRIUM:

V(x) = minimum

●

●

●

●

●

WHAT CAN GO WRONG

DYNAMICALLY

STABLE WITH STANDARD POLICIES ???

WHAT CAN GO WRONG

DYNAMICALLY

STABLE WITH STANDARD POLICIES ???

NO!

(Poisson traffic +

equi-saturation)

ORIGIN

SIGNAL

DESTINATION

(Poisson traffic +

equi-saturation) • ,

.

0 X1 s

s

X2

= equilibria

Non-unique

DEMAND

ORIGIN SIGNAL

DESTINATION

(Poisson traffic +


.

0 X1 s

s

X2 Bold lines = equilibria

Non-unique

T

T

ORIGIN SIGNAL

DEMAND

DESTINATION

PITCHFORK (Poisson traffic +


.

0 X1 s

s

X2

Bold lines = equilibria

Non-unique

PITCHFORK

BIFURCATION

ORIGIN SIGNAL

DESTINATION

SHORT AND LONG ROUTES

• ,

0 X1 s

2s

X2

Bold lines = equilibria

Non-unique

BIFURCATION

ORIGIN

DESTINATION

P0: BASIC IDEA

DEMAND

SET D

-C

b

SUPPLY –

FEASIBLE

SET S

n(D)

BASIC IDEA

DEMAND

SET D

-C

b

SUPPLY –

FEASIBLE

SET S

n(D)

-C is normal to D∩S

b is normal to S

P0:

BASIC IDEA

DEMAND

SET D

-C

b

SUPPLY –

FEASIBLE

SET S

n(D)


b is normal to S

-C = n(D) + b

P0:

BASIC IDEA

DEMAND

SET D

-C

b

SUPPLY –

FEASIBLE

SET S

n(D)


b is normal to S

-C = n(D) + b

-(C+b) = n(D)

P0:

BASIC IDEA

DEMAND

SET D

-C

b

SUPPLY –

FEASIBLE

SET S

n(D)


b is normal to S

-C = n(D) + b

-(C+b) = n(D)

-(C+b) is normal to D

P0:

BASIC IDEA

DEMAND

SET D

-C

b

SUPPLY –

FEASIBLE

SET S

n(D)


b is normal to S

-C = n(D) + b

-(C+b) = n(D)

-(C+b) is normal to D

EQUILIBRIUM

consistent with P0

P0:

P0 STAGE PRESSURES

sibi Basic P0

P0 STAGE PRESSURES

sibi Basic P0

Qi/Gi P0+vertical queue

IS Q/G RIGHT ?

sibi Basic P0

Qi/Gi P0 with vertical queue

Qip/Gi THIS TALK

Policy: At time “t” swap green-time

towards the stage with the higher pressure

IF Pressi(t) > Pressj(t)

THEN swap some green

from stage j to stage i



dGi (t)/dt = Pressi(t) - Pressj(t)

dGj (t)/dt = Pressj(t) - Pressi(t)



dG1(t)/dt = Press1(t) – Press2(t)

dG2(t)/dt = Press2(t) – Press1(t)

Exact Policy: At time “t” swap green-time

to exactly equalise stage pressures

Choose G(t) so that

Press1(t)= Press2(t)

Origin Destination

Exact p-policy: Green-times

satisfy Q1p/G1

= Q2p/G2

C

s2 = 2

s1 = 1

ONE possible p-dynamic

dG1(t)/dt = Q1 p(t)/G1(t) - Q2

p(t)/G2(t)

dG2(t)/dt = Q2 p(t)/G2(t) – Q1

p(t)/G1(t)


dG1(t)/dt = Q1 p(t)/G1(t) - Q2

p(t)/G2(t)

dG2(t)/dt = Q2 p(t)/G2(t) – Q1

p(t)/G1(t)

dX1(t)/dt = Q2(t)/(2G2(t)) - Q1(t)/G1(t)

dX2(t)/dt = Q1(t)/(G1(t)) – Q2(t)/2G2(t)


dG1(t)/dt = Q1 p(t)/G1(t) - Q2

p(t)/G2(t)

dG2(t)/dt = Q2 p(t)/G2(t) – Q1

p(t)/G1(t)

dX1(t)/dt = Q2(t)/(2G2(t)) - Q1(t)/G1(t)

dX2(t)/dt = Q1(t)/(G1(t)) – Q2(t)/2G2(t)

dQ1(t)/dt = X1(t – c) – G1(t)

dQ2(t)/dt = X2(t – c) – 2G2(t)

Green-time / route choice equilibria Equal p-pressures: Q1

p / G1= Q2 p / G2

Equal delays: Q1 / G1 = Q2 / (2G2)

Eliminate G.

Green-time / route choice equilibria Equal p-pressures: Q1

p / G1= Q2 p / G2

Equal delays: Q1 / G1 = Q2 / (2G2)

Eliminate G. To obtain:

a constraint on the queue vector Q.

Q2

Q1

Q2

Q1

ROUTEING / p-POLICY

EQUILIBRIA

p = 2

X2

X1

ROUTEING / p-POLICY

EQUILIBRIA

p = 2

X2

X1

Throughput

ROUTEING / p-POLICY

EQUILIBRIA

p = 2

???

X2

X1

2 Throughput

ROUTEING / p-POLICY

EQUILIBRIA

p = 2

X2

X1 1

2 Throughput

ROUTEING / p-POLICY

EQUILIBRIA

p = 2

X2

X1 1

6/5

2 Throughput

ROUTEING / p-POLICY

EQUILIBRIA

p = 2

X2

X1

p = 2 2

6/5

1

DYNAMICS

X2

X1

DYNAMICS

p = 2 2

6/5

1

Q2

Q1

DYNAMICS

p = 2 2

6/5

1

X2

X1

X-DYNAMICS

p = 1 (P0) 2

Throughput = 2 (MAX)

CONCLUSIONS:

Stated a p-evolution eqn. showing

how route-inflows, green-times and

queues evolve for all future time.

CONCLUSIONS:




p = 1 and p ≠ 1.

CONCLUSIONS:




p ≠ 1: FAILS to maximise capacity

p = 1: P0: maximises capacity.

CONCLUSIONS:

p ≠ 1: FAILS to maximise capacity

p = 1 (P0): maximises capacity.

Holmes:

All p ≠ 1 are eliminated, p = 1 or P0

remains; which must be the ONLY p

such that

“policy p is capacity maximising”

is the truth.

Capacity maximising traffic signal control policies

Automotive

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