Coordinated Platoon Routing in a Metropolitan Network Jeffrey - - PowerPoint PPT Presentation

Coordinated Platoon Routing in a Metropolitan Network

Jeffrey Larson Todd Munson Vadim Sokolov

Argonne National Laboratory

October 10, 2016

Computationally Enhanced Mobility

◮ Developing high-fidelity simulation tools to estimate the energy

impact of Connected and Automated Vehicles.

◮ Developing algorithms for optimally routing vehicles with

platooning capabilities.

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Computationally Enhanced Mobility

◮ Developing high-fidelity simulation tools to estimate the energy

impact of Connected and Automated Vehicles.

◮ Developing algorithms for optimally routing vehicles with

platooning capabilities.

POLARIS 2 of 15 .

Workflow

Optimization Model Vehicles Network POLARIS Autonomie

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Workflow

Optimization Model Vehicles Network POLARIS Autonomie

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Workflow

Optimization Model Vehicles Network POLARIS Autonomie

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Workflow

Optimization Model Vehicles Network POLARIS Autonomie

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Networks

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Networks

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Animation

Grid Chicago 5 of 15 .

Optimization Model - Model Parameters

Set Meaning V Vehicles to route I Network nodes E ⊆ I × I Network edges

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Optimization Model - Model Parameters

Set Meaning V Vehicles to route I Network nodes E ⊆ I × I Network edges Parameter Meaning Ov v ∈ V origin node Dv v ∈ V destination node T O

v

v ∈ V origin time T D

v

v ∈ V destination time C W

v

waiting cost for v ∈ V Ci,j cost for taking (i, j) ∈ E

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Optimization Model - Model Variables

Variable Meaning fv,i,j 1 if v travels on (i, j) qv,w,i,j 1 if v follows w on (i, j) ev,i,j Time v enters (i, j) wv,i Time v waits at i

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Optimization Model - Model Constraints

Node outflows must equal inflows. When platooning, enter times are equal. Platooning requires at least two vehicles. Only one vehicle can follow. T O

v

plus waiting time is the origin enter time. T D

v is the final enter time plus the time required to travel the final

edge plus waiting at the end. Intermediate enter times are equal plus the travel and waiting times. Can’t have nonzero enter time if there is no flow. Can’t have nonzero wait time if there is no flow. Platoon requires flow for the leader. Platoon requires flow for the followers.

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Optimization Model - Example Constraints

Can’t have nonzero enter time if there is no flow. ev,i,j ≤ Mfv,i,j; ∀v ∈ V , (i, j) ∈ E

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Optimization Model - Example Constraints

Can’t have nonzero enter time if there is no flow. ev,i,j ≤ Mfv,i,j; ∀v ∈ V , (i, j) ∈ E Enter times are equal when platooning. −M(1 − qv,w,i,j) ≤ ev,i,j − ew,i,j ≤ M(1 − qv,w,i,j)

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Optimization Model - Example Constraints

Can’t have nonzero enter time if there is no flow. ev,i,j ≤ Mfv,i,j; ∀v ∈ V , (i, j) ∈ E Enter times are equal when platooning. −M(1 − qv,w,i,j) ≤ ev,i,j − ew,i,j ≤ M(1 − qv,w,i,j) Objective: minimize

• v,i,j

Ci,j

• fv,i,j − η
• w

qv,w,i,j

• +
• v,i

C W

v wv,i

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Helping the MIP solver

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Helping the MIP solver

Consider v and w platooning on edge (i, j) only if max

• T O

v + MOv ,i, T O w + MOw ,i

• + Ti,j ≤ min
• T D

v − MDv ,j, T D w − MDw ,j

• where Ma,b is the minimum time required to reach b from a.

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Helping the MIP solver

Consider v and w platooning on edge (i, j) only if max

• T O

v + MOv ,i, T O w + MOw ,i

• + Ti,j ≤ min
• T D

v − MDv ,j, T D w − MDw ,j

• where Ma,b is the minimum time required to reach b from a.

Lemma

If vehicles use a fraction η less fuel when trailing in a platooning and ts is the shortest time for a vehicle to travel from its origin to destination, it will never travel a path longer than

1 1−ηts.

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Helping the MIP solver

Consider v and w platooning on edge (i, j) only if max

• T O

v + MOv ,i, T O w + MOw ,i

• + Ti,j ≤ min
• T D

v − MDv ,j, T D w − MDw ,j

• where Ma,b is the minimum time required to reach b from a.

Lemma

If vehicles use a fraction η less fuel when trailing in a platooning and ts is the shortest time for a vehicle to travel from its origin to destination, it will never travel a path longer than

1 1−ηts.

Lemma

There exists an optimal platoon routing in which no two vehicles split and then merge together.

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Problem Setup

◮ Vehicles all travel at free-flow speeds

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Problem Setup

◮ Vehicles all travel at free-flow speeds ◮ 10% savings for platooning

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Problem Setup

◮ Vehicles all travel at free-flow speeds ◮ 10% savings for platooning ◮ 25 vehicles

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Problem Setup

◮ Vehicles all travel at free-flow speeds ◮ 10% savings for platooning ◮ 25 vehicles

◮ Grid: random origin/destination pairs 11 of 15 .

Problem Setup

◮ Vehicles all travel at free-flow speeds ◮ 10% savings for platooning ◮ 25 vehicles

◮ Grid: random origin/destination pairs ◮ Chicago: origin/destination pairs drawn from the most common

morning commutes

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Problem Setup

◮ Vehicles all travel at free-flow speeds ◮ 10% savings for platooning ◮ 25 vehicles

◮ Grid: random origin/destination pairs ◮ Chicago: origin/destination pairs drawn from the most common

morning commutes

◮ Start times drawn uniformly from [0,100]

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Problem Setup

◮ Vehicles all travel at free-flow speeds ◮ 10% savings for platooning ◮ 25 vehicles

◮ Grid: random origin/destination pairs ◮ Chicago: origin/destination pairs drawn from the most common

morning commutes

◮ Start times drawn uniformly from [0,100] ◮ Destination times

T v

D = T v O + MOv ,Dv + p,

for some time p ≥ 0

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Problem Setup

◮ Vehicles all travel at free-flow speeds ◮ 10% savings for platooning ◮ 25 vehicles

◮ Grid: random origin/destination pairs ◮ Chicago: origin/destination pairs drawn from the most common

morning commutes

◮ Start times drawn uniformly from [0,100] ◮ Destination times

T v

D = T v O + MOv ,Dv + p,

for some time p ≥ 0

◮ No cost for waiting

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Problem Setup

◮ Vehicles all travel at free-flow speeds ◮ 10% savings for platooning ◮ 25 vehicles

◮ Grid: random origin/destination pairs ◮ Chicago: origin/destination pairs drawn from the most common

morning commutes

◮ Start times drawn uniformly from [0,100] ◮ Destination times

T v

D = T v O + MOv ,Dv + p,

for some time p ≥ 0

◮ No cost for waiting ◮ 10 replications

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Problem Setup

◮ Vehicles all travel at free-flow speeds ◮ 10% savings for platooning ◮ 25 vehicles

◮ Grid: random origin/destination pairs ◮ Chicago: origin/destination pairs drawn from the most common

morning commutes

◮ Start times drawn uniformly from [0,100] ◮ Destination times

T v

D = T v O + MOv ,Dv + p,

for some time p ≥ 0

◮ No cost for waiting ◮ 10 replications ◮ Running Gurobi until its optimality gap is less than 1e-8

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Example solution times

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Example solution times

20 40 60 80 100 120

Pauses

200 400 600 800 1000 1200 1400 1600

Seconds to solution

138 139 140 141 142 143 144 145

Fuel use

Objective value After 1 minute

Grid, waiting allowed at intermediate nodes

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Example solution times

20 40 60 80 100 120

Pauses

200 400 600 800 1000 1200 1400 1600

Seconds to solution

138 139 140 141 142 143 144 145

Fuel use

Objective value After 1 minute

Grid, no waiting allowed at intermediate nodes

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Example solution times

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Example solution times

20 40 60 80 100 120

Pauses

500 1000 1500 2000 2500 3000 3500 4000

Seconds to solution

1540 1560 1580 1600 1620 1640 1660 1680 1700

Fuel use

Objective value Lower bound After 1 minute

Chicago, waiting allowed at intermediate nodes

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Example solution times

20 40 60 80 100 120

Pauses

500 1000 1500 2000 2500 3000 3500 4000

Seconds to solution

1540 1560 1580 1600 1620 1640 1660 1680 1700

Fuel use

Objective value Lower bound After 1 minute

Chicago, no waiting allowed at intermediate nodes

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Example solution times

20 40 60 80 100 120

Pauses

100 200 300 400 500 600

Seconds to solution

7200 7300 7400 7500 7600 7700 7800 7900 8000 8100

Fuel use

Objective value Lower bound After 1 minute

Chicago, 100 vehicles, (20 from each of the 5 most common

• rigin/destination pairs), stopping at 1% optimality gap

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Example solution times

20 40 60 80 100 120

Pauses

100 200 300 400 500 600

Seconds to solution

7200 7300 7400 7500 7600 7700 7800 7900 8000 8100

Fuel use

Objective value Lower bound After 1 minute

Chicago, 100 vehicles, (20 from each of the 5 most common

• rigin/destination pairs), stopping at 1% optimality gap, using lemmas

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Lemma

max

• T O

v , T O w

• + MOv ,Dv ≤ min
• T D

v , T O w

• .

(1)

Lemma

Let v, w ∈ V satisfying Ov = Ow, Dv = Dw, and (1). Then if an

• ptimal solution has qv,w,i,j = 0 for any edge (i, j) ∈ E, there exists an
• ptimal solution with qv,w ′,i,j = 0 for all (i, j) all w ′ arriving later than w.

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Current work

◮ Non-free-flow speeds ◮ Graph-reduction techniques ◮ Larger problems ◮ http://www.mcs.anl.gov/~jlarson/Platooning/

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Workflow

Optimization Model Vehicles Network POLARIS Autonomie

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Workflow

Optimization Model Vehicles Network POLARIS Autonomie

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