Maximum-stability dispatch policy for shared autonomous vehicles - - PowerPoint PPT Presentation

Maximum-stability dispatch policy for shared autonomous vehicles

Michael W. Levin, Di Kang

Shared autonomous vehicles (SAVs)

SAV service currently in testing on public roads SAVs have safety driver

Motivation Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

2

Motivation Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

3

SAV : personal vehicle replacement rates 1 SAV : 10 personal vehiclesa 1 SAV : 9 personal vehiclesb 1 SAV : 3 personal vehiclesc

aDaniel J Fagnant and Kara M Kockelman. “The travel and environmental implications of shared autonomous

vehicles, using agent-based model scenarios”. In: Transportation Research Part C: Emerging Technologies 40 (2014), pp. 1–13.

bDaniel J Fagnant, Kara M Kockelman, and Prateek Bansal. “Operations of Shared Autonomous Vehicle Fleet

for Austin, Texas Market”. In: Transportation Research Record: Journal of the Transportation Research Board 2536 (2015), pp. 98–106.

cKevin Spieser et al. “Toward a systematic approach to the design and evaluation of automated

mobility-on-demand systems: A case study in Singapore”. In: Road Vehicle Automation. NY: Springer, 2014,

• pp. 229–245.

Motivation Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

3

Agent-based simulation

1

1Daniel J Fagnant and Kara M Kockelman. “Dynamic ride-sharing and fleet sizing for a system of shared autonomous

vehicles in Austin, Texas”. In: Transportation 45.1 (2018), pp. 143–158. Motivation Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

4

Agent-based simulation

2

2T Donna Chen, Kara M Kockelman, and Josiah P Hanna. “Operations of a shared, autonomous, electric vehicle fleet:

Implications of vehicle & charging infrastructure decisions”. In: Transportation Research Part A: Policy and Practice 94 (2016),

• pp. 243–254.

Motivation Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

5

SAV : personal vehicle replacement rates 1 SAV : 10 personal vehiclesa 1 SAV : 9 personal vehiclesb 1 SAV : 3 personal vehiclesc

aDaniel J Fagnant and Kara M Kockelman. “The travel and environmental implications of shared autonomous

vehicles, using agent-based model scenarios”. In: Transportation Research Part C: Emerging Technologies 40 (2014), pp. 1–13.

bDaniel J Fagnant, Kara M Kockelman, and Prateek Bansal. “Operations of Shared Autonomous Vehicle Fleet

for Austin, Texas Market”. In: Transportation Research Record: Journal of the Transportation Research Board 2536 (2015), pp. 98–106.

cKevin Spieser et al. “Toward a systematic approach to the design and evaluation of automated

mobility-on-demand systems: A case study in Singapore”. In: Road Vehicle Automation. NY: Springer, 2014,

• pp. 229–245.

Motivation Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

6

Queueing model

Passenger queueing model

Define a queue of waiting passengers at each zone: wrs(t). Conservation of waiting passengers: wrs(t + 1) = wrs(t) + drs(t) − min

  

• j∈A

yrs

rj(t), wrs(t)

  

where drs(t) are random variables with mean ¯ drs. yrs

rj(t) ≤ pr(t) is vehicles departing r for s to link j

Queueing model Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

8

Passenger queueing model

Define a queue of waiting passengers at each zone: wrs(t). Conservation of waiting passengers: wrs(t + 1) = wrs(t) + drs(t) − min

  

• j∈A

yrs

rj(t), wrs(t)

  

where drs(t) are random variables with mean ¯ drs. yrs

rj(t) ≤ pr(t) is vehicles departing r for s to link j

This defines a Markov chain on the state space N|Z|2.

Queueing model Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

8

Vehicle queueing model

xrs

j (t) is the number of vehicles on link j traveling from r to s

pr(t) is the number of vehicles parked at r

• j∈A
• (r,z)∈Z2

xrs

j (t) +

• r∈Z

pr(t) = F

Queueing model Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

9

Vehicle queueing model

xrs

j (t) is the number of vehicles on link j traveling from r to s

pr(t) is the number of vehicles parked at r

• j∈A
• (r,z)∈Z2

xrs

j (t) +

• r∈Z

pr(t) = F Conservation of link queues: xrs

j (t + 1) = xrs j (t) +

• i∈A

yrs

ij (t) −

• k∈A

yrs

jk(t)

• k∈Γ+

j

yrs

jk(t) ≤ xrs j (t)

Queueing model Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

9

Vehicle queueing model

xrs

j (t) is the number of vehicles on link j traveling from r to s

pr(t) is the number of vehicles parked at r

• j∈A
• (r,z)∈Z2

xrs

j (t) +

• r∈Z

pr(t) = F Conservation of link queues: xrs

j (t + 1) = xrs j (t) +

• i∈A

yrs

ij (t) −

• k∈A

yrs

jk(t)

• k∈Γ+

j

yrs

jk(t) ≤ xrs j (t)

Conservation of parked queues: pr(t + 1) = pr(t) +

• i∈A
• q∈Z

yqr

ir (t) −

• j∈A
• s∈Z

yrs

rj(t)

Queueing model Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

9

Markov decision process

Queues of passengers and vehicles define a Markov chain. wrs(t + 1) = wrs(t) + drs(t) − min

  

• j∈A

yrs

rj(t), wrs(t)

  

pr(t + 1) = pr(t) +

• i∈A
• q∈Z

yqr

ir (t) −

• j∈A
• s∈Z

yrs

rj(t)

xrs

j (t + 1) = xrs j (t) +

• i∈A

yrs

ij (t) −

• k∈A

yrs

jk(t)

Since vehicle movements can be controlled, this is a Markov decision process model.

Queueing model Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

10

Stability and passenger service

• (r,s)∈Z2 wrs(t) is the number of waiting passengers at time t

If demand is unserved, then

• (r,s)∈Z2 wrs(t) will increase over time

Queueing model Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

11

Stability and passenger service

• (r,s)∈Z2 wrs(t) is the number of waiting passengers at time t

If demand is unserved, then

• (r,s)∈Z2 wrs(t) will increase over time

Definition

The stochastic queueing model is stable if there exists some K < ∞ s.t. 1 T

T

• t=1
• (r,s)∈Z2

E [wrs(t)] ≤ K ∀T ∈ N Equivalently, ∃ Lyapunov function ν(w(t)) ≥ 0 s.t. E [ν(w(t + 1)) − ν(w(t))|w(t)] ≤ κ − ǫ|w(t)| for all w(t) for κ < ∞, ǫ > 0.

Queueing model Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

11

Assumptions

SAV travelers wait in the system until served

◮ If SAV travelers exited, the concept of stability would need to be

redefined.

Constant travel times for vehicles Entire SAV fleet can be centrally dispatched

Queueing model Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

12

Maximum-stability policy

Max-pressure policy with maximum stability

max 1 T

T

• τ=1
• (r,s)∈Z2

wrs(t)f rs(t + τ) s.t.

• s∈Z

f rs(t + τ) ≤ pr(t + τ) pr(t + τ + 1) = pr(t + τ) +

q∈Z

f qr t + τ − Φr

q

• −
• s∈Z

f rs(t + τ) +

q∈Z

• i∈A

xqr

i (t + τ − Φr i )

f rs(t + τ) ≥ 0

T is the planning horizon — how far we look ahead frs(t + τ) anticipates future vehicle dispatch pr(t + τ) anticipates future vehicle availability

Maximum-stability policy Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

14

Planning horizon analysis

max 1 T

T

• τ=1
• (r,s)∈Z2

wrs(t)f rs(t + τ) s.t.

• s∈Z

f rs(t + τ) ≤ pr(t + τ) pr(t + τ + 1) = pr(t + τ) +

q∈Z

f qr t + τ − Φr

q

• −
• s∈Z

f rs(t + τ) +

q∈Z

• i∈A

xqr

i (t + τ − Φr i )

f rs(t + τ) ≥ 0

T is the planning horizon — how far we look ahead T must be large enough to dispatch vehicles across the network. At least max

r

• Φr

q

• .

Maximum-stability policy Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

15

Stability region

What demand rates ¯ d ∈ D could be served by any SAV dispatch policy? We want to serve any ¯ d ∈ D with the max-pressure policy.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

16

Stability region

What demand rates ¯ d ∈ D could be served by any SAV dispatch policy? We want to serve any ¯ d ∈ D with the max-pressure policy. Average SAV flow rates from r to s are enough to serve average demand:

• i∈Γ+

r

¯ yrs

ri ≥ ¯

drs ∀(r, s) ∈ Z2

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

16

Stability region

What demand rates ¯ d ∈ D could be served by any SAV dispatch policy? We want to serve any ¯ d ∈ D with the max-pressure policy. Average SAV flow rates from r to s are enough to serve average demand:

• i∈Γ+

r

¯ yrs

ri ≥ ¯

drs ∀(r, s) ∈ Z2 Constraints on average SAV flow rates:

• q∈Z
• i∈Γ−

r

¯ yqr

ir =

• s∈Z
• j∈Γ+

r

¯ yrs

jr

∀q ∈ Z

• i∈Γ−

j

¯ yrs

ij =

• j∈Γ+

j

¯ yrs

jk

∀(r, s) ∈ Z2, ∀j ∈ Ao

• (r,s)∈Z2
• (i,j)∈A2

¯ yrs

ij ≤ F

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

16

Stability region — D

Proposition

If d / ∈ D, then the system cannot be stabilized by some ¯ y ∈ Y. Proof. For any SAV dispatch policy ∃ an (r, s) with an η > 0 s.t.

• i∈Γ+

r

¯ yrs

ri − ¯

drs ≥ η. Then on average wrs(t) will increase by η each time step.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

17

Stability region — D

Proposition

If d / ∈ D, then the system cannot be stabilized by some ¯ y ∈ Y. Proof. For any SAV dispatch policy ∃ an (r, s) with an η > 0 s.t.

• i∈Γ+

r

¯ yrs

ri − ¯

drs ≥ η. Then on average wrs(t) will increase by η each time step.

• (r,s)∈Z2
• (i,j)∈A2

¯ yrs

ij ≤ F

so if ¯ d / ∈ D, then a larger fleet size is needed to serve ¯ d.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

17

Passenger service rates

Proposition

The boundary of D is linear wrt F, i.e. if the fleet size increases to αF then demand of α¯ d can be stabilized. Proof. αF admits a linear increase of α in all other constraints defining the stable region.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

18

Passenger service rates

Proposition

The boundary of D is linear wrt F, i.e. if the fleet size increases to αF then demand of α¯ d can be stabilized. Proof. αF admits a linear increase of α in all other constraints defining the stable region. An increase in the SAV fleet size should result in a proportional increase in the number of passengers that can be served.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

18

Stability region — maximize service rate

max

• (r,s)∈Z2

¯ drs s.t.

• i∈Γ+

r

¯ yrs

ri ≥ ¯

drs ∀(r, s) ∈ Z2

• q∈Z
• i∈Γ−

r

¯ yqr

ir =

• s∈Z
• j∈Γ+

r

¯ yrs

jr

∀q ∈ Z

• i∈Γ−

j

¯ yrs

ij =

• j∈Γ+

j

¯ yrs

jk

∀(r, s) ∈ Z2, ∀j ∈ Ao

• (r,s)∈Z2
• (i,j)∈A2

¯ yrs

ij ≤F

Analytical method to find the theoretical maximum service rate.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

19

Stability region — let D0 be the interior of D

Average SAV flow rates from r to s are enough to serve average demand:

• i∈Γ+

r

¯ yrs

ri ≥ ¯

drs ∀(r, s) ∈ Z2 Constraints on average SAV flow rates:

• q∈Z
• i∈Γ−

r

¯ yqr

ir =

• s∈Z
• j∈Γ+

r

¯ yrs

jr

∀q ∈ Z

• i∈Γ−

j

¯ yrs

ij =

• j∈Γ+

j

¯ yrs

jk

∀(r, s) ∈ Z2, ∀j ∈ Ao

• (r,s)∈Z2
• (i,j)∈A2

¯ yrs

ij ≤F

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

20

Stability proof sketch

If ¯ d ∈ D0, then there exists some ¯ y such that ¯ drs −

• i∈A

¯ yrs

ri ≤ −ǫ

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

21

Stability proof sketch

If ¯ d ∈ D0, then there exists some ¯ y such that ¯ drs −

• i∈A

¯ yrs

ri ≤ −ǫ

Proposition

There exists a sequence (y(t)) such that lim

T→∞

1 T

T

• t=0

y(t) = ¯ y The max-pressure policy constructs a sequence ˆ y(t + τ) with limit ¯ y.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

21

Lemma

Suppose that there exists a T ∈ N and a κ1, κ2 < ∞ such that E [ν(w(t + T)) − ν(w(t + T + 1) + ν(w(t + 1)) − ν(w(t))|w(t)] ≤ κ1 E [ν(w(t + T + 1) − ν(w(t + T))|w(t)] ≤ κ2 − ǫ|w(t)| then the system is stable.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

22

Lemma

Suppose that there exists a T ∈ N and a κ1, κ2 < ∞ such that E [ν(w(t + T)) − ν(w(t + T + 1) + ν(w(t + 1)) − ν(w(t))|w(t)] ≤ κ1 E [ν(w(t + T + 1) − ν(w(t + T))|w(t)] ≤ κ2 − ǫ|w(t)| then the system is stable.

Lemma

Suppose that there exists a T ∈ N and a function ν(w(t)) such that E [ν(w(t + T)) − ν(w(t + T + 1) + ν(w(t + 1)) − ν(w(t))|w(t)] ≤ κ1 E

• 1

T

T

• τ=1

(ν(w(t + τ + 1) − ν(w(t + τ))) |w(t)

• ≤ κ2 − ǫ|w(t)|

then the system is stable.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

22

Lyapunov function ν(w(t)) =

• (r,s)∈Z2

(wrs(t))2

Proposition

∀¯ d ∈ D0 ∃M < ∞ such that if T > M then the max-pressure control using the planning horizon T yields E

  1

T

T

• τ=1
• (r,s)∈Z2

(wrs(t + τ + 1))2 − (wrs(t + τ))2 |w(t)

  ≤ κ − ǫ|w(t)|

For any η > 0, there exists a M s.t. if T > M then 1 T

T

• τ=1

ˆ y(t + τ) ≤ |¯ y − η1| If η < ǫ, ∃ǫ2 > 0 = ǫ − η such that E [ν(w(t + 1) − w(t))|w(t)] ≤ κ − ǫ2|w(t)|

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

23

Planning horizon analysis

For any η > 0, there exists a M s.t. if T > M then 1 T

T

• τ=1

ˆ y(t + τ) ≤ |¯ y − η1| We need η < ǫ.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

24

Planning horizon analysis

For any η > 0, there exists a M s.t. if T > M then 1 T

T

• τ=1

ˆ y(t + τ) ≤ |¯ y − η1| We need η < ǫ.

• i∈Γ+

r

¯ yrs

ri > ¯

drs ⇒

• i∈Γ+

r

¯ yrs

ri − ¯

drs > ǫ

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

24

Planning horizon analysis

For any η > 0, there exists a M s.t. if T > M then 1 T

T

• τ=1

ˆ y(t + τ) ≤ |¯ y − η1| We need η < ǫ.

• i∈Γ+

r

¯ yrs

ri > ¯

drs ⇒

• i∈Γ+

r

¯ yrs

ri − ¯

drs > ǫ The larger the time horizon, the closer demand can get to the boundary of the stable region.

Stability proof Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

24

Numerical results

Sioux Falls

Numerical results Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

25

Example — stable demand

20 40 60 80 100 120 5000 10000 15000 20000 25000 30000 35000 40000 45000

Unserved queue Time (s)

Numerical results Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

26

Example — stable demand

5 10 15 20 25 30 35 40 45 50 5000 10000 15000 20000 25000 30000 35000 40000 45000

Average queue length Time (s)

Numerical results Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

27

Example — unstable demand

50 100 150 200 250 300 5000 10000 15000 20000 25000 30000 35000 40000 45000

Unserved passengers Time (s)

Numerical results Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

28

Example — unstable demand

20 40 60 80 100 120 140 160 5000 10000 15000 20000 25000 30000 35000 40000 45000

Average queue length Time (s)

Numerical results Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

29

Maximum stable demand vs. fleet size

5000 10000 15000 20000 25000 30000 200 250 300 350 400 450 500

Maximum stable demand Fleet size

Symmetric demand CBD demand

OD demand proportions are constant.

Numerical results Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

30

Maximum stable demand vs. fleet size

2000 4000 6000 8000 10000 12000 200 250 300 350 400 450 500

Rebalancing trips Fleet size

Symmetric demand CBD demand

OD demand proportions are constant.

Numerical results Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

31

Effect of planning horizon T on maximum stable demand

23000 23200 23400 23600 23800 24000 24200 24400 24600 24800 1500 2000 2500 3000 3500 4000

Maximum stable demand Planning horizon

OD demand proportions are constant.

Numerical results Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

32

Conclusions

Stability analysis of SAVs Maximum-stability policy with proof Numerical results evaluating stable region Future work: Decentralized policy Ridesharing, electric vehicles Efficient heuristics, or evaluate stability of heuristic policies

Conclusions Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

33

Thank you Questions?

mlevin@umn.edu

Max-stability dispatch for SAVs

• M. W. Levin, D. Kang

34