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A Simple Asymptotically Optimal Energy Allocation and Routing Scheme in Rechargeable Sensor Networks

Shengbo Chen, Prasun Sinha, Ness Shroff, Changhee JooElectrical and Computer Engineering & Computer Science and Engineering

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Introduction[Rechargeable sensor networks]

Applications Environment monitoring: earthquake, structural, soil, glacial

Unattended operability for long periods

Opportunity Harvesting and storing renewable energy (like solar or wind)

Challenges Full battery means no opportunity to harvest renewable energy Unpredictable and time-varying renewable energy

Goal: develop control mechanism to maximize the total utility for a sensor network with energy replenishment

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Outline Model

Problem statement

Related Literature

Our approach

Simulation results

Conclusion

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Model[Rechargeable sensor node]

M

( 1) min max ( ) ( ),0 ( ),B t B t e t r t M

r(t) B(t) e(t)B(t+1)

M: Battery size

B(t): Battery level in time slot t

e(t): allocated energy in time slot t

r(t): harvested energy in time slot t

Rechargeable sensor node

Rate-power functionNondecreasing and strictly concaveAmount of data transmitted with spending units of energy

( )e

e e

( )e

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*

1

1( ) max ( )

s.t. Routing constraints

Energy constraints

Ts

ss t

J T U x tT

Problem Statement

Sensor network with renewable energy Assume the date rate is low

Ignore interference from other nodes

Problem: utility maximization

amount of data from source to destination in slot t

is a strictly concave utility function

( )sx t

1. Convex optimization problem: Joint energy allocation and routing

2. Requires full knowledge of the replenishment profile

3. Time coupling property: have to optimize all time slots simultaneously

Flow 1

Flow 2

, ,

:( , ) :( , ) : ,

:( , )

*

1

1( ) max ( )

s.t. ( ) ( ) ( ) 0 for all ,

( ) ( ( )) for all ,

e x

j i j L j i j L s f i d ds s

j i j L

Ts

ss t

d d sij ji

t t t

dij i

d

J T U x tT

t t x t d i

t e t t i

666666666666666666666666666666666666666666

sU

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Related Literature Finite horizon

[S. Chen, P. Sinha and N. B. Shroff], INFOCOM, 2011. [A. Fu, E. Modiano and J. Tsitsiklis], TON, 2003.

Dynamic programming Infinite horizon

[L. Lin, N. B. Shroff, and R. Srikant], TON, 2007. Asymptotically optimal competitive ratio

[Z. Mao, C. E. Koksal, N. B. Shroff ], TAC, 2011 Finite battery size

[M. Gatzianas, L. Georgiadis, and L. Tassiulas], TWC, 2010. Maximize a function of the long-term rate per link

[L. Huang, M. Neely], Mobihoc, 2011 Asymptotically optimal utility

Our focus: Infinite horizon

Lyapunov optimizati

on technique

Our contribution: develop a low-complexity solution

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Our approach Construct a fictitious infeasible energy allocation and routing

scheme

Prove that its performance forms an upper bound on

Develop a low-complexity online scheme

Prove that the performance achieved by the online scheme can get arbitrarily close to the upper bound as tends to infinity

*( )J T

T

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Assumption Replenishment process has a finite mean value Infinite battery capacity

Upper bound for the optimum Jensen’s Inequality: is an upper bound

1

1max ( ( ))

T

et

e tT

1

1lim ( )

T

Tt

r r tT

( )r

Single node case [Throughput maximization]

Spending energy at the average rate is the best

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Single node case (cont’d) [Throughput maximization]

Consider the energy allocation scheme In each time slot, the estimated average replenishment rate

The allocated energy in each slot

where is an arbitrary parameter

1

1ˆ( ) ( )

t

r t rt

ˆ ˆ(1 ) ( ), ( ) ( ) (1 ) ( ),( )

( ) ( ),

r t if B t r t r te t

B t r t Otherwise

Theorem 1: The scheme above achieves the throughput performance arbitrarily close to by choosing

to be sufficiently small as tends to infinity

( )rT

ˆ( ) tr t r

Intuition: spend energy at a rate close to the mean

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Upper bound on the optimum Consider a fictitious infeasible scheme

For each node i, energy allocation in each slot

Routing decision in each slot

*( )J T

( ) (1 )i ie t r

:( , ) :( , ) : ,

:( , )

( ) argmax ( )

s.t. ( ) ( ) ( ) 0 for all ,

( ) (1 ) for all

j i j L j i j L s f i d ds s

j i j L

s sub s

s

d d sij ji

dij i

d

x t U x t

t t x t d i

t r i

1. Energy allocation and routing decoupled

2. Time decoupled

3. Time homogeneous

Theorem 2: is upper bounded by *( )J T ( ) ( )ub ss ubcs

J T U x

( )s sub ubcx t x

Network Case[fictitious scheme]

Spend a little more energy than the

average harvested

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Consider the online scheme Energy allocation (same as the single node case)

The estimated average replenishment rate

The allocated energy in each slot

Routing decision in each slot

:( , ) :( , ) : ,

:( , )

max ( )

s.t. ( ) ( ) ( ) 0 for all ,

( ) ( ) for all

j i j L j i j L s f i d ds s

j i j L

ss

s

d d sij ji

dij i

d

U x t

t t x t d i

t e t i

1

1ˆ ( ) ( )

t

i ir t rt

ˆ ˆ(1 ) ( ), ( ) ( ) (1 ) ( ),( )

( ) ( ),i i i i

ii i

r t if B t r t r te t

B t r t Otherwise

Theorem 3: The scheme achieves the performance

arbitrarily close to

by choosing to be sufficiently

small as tends to infinity

( )ubJ T

Network Case (cont’d)[Online scheme]

T

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Distributed algorithm Duality based

At each time slot, source s generates data at rate by solving

Routing

Lagrange multipliers are updated as

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Simulation Setup Network topology:

100 nodes and three flows in 1×1 field Link available if distance is less than 0.2

Using real traces of solar energy and wind energy [3] June 5th, 2011-July 5th, 2011

[3]. “National Renewable Energy Laboratory,” http://www.nrel.gov.

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Simulation results

ESA: Infinite-horizon based scheme in [1][1] L. Huang, M. Neely, “Utility Optimal Scheduling in Energy Harvesting Networks,” in Proceedings of Mobihoc, May 2011.

minute minute

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Conclusion Study the joint problem of energy allocation and

routing to maximize total utility in a sensor network with energy replenishment.

Develop a low-complexity online solution that is asymptotically optimal with general energy replenishment profiles.

Evaluate the performance using real traces

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Simulation results for one node

ESA: Infinite-horizon based scheme in [1][1] L. Huang, M. Neely, “Utility Optimal Scheduling in Energy Harvesting Networks,” in Proceedings of Mobihoc, May 2011.

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Finite battery size Required battery size