atomic scheduling of appliance energy consumption in residential smart grid

Atomic Scheduling of ApplianceEnergy Consumption in Residential

Smart Grid

Kyeong Soo (Joseph) Kim(With S. Lee, T. O. Ting@XJTLU and X.-S. Yang@Middlesex)

Department of Electrical and Electronic EngineeringXi’an Jiaotong-Liverpool University

CeSGIC 1st International Workshop onSmart Grid Technology and Data Processing

19 June 2015

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OutlineIntroductionReview of Current Formulation of Appliance EnergyConsumption Scheduling

Atomic Scheduling of Appliance Energy ConsumptionStarting-Time-Based FormulationOptimal-Routing-Based FormulationSuccessive Convex RelaxationExamples

Numerical ResultsExperimental ParametersComparison of Bounds with True Optimal ValuesCost MinimizationPAR Minimization

Conclusions

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Next . . .IntroductionReview of Current Formulation of Appliance EnergyConsumption Scheduling

Conclusions

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Autonomous Demand-Side Management inSmart Grid

Gateway(Traditional)

Electricity

Greenfield

Bi-Directional

Communication

Power Plant

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Scheduling of Appliance EnergyConsumption

I A key to autonomous DSM inoptimizing energy production andconsumption.

I Based on two-way digitalcommunications between a utilitycompany and users through smartmeters at users’ premises.

I Typical objectivesI Peak-to-average ratio (PAR)I Total energy cost

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A Question on Scheduled EnergyConsumption

I Can a washing machinesuccessfully complete its job withthe energy consumption scheduledas follows?

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9am 3pm12am

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9am 3pm12am

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9am 3pm12am

9am 3pm2pm10am

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Conclusions

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Optimization VariablesI Based on energy consumption over equally-divided time

slots of a day (typically hourly time slots) asoptimization variables, i.e.,

xn ,[x0

n, . . . , xhn, . . . , x

H−1n

I User n ∈ N , {1, . . .,N};I Time slot h ∈ H , {0, . . .,H−1}.

I Because the optimization variables take continuousvalues, we can easily apply

I Convex optimization [1];I Distributed algorithms through concave n-person

games [2].

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xn ,[x0

n, . . . , xhn, . . . , x

H−1n

games [2].

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xn ,[x0

n, . . . , xhn, . . . , x

H−1n

games [2].

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xn ,[x0

n, . . . , xhn, . . . , x

H−1n

games [2].

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xn ,[x0

n, . . . , xhn, . . . , x

H−1n

games [2].

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xn ,[x0

n, . . . , xhn, . . . , x

H−1n

games [2].

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Feasible SetA feasible energy consumption scheduling set for user n

∣∣∣∣∣∣ ∑h∈Hn

xhn=En, γ

minn ≤xh

n≤γmaxn , ∀h∈Hn, xh

n=0, ∀h∈H\Hn

}whereI γmin

n : Minimum energy level;I γmax

n : Maximum energy level;I En: Total daily energy consumption;I Hn: Scheduling interval defined as follows:

Hn ,{h∣∣∣h = i mod H, ∀i∈

[αn, βn

]}with αn∈[0, H−1], βn∈[1, 2H−2], and 1≤βn−αn≤H−1.

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∣∣∣∣∣∣ ∑h∈Hn

xhn=En, γ

minn ≤xh

n=0, ∀h∈H\Hn

}whereI γmin

[αn, βn

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∣∣∣∣∣∣ ∑h∈Hn

xhn=En, γ

minn ≤xh

n=0, ∀h∈H\Hn

}whereI γmin

[αn, βn

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∣∣∣∣∣∣ ∑h∈Hn

xhn=En, γ

minn ≤xh

n=0, ∀h∈H\Hn

}whereI γmin

[αn, βn

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∣∣∣∣∣∣ ∑h∈Hn

xhn=En, γ

minn ≤xh

n=0, ∀h∈H\Hn

}whereI γmin

[αn, βn

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∣∣∣∣∣∣ ∑h∈Hn

xhn=En, γ

minn ≤xh

n=0, ∀h∈H\Hn

}whereI γmin

[αn, βn

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Optimal SchedulingThe optimal scheduling is formulated as an optimizationproblem for a given objective function φ(·) (e.g., totalenergy cost or PAR) as follows:

minimizexn∈Xn, ∀n∈N

φ (L(x))

whereI x , [x1, . . . , xN]: A vector of user energy consumption

vectors;I L(x) , [L0 (x) , . . . ,LH−1 (x)]: A vector of aggregate

loads across all users at each time slot, which aredefined as

Lh (x) ,∑n∈N

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φ (L(x))

Lh (x) ,∑n∈N

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φ (L(x))

Lh (x) ,∑n∈N

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φ (L(x))

Lh (x) ,∑n∈N

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Objective Functions

I For energy cost minimization:

φ (L(x)) =∑h∈H

Ch (Lh(x))

whereI Ch(·): A cost function for generating or distributing

electricity energy at a time slot h.

I For PAR minimization:

φ (L(x)) =

H maxh∈H

Lh(x)∑n∈N

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Objective Functions

I For energy cost minimization:

φ (L(x)) =∑h∈H

Ch (Lh(x))

whereI Ch(·): A cost function for generating or distributing

electricity energy at a time slot h.

I For PAR minimization:

φ (L(x)) =

H maxh∈H

Lh(x)∑n∈N

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Atomic vs. Non-Atomic Scheduling

αn,a βn,a

Gap Gap

γminn

γmaxn

γopn (·)

γminn

γmaxn

Examples of (a) non-atomic and (b) atomic scheduling.I γmin

n : Minimum energy levelI γmax

n : Maximum energy levelI γop

n (·): Operating energy level

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Non-Atomic Scheduling Example

Appliance 1

Appliance 2

Scheduled

Consumption?

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Non-Atomic Scheduling Example: Case 1

Appliance 1

Appliance 2

Scheduled

Consumption

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Appliance 1

Appliance 2

Scheduled

Consumption

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Appliance 1

Appliance 2

Scheduled

Consumption

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Conclusions

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Overview of Atomic Scheduling ProblemFormulation & Solution

Starting-Time-BasedFormulation

Optimal-Routing-BasedFormulation

Convex Relaxation

Successive ConvexRelaxation withFractional-Value

Dropping

ConvexOptimization

(Feasible Upper Bound)

CombinatorialOptimization

Boolean-ConvexOptmization

ConvexOptmization

(Lower Bound)

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Conclusions

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Starting-Time-Based FormulationI Optimization variables:

s , [s1, . . . , sN]

I A feasible set for user n:

Sn ,{sn

∣∣∣sn = i mod H, ∀i∈[αn, βn−δn+1]}

I Aggregate load across all users at each time slot h:

Lh(s) ,∑n∈N

γopn ((h − sn) mod H) IRn(sn)(h)

whereI IRn(sn)(h): An indicator function for a set Rn(sn).I Rn(sn): A range of user n’s appliance operation for sn

defined as follows:

Rn(sn) ,{h∣∣∣h = i mod H, ∀i∈ [sn, sn + δn − 1]

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s , [s1, . . . , sN]

Sn ,{sn

Lh(s) ,∑n∈N

defined as follows:

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s , [s1, . . . , sN]

Sn ,{sn

Lh(s) ,∑n∈N

defined as follows:

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s , [s1, . . . , sN]

Sn ,{sn

Lh(s) ,∑n∈N

defined as follows:

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s , [s1, . . . , sN]

Sn ,{sn

Lh(s) ,∑n∈N

defined as follows:

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s , [s1, . . . , sN]

Sn ,{sn

Lh(s) ,∑n∈N

defined as follows:

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Issues with Starting-Time-Based Formulation

I Because the feasible set is now discrete, we have toevaluate the objective function for all the elements in thefeasible set.

I The optimization by direct enumeration becomesimpractical for large N and H.

I When N=100 and H=24 with the worst case scenarioof αn=0, βn=23, and δn=1 for all n∈N , we need toevaluate the objective function 24100 times, which ison the order of 10138 times!

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Conclusions

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Why London Eye?

Does the London Eyehave something todo with the atomicscheduling?

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A Network, A Path, and Links

10111213

l9,10l10,11

l11,12

A network connectingthe source (S) and thedestination (D)through 24intermediate nodeswith a path (p9,3) andits constituent links(l9,10, l10,11, and l11,12).

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Mapping of Atomic Operations to Flows

10111213

f112f12

Mapping of allpossible atomicoperations of twoappliances into twogroups of flows(f 0

1 , . . . , f41 and

f 92 , . . . , f

122 ) over

multiple paths on thenetwork.

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Optimization Variables and Feasible SetI Optimization variables: Flow configurations of all users

defined asf ,

[f 1, . . . , f n, . . . , f N

]where

f n ,[f 0n , . . . , f

H−1n

I A feasible atomic energy consumption scheduling setfor user n:

∣∣∣∣∣∣∑s∈Sn

f sn=1, f s

n∈ {0, 1} , ∀s∈Sn, f sn=0, ∀s∈H\Sn

}where Sn is the feasible set of starting times for user nthat is already defined in starting-time-basedformulation.

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defined asf ,

[f 1, . . . , f n, . . . , f N

]where

f n ,[f 0n , . . . , f

H−1n

∣∣∣∣∣∣∑s∈Sn

f sn=1, f s

n∈ {0, 1} , ∀s∈Sn, f sn=0, ∀s∈H\Sn

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defined asf ,

[f 1, . . . , f n, . . . , f N

]where

f n ,[f 0n , . . . , f

H−1n

∣∣∣∣∣∣∑s∈Sn

f sn=1, f s

n∈ {0, 1} , ∀s∈Sn, f sn=0, ∀s∈H\Sn

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defined asf ,

[f 1, . . . , f n, . . . , f N

]where

f n ,[f 0n , . . . , f

H−1n

∣∣∣∣∣∣∑s∈Sn

f sn=1, f s

n∈ {0, 1} , ∀s∈Sn, f sn=0, ∀s∈H\Sn

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Atomic Optimal SchedulingI Atomic optimal scheduling for an objective function

of φ(·) is formulated as follows:

minimizefn∈Fn,∀n∈N

φ (L (f)) .

whereI L (f), [L0 (f) , . . . ,LH−1 (f)]: A vector of aggregate loads

across all flows at each time slot, which are defined as

Lh(f) ,∑n∈N

γopn ((h−s) modH)

∑s∈Sn

f snIRn(s)(h)

.I Note that for a convex objective function, this

problem becomes a Boolean-convex problem, since theoptimization variable f s

n is restricted to only 0 or 1.28 / 55

φ (L (f)) .

Lh(f) ,∑n∈N

∑s∈Sn

f snIRn(s)(h)

φ (L (f)) .

Lh(f) ,∑n∈N

∑s∈Sn

f snIRn(s)(h)

φ (L (f)) .

Lh(f) ,∑n∈N

∑s∈Sn

f snIRn(s)(h)

Conclusions

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Relaxed Atomic Optimal SchedulingI We can relax the atomic optimal scheduling problem

by replacing f sn∈ {0, 1}with 0≤f s

n≤1 in constraints asfollows:

minimizefn∈F̂n,∀n∈N

φ (L (f))

F̂n =

∣∣∣∣∣∣∑s∈Sn

f sn=1, 0 ≤ f s

n ≤ 1, ∀s∈Sn, f sn=0, ∀s∈H\Sn

I For a convex objective function, this problembecomes convex because F̂n is now a convex set. Itcan be solved efficiently, for instance, using thewell-known interior-point method [1].

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φ (L (f))

F̂n =

∣∣∣∣∣∣∑s∈Sn

f sn=1, 0 ≤ f s

n ≤ 1, ∀s∈Sn, f sn=0, ∀s∈H\Sn

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φ (L (f))

F̂n =

∣∣∣∣∣∣∑s∈Sn

f sn=1, 0 ≤ f s

n ≤ 1, ∀s∈Sn, f sn=0, ∀s∈H\Sn

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Relaxed vs. Original Scheduling Problems

I The relaxed atomic optimal scheduling problem isnot equivalent to the original problem.

I The elements of the optimal solution from the relaxedproblem can take fractional values (e.g., 0.75).

I The optimal solution of the relaxed problem,however, provides a lower bound on the optimalsolution of the original problem.

I The feasible set for the relaxed problem contains thefeasible set for the original problem.

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Successive Convex Relaxation

First, we solve the relaxed convex optimization problem.Then, carry out the following procedures:

1. Identify the maximum element of each user flowconfiguration vector (i.e., corresponding to f n) andexclude them in the following procedures.

2. Arrange in ascending order the remaining elementsthat are less than 1.

3. Drop the smallest element and add a zero constraintfor it.

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Successive Convex Relaxation (Cont.)

4. For the rest of the elements, drop them and add zeroconstraints from the smallest element up to ND

elements in total (including the one in step 3) as far asthe element is less than a dropping threshold (θD);otherwise, stop dropping and go to the next step.

5. If there remains only one nonzero element per userflow configuration vector, stop here (a solutionfound); otherwise, solve a new relaxed convexoptimization problem with augmented constraintsand repeat the whole procedure from step 1.

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Successive Convex Relaxation (Cont.)

4. For the rest of the elements, drop them and add zeroconstraints from the smallest element up to ND

elements in total (including the one in step 3) as far asthe element is less than a dropping threshold (θD);otherwise, stop dropping and go to the next step.

5. If there remains only one nonzero element per userflow configuration vector, stop here (a solutionfound); otherwise, solve a new relaxed convexoptimization problem with augmented constraintsand repeat the whole procedure from step 1.

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Successive Convex Relaxation: An ExampleConsider a simple case of N=2, H=4, and ND = 1.I Initial condition:

f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]

I After 1st step:

f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]

I After 2nd step:

f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]

Stop here. Solution found!

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f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]

I After 1st step:

f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]

I After 2nd step:

f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]

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f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]

I After 1st step:

f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]

I After 2nd step:

f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]

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f = [ 0.25 0.25 0.25 0.25 | 0.3 0.2 0.25 0.25 ]

I After 1st step:

f = [ 0.0 0.2 0.3 0.5 | 0.7 0.0 0.2 0.1 ]

I After 2nd step:

f = [ 0.0 0.0 0.2 0.8 | 0.9 0.0 0.1 0.0 ]

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Conclusions

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Energy Cost and PAR Minimization

I Energy cost minization

∑h∈H

Ch (Lh(f)) .

I PAR minimization

minimizeΓ,fn∈F̂n,∀n∈N

subject to Γ ≥ Lh(f), ∀h ∈ H .

I Note that PAR minimization is formulated as a relaxedlinear program by introducing a new auxiliary variableΓ.

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∑h∈H

Ch (Lh(f)) .

I PAR minimization

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∑h∈H

Ch (Lh(f)) .

I PAR minimization

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Conclusions

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Conclusions

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Appliance Energy ConsumptionRequirements

Appliance Parametersα [h] β [h] γop [kWh] δ [h]

Dish Washer 0 23 0.7200 2Washing Machine 0 23 0.4967 3(Energy Star)Washing Machine 0 23 0.6467 3(Regular)

Clothes Dryer 0 23 0.6250 4PHEV1 222 292 3.3000 3

1 Plug-in hybrid electric vehicle.2 Scheduling interval of 10 PM–5 AM.

I An appliance is randomly selected for each user.I Constant operating energy levels (γop) are assumed.

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Hourly Cost Function

We assume a simple quadratic hourly cost function as in[3], i.e.,

Ch (Lh) = ahL2h [cent]

{0.2 if h ∈ [0, 7],0.3 if h ∈ [8, 23].

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Hourly Cost Function

We assume a simple quadratic hourly cost function as in[3], i.e.,

Ch (Lh) = ahL2h [cent]

{0.2 if h ∈ [0, 7],0.3 if h ∈ [8, 23].

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Performance Measures and Parameter Values

I Performance measuresI Lower bound: LBI Upper bound with ND: UB (ND)I Gap: G , UB(D) − LBI Number of iterations

I Parameter values for successive convex relaxationI Dropping threshold (θD): 0.1I Maximum number of fractional-valued elements that

can be dropped per iteration (ND): 1, 2, 5, 10

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Conclusions

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Upper/Lower Bounds vs. True OptimalValues: Cost Minimization

2 3 4 5 6 7 8 9 100

5 · 10−2

Energy

Cost[U

UB(10)

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Upper/Lower Bounds vs. True OptimalValues: PAR Minimization

2 3 4 5 6 7 8 9 100

tedLoad

UB(10)

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Conclusions

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Cost Minimization: Upper/Lower Bounds

2 10 20 30 40 500

Energy

Cost[U

LBUB(1)UB(2)UB(5)UB(10)

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Cost Minimization: Gaps

2 10 20 30 40 500

7×10−2

UB(1)−LBUB(2)−LBUB(5)−LBUB(10)−LB

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Cost Minimization: Number of Iterations

2 10 20 30 40 500

Number

ofIterations

UB(1)UB(2)UB(5)UB(10)

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Conclusions

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Par Minimization: Upper/Lower Bounds

2 10 20 30 40 500

inAggregatedLoad

LBUB(1)UB(2)UB(5)UB(10)

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Par Minimization: Gaps

2 10 20 30 40 500

UB(1)−LBUB(2)−LBUB(5)−LBUB(10)−LB

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Par Minimization: Number of Iterations

2 10 20 30 40 500

Number

ofIterations

UB(1)UB(2)UB(5)UB(10)

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Conclusions

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ConclusionsI We have provided a new formulation of appliance energy

consumption scheduling based on the optimal routingframework.

I It guarantees the atomicity of resulting scheduledenergy consumption.

I We have also provided an efficient solution techniquebased on successive convex relaxation for aBoolean-convex problem resulting from a convexobjective function.

I It enables us to carry out systematic analysis of theoriginal problem with both upper and lower bounds.

I Possible extensions to the current work includeI Distributed atomic energy consumption scheduling;I Advanced techniques refining the feasible region at

each iterative step to reduce the total number ofiterations.

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References I

S. Boyd and L. Vandenberghe, Convex optimization.Cambridge, U.K.: Cambridge Universtiy Press, 2004.

J. B. Rosen, “Existence and uniqueness of equilibriumfor concave N-person games,” Econometrica, vol. 33,no. 3, pp. 520–534, Jul. 1965.

A.-H. Mohsenian-Rad, V. W. S. Wong, J. Jatskevich,R. Schober, and A. Leon-Garcia, “Autonomousdemand-side management based on game-theoreticenergy consumption scheduling for the future smartgrid,” IEEE Trans. Smart Grid, vol. 1, no. 3, pp.320–331, Dec. 2010.

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