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Peak-to-Average Ratio Constrained Demand-SideManagement with Consumer’s Preference in
Residential Smart GridYi Liu, Chau Yuen, Senior Member, IEEE, Shisheng Huang, Member, IEEE, Naveed Ul Hassan, Member, IEEE,
Xiumin Wang and Shengli Xie, Senior Member, IEEE
Abstract—In a smart grid network, demand-side managementplays a significant role in allowing consumers, incentivized byutilities, to manage their energy consumption. This can be donethrough shifting consumption to off-peak hours and thus reduc-ing the peak-to-average ratio (PAR) of the electricity system.In this paper, we begin by proposing a demand-side energyconsumption scheduling scheme for household appliances thatconsiders a PAR constraint. An initial optimization problem isformulated to minimize the energy cost of the consumers throughthe determination of the optimal usage power and operation timeof throttleable and shiftable appliances, respectively. We realizethat the acceptance of consumers of these load managementschemes is crucial to its success. Hence, we then introduce amulti-objective optimization problem which not only minimizesthe energy cost but also minimizes the inconvenience posed toconsumers. In addition to solving the proposed optimizationproblems in a centralized manner, two distributed algorithms forthe initial and the multi-objective optimization problems are alsoproposed. Simulation results show that the proposed demand-sideenergy consumption schedule can provide an effective approachto reducing total energy costs while simultaneously consideringPAR constraints and consumers’ preferences.
Index Terms—Demand-side management, peak-to-average ra-tio, energy consumption scheduling scheme, consumer’s prefer-ence, smart grid.
I. INTRODUCTION
GENERALLY, traditional power grids are used to dis-tribute power from a few central generators to a large
This research is partially supported by Singapore University of Tech-nology and Design (SUTD) under Energy Innovation Research Program(EIRP) (grant no. NRF2012EWT-EIRP002-045), programs of NSFC (grantnos. 61322306, 61333013, 61370159, 61273192, U1201253, U1035001, and61300212), Guangdong Province Natural Science Foundation of under GrantS2011030002886 (team project), and the Science and Technology Programof Guangzhou, China (grant no. 2014J2200097), P.D. Programs Foundationof Ministry of Education of China (grant no. 20130111120010), LahoreUniversity of Management Sciences (LUMS), Research Startup Grant.
Copyright(c) 2014 IEEE. Personal user of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected]
Yi Liu is with Guangdong University of Technology, Guangzhou, Chinaand Singapore University of Technology and Design, Singapore (e-mail:[email protected]).
Chau Yuen and Shisheng Huang are with Singapore Universityof Technology and Design, Singapore (e-mail: [email protected],shisheng [email protected]).
Naveed Ul Hassan is with Department of Electrical Engineering, SSE,LUMS, Lahore (email: [email protected]).
Xiumin Wang is with School of computer and information, Hefei Universityof Technology, Hefei, China (e-mail: [email protected]).
Shengli Xie (corresponding author) is with Guangdong University ofTechnology, Guangzhou, China (e-mail: [email protected]).
number of consumers. Although adequate to meet past chal-lenges, renewable integration, distributed generation, demandmanagement, have increasingly exposed the limitations of thetraditional grid systems. In contrast, the Smart Grid (SG)system uses two-way flows of electricity and information tocreate an automated and distributed advanced energy deliverynetwork [1]-[4]. By utilizing adaptive signal processing, statis-tical signal processing and modern information technologies,[5]-[8], the SG is able to distribute power in more efficientways and respond to wide ranging conditions and variationalchanges. Broadly stated, the residential SG could respondto events that occur anywhere in the grid that could impactgeneration, transmission, distribution, and consumption, andthen adopt the corresponding strategies to cope with them [9]-[13].
Made possible by advanced communication technologies,a key advantage of the SG is flexible bi-directional demandside management (DSM) for residential smart grid [14]-[17]. Benefiting from information exchange between supplyand demand, DSM is able to reduce peak electricity loadsand increase the reliability of the power grid. Utilities andsystem operators can effectively manage power generationand through incentives, encourage the shifting of high-energydemand household appliances to off-peak hours [18]. Costsensitive consumers are able to adjust their demand accordingto time differentiated electricity pricing and make appropriateconsumption scheduling decisions [19]-[22]. Recently, therehave been several studies detailing DSM approaches in resi-dential grid networks [23]-[29].
In [23], an incentive-based energy consumption schedulingscheme was proposed to schedule the power of individualappliances to reduce the energy cost and peak-to-averageratio (PAR). Both centralized optimization based on linearprogramming and decentralized game theoretic approacheshave also been proposed. PAR provides a measure on howpeak electricity consumption affects the system, particularlyin efficiency and reliability. A reduction in peak generallyimproves the electricity generation efficiency of the systemand this effect can be seen in [24]. If we further assumethat the overall energy consumption of the system remainsconstant, a reduction in PAR would also result in the reductionof the magnitude of peak energy demand. This reduction canincrease system reliability through increasing spare capacityin the event of supply tightness, which was one of themain contributing factors in the Californian electricity crisis
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of 2000 [25]. Inspired by [23], [26] proposed a demand-side management mechanism by considering the fixed powerconsumption pattern of household appliances. The proposedmechanism optimized the energy consumption schedule tosatisfy the specific requirements of the individual householdappliances. [27] presented demand-side management strategiesfor a large number of the household appliances of severaltypes. Besides the load shifting management of the shiftablehousehold appliances, another management strategy is thecontrol of energy usage by controlling throttleable appliances.These appliances are defined as appliances which have a fixedoperational period but flexible power consumption pattern [28][29]. In [30] [31] the authors focused on energy consumptionscheduling to reduce the energy cost by considering the con-sumer related consumption preferences. However, the specificdissatisfaction functions for both of shiftable appliances andthrottleable appliances are not considered. In addition, how toyield the dissatisfaction functions to fit various cases when thedifferent user has different preference is still an open issue.
The focus of this paper is on demand-side energy con-sumption scheduling scheme for both shiftable and throttleableappliances in terms of constrained PAR. Under this constraint,a uniform load demand during the day can be guaranteed. Inaddition, the consumers’ own preferred usage requirementsare addressed in our energy scheduling algorithm. Whenconsidering the reschedule of shiftable appliances to reduceelectricity costs, consumers may have a limited window inwhich they will accept an operational shift. We define thisdelay, operation delay, as a metric for measuring the amountof operational delay a consumer is willing to endure. Similarly,for throttleable appliances, the power gap is used to measurethe consumers’ willingness to endure reduced consumption;here, it is defined as the difference between the consumer’spreferred power and the scheduled power. Considering bothof the metrics in the optimization problem should allow forbetter consideration of the consumers’ usage preferences.
Hence, in this paper, we propose an improved energyconsumption* scheduling scheme for household appliancesunder the supply-demand framework of the residential smartgrid. Three types of appliances are considered: shiftable,throttleable and essential appliances, where essential appli-ances correspond to those appliances that are non-shiftableand non-throttleable. The initial objective of our proposedscheduling scheme is to seek the optimal response strategyfor each type of appliances in order to minimize the totalcosts associated with electricity consumption under the PARconstraint. An initial optimization problem of minimizingthe energy cost is formulated in order to find the optimalenergy consumption and operation time of the throttleableand shiftable appliances, respectively. Subsequently, a multi-objective optimization problem considering consumer prefer-ences is formulated, with objectives to minimize the energycost, operation delay and energy gap. In addition to solvingthe optimal problems in a centralized manner at residentialscheduler (RS) level, two distributed algorithms for the initial
∗Throughout this paper, unless otherwise indicated, the phrases “energyconsumption” and “power consumption” all refer to the same meaning.
optimization problem and the multi-objective optimizationproblem are also proposed. The distributed algorithms can findthe near-optimal schedules with minimal information exchangebetween the RS and consumers. Simulation results showthat the proposed scheduling scheme can achieve effectivescheduling under a PAR constraint considering different levelsof consumer preferences.
The rest of the paper is organized as follows. In SectionII, the system model is introduced. The initial energy costminimization problem for the residential smart grid systemis proposed and solved in both centralized and distributedformulations in Section III. Section IV studies the energycost minimization problem with consumer’s preference andprovides a distributed subgradient algorithm to solve thisproblem. Section V presents the simulation results of theproposed algorithms. Finally, we conclude the paper in SectionVI.
II. SYSTEM MODEL
A. Residential Grid Networks
We consider the residential smart grid in which the energyfrom the power grid is shared by several consumers (homes)through the power line, as shown in Fig.1. Each consumer isequipped with a smart meter, which not only distributes theelectricity from the power line to all appliances in each home,but also collects each consumer’s demand and preference in-formation. Through the communication line, the smart metersare capable of reporting the collected information to the RS. Ina centralized manner, the RS can globally optimize the energyconsumption and schedule all household appliances based onthe collected information. In a distributed manner, the RS canprovide the variation information of the entire grid networks toeach consumer for local scheduling process. Here, we conducta theoretical study by assuming that information of a day isknown before hand. We will later in the future work designalgorithm based on only historical or statistical information,and the result of this paper can be serve as a performancebound for a practical scenario.
The appliances in the residential grid network are cate-gorized into three types: shiftable, throttleable and essentialappliances. The scheduling process would only affect theshiftable and throttleable appliances. For shiftable appliances,such as dishwashers, washing machines and clothes dryers, theoperational period can be delayed while power consumptionfor these appliances would not be adjusted during operation.For throttleable appliances with fixed operational periodssuch as refrigerators and air conditioning units, their powerconsumption can be adjusted within a range during theirfixed operational period. For essential appliances such asTVs, electric stoves and lamps, which have a fixed powerrequirement and operational period, a continuous supply ofpower should be ensured throughout the optimization period.
B. Load Demand Description
Let N denote the set of consumers, where the numberof consumers is N , |N |. For each consumer n ∈ N ,let Mn = In
⋃Sn
⋃Rn denote the set of household
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Residential Scheduler
Power Grid
Smart
Meter
Appliance 1 Appliance 2 Appliance M
Consumer 1
(Home)Smart
Meter
Appliance 1 Appliance 2 Appliance M
Consumer N
(Home)Smart
Meter
Appliance 1 Appliance 2 Appliance M
Consumer 2
(Home)
Communication Line
Power Line
Fig. 1. The system model.
appliances, where In, Sn and Rn denote the set of essential,shiftable and throttleable appliances, respectively. Assumethat the number of household appliances in each consumeris M , |Mn|. Then, we define the energy consumptionscheduling vector for these three types of the appliances as
en,i , [e1n,i, · · · , et
n,i, · · · , eTn,i],
en,s , [e1n,s, · · · , et
n,s, · · · , eTn,s],
en,r , [e1n,r, · · · , et
n,r, · · · , eTn,r],
where scalar etn,i, et
n,s and etn,r denote the corresponding
energy consumption that is scheduled for essential appliancesi ∈ In, shiftable appliance s ∈ Sn and throttleable appliancer ∈ Rn by consumer n at unit time interval t, respectively.
Let Ln,t denote the total load of consumer n at each timeinterval t ∈ T , 1, ..., T, where T is the total number ofall unit time intervals. The load for consumer n is denoted byLn,tT
t=1 , Ln,1, · · · , Ln,T . Moreover, the total load ofthe nth consumer at t is obtained as
Ln,t =∑
i,s,r∈An
etn,i + et
n,s + etn,r, t ∈ T .
Based on these definitions, the total load across all consumersat time interval t ∈ T can be calculated as
Lt ,∑
n∈NLn,t.
Let Lp and La denote the peak load and average load ofthe residential smart grid, respectively. We have
Lp = maxt∈T
Lt
and
La =1T
∑
t∈T
Lt.
Hence, the PAR of the load demand, denoted by ΓPAR, canbe obtained as [23]
ΓPAR =Lp
La=
T maxt∈T Lt∑t∈T Lt
.
C. Energy Cost Function
In this subsection, we try to abstract a general energy costfunction from some practical electricity pricing schemes toreflect the cost of electricity consumption to the consumer.Without loss of generality, we consider three electricity pricemodels [32] [33], which represent three tariffs commonly usedby utilities: flat rate model (FRM), two-step piecewise model(TPM) and five-tier pricing model (FPM). The first modelassumes a constant price for consumers regardless of theirenergy consumption. The latter two models employ increasingtiered pricing schemes for which the base level has the lowestprice and the price for the consumers increases in one (TPM)and four (FPM) additional tiers.
Let <(Lt) denote the energy cost for the consumers underthe pricing scheme at each time interval t. Three energy costfunctions which can represent the cost tariffs are given by:
FRM:<(Lt) = aFRM
t Lt, (1)
where aFRMt is a constant.
TPM: The energy cost under this model can be approxi-mated by the quadratic function given in [26]
<(Lt) = aTPMt L2
t + bTPMt Lt + cTPM
t , (2)
where aTPMt > 0 and bTPM
t , cTPMt ≥ 0 at each time interval
t.FPM: The energy cost in this model can also be approxi-
mated by the quadratic function with different coefficients.
<(Lt) = aFPMt L2
t + bFPMt Lt + cFPM
t , (3)
where aFPMt > 0 and bFPM
t , cFPMt ≥ 0 at each time interval
t.Fig. 2 shows the actual price model and the approximated
cost functions for TPM and FPM. Assume that aTPMt = 0.2
cents, bTPMt = cTPM
t = 0; and aFPMt = 0.7 cents,
bFPMt = cFPM
t = 0. We observe that the quadratic functionhas appropriate approximations for the practical price models.Hence, in this paper, we employ the quadratic function as thecost function for residential energy consumption. Meanwhile,we consider a time varying electricity tariff model throughvarying the value of at for different time intervals.
III. INITIAL ENERGY COST MINIMIZATION PROBLEM
A. Energy Cost Minimization
In this subsection, our goal is to schedule the operationalperiod and power consumption of the appliances in order toachieve minimum energy costs. Since the RS has no impact onthe schedule of energy consumption for essential appliances,the essential load Ei =
∑t∈T
∑n∈N
∑i∈In
etn,i has constant
value in the objective function of our problem.For shiftable appliances, we mainly focus on designing
the energy consumption scheduler to systematically shift theoperational period of the appliance. In this case, the consumerneeds to pre-set the start αn,s ∈ T and end βn,s ∈ T of a timeperiod that the appliance can be scheduled. For each consumern and each shiftable appliance s, we denote the total energyconsumption as En,s. Then, we have
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0 5 10 15 20 25 30 35 400
1
2
3
4
Energy Consumption (kWh)
Tot
al D
aily
Cos
t (D
olla
rs)
TPMCost Function
0 5 10 15 20 25 30 35 400
5
10
15
Energy Consumption (kWh)
Tot
al D
aily
Cos
t (D
olla
rs)
FPMCost Function
Fig. 2. Two pricing Model and approximated cost function.
βn,s∑t=αn,s
etn,s = En,s (4)
etn,s = 0, ∀t ∈ T \Tn,s, (5)
0 < etn,s ≤ pmax
n,s , ∀t ∈ Tn,s, (6)
where pmaxn,s is the maximum power level for s shiftable
appliance in consumer n and Tn,s , [αn,s, ..., βn,s]. For eachshiftable appliance, the time period assigned by the RS needsto be larger than or equal to the time interval needed to finishthe operation.
For throttleable appliances, the energy consumption sched-uler does not aim to change the appliance’s operational period,but instead adjusts the operational power demand of eachthrottleable appliance within a predetermined range at eachoperation time interval t. We define the tolerance power levelptol
n,r and the preferred power level ppren,r as the minimum
and maximum power for each appliance r ∈ R for eachconsumer n ∈ N . Tolerance power refers to the electricalpower consumed by each appliance operating at a level thatbarely satisfies the consumer’s requirement. Preference powerrefers to the consumed power of the appliance used accordingto its typical usage pattern. For example, consumers mayprefer a 24C to a tolerable 26C of the air-conditioning unitand may not accept a thermostat level lower than 24C orhigher than 26C. In such a case, the consumed power of theair-conditioner at 26C is used as the tolerance power level andthe preferred power level is the consumed power by the air-conditioning unit at 24C. Hence we can define the following
P toln,r ≤et
n,r ≤ P pren,r , ∀t ∈ Tn,s. (7)
We denote the total energy consumption of all throttleableappliances in nth custormer as En,r. We can obtain
βn,r∑t=αn,r
etn,r ≤ En,r, (8)
etn,r = 0, ∀t ∈ T \Tn,r, (9)
where αn,r ∈ T and βn,r ∈ T are the beginning and the end ofthe time interval that throttleable appliances can be operated,Tn,r , [αn,r, ..., βn,r]. For each throttleable appliance, thetime interval is fixed and the appliance should finish itsoperation during its own time interval.
Therefore, the energy consumption scheduling problem canbe formulated in terms of minimizing the energy costs toall consumers and appliances, which can be expressed as thefollowing optimization problem:
(P1) mine
T∑t=1
<(Lt)
=T∑
t=1
at
( ∑
n∈N
(∑
i∈In
etn,i +
∑
s∈Sn
etn,s +
∑
r∈Rn
etn,r
))2
s.t. ΓPAR =maxt∈T Lt1T
∑t∈T Lt
≤ Γmax
βn,s∑t=αn,s
etn,s = En,s, et
n,s = 0, ∀t ∈ T \Tn,s,
βn,r∑t=αn,r
etn,r ≤ En,r, et
n,r = 0, ∀t ∈ T \Tn,r,
0 < etn,s ≤ pmax
n,s , ∀t ∈ Tn,s,
ptoln,r ≤ et
n,r ≤ ppren,r , ∀t ∈ Tn,r.
The explanation of the constraints in (P1) are as follows: thefirst constraint restricts the PAR of the load demand from allresidences less than Γmax. In this paper, we choose Γmax
carefully to guarantee the feasibility of solving (P1). It mightnot be possible to achieve every value of PAR constraint;e.g. Γmax = 1 means perfectly flat load profile which mightnever be achieved by any algorithm and by any amount of re-scheduling. Minimum energy cost obtained by optimal energyconsumption schedule with different values of Γmax are shownin Section V.A. The second equality and the third inequalityare the energy consumption constraints of the shiftable and thethrottleable appliances of the nth consumer in each operationinterval t, respectively. The last two inequalities are the hourlyenergy consumption constraints of the shiftable appliances and the throttleable appliance r of consumer n in eachoperation interval t, respectively.
B. The Dual of Energy Minimization Problem
We firstly introduce vector en = en,i, en,s, en,r, whichis formed by stacking up the energy consumption schedulingvectors for all appliances i, s, r ∈ Mn of consumer n.Then, we define a feasible energy consumption scheduling setcorresponding to nth consumer as follows:
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En =en,i, en,s, en,r|
βn,s∑t=αn,s
etn,s = En,s, e
tn,s = 0,∀t ∈ T \Tn,s,
βn,r∑t=αn,r
etn,r ≤ En,r, et
n,r = 0, ∀t ∈ T \Tn,r,
0 < etn,s ≤ pmax
n,s , ∀t ∈ Tn,s,
ptoln,r ≤ et
n,r ≤ ppren,r , ∀t ∈ Tn,r
.
In the proposed scheduling method, the RS validly controlsconsumer’s energy consumption only if en ∈ En. Therefore,we can simplify problem (P1) based on the following theo-rem.
Theorem 1. If Lt =∑
n∈N (∑
i∈Inetn,i +
∑s∈Sn
etn,s +∑
r∈Rnetn,r) and E = 1
T (Ei +∑
n∈N∑
s∈SnEn,s +
T∑
n∈N∑
r∈Rnppre
n,r ), the problem (P1) can be rewritten as
minen
T∑t=1
< (Lt)
s.t. Lt ≤ ΓmaxE, t = 1, · · · , T,
en ∈ En, n = 1, · · · , N.
(10)
The proof of Theorem 1 is given by Appendix A. Let vectore = [e1, · · · , eN ] denote the energy consumption schedule forall consumers. Since (10) is a quadratic programming problem,we have the following expression of (10) as
mine∈E
12e′Qe
s.t. Ae ≤ b,(11)
where Q =
a1, 0, · · · 00, a2, · · · 0...
......
...0, 0, · · · aT
TNM×TNM
, at ,
atINM×NM , I is the unit matrix.
A =
I1×NM , 01×NM , · · · 01×NM
01×NM , I1×NM , · · · 01×NM
......
......
01×NM , 01×NM , · · · I1×NM
T×TNM
,
and b = (ΓmaxE)IT×1, M is the number of the appliancesin each consumer. Then, the dual function of (11) is
q(µ) = infe∈E
12e′Qe + µ′(Ae− b)
. (12)
where µ = [µ1, · · · , µT ]′ is a T × 1 vector. The infimum isattained for e = −Q−1Aµ, which should be checked whethere ∈ E is satisfied. In case of e 6∈ E, the exhausted method isused to find infimum e ∈ E (We search the values of e downto the second decimal point which may suggest the convincedprecision). A straightforward calculation after substituting ein (12), yields
q(µ) = −12µ′AQ−1A′µ− µ′b.
According to the duality theory [34], the dual problem can bewritten as
minµ
12µ′Pµ + b′µ
s.t. µ ≥ 0,(13)
where P = AQ−1A′. Note that the dual problem is alsoa quadratic program, but it has simpler constraints than theprimal problem (P1). It is easy to solve (13) in a centralizedfashion using convex programming techniques [34] [35]. Sincethe primal problem is a convex quadratic program whichhas a finite optimal value, both the primal and the dualprograms have an optimal solution and their optimal valuesare equal [34]. If µ∗ is the dual optimal solution, let e∗ denotethe optimal solution of the primal problem (P1), then thefollowing holds,
e∗ = −Q−1Aµ∗.
C. Distributed Energy Scheduling Algorithm
In the previous section, the proposed energy cost minimiza-tion problem (P1) can be solved by the RS in a centralizedfashion. In order to do global optimization, the RS needs toknow the energy consumption information of all appliancesof each consumer. Hence, in order to preserve the consumer’sprivacy, we propose a distributed approach in solving (P1).We solve the problem at the level of the smart meter in eachconsumer’s household without the need to reveal its individualappliance consumption profiles. In addition, we propose thatonly the total energy consumption value be reported to the RS,minimizing the amount of information exchanged among thesmart meters and the RS.
For consumer n, the optimal schedule can be determinedby solving the following local optimization problem:
minen∈En
12e′nQnen
s.t. Anen ≤ bn,(14)
where Qn =
a1, 0, · · · 00, a2, · · · 0...
......
...0, 0, · · · aT
TM×TM
∈ Q, at =
atIM×M ,
An =
I1×M , 01×M , · · · 01×M
01×M , I1×M , · · · 01×M
......
......
01×M , 01×M , · · · I1×M
T×TM
∈ A, M
is the number of the appliances of each consumer and bn =(ΓmaxE)IT×1. The dual function of (14) is
qn(µ) = infen∈En
12e′nQnen + µ′(Anen − bn)
. (15)
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Algorithm 1: Distributed algorithm executed by RS andnth consumer
1: Repeat;2: At random time instances Do;At the RS3: Randomly initializes µ = [µ1, · · · , µT ]′;4: Broadcasts µ to all consumers;5: If the load value Ln,tT
t=1 from nth consumer is receivedThen;6: Calculates P according to P = AQ−1A′;7: Updates µ according to (16);8: End;At the residential smart meters9: If vector µ is received Then;10: Calculates the values of en according to en = −Q−1
n Anµ;11: If en changes compared to current schedule12: Updates en according to the new value;13: Calculates Ln,tT
t=1 according to en and sends thevalue to the RS;14: End;15: End;16: End;17: Until no consumer sends any new schedule to the RS.
where µ is a T ×1 vector. Both the local primal problem (14)and the local dual problem (15) have an optimal solution andtheir optimal values are equal [34]. Let e∗n denote the optimalsolution of the local primal problem (14), then the followingholds,
e∗n = −Q−1n Anµ∗.
To obtain the feasible solution, we use the same constraintof ΓPAR in the local optimization problem as in the globaloptimization problem (P1). If the energy consumption of eachconsumer individually satisfies PAR constraint then it alsoensures that global PAR constraint is also satisfied. Next, weintroduce the update process of µ at RS when en is receivedfrom user n. The update process of µ at RS can guaranteethe global ΓPAR constraint is still satisfied in the proposeddistributed algorithm. Let µt and bt denote the tth column ofµ′ and b′, respectively. Let ptt denote the tth diagonal elementof P, where P = AQ−1A′. We have the following iteration(the derivation is presented in the Appendix B),
µt , max
0, µt − 1
ptt
(bt +
T∑
k=1
ptkµk
). (16)
We propose Algorithm 1 to solve problem (14) and (16)at consumers and RS respectively, in a distributed fashion.The related information exchange between the RS and eachconsumer are illustrated in Fig. 3. At the RS, the algorithmstarts with some random initial conditions, i.e., the RS assumesa random vector µ. This assumption is required since atthe beginning, the RS has no prior information about theconsumers. In line 4, the RS broadcasts the value of the µto all residential smart meters. The loop in line 5 to 8 is forthe RS to update the µ once en from the nth consumer isreceived. At the residential smart meter, if µ is received, eachsmart meter individually calculates its own version of localenergy consumption scheduling vector en in Line 10. The
consumers locally solve the local schedule according to thenew announced µ. Then, each consumer updates the RS withits new value of en, which is the total local power consumptionper time interval. In this way, residences do not need to revealtheir individual appliance consumption profiles. Finally, theloop in Lines 1 to 17 is executed until the algorithm converges.
Residential Scheduler
Smart
Meter
Consumer 1
Smart
Meter
Consumer 2
Smart
Meter
Consumer N
Fig. 3. Information Exchange between RS and consumers.
D. Convergence
The distributed Algorithm 1 presented in the previoussubsection allows consumers coordinate with RS to obtainthe optimal energy scheduling. In this subsection, we willprove the convergence of the proposed distributed algorithm toillustrate its usability. The assessment is based on the followingtheorem.
Theorem 2. RS starts from any randomly selected initialconditions. Algorithm 1 converges to its fixed point, if thebroadcast of the iteration vector from RS and the updatesof the individual energy consumption scheduling vectors fromconsumers are accomplished successfully*.
Proof: After successfully obtaining the iteration vector foreach consumer, the optimal energy schedule can be achievedby solving the optimization problem (14). From Algorithm1, the energy cost in the system either decreases or remainsunchanged every time a consumer updates its energy consump-tion schedule. Since the energy cost is bounded below (e.g.,the energy cost is always nonnegative), the convergence tosome fixed point is evident. On the other hand, at the fixedpoint of Algorithm 1, no consumer can reduce its cost bydeviating from the fixed point when operating at its optimalenergy schedule. This directly indicates that the fixed point isthe optimal point among all the consumers.
The proposed scheduling scheme is usable by operatorsto reduce peak loads through multi-tier electricity pricing.In this case, consumers are able to reduce their costs byscheduling the operation period and power of the residentialappliances. However, this scheduling process ignores the con-sumer’s usage preferences for both shiftable and throttleableappliances which may not be suitable in a practical residentialenvironment. The results from this optimization should thenbe used to serve as a baseline for comparison purposes.
∗The complete information exchanging between smart meters and RScan be guaranteed by wire or wireless communication technologies [6]-[8].The energy consumption scheduling for residential network under incompleteinformation exchanging scenario is beyond the scope of this paper.
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IV. ENERGY COST MINIMIZATION WITH CONSUMER’SPREFERENCE
In this section, we propose an alternative energy con-sumption scheduling scheme that considers consumers’ usagepreference. In this scheme, we are not only interested inminimizing energy cost, but also consider the preference ofthe consumer when scheduling household appliances. A multi-objective optimization problem is formulated and firstly solvedin a centralized manner. A subgradient method based dis-tributed algorithm is then proposed to solve the minimizationproblem in a distributed manner.
A. Problem Formulation
For shiftable appliances, we assume that the consumers havea specific required operational interval [Dn,s, Un,s], whichdenotes the range of the operation time interval for the sthshiftable appliance of the nth consumer. Without loss ofgenerality, we have Dn,s ≤ αn,s < βn,s ≤ Un,s, whereαn,s ∈ T and βn,s ∈ T are the start and end of a timeinterval that appliance s has been scheduled. Let en,s denotethe energy schedule of the shiftable appliance s in consumer n.The operation delay of the sth shiftable appliance in consumern is denoted by
On,s(en,s) = arg mintet
n,s > 0 −Dn,s.
Obviously, the operation delay should be minimized as wellas the energy cost of the shiftable appliance.
For the throttleable appliances, the power gap of the rththrottleable appliance for nth consumer is denoted as Gn,r,which reflects the difference between the consumer’s preferredpower P pre
n,r and the scheduled power en,r, as adjusted by theRS. We then have
Gn,r(en,r) =T∑
t=1
(P pre
n,r − etn,r
).
To maximize a consumer’s comfort and satisfaction, the powergap also needs to be minimized.
We can therefore incorporate the operation delay, energy gapand energy cost minimization into an optimization problem tofind an optimal scheduling vector en = en,i, en,s, en,r, n ∈N . An energy consumption schedule of appliances can beformulated in terms of minimizing the energy costs, operationdelay and the energy gap of all consumers, which can beexpressed as the following optimization problem:
(P2) mine
Ω1
T∑t=1
< (Lt) + Ω2
∑
n∈N
∑
s∈Sn
On,s(en,s)
+ Ω3
∑
n∈N
∑
r∈Rn
Gn,r(en,r)
s.t. ΓPAR =maxt∈T Lt1T
∑t∈T Lt
≤ Γmax
βn,s∑t=αn,s
etn,s = En,s, et
n,s = 0, ∀t ∈ T \Tn,s,
βn,r∑t=αn,r
etn,r ≤ En,r, et
n,r = 0, ∀t ∈ T \Tn,r,
0 < etn,s ≤ pmax
n,s , ∀t ∈ Tn,s,
ptoln,r ≤ et
n,r ≤ ppren,r , ∀t ∈ Tn,r.
where e = [e1, · · · , eN ] denote the energy consumptionschedule for all consumers, Lt =
∑n∈N (
∑i∈In
etn,i +∑
s∈Snetn,s+
∑r∈Rn
etn,r), Tn,s , Dn,s, · · · , Un,s, Tn,r ,
[αn,r, ..., βn,r] where αn,r ∈ T and βn,r ∈ T are the fixedstart and end times of throttleable appliances.. Ω1,Ω2 and Ω3
are the weight which can be adjusted based on consumers’preference. Given Ω1,Ω2,Ω3, and following Theorem 1 asdescribed in Section III-B, we have simplification of (P2) as
minen
Ω1
T∑t=1
< (Lt) + Ω2
∑
n∈N
∑
s∈Sn
On,s(en,s)
+ Ω3
∑
n∈N
∑
r∈Rn
Gn,r(en,r)
s.t. Lt ≤ ΓmaxE, t = 1, · · · , T,
en ∈ En, n = 1, · · · , N.
(17)
B. The Dual of Energy Minimization Problem
Let φt, t = 1, · · · , T denote the Lagrange multiplier ofproblem (17), the Lagrangian function for (17) is
L(φ, en) = Ω1
T∑t=1
< (Lt) + Ω2
∑
n∈N
∑
s∈Sn
On,s(en,s)
+Ω3
∑
n∈N
∑
r∈Rn
Gn,r(en,r) +T∑
t=1
φt (Lt − Γmax) .
(18)
The dual function and the dual problem take the form
q(φ) = minen∈En
L(φ, en). (19)
The Lagrange multiplier φt can be interpreted as the pricecharged by the utility company at t if the consumed energyfor all consumers exceed the constraint ΓmaxE. Hence, theunits of Lagrange multipliers can be interpreted to be monetaryunits/kWh. Moreover, the dual problem can be written as
q∗ = maxφ≥0
q(φ). (20)
8
Note that (20) is a convex problem. It can be solved in acentralized manner by RS. Because problem (17) is convex andhas zero duality gap, let e∗ denote the optimal solution of (17),and φ∗ denote the optimal Lagrange multipliers correspondingto the dual solution [34]. Variables e∗ are also the minimizersin (19) for φ = φ∗, i.e., q(φ∗) = L(φ∗, e∗n).
For the same reasons as the previous section, we thendevelop a distributed algorithm for that can solve (P2) in adistributed manner. The optimization problems are assignedto the smart meters of individual consumers. These metersare then, coordinated through the RS to obtain the jointoptimal schedule. To achieve this, the distributed iterations aredescribed in Section IV-C, and their convergence is establishedin Section IV-D.
C. Distributed Energy Scheduling Algorithm with Consumer’sPreference
In this subsection, a distributed energy scheduling algorithmconsidering consumers’ preferences is proposed. For any con-sumer n ∈ N , the optimal schedule can be determined bysolving the following local optimization problem:
en(l) ∈ arg minen∈En
Ω1
T∑t=1
<(Ln,t) +T∑
t=1
φt(l)(Ln,t − ΓmaxE)
+Ω2
∑
s∈Sn
On,s(en,s) + Ω3
∑
r∈Rn
Gn,r(en,r),
n = 1, 2, · · · , N, (21)
Then, we develop a subgradient method to solve the localscheduling problem (21). The iteration utilizes the Lagrangeminimization function (19) from the dual problem and min-imizes with respect to en for each consumer n. Specifically,the subgradient method consists of the following iterations, in-dexed by l = 1, 2, · · · and initialized with arbitrary φt(l) ≥ 0:
φt(l + 1) =
φt(l) + λl
[(Lh(l)− ΓmaxE)
]+
,
t = 1, 2, · · · , T, (22)
where λl is the step size and ·+ , max0, ·.Then, we propose distributed Algorithm 2 to solve problem
(21) and (22) at the RS and consumers, respectively. Theflow of the algorithm is as follow. Similar to Section III,the RS takes responsibility for the updating and broadcast-ing Lagrange multipliers φt(l)T
t=1 to all the consumer’ssmart meters. At each consumer, upon receiving the Lagrangemultipliers φt(l)T
t=1, the associated smart meter solves itsown version of local optimization problem (21). The relatedinformation exchange between RS and each consumer areillustrated in Fig. 4. Finally, the loop in Lines 1 to 16 isexecuted until the algorithm converges.
D. Convergence
In this subsection, we will prove the convergence of Algo-rithm 2. According to Theorem 2, i.e., the broadcast of theiteration vector from the RS and the updates of the individual
Residential Scheduler
Smart
Meter
Consumer 1
Smart
Meter
Consumer 2
Smart
Meter
Consumer N
Fig. 4. Information Exchange between RS and consumers.
Algorithm 2: Distributed algorithm executed by RS andnth consumer
1: Repeat;2: At random time instances Do;At the RS3: Randomly initializes λl and φt(l)T
t=1;4: Broadcasts φt(l)T
t=1 to all consumers;5: If Ln,t(l)T
t=1 from nth consumer is received Then;6: Updates φt(l + 1)T
t=1 according to (22);7: End;At the residential smart meter8: If φt(l)T
t=1 is received Then;9: Solves local problem (21) using Convex Programming [34];10: If en changes compared to current schedule11: Updates en according to the new solution;12: Calculates Ln,t(l)T
t=1 according to en and sends itsvalue to the RS;13: End;14: End;15: End;16: Until no consumer sends any new schedule to the RS.
energy consumption scheduling vectors from the consumersare successful. We note that the subgradient method will workif a sufficiently small step size β is used. This shown by thefollowing theorem.
Theorem 3. The proposed Algorithm 2 will converge if everyφt(l) and dual optimal solution φ∗ have
‖ φt(l + 1)− φ∗ ‖<‖ φt(l)− φ∗ ‖,for all step size λl such that
0 < λl <2q(φ∗)− q(φt(l))‖ Lh(l)− ΓmaxE ‖2 (23)
The proof of Theorem 3 is given in Appendix C. Theorem 3can be used to establish convergence and rate of convergenceresults for the subgradient method with step size rules satis-fying (23). However, unless we know the dual optimal valueq(φ∗), which is rare, the range of step size (23) is unknown.In this paper, we use the step size formula
λl =αl(qt(l)− q(φt(l)))‖ Lh(l)− ΓmaxE ‖2
where qt(l) is an approximation to the optimal dual value and0 < αl < 2. To estimate the optimal dual value qt(l), we canuse the best current dual value
qt(l) = max0≤i≤l
qt(φ(i))
9
TABLE ITHE EXCHANGING INFORMATION BETWEEN RS AND CONSUMER n IN
DIFFERENT ENERGY SCHEDULING SCHEMES.
Scheduling Type Centralized Distributed
Nonpreferet
n,i, etn,s, et
n,rTt=1,
µ, Ln,tTt=1where i, s, r ∈ Mn
PreferΩ1,Ω2,Ω3, φt(l)T
t=1,et
n,i, etn,s, et
n,rTt=1, Ln,t(l)T
t=1where i, s, r ∈ Mn
as an upper bound. Meanwhile, any primal value correspond-ing to a primal feasible solution can be used. The primalfeasible solutions are naturally obtained in the course ofthe Algorithm 2 if the energy scheduling vector from allconsumers are achieved successfully. Here, we employ thefollowing way to choose αl and qt(l) in the step size formula:αl = 1 for all l and q(l) is given by
q(l) = qt(l) + δ(l),
where qt(l) is the best current dual value and δ(l) is a positivenumber, which is increased by a certain factor if the previousiteration was a “success”, i.e., it proved the best current dualvalue, and is decreased by some other factor otherwise. Thus,δ(l) may be viewed as an “aspiration level” of improvementthat we hope to attain in subsequent iterations. If upper boundsqt(l) to the optimal dual value are available, then a naturalimprovement to the above equation is
q(l) = minqt(l), qt(l) + δ(l).We refer to [36] for related convergence analysis. We use avariation of the method, is to choose
q(l) = (1 + ψ(l))qt(l),
where ψ(l) is the number greater than zero, which is increasedor decreased based on algorithm progress. This variationrequires that qt(l) > 0.
Discussion: In this paper, we considered schedulingschemes that explored combinations of two design parame-ters - whether the scheduling scheme considered consumerpreferences and whether the scheduling should be centralizedor distributed. Centralized and distributed scheduling schemeswhich do not consider consumer preferences are Centralized-Nonprefer and Distributed-Nonprefer respectively. Similarly,centralized and distributed scheduling schemes which considerconsumer preferences are named as Centralized-Prefer andDistributed-Prefer, respectively. Table I shows the informationexchanged between RS and consumer n in these four proposedscheduling schemes. It can be seen that the informationexchange in centralized scheduling schemes are more thanthat in the distributed scheduling schemes. This is because thedistributed schemes only need the total energy consumptionof each consumer instead of the detailed individual applianceconsumption information of all consumers in the centralizedschemes. More importantly, this implies that the privacy of theconsumer’s energy usage is carefully protected in distributedschemes.
V. SIMULATION RESULTS
In our considered residential smart grid system, we set aoptimization period as 24 hours (i.e. T = 24) and the lengthof each time interval t ∈ T is 1 hour. We assume that eachconsumer (home) has between 10 to 20 essential applianceswith strict energy consumption scheduling constraints. Suchappliances may include electric stoves (daily usage E: 1.89kWh for self-cleaning and 2.01 kWh for regular) and lighting(daily usage for 10 standard bulbs E: 1.00 kWh) [26]. Eachhome also has between 10 to 20 appliances with shiftableand throttleable operations. These are the appliances that thesmart meter can schedule due to soft energy consumptionscheduling constraints. The shiftable appliances may includedishwashers (daily usage E: 1.44 kWh, [D, U ] = [18, 22]),washing machines (daily usage E: 1.49 kWh for energy-start,1.94 kWh for regular, [D, U ] = [18, 24]), clothes dryers (dailyusage E: 2.50 kWh, [D, U ] = [1, 8]), and PHEVs (daily usageE: 9.9 kWh, [D, U ] = [1, 8]) [26]. The throttleable appliancesmay include the air-conditioning unit (with a preferred andtolerance demands of 750W and 600W, respectively) and therefrigerator (with preferred and tolerance demands of 140Wand 112W respectively). In addition, for a comparison with[26], we adopt the same pricing model. The parameters forthe pricing scheme are bt = c = 0 for all t ∈ T and at = 0.3cents during daytime hours, i.e., from 8:00 in the morning to12:00 at night and at = 0.2 cents during the night, i.e., from12:00 at night to 8:00 AM the day after.
A. Performance Analysis for Different Scheduling Schemes
In this subsection, we seek to compare the performancesof the different scheduling schemes. The common compara-tor is the energy cost involved in each of the scheme.In addition to the four defined schemes from the sec-tion before, a baseline scheme (no-scheduling) where noscheduling of appliances is carried out and the scheme pro-posed in [26], Automatic Demand-Side Management (ADM)are compared. As a summary, the four previously definedschemes are Centralized-Nonprefer, Distributed-Nonprefer,Centralized-prefer and Distributed-prefer. In the Prefer cases,we set the weights for the consumer’s usage preference asΩ1 = 1,Ω2 = 600,Ω3 = 1. We will evaluate the performanceof the proposed scheduling scheme under different combina-tions of the weights in the next subsection.
Fig. 5 and Fig. 6 show the energy costs in the differentschemes with Γmax = 2 and Γmax = 4 respectively. Itis shown that the energy costs achieved by the proposedschemes (Centralized-Nonprefer and Centralized-Prefer) areless than those of the baseline schemes of no-scheduling andADM [26]. It is obvious that there should be cost savingswhen we adopt scheduling strategies for appliances. Thosecosts savings advantage over ADM is attributable to the factthat the proposed schemes are not only able to schedule theshiftable appliances to the low electricity price periods, butalso reduce the power draw of the throttleable appliances whilethe price is high. We also note that the energy cost in theCentralized-Prefer case is higher than that in the Centralized-Nonprefer case. That is, the additional objectives involved in
10
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
Number of Consumers
Ene
rgy
Cos
t (D
olla
rs)
Centralized−NonpreferCentralized−PreferADMNo−schedulingDistributed−NonpreferDistributed−Prefer
Fig. 5. Energy cost in terms of different energy scheduling schemes withΓmax = 2.
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
Number of Consumers
Ene
rgy
Cos
t (D
olla
rs)
Centralized−NonpreferCentralized−PreferADMNo−schedulingDistributed−NonpreferDistributed−Prefer
Fig. 6. Energy cost in terms of different energy scheduling schemes withΓmax = 4.
the Centralized-Prefer case results in the reduction of costsavings to improve consumers’ satisfaction through reductionin controlling of shiftable and throttleable appliances duringhigh price periods. This translates to shorter operation delaysand lower energy gaps than that in the Centralized-Nonpreferscheme as shown in Fig. 7 and Fig. 8, respectively. Moreover,we observe that the energy costs in the distributed schedulingschemes are close but higher than that in the centralizedscheduling cases. That is, the distributed schemes only use thelocal information to obtain the optimal energy schedule whichmay lead to the suboptimal schedule for each of the consumer.In contrast, centralized schemes are able to achieve optimalscheduling of energy consumption through the knowledge ofglobal information.
In addition, Fig. 5 and Fig. 6 show that the energy cost inΓmax = 2 scenario is higher than that in Γmax = 4 scenario.This is because a lower Γmax indicates the stricter constraintof energy consumption allowable for consumers in each hour.In this case, some cost reduction driven operations in highprice periods may be unable to shift to lower periods due toa lower peak allowed for electricity consumption.
Next, we show the resulting PAR and daily cost for 10
0 1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
30
35
40
45
Number of Cosumers
Ope
ratio
n D
elay
(ho
urs)
Centralized−NonpreferCentralized−Prefer
Fig. 7. The operation delay in both of the Centralized-Nonprefer and theCentralized-prefer scenarios with Γmax = 4.
0 1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
30
35
40
45
Number of Consumers
Ene
rgy
Gap
(kW
)
Centralized−NonpreferCentralized−Prefer
Fig. 8. The Energy gap in both of the Centralized-Nonprefer and Centralized-prefer scenarios with Γmax = 4.
TABLE IITHE RESULTING PAR AND DAILY COST FOR 10 CONSUMERS.
Schemes PAR Daily Cost(dollars)
No-scheduling 2.7 63.55ADM [26] 2.4 50.83Centralized-Prefer 2.2 42.47Centralized-NonPrefer 2.1 34.05Distributed-Prefer 2.5 44.39Distributed -NonPrefer 2.4 38.72
consumers for all compared schemes in Table II. The proposedcentralized schemes can achieve a lower cost and lower PARthan the ADM scheme. That is, the centralized scheme canobtain better scheduling performance by knowing the globalinformation of all users. Then, for the proposed distributedschemes, we can see that the distributed-nonprefer scheme hasthe same PAR as the ADM scheme (but lower cost) while thedistributed-prefer scheme has slightly higher PAR as comparedto ADM. This indicates that the proposed distributed schemewithout considering the user’s preference can obtain betterperformance as ADM (same PAR but lower cost). In the casewhen the user’s preference is considered, the peak loads mayincrease which leads to the higher PAR.
11
B. Performance Analysis with Scheduling Weight
When considering schedules that account for user pref-erences, the performance of the appliance schedule willbe influenced by the different combination of the weights(Ω1,Ω2,Ω3). Given Γmax = 4, to illustrate the influences, wedefine four weight combinations as: balanced criteria (Ω1 =1,Ω2 = 1,Ω3 = 1), cost-sensitive criteria (Ω1 = 600,Ω2 =1,Ω3 = 1), delay-sensitive criteria (Ω1 = 1,Ω2 = 600,Ω3 =1) and gap-sensitive criteria (Ω1 = 1,Ω2 = 1,Ω3 = 600).
Fig. 9 shows the energy cost in terms of the different weight-ing criteria of consumer preferences. Note that the energy costunder the balanced criteria and cost-sensitive criteria are nearlythe same. This indicates that the Ω1 is dominant of energycost in (17). However, in the delay-sensitive and gap-sensitivecases, the energy cost is higher than that in the former twocases. It is important to note that the consumer will incurhigher energy costs than balanced and cost-sensitive criteria ifthey emphasize the need to minimize the operation delay (withlarger Ω2) or to reduce the energy gap (with larger Ω3). Also,we observe that the energy cost in the delay-sensitive criteriais higher than that in the gap-sensitive criteria. This is becausethe shiftable appliances has higher daily energy usage than thatof the throttleable appliances in our simulation settings. Hence,when consumers focus on minimizing the operation delay ofthe shiftable appliances, more energy cost will be caused thanthat in the gap-sensitive case in which consumers devote toreduce the energy gap of the throttleable appliances.
The operation delay and energy gap in terms of differentweight criteria are shown in Fig. 10 and Fig. 11, respectively.In Fig. 10, the operation delay has the lowest value inthe delay-sensitive criteria when that delay minimization isthe dominant criterion in the multiple objective optimizationproblem (17). Similarly, in Fig. 11, the energy gap can achievethe minimum value under the gap-sensitive criteria (17).
1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
30
35
40
45
Number of Cosumers
Ene
rgy
Cos
t (D
olla
rs)
Balanced criteriaDelay−sensitive criteriaGap−sensitive criteriaCost−sensitive criteria
Fig. 9. Energy cost in terms of different combinations of the weight.
C. Convergence of Distributed Algorithms
The total energy costs while Algorithm 1 and Algorithm 2proceed along their distributed iterations are shown in Fig. 12and Fig. 13, respectively. In Fig. 12, the distributed iterationsare operated in terms of different values of Γmax, whereΓmax = 2 and Γmax = 4. We observe that when Algorithm
1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
30
35
40
45
Number of Cosumers
Ope
ratio
n D
elay
(ho
urs)
Balanced criteriaDelay−sensitive criteriaGap−sensitive criteriaCost−sensitive criteria
Fig. 10. The operation delay in terms of different combinations of the weight.
1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
30
35
40
Number of Cosumers
Ene
rgy
Gap
(kW
)
Balanced criteriaDelay−sensitive criteriaGap−sensitive criteriaCost−sensitive criteria
Fig. 11. The energy gap in terms of different combinations of the weight.
1 is running, the energy cost decreases until the algorithmconverges after about 22 iterations. Similar trends can berevealed as the consumers and RS jointly run Algorithm 2 asshown in Fig. 13. Also, in both of the Γmax = 2 and Γmax = 4cases, it is seen that the proposed distributed Algorithm 2converges quickly. Steady state is also reached after around22 iterations when the energy cost is minimized.
VI. CONCLUSION AND FUTURE WORK
With the constraint of PAR, we presented a demand-sidemanagement system based on energy consumption scheduling
0 5 10 15 20 25 30 3535
40
45
50
55
60
65
Iteration Number
Ene
rgy
Cos
t (D
olla
r)
No−scheduling, Γmax
=2
No−scheduling, Γmax
=4
Distributed Algorithm 1, Γmax
=2
Distributed Algorithm 1, Γmax
=4
CompleteConvergence
Fig. 12. The energy cost along the iterations of Algorithm 1.
12
0 5 10 15 20 25 30 35 4044
46
48
50
52
54
56
58
60
62
64
Iteration Number
Ene
rgy
Cos
t (D
olla
r)
No−scheduling, Γmax
=2
No−scheduling, Γmax
=4
Distributed Algorithm 2, Γmax
=2
Distributed Algorithm 2, Γmax
=4
Complete Convergence
Fig. 13. The energy cost along the iterations of Algorithm 2.
for the residential smart grid. By scheduling of shiftableappliances and adjusting the power of throttleable appliances,the residential scheduler is able to minimize the energy costaccording to a multi-tier and time-varying pricing structure.Taking the consumer’s preferences into account, we alsoextended our scheduling scheme to consider instances wherethe shiftable appliances and the throttleable appliances havespecific operational periods and preferred power consumptionlevels. It can also be applied by the consumer to balance thetradeoff between appliance usage preferences and economicbenefits. Moreover, distributed algorithms are proposed tosolve the energy cost minimization problem. By requiringlimited message exchanges between consumers and residentialscheduler, the proposed distributed algorithms do not need toreveal the consumers’ energy consumption profile, which isimportant in protecting the privacy of consumers.
In the future, we plan to extend our work in several direc-tions. First, we can design dynamic pricing or other incentiveschemes that can better manage the demand response. Second,our model system can be extended to scenarios where thestability of the power supply may be influenced by renewableenergy sources in the system. In that case, new constraintscan be added in the optimization problem to cover the un-stable and intermittent characteristics of renewable energysources. Finally, we intend to design a real-time distributedenergy management strategy for consumers to schedule theirconsumption individually when they only know the localknowledge of current demand conditions and the providedprice plan.
APPENDIX APROOF OF THEOREM 1
Theorem 1. If Lt =∑
n∈N (∑
i∈Inetn,i +
∑s∈Sn
etn,s +∑
r∈Rnetn,r) and E = 1
T (Ei +∑
n∈N∑
s∈SnEn,s +
T∑
n∈N∑
r∈Rnppre
n,r ), the problem (P1) can be rewritten as
minen
T∑t=1
< (Lt)
s.t. Lt ≤ ΓmaxE, t = 1, · · · , T,
en ∈ En, n = 1, · · · , N.
(24)
Proof: Since Lt =∑
n∈N (∑
i∈Inetn,i +
∑s∈Sn
etn,s +∑
r∈Rnetn,r), the last constraint in (P1) can be written as
maxt∈T
Lt ≤ Γmax1T
∑
t∈T
Lt
=1T
∑
t∈T
∑
n∈N(∑
i∈In
etn,i +
∑
s∈Sn
etn,s +
∑
r∈Rn
etn,r).
We observe that the following relations hold,∑
t∈T
∑
n∈N
∑
i∈In
etn,i = Ei,
∑
t∈T
∑
n∈N
∑
s∈Sn
etn,s =
∑
n∈N
∑
s∈Sn
En,s,
∑
t∈T
∑
n∈N
∑
r∈Rn
etn,r ≤ T
∑
n∈N
∑
r∈Rn
ppren,r .
Defining E as E = 1T (Ei +
∑n∈N
∑s∈Sn
En,s +T
∑n∈N
∑r∈Rn
ppren,r ), we have
maxt∈T
Lt ≤ ΓmaxE.
Moreover, it is obvious that the following inequality holds:
Lt ≤ maxt∈T
Lt ≤ ΓmaxE, t = 1, · · · , T.
Finally, by substituting the inequality in (24), the result fol-lows.
APPENDIX BDERIVATION OF THE µ ITERATION
Let xt denote the tth column of A′. Since Q is symmetricand positive definite, the tth diagonal element of P, givenby ptt = x′tQ
−1xt, is positive. This means that for every t,the dual function is strictly convex along the tth coordinate.Therefore, the strict convexity is satisfied and it is possibleto use the coordinate ascent method [36]. Because the dualfunction is quadratic, the minimization with respect to µ alongeach coordinate can be done analytically, and the iteration canbe written explicitly.
The first partial derivation of the dual function with respectto µt is given by
bt +T∑
k=1
ptkµk,
where ptk are the corresponding elements of the matrix P.Setting the derivative to zero, we see that the unconstrainedminimum of the dual along the tth coordinate starting from µis attained at µt given by
µt =− 1ptt
bt +
T∑
k 6=t
ptkµk
= µt − 1
ptt
(bt +
T∑
k=1
ptkµk
).
Taking into account the nonnegativity constraint µt ≥ 0, wesee that the tth coordinate update has the form
µt , max0, µt = max
0, µt − 1
ptt
(bt +
T∑
k=1
ptkµk
).
13
APPENDIX CPROOF OF THEOREM 3
Theorem 3. The proposed Algorithm 2 will converge ifevery φt(l) and dual optimal solution φ∗ have
‖ φt(l + 1)− φ∗ ‖<‖ φt(l)− φ∗ ‖,for all step size λl such that
0 < λl <2q(φ∗)− q(φt(l))‖ Lt(l)− ΓmaxE ‖2 (25)
Proof: Let gl = Lt(l)− ΓmaxE, we have
‖ φt(l) + λlgl − φ∗ ‖2=‖ φt(l)− φ∗ ‖2 −2λl(φ∗ − φt(l))′gl + (λl)2 ‖ gl ‖2,
and by using the subgradient inequality,
(φ∗ − φt(l))′gl ≥ q(φ∗)− q(φt(l)),
we obtain
‖ φt(l) + λlgl − φ∗ ‖2≤‖ φt(l)− φ∗ ‖2 −2λl
(q(φ∗)− q(φt(l))
)′gl + (λl)2 ‖ gl ‖2,
We can verify that for the range of step size of (25) the sum oflast two terms in the above relation is negative. In particular,with a straightforward calculation, we can write this relationas
‖ φt(l) + λlgl − φ∗ ‖2
≤‖ φt(l)− φ∗ ‖2 −γ(l)(2− γ(l)
)(q(φ∗)− q(φt(l))
)′‖ gl ‖2 ,
(26)
where γ(l) = λl‖gl‖2q(φ∗)−q(φt(l))
. If the step size λl satisfies (25),then 0 < γ(l) < 2, so (26) yields
‖ φt(l) + λlgl − φ∗ ‖<‖ φt(l)− φ∗ ‖ .
We now note that since φ∗ > 0 and the projection operationis non-expansive [36], we have
‖ φt(l) + λlgl+ − φ∗ ‖<‖ φt(l) + λlgl − φ∗ ‖ .
Because φt(l + 1) = φt(l) + λlgl+, by combining the lasttwo inequalities, the result follows.
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Yi Liu received his Ph.D. degree from South ChinaUniversity of Technology (SCUT), Guangzhou,China, in 2011. After that, he worked in the Instituteof Intelligent Information Processing at GuangdongUniversity of Technology (GDUT). In 2011, hejoined the Singapore University of Technology andDesign (SUTD) as a post-doctoral. His research in-terests include cognitive radio networks, cooperativecommunications, smart grid and intelligent signalprocessing.
Chau Yuen received the B.Eng. and Ph.D. degreesfrom Nanyang Technological University, Singapore,in 2000 and 2004, respectively. He was a Post DocFellow at Lucent Technologies Bell Labs (MurrayHill) during 2005, and a Visiting Assistant Professorof Hong Kong Polytechnic University in 2008. From2006-2010, he worked at the Institute for InfocommResearch (Singapore) as a Senior Research Engi-neer. He has published over 150 research papersin international journals or conferences. His presentresearch interests include green communications,
massive MIMO, Internet-of-thing, machine-to-machine, network coding, anddistributed storage. He joined Singapore University of Technology and Designas an assistant professor in June 2010. He also serves as an Associate Editorfor the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY.
Shisheng Huang received his B.Eng. from theNational University of Singapore in 2006 and sub-sequently, his Ph.D. from Purdue University, WestLafayette, IN in 2011. Currently, he is a researchfellow with the SUTD-MIT International DesignCentre. His research focuses on systems analysisusing simulation and modelling approaches; with ap-plications in energy systems, transportation systemsand the smart grid.
Naveed UL Hassan received the B.E. degree inAvionics Engineering from College of Aeronauti-cal Engineering, Pakistan in 2002. In 2006 and2010 he received M.S. and Ph.D. degrees both inelectrical engineering with specialization in digitaland wireless communications from Ecole SuperieuredElectricite (Supelec), France. Since 2011, he isworking as an Assistant Professor in the departmentof Electrical Engineering at Lahore University ofManagement Sciences (LUMS), Pakistan. He was avisiting Assistant Professor at Singapore University
of Technology and Design (SUTD) in 2012 and 2013. He has severalyears of research experience and has authored/co-authored several researchpapers in refereed international journals and conferences. His major researchinterests include cross layer design and resource optimization in wirelessnetworks, demand response management in smart grids, indoor localizationand heterogeneous networks.
Xiumin Wang received her B.S. from the Depart-ment of Computer Science, Anhui Normal Univer-sity, China, in 2006. She received her Ph.D. degreefrom both the Department of Computer Science,University of Science and Technology of China,Hefei, China, and the Department of Computer Sci-ence, City University of Hong Kong, under a jointPhD program. She is currently an Assistant Professorin School of Computer and Information, Hefei Uni-versity of Technology. Her research interests includewireless networks, resource allocation, and network
coding.
Shengli Xie received the M.S. degree in mathemat-ics from Central China Normal University, Wuhan,China, in 1992, and the Ph.D. degree in automaticcontrol from the South China University of Tech-nology, Guangzhou, China, in 1997. He was a vicedean with the School of Electronics and InformationEngineering, South China Univeristy of Technology,China, from 2006 to 2010. Currently, he is the Di-rector of both the Institute of Intelligent InformationProcessing (LI2P) and Guangdong key Laboratory ofInformation Technology for the Internet of Things,
and also a professor with the School of Automation, Guangdong Universityof Technology, Guangzhou, China. He has authored or co-authored fourmonographs and more than 100 scientific papers published in journalsand conference proceedings, and was granted more than 30 patents. Hisresearch interests broadly include statistical signal processing and wirelesscommunications, with an emphasis on blind signal processing and Internet ofthings.