770 ieee transactions on smart grid, vol. 3, no. 2, …770 ieee transactions on smart grid, vol. 3,...

770 IEEE TRANSACTIONS ON SMART GRID, VOL. 3, NO. 2, JUNE 2012

Residential Load Control: Distributed Scheduling andConvergence With Lost AMI MessagesNikolaos Gatsis, Student Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE

Abstract—This paper deals with load control in a multiple-res-idence setup. The utility company adopts a cost function repre-senting the cost of providing energy to end-users. Each residentialend-user has a base load, two types of adjustable loads, and pos-sibly a storage device. The first load type must consume a specifiedamount of energy over the scheduling horizon, but the consump-tion can be adjusted across different slots. The second type doesnot entail a total energy requirement, but operation away froma user-specified level results in user dissatisfaction. The researchissue amounts to minimizing the electricity provider cost plus thetotal user dissatisfaction, subject to the individual constraints ofthe loads. The problem can be solved by a distributed subgradientmethod. The utility company and the end-users exchange infor-mation through the Advanced Metering Infrastructure (AMI)—atwo-way communication network—in order to converge to theoptimal amount of electricity production and the optimal powerconsumption schedule. The algorithm finds near-optimal sched-ules even when AMI messages are lost, which can happen in thepresence of malfunctions or noise in the communications network.The algorithm amounts to a subgradient iteration with outdatedLagrange multipliers, for which convergence results of wide scopeare established.

Index Terms—Advanced metering infrastructure, demand-sidemanagement, distributed algorithms, energy consumption sched-uling, smart grid.

I. INTRODUCTION

D EMAND-SIDE management (DSM) is instrumentalin transforming today’s aging power grid into a more

reliable and economically operated smart grid. DSM manifestschanges in electric usage by the consumers [1], and has amuch needed positive impact towards smoothing out the peakdemand, increasing the system reliability, reducing genera-tion cost—especially at peak times—and meeting pollutionmandates.DSM can be effected by load control in response to smart

time-based, or time-varying, pricing schemes. These schemesare judiciously controlled by the utility companies to elicitdesirable energy usage. Load control through pricing has alsobeen termed demand response or load response, among others;see, e.g., [2], [3]. Residential loads have the potential to offersignificant benefits to this end, because they consist of loadsthat can for instance be adjusted—e.g., an air conditioningunit (A/C)—or be deferred for later. The advent of smart grid

Manuscript received May 22, 2011; revised September 11, 2011; acceptedOctober 17, 2011. Date of publication March 13, 2012; date of current ver-sion May 21, 2012. This work was supported by NSF grants CCF-1016605 andECCS-1002180; and grant NPRP 09-341-2-128. The material in this paper waspresented in part at the 45th Annual Conference on Information Sciences andSystems, Baltimore, MD, March 2011. Paper no. TSG-00186-2011.The authors are with the Department of Electrical and Computer Engineering,

University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSG.2011.2176518

Fig. 1. Electricity distribution network and AMI.

technologies has also made available energy storage devices(batteries) at the residential level [4]. These can be charged anddischarged throughout the day according to residential needs,and constitute an additional device for load control.DSM is facilitated by deployment of the advanced metering

infrastructure (AMI), which comprises a two-way communica-tion network between utility companies and end-users (Fig. 1)[5], [6]. Smart meters installed at end-users’ premises are theAMI terminals at the end-users’ side. These measure not justthe total power consumption, but also the power consumptionprofile throughout the day, and report it to the utility companyat regular time intervals. The utility company sends pricing sig-nals to the smart meters through the AMI, for the smart metersto adjust the power consumption profile of the various residen-tial electric devices, in order to minimize the electricity bill andmaximize the end-user satisfaction.This paper deals with optimal energy scheduling for load con-

trol of multiple residential loads, which comprise various typesof devices. Economical operation of the utility company is ac-counted for. Distributed algorithmswhich leverage the AMI andhave guaranteed convergence to optimal schedules even underAMI outages are also developed.Energy scheduling problems for multiple residences consider

end-users jointly maximizing the satisfaction from power con-sumption offset by the total cost of electricity from all resi-dences, which is the social welfare [7], [8]. The cost of elec-tricity and the power requests are known ahead of the schedulinghorizon. In the aforementioned setup, the cost of electricity isminimized without accounting for user satisfaction in [9], whilean extension where users have limited knowledge about theother users’ power requests is studied in [10]. Energy sched-uling with distributed storage in a game-theoretic (multiuser)setup is pursued in [11], where each end-user maximizes its in-dividual welfare. Scheduling for a single end-user with storageis addressed in [12]–[14].The present work deals with social welfare maximization for

energy scheduling between a utility company and residentialend-users. It differs from [7]–[10], which focus on a single typeof adjustable load, and typically assume convenient forms ofobjective functions, such as strictly convex or differentiable.Any convex objective can be accommodated in the presentwork—not necessarily strictly convex—such as piecewise

1949-3053/$31.00 © 2012 IEEE

GATSIS AND GIANNAKIS: RESIDENTIAL LOAD CONTROL: DISTRIBUTED SCHEDULING AND CONVERGENCE 771

linear cost of electricity. Each residential end-user has twoclasses of adjustable loads, as well as a storage device. The firstclass must consume a specified total amount of energy over thescheduling horizon, but the consumption can be adjusted acrossdifferent slots. An example is plug-in hybrid electric vehicle(PHEV) charging. The second class has adjustable powerconsumption without a total energy requirement, but operationof the load at reduced power results in dissatisfaction of theend-user, with A/C being an example. These two classes reflectthe two types of load that can offer residential load response,and it is therefore important to be captured jointly.The resulting optimization problem is solved through a dis-

tributed subgradient algorithm. The algorithm entails exchangeof information among the utility company and the end-users,and has similar communication requirements as the ones in [7]and [8]. The utility company sends out Lagrange multipliers,which are readily interpreted as pricing signals, and eachresidence sends back total hourly energy consumption—butnot individual appliance consumption. It is established that thedistributed algorithm converges to optimal schedules, even ifthere are lost AMI messages in any of the two directions. Inorder to establish this result, the overall algorithm is cast ina very general setup as an asynchronous subgradient methodwith outdated Lagrange multipliers. General convergenceresults are established for the Lagrange multipliers and primaloptimization variables, contributing to the related optimizationliterature [15]–[17]. A related asynchronous algorithm has beendeveloped for wireless networking [18], but the convergenceresults in the present work are more general.The rest of this paper is organized as follows. Section II

lays out the multiple-residence load scheduling problem. Adistributed iterative solver is developed in Section III, andits convergence is established under lost AMI messages.Section IV presents numerical tests, and Section V concludesthis paper with pointers to future directions.

II. COOPERATIVE LOAD CONTROL FORMULATION

Consider residences provided with electricity from thesame utility company. Each residence has a smart meter thatcommunicates with the various devices per residence, and alsowith the utility company through the AMI. It is supposed thatthe cost structure of energy provided by the utility company isdetermined in advance for a given time period . Eachtime slot of the scheduling horizon can represent, e.g., one hour,with corresponding to one day.

A. Residential Appliances

A base residential nondeferrable load is considered alongwith two classes of devices with adjustable power, denotedrespectively as and , where indexesthe residences. The base load at slot is denoted by

, and can be, e.g., lights or computers.The devices of classes and are generically indexed

by , and denotes the power consumption of device overslot . Note that the term power here represents essentially con-sumption over the fixed duration of the slot, and therefore hasunits of kilowatt hours. The particular characteristics of thoseclasses are as follows:• Class contains devices with a prescribed energyrequirement that has to be completed between slots

Fig. 2. (a) Example of disutility function . The feasible range is, while the value represents a desirable set point with min-

imal dissatisfaction. (b) Example of cost function . The cost is piecewiselinear, consisting of line segments, with slopes , and breakpoints

.

(start time) and (termination time). An exampleis PHEV charging, where the user may specify the chargingto start, e.g., at midnight, and finish by a morning hour.Power consumption vectorsacross slots are constrained to be in the set

(1)

The sought power consumption variables belonging to theinterval (with ) for each slot inorder to meet over the horizon, will be the result ofthe optimization formulation.

• Class includes devices operating with power in, but without a total energy requirement.

Instead, a disutility function is introduced tocapture dissatisfaction of the end-user for operating awayfrom a nominal point. The premise is that the end-usermay choose to operate away from a nominal point, ifthis can reduce the electricity bill, as determined by theoptimization formulation. An example from this class is anA/C unit. The disutility function is selected to be convex,and may vary with time to reflect the variable importanceof operating the device across time. A disutility functionexample is illustrated in Fig. 2(a). The power consumptionvectors for class 2 devices areconstrained to be in the set

(2)

When a time range over which the devicewill be operated is given, it holds that for

or .The notation collects the power

consumptions of all devices of residence . Note that the fea-sible set where belongs to is convex.

B. Residential Storage Device Model

Residence is also allowed to have a battery. Let, be the state of charge of the battery (i.e., energy


available) at the end of slot ; and the capacity of the bat-tery, so that . The available energy at the beginningof the horizon is denoted by .The battery can be either charged or discharged during slot. In the latter case, the energy stored in the battery is suppliedto the residential appliances. The optimal decision will be theresult of the optimization formulation. Let ,denote the energy drawn from or provided to the battery at slot, where in the former case, and in the lattercase. The charge/discharge variables as well as the energy storedin the battery are related by the dynamical equation

(3)

Variables are constrained in three different ways:i) Variables are limited by maximum charge anddischarge rates, i.e., .

ii) The battery-supplied energy is no more than the currentenergy consumption, i.e., for all and, where the summation over includes the base load

and all devices of classes 1 and 2 of end-user .iii) Each battery has efficiency , meaning that if

is stored at the end of slot , the discharge at slotis limited by .

The set of feasible per residence isgiven by the stated constraints collected in the following set:

(4)

In (4), a constraint for the final state of charge , whichis available at the beginning of the next horizon, has beenincluded; while the initial state is known. The ’s and

’s are coupled for end-user through the constraint, and this is indicated by the dependence of

on . Note also that is a convex set.

C. Social Welfare Maximization

Let denote the pricing function representing the costof electricity over slot . This is the cost that the utility incursin order to provide electricity to the end-user. For instance, thiscost can be determined by the company’s bidding to the whole-sale market, or it can represent distribution network operatingcosts; see, e.g., [6]–[10], [19], [20].Consider also a nonadjustable base load from other end-

users in the system—due to, e.g., end-users not participatingin the DSM program, or a commercial load. Then, the cost ofelectricity over slot is given by, where the summation over is over all residences. Letdenote a variable upper-bounding, which is interpreted as the power provided by the utility

company, and define . Let also and collectall and , respectively, for all . Recall that andfor all and are not optimization variables, and hence theyare not included in or . The multiresidential load control

task amounts to minimizing the total cost of electricity as wellas the total dissatisfaction of the end-users, that is,

(5a)

(5b)

(5c)

(5d)

A constraint ensuring a safety cap upon the total powerconsumption has been introduced in (5c), taking into accountsecurity and reliability considerations for the distribution net-work from the utility company to the residences. Note that themodel includes as special case the situation where there are nobatteries at all; in this case, variables and as well as theconstraint will simply not be present.The function is chosen to be convex, continuous, and

strictly increasing (see Fig. 2(b) for an example). Because of thelatter, (5b) holds with equality at the optimal point, matchingthe power supply with the demand. Moreover, (5) is a convexoptimization problem. It is supposed that there are feasible, and , so that (5b) holds as strict inequality. This is the

standard Slater’s constraint qualification, ensuring zero dualitygap and existence of optimal Lagrange multipliers.The objective in (5a) is the opposite of social welfare. There-

fore, problem (5) amounts to maximizing social welfare of end-users and the utility company. Pricing interpretations related to(5) are described in the ensuing subsection, while a sensitivityanalysis of the social welfare with respect to the parameters ofthe cost function is provided in Appendix A.

D. Lagrangian Duality and Economic Interpretation

Let denote the Lagrange multiplier corresponding to (5b),and . Then, keeping the rest of the constraintsimplicit, the Lagrangian function for (5) is

(6)

The dual function and the dual problem take the form

(7)

(8)

Because problem (5) is convex and has zero duality gap, stan-dard duality theory can be used to interpret how the Lagrangemultipliers act as pricing signals coordinating the utility com-pany with the end-users; see also [21, Sec. 5.1.6] describing La-grange multipliers as a coordination mechanism and [7] for arelated DSM application.


Specifically, let , and denote the optimal solution of(5), and the optimal Lagrange multipliers corresponding tothe dual solution [cf. (8)]. Variables , and are also theminimizers in (7) for , i.e., ;see, e.g., [22, Prop. 6.2.5]. Using the fact that the primal anddual optimal values are the same, the following holds:

(9)

where the last equality follows upon straightforward rearrange-ments of the terms in the Lagrangian function.The Lagrange multiplier can be interpreted as the price

charged from the utility company at slot . The term “net costfor utility” in (9) represents the cost the utility incurs to pro-vide electricity, minus the revenue from selling this electricityto end-users. Therefore, the units of Lagrange multipliers canbe interpreted to be monetary units/kWh. The term “aggregatecost for residence ” in (9) represents the payment to the utilitycompany plus the disutility experiencedby the end-user. It is instrumental in this interpretation to recallthat the total power consumed at slot , namely,

, is exactly the electricity provided by the utility, ;i.e., (5b) holds as equality.

III. LOAD CONTROL ALGORITHM

This section develops a distributed algorithm for solving (5).From a high-level view, simple optimization tasks are assignedto the residences and the utility company, which are coordinatedthroughAMI signaling to obtain the jointly optimal schedule. Tothis end, the distributed iterations are described in Section III-A,and their convergence in the presence of lost AMI messages isestablished in Section III-B.

A. Distributed Scheme

The multiresidence load control problem (5) is separable,meaning that the objective (5a) and the constraint (5b) com-prise sums of functions that depend only on some (but not all)optimization variables. In particular, there are groups ofvariables, namely, and the power consumption variablesand for each residence . This per-residence separabilityimplies that the problem is amenable to dual decompositionthrough the subgradient method developed next; see also [21,Sec. 5.1.6] on separable problems, and [22, Sec. 8.2] for thesubgradient method.1) Subgradient Iterations: The Lagrangian minimization

(7)—which defines the dual function and is also part of thesubgradient iterations—is easily seen to be decomposable

Fig. 3. Exchange of information between utility company and residence .

to individual minimizations with respect to , and also withrespect to the and for each residence . Specifically,the subgradient method consists of the following iterations, in-dexed by and initialized with arbitrary :

(10a)

(10b)

(10c)

where is the stepsize, and . Recall alsothat the summation over involves the base load (for which

), and all appliances at residence .The form of the updates (10) readily suggests a distributed

implementation. In particular, the utility broadcasts at every it-eration the value of the Lagrange multipliers toall residential smart meters through the AMI. These Lagrangemultipliers are needed to solve (10b) with respect to andat the smart meter of residence . Each residence sends backto the utility company the values of ,which correspond to the total power consumption per hour. Inthis way, residences do not reveal the individual appliance con-sumption or battery storage profiles. The exchange of informa-tion is illustrated in Fig. 3.The minimization (10a) with respect to takes place at the

utility company at every iteration . The Lagrange multiplier up-dates (10c) also take place at the utility, which has knowledge ofthe nonadjustable load , and then the process is repeated. Thedistributed load control algorithm runs ahead of the schedulinghorizon. If the parameters change at a slot during the horizon,then problem (5) with the new parameters can be solved for theremaining slots.The minimizations in (10a) and (10b) are elaborated next.2) Lagrangian Minimization: The problems in (10a) and

(10b) are convex. Therefore, they can be solved efficiently andlocally at the utility company or at the smart meter of residence. More details on the particular algorithms that can be used

are provided next.a) At the Utility Company: The minimization in (10a) in-

volves a single variable and a box constraint; therefore, it is easy


TABLE IPOWER CONSUMPTION SCHEDULING FOR CLASS 1 DEVICES

to solve in closed form. Specifically, if the cost function has aderivative with inverse , then the solution is

(11)

If the cost is piecewise linear as in Fig. 2(b), and without loss ofgenerality , then the solution is

(12)

b) At the Residential Smart Meter: In the absence of astorage element at residence , the minimization (10b) decou-ples into per device minimizations as follows:

(13a)

(13b)

The minimization in (13b) is analogous to the one in (10a). Theminimization in (13a) can be solved easily using the algorithmlisted in Table I.In the presence of a storage element, the minimization (10b)

couples all the devices and the storage element, due to the con-straint [cf. (4)]. Note that all constraintsare linear. The problem is convex, and can be solved by an in-terior point algorithm. In particular, it is a linear or a quadraticprogram, if the disutility function is respectively linear orquadratic.3) Convergence: Convergence of iterations (10) can be ob-

tained for the following three stepsize rules, each having desir-able properties that will be delineated shortly:

(S1) Constant stepsize:(S2) Nonsummable but square-summable stepsize: Thereexist sequences and such that (a)

; (b) as and; and (c) and .

(S3) Stepsize given by harmonic series: This is a specialcase of (S2) with , and

are constants.In order to recover the optimal primal variables from the sub-

gradient method, the following running averages of the iteratesand are formed alongside (10). These have con-

stant weights

(14)

(15)

or weights proportional to the stepsize

(16)

(17)

where . It is also useful to consider the dualaverage , in the same fashion as (14) or(15). It should be emphasized that the running averages can beefficiently computed in a recursive fashion, that is,

(18)

(19)

and similarly for , and .Convergence of the subgradient method with the stepsize

rules (S1)–(S3) has been studied in the literature. The re-lated results are summarized next for convenience. The exactstatements can follow as special cases of the statements inSubsection III.B, which deals with an asynchronous version ofthe algorithm.i) Stepsize (S1): The dual averages converge to a pointwith dual value within a ball of ; and the radius of theball is proportional to the stepsize; see, e.g., [23, Sec. 4]and references therein. The primal averages (14) and (15)become asymptotically feasible, and converge to a pointwhere the primal value is within a ball of the optimal [17].

ii) Stepsize (S2): The dual iterates converge to a dualoptimal solution [22, Prop. 8.2.6]. The primal averages(16) and (17) become asymptotically feasible and optimal[16, Th. 1].

iii) Stepsize (S3): All the results for stepsize (S2) hold inthis case. In addition, the primal averages (14) and (15)become asymptotically feasible and optimal [16, Th. 2].

Primal averaging is necessary if the primal objective is notstrictly convex. This can be the case here for two practically rel-evant reasons: a) the objective can include functions which arenot strictly convex in their argument, as with, e.g., a piecewiselinear [cf. Fig. 2(b)]; and b) the objective is not a func-tion of all optimization variables, namely, it does not involve the

’s for or the ’s.


TABLE IIEXAMPLE OF AMI MESSAGE PROGRESSION, WITH X DENOTING A LOSTMESSAGE. THE LAGRANGE MULTIPLIERS RECEIVED AT AND THE POWER

CONSUMPTION DATA TRANSMITTED FROM RESIDENCE ARE SHOWN, USINGTHE SHORTHAND NOTATION

The constant (S1) as well as the diminishing stepsize rules(S2) and (S3) have their own merits. Specifically, although con-vergence with constant stepsize is within a ball (near-optimal),it is typically faster than convergence with a diminishing step-size. Note further that the diminishing stepsize (S3) allows re-covery of the primal variables with the averaging scheme (14)and (15), which can be preferable for two reasons: a) it is thesimplest averaging scheme; and b) the scheme in (14) and (15)uses constant weights, and therefore has the potential to con-verge faster than the one in (16) and (17), which uses vanishingweights.

B. Convergence With Lost AMI Messages

The distributed scheduling scheme presented in the previoussubsection features two-way communication between the utilitycompany and the residential smart meters. In practice, the fol-lowing can happen to an AMI message, or

, at slot :• It might not be transmitted, e.g., if there is a malfunction.• It might not be received due to, e.g., noise in the commu-nications network, especially if there is a wireless networkconnecting the residential smart meters with the utilitycompany.

Both situations will be referred to as lost AMI messages, andare modeled next with outdated Lagrange multipliers. It turnsout that the algorithm still converges in this case.1) Outdated Lagrange Multipliers: Lost AMI messages

cause power consumption data used for theLagrange multiplier updates in (10c) to correspond to outdatedLagrange multipliers. Specifically, iterates cor-responding to the most recent Lagrange multipliers availableare used in (10c), as described next.• Suppose that the message for residence atslot is lost. Let be the index of the mostrecent Lagrange multiplier available. Then, the messagetransmitted to the utility company is

, which corresponds to the minimization ofthe Lagrangian in (10b) with instead of .See also Table II for an example.

• Suppose on the other hand that the messagefor residence at slot is received, but the transmittedmessage is lost. Then, the mostrecent received message will be used instead, which effec-tively corresponds to the Lagrangian minimizer at an out-dated Lagrange multiplier .

In a nutshell, if messages in either communicationdirection are lost, the algorithm is effectively using

asthe update direction in (10c). The function ,where , denotes the index of the Lagrangemultiplier with the following property: Lagrange multiplier

is used in the Lagrangian minimization at residenceto yield the variables and for the update of at

slot .Finally, the running averages (15) and (17) are formed by the

iterates and , which are actually used inthe Lagrange multiplier updates.2) Convergence: It turns out that the algorithm with outdated

multipliers converges, as shown next. In fact, the results areestablished in a generic setup, and are therefore useful moregenerally in optimization theory. The proofs as well as technicalelaborations are presented in the Appendix.The main results will be stated for the following proto-

type separable optimization problem with variables and:

(20a)

(20b)

(20c)

Variable corresponds to in (5), while collectsand . The association of functions and

, as well as of the sets and with theircorresponding particular cases in the load control problem (5)is straightforward.The Lagrangian function, the dual function, and the dual

problem corresponding to (20) are [cf. (6), (7), and (8)]

(21)

(22)

(23)

The subgradient method with outdated multipliers takes theform

(24)

where

(25a)

(25b)

with for all ; and for vectors, the oper-ator is applied entrywise. Iteration (24) constitutes anasynchronous subgradient algorithm, because the subgradient

consists of componentswhich correspond to “old” Lagrange multipliers. Iffor all and , then the algorithm becomes the standard(synchronous) subgradient method [cf. (10)].The running averages are formed by the available primal it-

erates used to update the Lagrange multipliers. Hence they are


as in (14) and (16) for the variable, while (15) and (17) arerespectively replaced by

(26)

(27)

Clearly, running averages (26) and (27) reduce to the ones in(15) and (17) when for all .The following two conditions are adopted for the generic

problem (20) and iterations (24).C1. Sets and are convex, closed, and bounded.Functions and are convexand continuous over their domain, which includes and

. There are and ,so that (20b) holds as strict inequality.C2. There exist integers , so that

The first condition asserts boundedness of the subgradients,and is clearly satisfied by the setup of the load control problem(5). More precisely, with denoting the Euclidean norm,there are finite bounds so that for alland ,

(28)

For the load control problem (5), such bounds are

(29)

(30)

where .The second condition states that the delay is upper

bounded by a finite number. Note that the condition allows forlost messages to happen infinitely often. In the ensuing proposi-tions, setting , yields the correspondingresults for the synchronous algorithm (cf. Section III-A3).In the following results, denotes

the distance of a point from a closed convex set . The firsttwo propositions deal with convergence of Lagrange multiplieriterates with constant and diminishing stepsizes, respectively.Proposition 1: Under C1, C2, and with stepsize (S1), the fol-

lowing hold:a) The sequence satisfies

(31)

b) Define the set of near-optimal dual solutions as

(32)

The sequence is bounded, and the sequencehas limit points, all of which are in the set .

Moreover, it holds that

(33)

The previous proposition states essentially that the dual aver-ages converge to a point which corresponds to a near-op-timal dual value. Proximity to the optimal value is quantifiedby , which decreases linearlywith the stepsize, and also decreases as the delays decrease.This “discrepancy” can be made as small as desired by choosinga sufficiently small stepsize .Proposition 2: Under C1 and C2, and with stepsize (S2) or

(S3), the sequence converges to an optimal dual solution.Proposition 2 asserts that the iterates converge to an

optimal Lagrange multiplier vector . This result extends [15,Prop. 6.1], which only deals with convergence of butnot of the iterates themselves.Interestingly, running averages of the primal variables also

converge under stepsizes (S1)–(S3), as asserted respectively inthe following three propositions.Proposition 3: Under C1, C2, and with stepsize (S1), the fol-

lowing hold:a) The primal averages in (14) and in (26)satisfy

i) (34)

ii)

(35)

iii) (36)

b) Define the set of near-optimal solutions of (20) as

(37)

The sequence has limit points,all of which are in the set . Moreover, it holds that

(38)


Fig. 4. Upper and lower limits for must-run residential load as a fraction of 3kWh; and commercial load as a fraction of 980 kWh.

The last proposition establishes that the primal averages be-come asymptotically feasible and asymptotically near-optimal.Specifically, the term inside the norm in part a)–i) is the vio-lation of the constraint ,and this converges to zero. Moreover, near-optimality is mea-sured by the quantity , whichdecreases with the stepsize and the delays . Note that thesame quantity appeared in Proposition 1, and hence, there is asymmetry between the primal and dual convergence results withconstant stepsize. Proposition 3 generalizes results of [17, Sec.4.2], which deals with a synchronous subgradient algorithm.Proposition 4: Under C1, C2, and with stepsize (S2), the fol-

lowing hold:a) The primal averages in (16) and in (27)satisfy

i) (39)

ii) (40)

b) The sequence has limit points.All such limit points are in the set of optimal solutions of(20), which is denoted by . Moreover, it holds that

(41)

The previous proposition reveals that the primal averageswith weights proportional to the stepsize will be asymptoti-cally feasible and asymptotically optimal. It extends [16, Th.1], which deals with a synchronous case.Proposition 5: Under C1, C2, and with stepsize (S3), all re-

sults of Proposition 4 hold for the primal averages in (16)and in (27). Moreover, the following also hold:

TABLE IIIPARAMETERS OF RESIDENTIAL DEVICES FOR TEST CASE 1. END-USERS 1–5HAVE A CLASS 1 DEVICE, AND END-USERS 1–3, 5, AND 6 HAVE A CLASS 2DEVICE. THE UNITS OF , AND ARE KILOWATT HOURS

a) The primal averages in (14) and in (26)satisfy

i) (42)

ii) (43)

b) The sequence has limit points,all of which are in the set (defined in Proposition 4).Moreover, it holds that

(44)

This proposition asserts that the primal averages with con-stant weights are asymptotically feasible and asymptotically op-timal. It extends [16, Th. 2], which deals with a synchronouscase.

IV. NUMERICAL TESTS

Two sets of numerical results are presented; one showingproperties of the optimal load control and scheduling in a6-user scenario (Section IV-A), and the other highlightingeconomic interpretations in a considerably larger scale scenario(Section IV-B).The time horizon in both tests is 24 h, corresponding to 8

A.M., 9 A.M., 10 A.M., etc., until 7 A.M. of the next day. Eachend-user has a base load , which is drawn randomly from auniform distribution with limits shown in Fig. 4 as fraction of3 kWh. The latter value corresponds to 50% of a typical peakresidential load of 6 kW [20, Figs. 2.5–2.7]. On top of the baseload there is adjustable residential load. The detailed setup andresults for each test case are described next.

A. Test Case 1: Load Control and Scheduling

For the scenario with 6 end-users, the characteristics of theresidential devices are listed in Table III. The class 1 devicecan be for instance a PHEV, which has battery of 10 kWh for a


Fig. 5. Schedule for the 6 end-users.

5-mile drive [24], and has to be charged during the night. A typ-ical value for maximum charging rate is 1.2 kW [25], giving amaximum hourly consumption of 1.2 kWh. The disutility func-tion selected for the class 2 device is .This can, e.g., be a room A/C unit that is to be operated duringthe day and in the evening, and has cooling capacity 10 000BTU and energy efficiency ratio (EER) 10 [26], entailing a max-imum hourly power consumption in the range of 1 kWh. The se-lected consumption parameters yield a daily residential powerconsumption in the order of 10 kWh to 30 kWh, which is rep-resentative of a household in the U.S. [27]. The cost functionis for all (similar to [9]), andkWh. There is no additional base load . The stepsize is

, which satisfies (S3), while averages (14) and (15)are used.1) Load Scheduling: The optimal schedules for the 6 resi-

dences are depicted in Fig. 5(a)–(f), and they verify the intu-ition. Specifically, the power consumption of the class 1 devicetakes its largest value between hours 1 A.M. and 5 A.M. for allend-users, in a fashion complementary to the base load. Thepower consumption for class 1 device of end-user 3 increasessharply after 10 P.M., as opposed to the more gradual increasefor the other end-users. The reason is that is the highestamong the ’s. It is also interesting to note that the powerconsumption of class 1 device for end-user 1 increases from 6A.M. to 7 A.M., while it is constrained to be zero at 7 A.M. for theremaining end-users (cf. Table III). It can be deduced that thedrop at hour 6 A.M. and then the rise at hour 7 A.M. for end-user1 helps to smooth out the total power consumption across allusers.2) Adding a Battery: Now suppose that end-user 1 also has

a storage device with capacity kWh as in [11],and remaining parameters kWh, kWh,

1 kWh, and 1 kWh.The total hourly power consumption for end-user 1 is shown

in Fig. 6. The battery is charged at the hours where the con-sumption is seen to have increased over the one without bat-tery. A clear effect of adding the storage device is that the peak

Fig. 6. Power consumption of end-user 1 with and without battery.

TABLE IVRESULTS FOR TEST CASE 1

power consumption of end-user 1 is reduced. Specifically, thebattery is discharged during the hours 6 P.M.–9 P.M., when thepeak demand occurs. Moreover, Table IV lists the optimal ob-jective value and its constituent costs. As the battery increasesthe scheduling degrees of freedom of the system, the resultingcosts are smaller.3) Asynchronicity Effects: Asynchronicity is now introduced

(in the setup without battery) by setting, for an integer . This has essen-

tially the effect that only the minimizers at slots


Fig. 7. Objective value evaluated at running averages,, for different delays.

TABLE VPARAMETERS OF RESIDENTIAL DEVICES FOR TEST CASE 2

, are used in the updates for all users (theunits of are slots). Therefore, condition C2 is satisfied. Set-ting recovers the synchronous case.The evolution of the objective value evaluated at the running

averages and is depicted in Fig. 7 for different valuesof . According to Proposition 5, convergence to the optimalvalue occurs under all delays, which is verified in the figure.Note though that convergence under larger delays in the sub-gradient updates is characterized by a higher overshoot and alarger lag.

B. Test Case 2: Scenario With Distribution System Data

A system with 420 end-users and an additional commercialbase load is tested. The hourly commercial load is depictedin Fig. 4, where the maximum corresponds to 980 kWh. The pa-rameters of the residential devices are listed in Table V, similarto the Test Case 1. Every end-user has a class 2 device, while80% of the end-users (randomly selected) have a class 1 device.Finally, the peak base residential load and the commercial loadcoincide at 8 pm (cf. [20, Fig. 2.7]).The parameters are selected as follows. The nominal total ac-

tive power load across the three phases in the IEEE 123-nodetest feeder is approximately 3500 kW [28]. Supposing a peakresidential load of 6 kW, 420 residential end-users correspond to72% of the peak load, while the remaining 28% is the peak com-mercial load of 980 kWh. Moreover, the class 2 device can bethought to represent an A/C unit. In this case, corresponds

Fig. 8. Optimal Lagrange multipliers.

Fig. 9. Total power consumption elicited from the optimal load control schemeand a fixed pricing scheme.

to roughly 40% of the nominal peak residential load of 6 kW.The disuitility function has the form .Moreover, the parameters , and , follow a typicaldaily profile [20, Fig. 2.4].The cost function is for all —if

the coefficient has units cents/(kWh) , then this costfunction gives an incremental cost of approximately 49 $/MWhat 3500 kWh, which falls in the range of values used in [6].The remaining parameters are kWh and

, while running averages (14) and (15) are used.The resulting optimal Lagrange multipliers are

shown in Fig. 8. It is interesting to observe that the magnitudeof the Lagrange multipliers follows the variation of the totalload across the 24 hours, which is depicted in Fig. 9 (thick graycurve). For example, they peak at 6 P.M., which is also whenthe highest power consumption occurs.The results of the proposed formulation are also compared

with a schemewhere the residential loads are not controlled, andthe residential power consumption is elicited by a flat pricingscheme. Specifically, with reference to (9), suppose thatare substituted by other quantities, , the same for all , playingthe role of prices. In this case, the power consumed by the end-users is computed as follows. For class 1 devices, the powerconsumption at slots is


Fig. 10. Objective value (opposite of social welfare) for the optimal load con-trol scheme and a fixed pricing scheme.

TABLE VIRESULTS FOR TEST CASE 2

, and 0 otherwise. For class 2 devices, the power isobtained from the minimization in (13b) with instead of .The power provided matches the total power consumption; thatis, .Fig. 10 shows the objective value—as defined in the left-hand

side of (9)—for the two schemes. Recall that the objective valueis to beminimized, as it represents the opposite of the social wel-fare. A range of prices covering the magnitudes of the optimalLagrange multipliers is used. It can be seen that the optimal loadcontrol scheme is the one with the largest social welfare for thewhole range of fixed prices.It is noted in Fig. 10 that the objective value is minimized

approximately for . The value is also used to com-pute the total power consumption depicted with the thin blackline in Fig. 9. The power consumption resulting from the op-timal load control scheme is much smoother than the one fromthe fixed pricing scheme; compare for instance the values of thepeak power consumptions. Smoothing out the aggregate load isa major attractive feature of (5).Table VI lists the objective value and its constituent costs,

as well as the total power consumption and the load factor, ob-tained from the two schemes by averaging 10 Monte Carlo real-izations. For the fixed pricing scheme, the value of minimizingthe objective value (in a fashion similar to Fig. 10) was chosen.The load factor is defined as the ratio , andthe closer it is to 1, the smoother the power consumption is[20, p. 58]. Observe that all costs are smaller under the optimalscheme, while the load factor is improved, although the totalpower consumption has increased.

V. CONCLUSIONS AND FUTURE DIRECTIONS

This paper presented a formulation for load control amongmultiple residences and the electricity provider. Two differenttypes of residential devices offering load response are consid-ered. The first type must consume a specified amount of energyover a prescribed horizon, but the consumption can be adjusted

across the horizon. The second type does not have a total energyrequirement, but operation of the device provides satisfaction tothe end-user, resulting in elastic power consumption. Dual de-composition yielded a distributed algorithm. The utility com-pany and the end-users exchange through the AMI Lagrangemultipliers and hourly consumption data in order to converge tothe optimal schedule. The algorithm was shown to find near-op-timal schedules even when AMI messages are lost, by estab-lishing convergence of the subgradient iterations with outdatedLagrange multipliers.An interesting future direction is to account for loads whose

starting time can be variable. A mixed integer program arises inthis case, which is challenging to solve in the distributed setupat hand. A further extension is to allow additional interactionsamong end-users, especially when possessing storage devicesso that they can sell electricity back to other end-users, or, tothe grid. Finally, a general direction is to consider distributedschemes with alternative delay patterns—even unboundedones—applicable to distributed load control for, e.g., very wideareas or emergency response purposes.

APPENDIX ASENSITIVITY ANALYSIS

This Appendix studies how the objective value in (5a) willchange, if the parameters of the optimization problem change.For instance, if the cost function is andthe parameter increases to , then how much does theoptimal social welfare change?Explicit formulas for the directional derivatives of the op-

timal objective value in (5) with respect to changes in certainparameters of the cost functions and areprovided. These are useful in three respects. First, the signof the derivative is enough to assess if the objective value isincreasing or decreasing in the parameter. Second, by com-paring the derivatives with respect to different parameters, it ispossible to identify the most influential parameters. And third,the value of the derivative multiplied by the parameter changeapproximately gives the change in the objective value. Thesensitivity analysis presented here relies on the formulas of [29,Coroll. 2.3.8] and [30].For simplicity in exposition, no storage devices are consid-

ered. Since the objective is the negative of the social welfare,the notation is used for the optimal objective value of (5).Now, suppose that the cost depends on a scalar param-eter ; this is denoted by . For example, if isquadratic, could be the coefficient of . As the optimalvalue of (5) depends on , this is denoted as .All assumptions on problem (5) stated so far (convexity,

Slater constraint qualification, etc.) hold throughout. Supposethat and are the solutions to (5) for a given nominalvalue of the parameter. Two sensitivity results will be givennext, depending on whether is differentiable (e.g.,linear or quadratic) or not (e.g., piecewise linear).Suppose that is continuously differentiable in bothand . Under a certain regularity condition, which will be

explained shortly, the derivative of at is given by

(45)


Similarly, if instead of , one is interested in changes in aparameter of the disutility function , then the fol-lowing holds under differentiability of :

(46)

In order to state the regularity condition, view every con-straint of problem (5) (i.e., (5b) as well as

) as func-tion of and , and consider the gradient vector of each ofthose. The regularity condition requires that the gradient vec-tors corresponding to active constraints (i.e., constraints holdingas equality) are linearly independent [29, CQ3, p. 24]. It canbe shown to hold in the present setup, iffor all , and if for every class 1 device there is a such that

, which are mild conditions.Consider now a piecewise linear . Specifically, sup-

pose that (cf. Fig. 2(b)).To analyze the sensitivities with respect to and , a stan-dard linear programming trick is applied. Auxiliary variables

are introduced, so that the cost isreplaced by , while the following constraints are added toproblem (5):

(47)

Let be Lagrange multipliers corresponding to (47). Undera regularity condition, the derivatives of the objective value atnominal values and are

(48)

The regularity condition for this case is the Mangasarian-Fro-movitz constraint qualification ([29], CQ1, p. 23), which holdsunder the same conditions as in the differentiable case.

APPENDIX BPROOFS

Note first that the primal problem (20) has an optimalsolution, as a consequence of Weierstrass’s theorem, becausethe feasible set is nonempty and compact, and the objectiveis continuous (cf. C1). Moreover, the dual function isconcave. Condition C1 implies that it is also finite everywhere;therefore, it is also proper, and continuous everywhere [22,Props. 6.2.1, 1.2.2, 1.4.6]. Moreover, convexity and Slater’sconstraint qualification in C1 imply that there is no dualitygap, i.e., , and that the set of optimal solutions ofthe dual problem (23) is nonempty and compact [22, Prop.6.4.3]. The set of optimal dual solutions is exactly what ismore commonly known as optimal Lagrange multipliers; see,e.g., [22, Props. 6.1.2 and 6.2.3].The subgradient method in Section III seeks to solve the dual

problem (23). The following lemma asserts that the update di-rection used in the multiplier updates (24) is in fact an -sub-gradient; see, e.g., [22, Sec. 4.3] for pertinent definitions.

Lemma 1: Under C1 and C2, it holds for all andthat

(49)

where in the case of constant stepsize (S1),

(50)

or in the case of a diminishing stepsize (S2) or (S3),

(51)

and as .The left-hand side of (49) contains the update direction used

in (24), and is exactly the definition of an -subgradient of thedual function at . It asserts that is constant if a constantstepsize is used, and vanishing if a vanishing stepsize is em-ployed. A result similar to the one in Lemma 1 has been devel-oped for an asynchronous subgradient algorithm and only fordiminishing stepsizes in [15]. The present result also covers thecase of constant stepsize, and explicitly gives the value of asa function of the maximum delays.

Proof of Lemma 1: The dual function can be writtenas a sum as follows [cf. (22)]:

(52)

Let denote the -th summand in the second term in (52).With reference to (49), it will be shown that

(53)

from which (49) follows readily.The left-hand side of (53) takes the following form, after ap-

plying the definition of the subgradient at :

(54)

(55)

Adding and subtracting the same terms in the right-handside of (55), and applying the definition of the subgradient at


, it follows that

(56)

(57)

Applying the Cauchy-Schwartz inequality to the right-hand sideof (57) leads to

(58)

From the subgradient iteration (24) at andthe nonexpansive property of the projection [22, Prop. 2.2.1], itfollows easily that

(59)

Combining (58) with (59) and the bounds in (28), it is deducedthat

(60)

Recalling also that (cf. C2), it holds for aconstant stepsize that

(61)

For diminishing stepsizes, the fact that the stepsize sequenceis majorized by the sequence , and also the condition

imply that

which vanishes because as .

The fact that the asynchronous subgradient is an -subgra-dient with special dependence of on the stepsize will be in-strumental in proving convergence of the Lagrange multiplieriterations. Next, Proposition 1 on the constant stepsize is shown,followed by the proof of Proposition 2 on the diminishing step-size.

Proof of Proposition 1: The proof relies on application of[23, Th. 4.1]. The function of [23] corresponds to in thepresent setup, and the set to . The theorem can beapplied because the dual function is a proper concave function,continuous everywhere, and moreover, the set of optimal La-grange multipliers is nonempty and compact, as explained ear-lier. This falls into the coercive case [23, p. 809].The claims of Proposition 1 are deduced from [23, Th. 4.1]

as follows:a) Apply parts i) and iv). The quantity corresponds to

in the present case, byapplication of Lemma 1.

b) Part ii) implies that the sequence is bounded. Theremaining claims follow readily from part iv).

Proof of Proposition 2: Convergence can be shown using[23, Th. 3.3] or [31, Th. 8]. The essential requirement in orderto apply those results is to show that

(62)

The latter can be shown using (51) and the properties of thesequence majorizing . Specifically, it holds for

(and an integer if needed) that

(63)

(64)

where (63) holds because , while (64) follows from thefact that is monotonically decreasing. The series in (64) isfinite, because is square-summable. Therefore, (62) holdstoo.Next, attention is turned to the recovery of primal variables.

Two lemmas will be useful in proving Propositions 3–5. The fol-lowing lemma provides an upper bound on the objective func-tion evaluated at the primal averages.Lemma 2: The following holds for the running averages (14)

and (26) under C1 and C2:

(65)

where is given by (50) if the constant stepsize (S1) is used,or by (51) if the diminishing stepsize (S3) is used.


Similarly, the following holds for the running averages (16)and (27) under C1 and C2 and for the stepsize (S2):

(66)

where is given by (51).Proof of Lemma 2: It holds due to the convexity of

that

(67)

Adding and subtracting identical terms to the right-hand side of(67) leads to

(68)

Consider the -th summand in the second term on the right-hand side of (68). Using the definition of the functionand (53), it holds that

(69)

where is given by (50) or (51).Introducing (69) into (68), and using the fact that, establishes (65). The proof of (66) is analogous.The preceding proof adapts methods used in the proof of [17,

Prop. 1(b)]. The additional difficulty here lies on the fact thatthe subgradients used in the updates correspond to outdated La-grange multipliers. To overcome this issue, the preceding proofis constructed in order to leverage Lemma 1.The following lemma provides a lower bound on the objec-

tive function evaluated at the primal averages.

Lemma 3: The following holds under C1 and C2, and anystepsize rule:

(70)

where is any dual optimal solution.Inequality (70) holds also if and are replaced byand , respectively.

This result follows readily from [17, Prop. 1(c)].We prove Proposition 3 next.Proof of Proposition 3: a)-i) Convexity of the constraint

function implies that

(71)

It is deduced from the Lagrange multiplier updates (24) thatfor all ,

(72)

It follows from (71) after substituting (72) that

(73)

from which (34) follows because is bounded.a)-ii) The Lagrange multiplier updates (24) using the nonex-

pansive property of the projection imply that

(74)

and therefore,

(75)

Upon substituting (zero duality gap), (50), and (75)into (65), and taking on both sides of the resulting in-equality, (35) follows.a)-iii) The result follows by taking in (70), using part

a)-i), and the fact that the dual optimal set is bounded.b) Since the sets and , are compact,

the sequence has limit points in


. Let be one such limit point.Due to the continuity of the projection and the norm ,part a)-i) implies that

(76)

or equivalently, satisfies (20b).Moreover, satisfies

(77)

The latter holds due to part a)-ii) and the fact that is thesupremum among all subsequential limits of a sequence; see,e.g., [32, Def. 3.16]. It also follows in a similar way from parta)-iii) that the left-hand side of (77) is lower-bounded by . Itis therefore concluded that .In order to show (38), note that there is a subsequence of

indexed by so that [see, e.g.,[32, Th. 3.17(a)]]

(78)

The right-hand side of (78) is zero (restricting to a further subse-quence if necessary) due to the continuity of the distance func-tion and the fact that the limit of every convergent subsequenceis in . Equation (38) follows readily.The proof of Proposition 4 is completely analogous to the

preceding proof, and it is presented next noting only the pointsof difference.

Proof of Proposition 4: a)-i) Similar to (73), it can beshown that

(79)

a)-ii) Following the steps in the proof of Proposition 3a)-ii),it can be shown that

(80)

It holds using Toeplitz’s lemma—see, e.g., [23, Lemma2.2]—that

(81)

where it was used that and (cf. Lemma1), as . Taking on both sides of (80) and using(81), it is deduced that

(82)

A bound as in part a)-iii) of Proposition 3 holds also forand , by taking the in (70). Uponcombining the latter two results, (40) follows readily.b) This part is analogous to the one in Proposition 3.It should be noted that Proposition 4 could also follow di-

rectly from [31, Th. 20]. The proof presented here is an alterna-tive approach, based on the methods of Proposition 3.Finally, the proof of Proposition 5 follows.Proof of Proposition 5: a)-i) As with (72), it is deduced

from the Lagrange multiplier updates that for all ,

(83)

Using (83) into (71), it follows that

(84)

The ensuing claim will be proved next, from which (42) fol-lows due to the continuity of the projection and norm functions.Claim: It holds that

(85)

To prove the claim, recall that the sequence convergesto an optimal dual solution (cf. Proposition 2). The quantityof which the limit is taken in (85) is equivalently written in thefollowing form involving the particular :

(86)

Substituting and introducing the triangleinequality imply that

(87)

Upon rearranging the right-hand side of (87), it is deduced that

(88)


Each term in the right-hand side of (88) has limit zero as(the first one due to Toeplitz’s lemma). Therefore, the

claim holds.a)-ii) Equation (65) will be employed to show that a bound as

in (82) holds for and . Combining the latter with theon (70) will lead to (43).

It holds from the Lagrange multiplier updates in a fashionsimilar to (74) and (75) that

(89)

The second term has limit zero as , while in a fashionsimilar to part i), the following is true for the first term.Claim: It holds that

(90)

The desired relationship follows by taking the in (65)and using the previous results.b) The proof is identical to the one of Proposition 4.

REFERENCES

[1] C. W. Gellings and J. H. Chamberlin, Demand-Side Management:Concepts and Methods. Lidburn, GA: Fairmont Press, 1988.

[2] “Demand response,” U.S. Dept. Energy, Office of Electricity De-livery and Energy Reliability [Online]. Available: http://www.oe.energy.gov/demand.htm

[3] International EnergyAgency,The Power to Choose: Demand Responsein Liberalised Electricity Markets. Paris, France, OECD Publishing,2003.

[4] “Grid 2030: A national vision for electricity’s second 100 years,” U.S.Dept. Energy, Office of Electric Transmission and Distribution, Jul.2003 [Online]. Available: http://www.oe.energy.gov/Documentsand-Media/Electric_Vision_Document.pdf

[5] “The smart grid: An introduction,” U.S. Dept. Energy, 2008 [Online].Available: http://www.oe.energy.gov/SmartGridIntroduction.htm

[6] A. M. Giacomoni, S. M. Amin, and B. F. Wollenberg, “Reconfig-urable interdependent infrastructure systems: Advances in distributedsensing, modeling, and control,” in Proc. Amer. Control Conf., SanFrancisco, CA, Jun.–Jul. 2011.

[7] P. Samadi, A.-H. Mohsenian-Rad, R. Schober, V. W. S. Wong, andJ. Jatskevich, “Optimal real-time pricing algorithm based on utilitymaximization for smart grid,” in Proc. 1st IEEE Int. Conf. Smart GridCommun., Gaithersburg, MD, Oct. 2010, pp. 415–420.

[8] L. Chen, N. Li, S. H. Low, and J. C. Doyle, “Twomarket models for de-mand response in power networks,” in Proc. 1st IEEE Int. Conf. SmartGrid Commun., Gaithersburg, MD, Oct. 2010, pp. 397–402.

[9] A.-H. Mohsenian-Rad, V. S. W. Wong, J. Jatskevich, R. Schober, andA. Leon-Garcia, “Autonomous demand side management based ongame-theoretic energy consumption scheduling for the future smartgrid,” IEEE Trans. Smart Grid, vol. 1, no. 3, pp. 320–331, Dec. 2010.

[10] S. Caron and G. Kesidis, “Incentive-based energy consumption sched-uling algorithms for the smart grid,” in Proc. 1st IEEE Int. Conf. SmartGrid Commun., Gaithersburg, MD, Oct. 2010, pp. 391–396.

[11] P. Vytelingum, T. D. Voice, S. D. Ramchurn, A. Rogers, and N. R. Jen-nings, “Agent-based micro-storage management for the smart grid,”in Proc. 9th Int. Conf. Autonomous Multiagent Syst., Toronto, ON,Canada, May 2010, pp. 39–46.

[12] B. Daryanian, R. E. Bohn, and R. D. Tabors, “Optimal demand-sideresponse to electricity spot prices for storage-type customers,” IEEETrans. Power Syst., vol. 4, no. 3, pp. 897–903, Aug. 1989.

[13] L. Exarchakos, M. Leach, and G. Exarchakos, “Modelling electricitystorage systems management under the influence of demand-side man-agement programmes,” Int. J. Energy Res., vol. 33, no. 1, pp. 62–76,Jan. 2009.

[14] X. Guan, X. Zu, and Q.-S. Jia, “Energy-efficient buildings facilitatedby microgrid,” IEEE Trans. Smart Grid, vol. 1, no. 3, pp. 243–252,Dec. 2010.

[15] K. C. Kiwiel and P. O. Lindberg, “Parallel subgradient methods forconvex optimization,” in Inherently Parallel Algorithms in Feasibilityand Optimization, D. Butnariu, Y. Censor, and S. Reich, Eds. Ams-terdam, The Netherlands: Elsevier Science B.V., 2001, pp. 335–344.

[16] T. Larsson, M. Patriksson, and A.-B. Strömberg, “Ergodic, primalconvergence in dual subgradient schemes for convex programming,”Math. Program., vol. 86, no. 2, pp. 283–312, 1999.

[17] A. Nedić and A. Ozdaglar, “Approximate primal solutions and rateanalysis for dual subgradient methods,” SIAM J. Optim., vol. 19, no.4, pp. 1757–1780, 2009.

[18] K. Rajawat, N. Gatsis, and G. B. Giannakis, “Cross-layer designs incoded wireless fading networks with multicast,” IEEE/ACM Trans.Netw., vol. 19, no. 5, pp. 1276–1289, Oct. 2011.

[19] A. J. Wood and B. F. Wollenberg, Power Generation, Operation, andControl, 2nd ed. New York: Wiley, 1996.

[20] H. L. Willis, Power Distribution Planning Reference Book, 2nd ed.New York: CRC, 2004.

[21] D. P. Bertsekas, Nonlinear Programming. Belmont, MA: AthenaScientific, 1999.

[22] D. P. Bertsekas, A. Nedić, and A. Ozdaglar, Convex Analysis and Op-timization. Belmont, MA: Athena Scientific, 2003.

[23] K. C. Kiwiel, “Convergence of approximate and incremental subgra-dient methods for convex optimization,” SIAM J. Optim., vol. 14, no.3, pp. 807–840, 2004.

[24] “Charging plug-in hybrid and all-electric vehicles at home,” U.S.Dept. Energy, Office of Energy Efficiency and Renewable Energy,Alternative Fuels and Advanced Vehicles Data Center, Jan. 28, 2011[Online]. Available: http://avt.inel.gov/pdf/phev/phevInfrastructur-eReport08.pdf

[25] K.Morrow, D. Karner, and J. Francfort, “Plug-in hybrid electric vehiclecharging infrastructure review,” U.S. Dept. Energy Vehicle Technolo-gies Program—Advanced Vehicle Testing Activity, Final Rep., Nov.2008 [Online]. Available: http://avt.inel.gov/pdf/phev/phevInfrastruc-tureReport08.pdf, Battelle Energy Alliance, Contract No. 58517

[26] “Appliance energy data,” Federal Trade Commission [Online]. Avail-able: http://www.ftc.gov/appliancedata

[27] “End-use consumption of electricity 2001,” [Online]. Avail-able: http://www.eia.gov/emeu/recs/recs2001/enduse2001/en-duse2001.html Apr. 20, 2009, U.S. Dept. Energy, U.S. EnergyInformation Administration

[28] IEEE Distribution Planning Working Group Report, “Radial distribu-tion test feeders,” IEEE Trans. Power Syst., vol. 6, no. 3, pp. 975–985,Aug. 1991.

[29] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Non-linear Programming. New York: Academic, 1983.

[30] E. Castillo, A. Conejo, R. Minguez, and C. Castillo, “A closed for-mula for local sensitivity analysis in mathematical programming,”Eng.Optim., vol. 38, no. 1, pp. 93–112, Jan. 2006.

[31] T. Larsson, M. Patriksson, and A.-B. Strömberg, “On the convergenceof conditional -subgradient methods for convex programs and con-vexconcave saddle-point problems,” Eur. J. Oper. Res., vol. 151, no.3, pp. 461–473, 2003.

[32] W. Rudin, Principles of Mathematical Analysis, 3rd ed. New York:McGraw-Hill, 1976.

Nikolaos Gatsis (S’04) received the Diploma degreein electrical and computer engineering from the Uni-versity of Patras, Patras, Greece, in 2005 with honors.Since September 2005, he has been working towardthe Ph.D. degree with the Department of Electricaland Computer Engineering, University ofMinnesota,Minneapolis.His research interests are in optimization methods

for wireless communication networks and the smartpower grid, with focus on resource management.


Georgios B. Giannakis (F’97) received the Diplomadegree in electrical engineering from the NationalTechnical University of Athens, Greece, in 1981and the M.Sc. degree in electrical engineering, theM.Sc. degree in mathematics, and the Ph.D. degreein electrical engineering from the University ofSouthern California, Los Angeles, in 1983, 1986,and 1986, respectively.Since 1999 he has been a Professor with the Uni-

versity of Minnesota, Minneapolis, where he nowholds an ADC Chair in Wireless Telecommunica-

tions in the ECE Department, and serves as director of the Digital TechnologyCenter. His general interests span the areas of communications, networking andstatistical signal processing—subjects on which he has published more than

300 journal papers, 500 conference papers, two edited books, and two researchmonographs. Current research focuses on compressive sampling, cognitiveradios, network coding, cross-layer designs, mobile ad hoc networks, the smartgrid, wireless sensor and social networks. He is the (co-) inventor of 20 patentsissued.Dr. Giannakis is the (co-) recipient of seven paper awards from the IEEE

Signal Processing (SP) and Communications Societies, including the G. Mar-coni Prize Paper Award in Wireless Communications. He also received Tech-nical Achievement Awards from the SP Society (2000), from EURASIP (2005),a Young Faculty Teaching Award, and the G.W. Taylor Award for DistinguishedResearch from the University of Minnesota. He is a Fellow of EURASIP, andhas served the IEEE in a number of posts, including that of a Distinguished Lec-turer for the IEEE-SP Society.