1976 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4 ... · Bus admittance matrix for DC power...

1976 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013

Decomposition Algorithms for Market Clearing WithLarge-Scale Demand Response

Nikolaos Gatsis, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE

Abstract—This paper is concerned with large-scale integrationof demand response (DR) from small loads such as residentialsmart appliances intomodern electricity systems. These applianceshave intertemporal consumption constraints, while the satisfac-tion of the end-user from operating them is captured throughutility functions. Incorporation of the appliance scheduling flex-ibility and user satisfaction in the system optimization points tomaximizing the social welfare. In order to solve the resultant verylarge optimization problem with manageable complexity, dualdecomposition is pursued. The problem decouples into separatesubproblems for the market operator and each aggregator. Eachaggregator addresses its local optimization aided by the end-users’smart meters. The subproblems are coordinated with carefullydesigned information exchanges between the market operator andthe aggregators so that user privacy is preserved. Numerical testsillustrate the benefits of large-scale DR incorporation.Index Terms—Aggregators, decomposition algorithms, demand

response, demand-side management, pricing.

NOMENCLATURE

A. Constants, Sets, and Indices

Number of scheduling periods, periodindex.

Number of buses, bus indices.

Number of lines.

Number of generators, generator index.

Number of aggregators, aggregatorindex.

Number of end-users corresponding toaggregator , end-user index.

Set of smart appliances of residentialend-user of aggregator , and smartappliance index.

Generator ’s minimum and maximumoutput.

Generator ’s ramp-up and ramp-downlimits.

Generator ’s initial power output.

Manuscript received October 02, 2012; revised February 08, 2013; acceptedApril 01, 2013. Date of publication July 12, 2013; date of current versionNovember 25, 2013. This work was supported by NSF grant 1202135; and bythe Institute of Renewable Energy and the Environment (IREE) under GrantRL-0010-13, University of Minnesota. Paper no. TSG-00684-2012.The authors are with the Digital Technology Center and the Department of

Electrical and Computer Engineering, University of Minnesota, Minneapolis,MN (email: [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.2013.2258179

Convex and compact feasible set forpower consumption of smart appliancecorresponding to end-user of

aggregator over the entire schedulinghorizon ( , and denotesthe nonnegative reals).

Total energy requirement of smartappliance .

Start and end times of smart appliance .

Minimum and maximum powerconsumption of smart appliance .

Maximum power provided by aggregator.

Parameters of end-user utility functions.

Vector of base load on all busesat .

Reactance of line .

Bus admittance matrix for DC powerflow.

Matrix relating power flows tobus angles.

Bus/generator indicator matrixif generator is

on bus .

Bus/aggregator indicator-matrixif aggregator

is on bus .

Vectors of line flow limits.

Lower and upper bounds for multipliers.

Algorithm iteration index and auxiliaryindex.

Optimal value of market clearing.

Upper bound on optimal value atiteration .

Tolerance for termination criterion.

B. Variables

Output of generator at period .

Consumption of smart appliance at .

1949-3053 © 2013 IEEE

GATSIS AND GIANNAKIS: DECOMPOSITION ALGORITHMS FOR MARKET CLEARING WITH LARGE-SCALE DEMAND RESPONSE 1977

Total power provided by aggregator toits end-users at .

Bus angle at period

Vector collecting for all .


Vector collecting .

Vector collecting for all


Vector collecting for all and ,i.e., collects the power consumptionsof all smart appliances corresponding toaggregator .

C. Functions

Convex cost function of generator .

Concave utility function of smartappliance and end-user .

Concave utility function of smartappliance and end-user at period .

Partial Lagrangian function of marketclearing formulation.

Summands of the previous Lagrangianfunction .

Dual function of market clearingformulation.

Summands of the previous function.

Full Lagrangian function.

D. Lagrange Multipliers

Multiplier corresponding to the balanceequation for bus at period .

Multiplier corresponding to the balanceequation for aggregator at period .




Variables of master programs.

Auxiliary variables of master programs.

Vectors collecting the previous variables.

I. INTRODUCTION

D EMAND response (DR) is an important resource man-agement task promising to enable interaction of end-users

with the grid of the future. DR amounts to adaptation of theend-user power consumption in response to time-varying en-ergy pricing. In the context of modern power markets, DR hasbeen proposed as an alternative form of generation or load bid

[1], [2]. DR aggregators will sign up groups of individual resi-dential and commercial loads to offer large enough DR bids intothe market. Bidirectional communication between aggregatorsand end-users will be provided by the advanced metering infra-structure (AMI) [3], with the smart meters installed at end-users’premises being the AMI terminals.This paper is concerned with large-scale incorporation of DR

into an energy market. The context includes DR from end-userswith small-scale electricity consumption, such as residentialend-users. Residential loads participating in DR programs areair conditioning (A/C) units, heaters, pool pumps, or plug-inhybrid electric vehicle (PHEV) charging. The major researchchallenges are the high computational complexity associatedwith coordination of a very large number of end-users, and alsothe incorporation of various intertemporal constraints of theirloads needed to obtain system-wide benefits. To this end, thepresent work advocates optimal decomposition methodologiesof manageable complexity to develop algorithms for DR co-ordination across the grid that preserve end-user preferences,constraints, and privacy, while relying upon decentralizedcommunication protocols between the market operator, theaggregators, and the end-users.Approaches to incorporate DR bidding have focused either on

large customers able to shed load, or, on DR bidding from aggre-gators [4], [5]. However, the intertemporal load shifting capabil-ities have not been leveraged in [4] and [5]. Recently, aggregatorbidding strategies for charging a PHEV fleet have been also pro-posed [6]–[8]. These works model a single aggregator as a pricetaker, and offer bidding strategies to maximize the aggregator’sprofit. The present work on the other hand, investigates the ef-fect of multiple aggregators on achieving system-wide bene-fits such as social welfare maximization, and reduction of thesystem marginal prices.The intertemporal constraints of the demand side and their

impact on system performance indices have been investigatedfor demand-side bidding by introducing parameters indicatingthe load shifting and recovery possibilities [9]–[13]. In a re-lated approach, a market where the demand is allowed to bidfor total energy across a horizon is studied in [14]. Price elas-ticity matrices (PEMs) representing the willingness of loads toshift their consumption depending on the prices have been uti-lized for market clearing in [15] and [16]. The process iteratesbetween price determination from the market clearing formula-tion, demand adjustment using the PEMs, and feeding back theadjusted demand into the market clearing algorithm.The aforementioned works [9]–[16] focus on large-scale

customers. However, the ideas of aggregating many individualsmall-scale user preferences and constraints, and decomposingthe resultant large-scale optimization problem (e.g., involvinggeneration and end-user coordination) have not been explored.Recently, a method to aggregate multiple loads with intertem-poral constraints based on polytope addition techniques wasadvocated in [17], where the aggregated load control capabilitywas utilized to accommodate fluctuations of the generation.The difference with [17] is that in addition to the intertem-poral scheduling constraints, utility functions capturing theindividual user satisfaction are adopted here, which makes theoverall optimization more involved. Incorporation of aggre-gated end-user preferences into power system scheduling hasbeen pursued through dual decomposition in [18]. The mainlimitations of [18] are: a) algorithm convergence is not fullyaddressed; b) end-users must announce entire demand-pricefunctions to the aggregators, which may raise privacy concerns;


and c) the transmission network is not accounted for, whileonly strictly convex costs are considered.There is also a large body of literature dealing with distributed

demand response, see e.g., [19]–[24]. These works focus on a setof end-users served by a single load-serving-entity, and consideronly the demand-supply power balance, without accounting forthe power network. The distribution network is introduced in[25]. The decomposition techniques developed in all aforemen-tioned works however are not tailored to power systems withmultiple generators and aggregators.The present work postulates a set of DR aggregators per bus

of the transmission network. Each aggregator controls loads ofseveral end-users. Each end-user has preferences about its con-trollable load operation, captured by utility functions and con-straint sets. The objective is social welfare maximization forday-ahead system scheduling, while transmission network con-straints are included in the form of DC power flows. The diffi-culty here is that small end-users cannot participate directly inthe market because a) individual end-users may not be willingto reveal their utility functions as well as individual schedulingconstraints, and b) their power consumption is small (in theorder of kW), and it would be a burden for the market operatorto solve an optimization problem directly coordinating a signif-icantly large number of customers.Seasoned to cope with these issues, the approach here ap-

plies dual decomposition to the optimization problem after in-troducing carefully selected auxiliary variables representing thetotal DR-controllable power consumption at each bus, and alsoadditional coupling constraints. Leveraging Lagrangian relax-ation of the coupling constraints, the large-scale optimizationdecomposes into manageable optimization problems of favor-able structure. The market operator and each aggregator is as-signed one of these problems. The aggregator solves its problemin coordination with the users’ smart meters. The problems thatthe market operator and aggregators solve need to be coordi-nated, and this is accomplished through properly designed in-formation exchanges, which do not need to reveal the end-userpreferences.For any algorithm based on dual decomposition, part of the

design is a) to bring up the constraints that will be consideredcoupling, and therefore associated with Lagrange multipliers,and b) to select a suitable algorithm for multiplier update.The former design consideration dictates how different taskswill be assigned to network entities, while the latter deter-mines messages exchange and the convergence speed. Thecontributions of the present paper can then be summarizedas follows: 1) The scheduling problem is formulated in away that dual decomposition yields separate problems for themarket operator and each aggregator. 2) As a result, an enticingfeature of the proposed method is that the market operatorcan account for the transmission network and simultaneouslyrely on current state-of-the-art algorithms for e.g., DC optimalpower flow (OPF), without modification. 3) The operation ofthe aggregator preserves user privacy, and does not requirerevealing price-demand functions or other user preferences. 4)The cutting plane method with disaggregated cuts (see e.g.,[26, Ch. 7]) is advocated as an attractive means of updatingthe multipliers. This choice is tailored to the structure of theproblem, and yields faster convergence than standard cuttingplane or subgradient methods.The remainder of this paper is organized as follows. Section II

presents the optimization problem for market clearing with DR

from a large number of residential end-users. The decompo-sition algorithm is developed in Section III, alongside systempricing and economic considerations. Numerical tests are inSection IV, and Section V concludes the paper. The Appendixpresents a short review of cuttinng plane methods.

II. MARKET CLEARING FORMULATION

Before the optimization formulation is presented, it is in-structive to detail the end-user preferences, and their modelingthrough utility functions and constraint sets. Specifically, it ispostulated that user corresponding to aggregator has a setof smart appliances. The power consumption of smart appli-ance across the horizon is constrained to be in a set

. This set models the possible intertemporal operationalconstraints, and is assumed convex, closed, and bounded. More-over, a user may draw satisfaction from using the appliance atdifferent power levels. This is captured by postulating a utilityfunction , assumed to be concave in . Note thatin general, the utility function depends jointly on the power con-sumption across different periods; thus, it is a function of theentire vector .This smart appliance consumption model is very general, and

can accommodate various cases of interest. To make the exposi-tion concrete, three examples for and are givennext.Example 1: Consider appliances that need to consume a spe-

cific amount of energy over a horizon in order to complete thedesired tasks, but their actual consumption is allowed to varyfrom period to period. Specific cases include charging a PHEVor operating a pool pump. The prescribed total energy mustbe consumed between a start time and an end time ,while the consumption must remain within bounds and

per period. Set then takes the form

(1)

Moreover, the utility function can be simply selected to be zerohere.Example 2: The effect of charging profile to battery lifetime

has been the theme of several studies; see e.g., [27] and [28]concerning lithium-ion batteries, which are popular choices forPHEVs. To minimize impacts on the battery lifetime, avoidingcharging at full power and postponing the start of charging havebeen advocated. Such charging profile characteristics can be in-corporated here by judicious selection of , in addi-tion to imposing constraint (1).Specifically, to avoid charging at full power, can

be selected as

(2)


so as to encourage larger deviations from . To postpone thestart of charging, the utility can be chosen as

(3)

with weights increasing in to encourage smaller atthe beginning of the horizon, and larger at later slots.Example 3: Appliances with desired set points of operation

can also be considered. Here too, the appliance is to be operatedbetween and , and the utility function is written as thesuperposition of per-period utilities; that is,

(4)

Each of these functions attains its maximum at the given desiredoperating point, which yields maximum satisfaction or comfortto the end-user, and can be variable across the horizon. The set

simply takes the form

(5)

Note that contrary to Example 1, the particular form of in(5) does not entail coupling across periods.It is interesting that in all examples, is a polyhedron,

meaning that it is described by a set of linear equalities andinequalities. This turns out to be true for many cases of interest,including e.g., thermostatically controlled loads; see [19] fordetails and additional examples. Further, it is worth mentioningthat user may also have a base load which is not schedulablebut constant; this can be included in the model by simply taking

to be a singleton for a particular .The aim is to formulate an optimization problem for system

scheduling and dispatch, which takes into account the end-users.In a nutshell, the intertemporal constraints in allow for loadshifting, while the utilities allow for load adjustment(increase or decrease). It is exactly these features that modelthe DR capabilities of residential loads, and these need to beintegrated in the system scheduling.Following the standard approach in power systems, the focus

here is to maximize the system social welfare for day-aheadsystem scheduling. Therefore, the optimization is based on theDC optimal power flow (OPF), and stands as follows:

(6a)

(6b)

(6c)

(6d)

(6e)

(6f)

Fig. 1. Power system example featuring 6 buses, 3 generators, 4 aggregators,and base loads at three of the buses.

(6g)

(6h)

(6i)

The objective in (6a) is to minimize the negative social wel-fare. Equality (6b) amounts to the per bus balance. Inequalities(6c) and (6d) are the standard generator output and ramp limitconstraints. Network line flow constraints are accounted for in(6e). Taking bus 1 as reference without loss of generality, itsbus angle is constrained to zero in (6f). Constraint (6g) giveslower and upper bounds on the energy provided from aggre-gators. Equality (6h) amounts to the aggregator balance equa-tion; that is, energy allocated to the aggregator is consumed byits end-users. Finally, (6i) is the smart appliance constraint. Thefollowing example clarifies howmatrices , and areconstructed.Example 4: Consider the power system in Fig. 1, whose

topology is an adaptation of the Western System CoordinatingCouncil system [29]. Noting that , and

, matrices and take the following form:

(7)

With denoting the reactance of line , matrixhas elements [30]

(8)

where by convention, if there is no line betweenbuses and . Finally, matrix has dimensions , sothat if line connects buses and , the entriesof are

(9)

If reactances are expressed in p.u. (per unit), then the right-hand sides of (8) and (9) need to be multiplied with the p.u. base(e.g., 100 MVA), in order for matrices and to have theircorrect values when used in (6b) and (6e).


Problem (6) maximizing the system social welfare presentsa principal method for incorporation and coordination of DRfrom small-scale users. However, one faces two chief challengeswhen it comes to solving (6):i) Functions and sets are private, andcannot be revealed to the market operator, who wouldotherwise be typically responsible for solving (6).

ii) Even if and were to be made known,including all as variables would render the overallproblem intractable, due to the sheer number of variables.(Recall that residential consumptions are in the order ofkWh, while generation in the order of MWh.)

The aggregator holds a critical role in successfully copingwith these two challenges through an appropriate optimizationdecomposition and an iterative market clearing process, as de-tailed in the ensuing section.

III. DECOMPOSITION ALGORITHM

A. Dual Decomposition

Dual decomposition is used here in order to decouple theproblem into simpler ones that will be solved by the marketoperator and the aggregators relying on the AMI. Dual decom-position is a general method which finds several applicationsin power systems, among others [31]. However, there are twodesign choices that must be adapted to the problem at hand: i)selection of the coupling constraints, with which Lagrange mul-tipliers need to be associated; and ii) an efficient method to up-date the multipliers.The only coupling constraints considered will be (6h); the re-

maining constraints will be kept implicit. Let be the Lagrangemultiplier corresponding to (6h). Then, the (partial) Lagrangianfunction is

(10)

Upon straightforward re-arrangements, the Lagrangian func-tion can be written as

(11)

where

(12)

(13)

The dual function is defined as

–

(14)

The dual decomposition method iterates between two steps:S1) Lagrangian function minimization given the current mul-tipliers, and S2) multiplier update, using the results of the La-grangian minimization. It is clear from (11) that the Lagrangianminimization can be decoupled into minimizations,where one is performed by the market operator, and each of theremaining ones by the aggregators.Specifically, letting index iterations, and

given the multipliers , the market operator at iterationminimizes [cf. (12)] subject to the constraints(6b)–(6g); that is,

(15a)

(15b)(15c)

(15d)(15e)(15f)(15g)

The last line in (15) emphasizes that the previous optimizationreturns the solution denoted as forall . It is worth stressing that (15) is a standard DC OPFproblem which includes generator costs, and also “supplyoffers” from the aggregators through the objective term

. Therefore, this problem is tractableusing today’s methods.Turning attention to the remaining Lagrangian minimiza-

tions, each aggregator must solve per iteration an optimiza-tion problem with objective , and constraints (6i)for all and . It is easily seen from (13) that this problemdecouples per residential end-user, yielding the followingminimization per

(16a)

(16b)

The last line stresses that the optimization returns forall smart appliances. Optimization problem (16a-16b) is easy


because it is convex, and can be handled efficiently by the smartmeters installed at the end-user premises. In fact, for most ex-amples of Section II, the problem can be solved in closed form.The multipliers needed to this end can be transmitted tothe user’s smart meter using the AMI.Notice that summing the optimal values of (15) and (16)

yields the dual value at , that is,

(17)

Having described the Lagrangian minimization subproblems,the next step is to design a method for updating the multipliersusing the solutions of (15) and (16).

B. Multiplier Update

The choice of the multiplier update method is crucial, be-cause fewer update steps imply less communication betweenthe market operator and the aggregators. A popular method ofchoice in the context of dual decomposition is the subgradientmethod, which is typically slow. Possible alternatives exhibitingfaster convergence are methods using multiple subgradients,such as cutting plane or bundle methods [26, Ch. 7]. For con-creteness, two versions of the cutting plane method are adoptedand compared here; one is the standard cutting plane method(CPM) [26, Sec. 7.2] and the other uses the concept of disag-gregated cuts [26, p. 409]. The two methods are reviewed inthe Appendix for completeness, while the present subsection fo-cuses on giving the multiplier update rules, and describing theimplementation of the algorithm. It is worth pointing out thatthe CPM with disaggregated cuts is better suited to the problemat hand yielding faster convergence, because it exploits the factthat the dual function can be written as a sum of separate terms[cf. (14)]. Numerical tests in Section IV illustrate differences interms of convergence speed.The two methods utilize lower and upper bounds andso that the optimal multipliers lie strictly between these

bounds. In practice, sufficiently small or large numbers can beselected for the lower and upper bounds, respectively. At iter-ation , the CPM with disaggregated cuts amounts to solvingthe following problem:

(18a)

(18b)(18c)

(18d)

Fig. 2. Information exchanges between aggregator, market operator, and smartmeter.

where in (18a) denotes transpose.The previous problem is typically called the master program,

and its variables are the coefficients , and , wherethe latter are collected in vectors and , respectively, for

and . One recognizes readily that(18) is a linear program with special structure due to its con-straints; therefore, it can be efficiently solved. Moreover, notethat the solutions of (15) and (16) enter (18) through the terms

and in (18a), and like-wise for and in (18d). The main pur-pose of (18) is to yield the multipliers , which will beused in the next iteration for (15) and (16). These multipliers areobtained as the optimal multipliers corresponding to (18d). Toobtain these multipliers, problem (18) must be solved by an al-gorithm that yields not only the optimal solution, but also theoptimal Lagrange multipliers (e.g., primal-dual interior pointmethods). The algorithm is initialized with arbitrary ,which are used in the minimizations (15) and (16).Problem (18) that yields the updated multipliers

can be solved at the market operator. To this end, the fol-lowing quantities are needed from each aggregator per iteration

and . To obtain thelatter, the user’s smart meter must transmit to the aggregatorthe sums and usingthe AMI. The latter among these is the scheduled total powerconsumption at period , and the former is a single scalarnumber. Then, the total consumption of all users is formedat the aggregator level as , and along with thescalar quantity , they are both transmittedto the market operator, in order to solve (18). This informationexchange is depicted in Fig. 2. The upshot here is that theproposed decomposition and solution method respects userprivacy, as and are never revealed.When applied to (6), the method is guaranteed to converge to

the optimal multipliers as . In fact, if (6) is a linearprogram, the method terminates in a finite number of steps. Thisis the case when and are piecewise linearfunctions, and all are polyhedral constraint sets (cf. thediscussion after Example 3).It is worth noting that the optimal value of (18) is an approx-

imation—in fact an upper bound—of the optimal valueof (6), as explained in the Appendix. Therefore, termination ofthe algorithm should in practice be based on the proximity be-tween the current upper bound and the dual value atthe latest Lagrange multipliers , which is an esti-mate of the duality gap. For instance, for a prescribed , termi-nation can be declared when

(19)


To obtain power generation and consumption schedules, thefollowing linear combinations need to be formed:

(20a)

(20b)

(20c)

(20d)

These quantities will converge to the optimal solution of (6).The overall algorithm is summarized in Table I.The standard CPM operates in a fashion similar to the CPM

with disaggregated cuts. The main difference is the master pro-gram, which takes the following form:

(21a)

(21b)(21c)

(21d)

Compared to (18), problem (21) has fewer equations (versus ) and fewer variables ( versus

). The overall algorithm proceeds as sum-marized in Table I, where (21) is used in Step 27. In order toobtain the primal variables, the coefficients are usedin all linear combinations of (20).Having described how to optimally solve (6), the next sub-

section deals with pricing issues.

C. Pricing Considerations

The algorithm of Section III-B returns the optimal Lagrangemultipliers for constraint (6h). In addition, the optimal mul-tiplier corresponding to the nodal balance equation (6b) rep-resents the system marginal prices. The two multiplier vectorsare related as explained in the next proposition.Proposition 1: Suppose that holds

for the optimal and for all and . With and

TABLE IITERATIVE ALGORITHM. THE ABBREVIATIONS MO, AGG, SM ARE USED FOR

THE MARKET OPERATOR, AGGREGATOR, AND SMART METER

denoting the optimal multiplier vectors for constraints (6h) and(6b), it holds that

(22)

Proof: First note that conditionimplies that constraint (6g) can be dropped from problem (6)without altering the optimal solution or the optimal value of theproblem. Next, consider the Lagrangian function for (6) whereconstraint (6b) is dualized withmultipliers , in addition to (6h)(cf. (10)). The Lagrangian function takes the following form:

(23)

After straightforward rearrangements, the Lagrangian becomes


TABLE IIGENERATOR PARAMETERS. PARAMETERS ARE IN $/MWH; THE REST ARE

IN MW

(24)

To obtain the dual function, the Lagrangian function mustbe minimized with respect to all primal variables, that is,

. The only term in (24) that involvesis the last summand, where enters linearly, whileis unconstrained. It is well known that the minimum of a linearfunction is unless the function is identically 0. Therefore,the dual function is finite only if holds, whichleads to (22) .Equation (22) provides an easy way to obtain system prices

from . In fact, using the properties of matrix , (22) im-plies that for all aggregators sitting on bus . Whenapplied to the power system of Example 4, Proposition 1 im-plies that , and . It isalso worth mentioning that condition isnot restrictive, because the energy provided from the aggregatorwill always be nonzero, and a sufficiently large upper bound canbe selected from historical records related to that bus.The system prices are used to obtain the profits of generators

and payments of the aggregators in the standard way. Specifi-cally, the profit of a generator will be ,where here indicates the bus where generator is situated.Similarly, aggregator’s payment to the market operator willbe , as per the price relation (22). Note thatwithout DR, it is possible to think of aggregators as conven-tional load-serving entities that need to purchase energy for theircustomers, and their demand is inelastic. In this case, they needto purchase fixed , which is not a variable. From an op-timization-theoretic point of view, allowing for to bean optimization variable through DR, enlarges the feasible set,which in turn leads to higher social welfare. Also the systemprices are expected to be smaller. This implies that the aggre-gators incur smaller payments, and can afford passing part ofthe associated savings on to the end-users. The positive effectof DR is demonstrated numerically in Section IV.

IV. NUMERICAL TESTS

The effectiveness of formulation (6) and the decompositionalgorithm are illustrated on the system of Fig. 1, where each ofthe 4 aggregators serves 1000 residential end-users.The scheduling horizon consists of twenty-four 1-hour pe-

riods, starting from the hour ending at 1 A.M. until the hourending at 12 A.M.. The generator cost function has the form

for all and . All generator parameters arelisted in Table II.The network reactances have values

p.u., at a base of 100

Fig. 3. Upper and lower limits for random residential non-schedulable load;and system base load. The former limits are scaled to 5 kWh, while the latter to50 MWh.

TABLE IIIPARAMETERS OF RESIDENTIAL APPLIANCES. ALL LISTED HOURS ARE THE

ENDING ONES; W.P. MEANS WITH PROBABILITY

MVA. Flow limits were not imposed, so that congestion effectsare not prevalent.Each end-user has a PHEV to be charged overnight, and a

pool pump to be operated during the day. Because the night in-terval (say, 6 P.M. to 6 A.M. of the next day) is split into twoparts at the beginning and at the end of the scheduling horizon,55% of the PHEV capacity will be charged during the first part,and the remaining 45% during the second part. Effectively, theend-user has 3 appliances as in Example 1, while the utilityfunction is selected to be zero. The PHEV parameters arerandomly chosen using values from [32], and the pool pumpparameters from [33]; all details are given in Table III. More-over, each residential end-user has a non-schedulable base load.It is chosen randomly between upper and lower limits with dailyvariation depicted in Fig. 3 and scaled to 5 kWh, following [34,Sec. 2.2]. The upper bound on each aggregator’s consumptionwas set to MW.Finally, there is a system total base load depicted also in

Fig. 3, which follows variation of the total MISO actual loadfor September 25, 2012, [35], scaled to a peak 50 MW. Thisload is equally split among buses 4, 5, and 6, as shown in Fig. 1.The algorithms of Section III-A are used for the solution, with

, and for the termination crite-rion (19). Note that the problem here involves 4,000 end-users,


Fig. 4. Convergence of the master program objective value and dual value (de-noted as in the caption).

Fig. 5. Convergence of the Lagrange multipliers.

corresponding to roughly variables for thedemand side (with the setup of Table III). Fig. 4 depicts the evo-lution of the master program objective and the dualvalue for the standard CPM and the CPM withdisaggregated cuts. Recall that the algorithm terminates whenthese values are close, as per (19). It is clearly seen that the stan-dard version requires at least three times more iterations than theone with disaggregated cuts. The punchline here is that the se-lection of the multiplier update method is important in order toensure faster convergence. Practically, faster convergence im-plies fewer communication between the market operator andthe aggregators (cf. Table I). Fig. 5 depicts the Lagrange mul-tiplier sequence , also illustrating differences in conver-gence speed.The results of the algorithm are compared to a case where

there is no DR. To obtain the results for this case, the powerconsumption from all adjustable appliances is derived as

(25)

which shows the consumption is distributed uniformly over theallowable operation interval. The total power consumption isthen computed from (6h), and the resultant values of areused to solve (6a)–(6f).Fig. 6 depicts the resultant total system load, where it is

shown that accounting for residential DR offers the benefit of

Fig. 6. Total system load with and without DR.

Fig. 7. Marginal prices (Lagrange multipliers).

smoothing out the total load, and in particular it reduces thepeak load by over 10 MWh. The effect on prices is illustratedin Fig. 7. Because there are no congestion effects, all buseshave the same ’s, which are depicted in Fig. 7. There are twoperiods where the system marginal price is higher without DRthan with DR, namely the periods 5 A.M. to 6 A.M. and 8 P.M.to 9 P.M.. The reason is that the higher system load at thosetwo periods requires additional generation. To better illustratethis situation, the power production per generator is depictedin Figs. 8 and 9. For instance, for the period 8 P.M.–9 P.M.,Generators 1 and 2 already operate at their capacity in Fig. 9.Generator 3 also contributes to support the load, setting theprice at 50 $/MWh, in contrast to the situation in Fig. 8.The previous remarks are further substantiated by Tables IV

and V, where the various costs and payments are listed. Specif-ically, the generation costs and profits with and without DR aregiven in Table IV. It is immediately seen that the generator costsare smaller when DR is accounted for. On the other hand, thegenerator profits are smaller. This is explained by the fact thatthe system marginal prices are smaller with DR. Table V liststhe aggregator payments to the market. It is clearly seen that thetotal payments are over $3000 less with DR than without DR.Part of these savings can be passed on to the end-users throughappropriate pricing and rebate schemes.Finally, the load factor for the two scenarios is evaluated (see

[34, Sec. 2.2] for definition). With DR, the load factor is 0.7039,while without DR it drops to 0.6435. A load factor closer to 1means smoother total load (smaller peak and higher valleys).


Fig. 8. Generator output with DR.

Fig. 9. Generator output without DR.

TABLE IVCOSTS AND LOAD FACTOR WITH AND WITHOUT DR

TABLE VAGGREGATOR PAYMENTS WITH AND WITHOUT DR

This desirable effect is therefore enabled here by the incorpo-ration of DR, and in particular, by the user load intertemporalshifting availability.

V. SUMMARIZING REMARKS AND FUTURE DIRECTIONS

This work is motivated by the vision to incorporate residen-tial DR in a large scale. Accounting for the intertemporal con-straints, as well as user scheduling preferences and satisfaction,leads to social welfare maximization. The dual decomposition

method relies on dualizing only the aggregator balance equalityconstraint in order to separate problems for the market oper-ator and each aggregator. In a nutshell, the approach has the fol-lowing desirable characteristics: a) it allows the market operatorto integrate DR resources in a large-scale fashion; b) it admits ascalable distributed solution tapping into the two-way commu-nication network in smart grids; and c) it captures end-user pref-erences while respecting privacy concerns. It is interesting topursue extensions where residential DR is incorporated in jointenergy and reserve markets, and also in market clearing for-mulations that include deterministic or stochastic security con-straints.

APPENDIXREVIEW OF CUTTING PLANE METHODS

The purpose of this Appendix is to provide a short reviewon cutting plane methods, in order to motivate the ideas ofSection III; see [26, Ch. 7] or [36, Ch. 7] for detailed discus-sions.Consider the following prototype convex optimization withlinear constraints:

(26a)

(26b)

(26c)

Problem (26) is separable, in the sense that the objective andconstraints are sums of terms, and each of these terms dependson different optimization variables. Constraint (26b) corre-sponds to (6h). Set captures constraints (6b)–(6g), while

, corresponds to (6i).Dualizing constraint (26b) with multiplier vector , the dual

function is written as where isdefined as the minimum of the partial Lagrangian function

(27)

The dual problem is to maximize the dual function with respectto the Lagrange multipliers:

(28)

where strong duality is supposed to hold here. Let andbe lower and upper bounds so that an optimal solution

of (28) is included in the region .Cutting plane methods in general tackle (28) by solving a

sequence of problems, where each problem is to maximize apiecewise linear overestimator of .Specifically, suppose that the method has so far generated the

iterates after steps. Let be the primal mini-mizer in (27) corresponding to . Observe that the vector de-fined as is a subgradient of functionat point , and it therefore holds that

(29)


Clearly, the right-hand side of (29) is a linear overestimatorof . The minimum of the right-hand side of (29) over

is a concave and piecewise linear overestimatorof . The CPMwith disaggregated cuts maximizes the sumof these overestimators. Mathematically, the problem can be ex-pressed as

(30a)

(30b)

(30c)

(30d)

The solution of this problem yields the next iterate ,while its optimal value is denoted as for reasons that willbe clear shortly.The purpose of adding constraint (30c) is to ensure that (30)

has an optimal solution. Notice that (30) is a linear program,for which one can solve equivalently its dual problem. To thisend, let denote Lagrange multipliers corresponding to(30b), and corresponding to (30c). The dual function canbe written after rearrangements as

(31)

The dual problem is to minimize (31) over the Lagrange multi-pliers , and .It holds by definition that

. Moreover, because the maximization over andis unconstrained, the dual function is finite

only when all terms in parentheses in (31) are zero. Taking intoaccount the previous considerations, the dual problem of (30)takes the form

(32a)

(32b)

(32c)

(32d)

Problem (32) corresponds exactly to the master program (18).Assigning Lagrangemultipliers to (32c), it is not hard to verifythat the dual of (32) is (30). This fact confirms that the nextiterate can be obtained either as solution of (30), oras optimal Lagrange multipliers corresponding to (32c).

Since (30) is obtained as an overestimator of , and be-cause of (28), it is deduced that

(33)

which forms the basis for the termination criterion in (19), andalso reveals that (30) and (32) yield an upper bound on .Finally, the standard CPM does not consider overestimators

for every summand , but rather for the entire . Con-straint (30b) is replaced by

(34)

and the objective is to maximize . Assigning multipliersto (34), one is led to (21). The punchline is that the CPM withdisaggregated cuts takes advantage of the separability of (26).

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Nikolaos Gatsis (S’04) received the Diploma degreein electrical and computer engineering from theUniversity of Patras, Patras, Greece, in 2005 withhonors. He received the Ph.D. degree in electricalengineering with minor in mathematics from theUniversity of Minnesota, Minneapolis, MN, USA,in 2012. His research interests are in the areas ofsmart power grids, renewable energy management,wireless communications, and networking, with anemphasis on optimization methods and resourcemanagement.

Georgios B. Giannakis (F’97) received his Diplomain electrical engineering from the National TechnicalUniversity of Athens, Greece, in 1981. From 1982to 1986 he was with the University of SouthernCalifornia (USC), Los Angeles, CA, USA, where hereceived his M.Sc. degree in electrical engineeringin 1983, M.Sc. degree in mathematics in 1986, andPh.D. degree in electrical engineering in 1986.Since 1999 he has been a professor with the Uni-

versity of Minnesota, Minneapolis, MN, USA, wherehe now holds an ADC Chair in Wireless Telecommu-

nications in the ECE Department, and serves as director of the Digital Tech-nology Center. His general interests span the areas of communications, net-working, and statistical signal processing—subjects on which he has publishedmore than 350 journal papers, 580 conference papers, 20 book chapters, twoedited books, and two research monographs (h-index 104). Current research fo-cuses on compressive sensing, cognitive radios, cross-layer designs, wirelesssensors, social and power grid networks. He is the (co-) inventor of 21 patentsissued, and the (co-) recipient of 8 best paper awards from the IEEE Signal Pro-cessing (SP) and Communications Societies, including the G. Marconi PrizePaper Award inWireless Communications. He also received Technical Achieve-ment Awards from the SP Society (2000), from EURASIP (2005), a Young Fac-ulty Teaching Award, and the G. W. Taylor Award for Distinguished Researchfrom the University of Minnesota. He is a Fellow of EURASIP, and has servedthe IEEE in a number of posts, including that of a Distinguished Lecturer forthe IEEE-SP Society.

1976 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4 ... · Bus admittance matrix for DC power...

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Transcript of 1976 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4 ... · Bus admittance matrix for DC power...