IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 5, NO. 2, … · 2014-03-24 · Residential Distribution Systems Emiliano Dall’Anese, Member, IEEE, Sairaj V. Dhople, Member, IEEE,

Optimal Dispatch of Photovoltaic Inverters inResidential Distribution Systems

Emiliano Dall’Anese, Member, IEEE, Sairaj V. Dhople, Member, IEEE, andGeorgios B. Giannakis, Fellow, IEEE

Abstract—Low-voltage distribution feeders were designed tosustain unidirectional power flows to residential neighborhoods.The increased penetration of roof-top photovoltaic (PV) systems hashighlighted pressing needs to address power quality and reliabilityconcerns, especially when PV generation exceeds the householddemand. A systematic method for determining the active- andreactive-power set points for PV inverters in residential systems isproposed in this paper, with the objective of optimizing the opera-tion of the distribution feeder and ensuring voltage regulation.Binary PV-inverter selection variables and nonlinear power-flowrelations render the optimal inverter dispatch problem nonconvexand NP-hard. Nevertheless, sparsity-promoting regularization ap-proaches and semidefinite relaxation techniques are leveraged toobtain a computationally feasible convex reformulation. Themeritsof the proposed approach are demonstrated using real-world PV-generation and load-profile data for an illustrative low-voltageresidential distribution system.

Index Terms—Distribution networks, inverter control, optimalpower flow (OPF), photovoltaic (PV) systems, sparsity, voltageregulation.

I. INTRODUCTION

T HE installed residential photovoltaic (PV) capacityincreased by 61% in 2012, driven in large by falling

prices, increased consumer awareness, and governmental incen-tives [1]. The proliferation of residential-scale roof-top PVsystems presents a unique set of challenges related to powerquality and reliability in low-voltage distribution systems. Inparticular, overvoltages experienced during periods when PVgeneration exceeds the household demand, and voltage sagsduring rapidly varying irradiance conditions have become press-ing concerns [2], [3].

Efforts to ensure reliable operation of existing distributionsystems with increased behind-the-meter PV generation aretherefore focused on the possibility of PV inverters providingancillary services [4]. This setup requires a departure fromcurrent interconnection standards [5]. Organizations across in-dustry are endeavoring to address the issue and bring consistencyto grid-interactive controls [6], [7]. For instance, by appropriate-ly derating PV inverters, reactive power generation/consumption

based on monitoring local electrical quantities has been recog-nized as a viable option to effect voltage regulation [4], [8]–[12].However, such reactive power control (RPC) strategies typicallyyield low power factors (PFs) at the feeder input and highnetwork currents, with the latter translating into power lossesand possible conductor overheating [13]. To alleviate theseconcerns, an alternative is to curtail the active power producedby the PV inverters during peak irradiance hours [13], [14]. Forexample, in [13], the amount of active power injected by a PVinverter is lowered whenever its terminal voltage magnitudeexceeds a predefined limit. The premise for these active powercurtailment (APC) strategies is that the resistance-to-reactanceratios in low-voltage distribution networks render the voltagesvery sensitive to variations in the active power injections[13], [15]. Of course, pertinent questions in this setup includewhat is the optimal amount of power to be curtailed and bywhichPV systems in the network.

A systematic and unified optimal inverter dispatch (OID)framework is proposed in this paper, with the goal of facilitatinghigh PV penetration in existing distribution networks. The OIDtask involves solving an optimal power flow (OPF) problem todetermine PV inverter active- and reactive-power set points, sothat the network operation is optimized according towell-definedcriteria (e.g., minimizing power losses), while ensuring voltageregulation and adhering to other electrical network constraints.The proposed OID framework provides increased flexibilityover RPC or APC alone, by invoking a joint optimization ofactive and reactive powers. Indeed, although [4], [8]–[10], [12],and [13] (and pertinent references therein) demonstrated thevirtues of RPC and APC in effecting voltage regulation, thesestrategies are based on local information and, therefore, theymay not offer system-level optimality guarantees. On the otherhand, by promoting network-level optimization, the OPF-basedformulation proposed in this work inherently ensures system-level benefits.

Unfortunately, due to binary PV-inverter selection variablesand nonlinear power-balance constraints, the formulated prob-lem is nonconvex andNP-hard. Nevertheless, a computationallyaffordable convex reformulation is derived here by leveragingsparsity-promoting regularization approaches [16], [17] andsemidefinite relaxation techniques [18]–[21]. Sparsity emergesbecause the proposed framework offers the flexibility of con-trolling only a (small) subset of PV inverters in the network. Thisallows one to discard binary optimization variables and effectinverter selection by using group-Lasso-type regularizationfunctions [16], [17]. To cope with nonconvexity due to bilinearpower-balance constraints, the semidefinite relaxation approach

Manuscript received July 14, 2013; revisedOctober 13, 2013andNovember 21,2013; accepted November 22, 2013. Date of publication January 22, 2014; date ofcurrent version March 18, 2014. This work was supported by the Institute ofRenewable Energy and the Environment (IREE) under Grant RL-0010-13,University of Minnesota.

The authors are with the Digital Technology Center and the Department ofElectrical and Computer Engineering, University of Minnesota, Minneapolis, MN55455 USA (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available online athttp://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSTE.2013.2292828

1949-3029 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 5, NO. 2, APRIL 2014 487

[22] proposed in [18] and [19] for the OPF problem in balancedtransmission systems, and further extended to balanced andunbalanced distribution systems in [20], [23], and [21], respec-tively, is employed here too. This approach has the well docu-mented merit of finding the globally optimal solution of OPFproblems in many practical setups [20], [21]. The resultantrelaxed OID is a semidefinite program (SDP), and it can beefficiently solved via primal–dual iterations or general purposeSDP solvers [24], with a significantly lower computationalburden compared to off-the-shelf solvers for mixed-integernonlinear programs (MINPs) [25]. Effectively, the OID task canbe implemented in real time, based on short-term forecasts ofambient conditions and loads. Subsequently, state-of-the-artadvanced metering infrastructure can be used to relay set pointsto the inverters [15]. Toward this end, initiatives are currentlyunderway to identify and standardize communication architec-tures for next-generation PV inverters [26].

To summarize, the main contributions of this work are asfollows: 1)A joint network-wideoptimizationofPV-inverter realand reactive power production improves upon conventionalstrategies that solely focus on either real [13] or reactive power[4], [8]–[11]. 2) Sparsity-promoting regularization approachesare proposed for the OPF problem to offer PV-inverter selectioncapabilities. Although existing approaches either require control-ling all the PV inverters or assume that nodes providing ancillaryservices are preselected [4], [8]–[11], [13], [21], the formulatedproblem identifies optimal PV inverters for ancillary servicesprovisioning. 3) A third contribution is represented by the RPCandAPC strategies outlined in Section IV-C. In fact, themethodsin [4], [8]–[10], and [13] are based on local information andassume that nodes providing ancillary services are preselected. Incontrast, the RPC and APC tasks are cast here as instances of theOPF problem, and they offer inverter selection capabilities. OPF-type RPC strategies are proposed in [11] and [27]; however,inverters providing reactive support are preselected. 4) The OIDstrategy is tested using real-worldPV-generation and load-profiledata for an illustrative low-voltage residential distribution systemand its performance is compared with RPC and APC.

The remainder of this paper is organized as follows. InSection II, the network and PV inverter models are described,and theRPC,APC, andOID strategies are introduced. Section IIIoutlines the proposed OID problem, and Section IV describes itsrelaxed convex reformulation. Case studies and implementationdetails are presented in Section V, and Section VI concludes thepaper and outlines future directions.

II. PRELIMINARIES AND SYSTEM MODELS

A. Notation

Upper-case (lower-case) boldface letters will be used for

matrices (column vectors), T for transposition, for com-

plex-conjugate, H for complex-conjugate transposition,Rand I denote the real and imaginary parts of a complex

number, respectively, is the imaginary unit, isthe matrix trace, is the matrix rank, denotes themagnitude of a number or the cardinality of a set,

T , , and stands for the

Frobenius norm. Finally, denotes the identity matrix,and , denote the matrices with all zeroesand ones, respectively.

B. Distribution-Network Electrical Model

Consider a low-voltage radial feeder comprising nodescollected in the set N and overhead linesrepresented by the set of edges E N N . Node0 is taken to be the secondary of the step-down transformer.Let U, H N collect nodes that correspond to utility poles andhouses with roof-top PV systems, respectively. For example,consider the residential network in Fig. 1, which is adopted from[3] and [13], and is utilized in the case studies presentedsubsequently. In this network, one has Uand H (the latter corre-sponding to houses ).

The distribution lines E are modeled as -equivalentcircuits. The series and shunt admittances of line are givenby and , where ,

, , and denote the line resistance, inductance, shuntcapacitance, and angular frequency, respectively [28]. Thedistribution-network admittance matrix is denoted by

C , and its entries are defined as

NE

where N N E denotes the set of nodesconnected to the th node through a distribution line. Let

C denote the phasors for the line-to-ground voltageand the current injected at node N , respectively. Collectingthe currents injected in all nodes in the vector

T C and the node voltages in the vectorT C , it follows that Ohm’s law can

be rewritten in matrix–vector form as

C. Residential PV Inverter and Load Models

Given prevailing ambient conditions, let denote the maxi-mum active power that can be generated by the PV inverter(s)

Fig. 1. Example of low-voltage residential network with a high PV penetrationadopted from [3] and [13]. Node 0 corresponds to the secondary of the step-downtransformer, while set U collects nodes corresponding tothe distribution poles. Homes are connected to nodes in the setH .

488 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 5, NO. 2, APRIL 2014

located at node H. Henceforth, is referred to as theavailable active power at node . The H vector collectingthe available active powers H is denoted by . Letand denote the active power injected and the reactive powergenerated/consumed by the PV inverter at node , respectively.The H vectors collecting H and H aredenoted by and , respectively. For conventional grid-tiedresidential-scale inverters that do not offer energy storage capa-bilities and operate at unity PF, it follows that and

. Nevertheless, since strategies where PV inverters areallowed to curtail their power output will be considered inSection II-D, it is useful to denote the active power curtailedby inverter H as

Furthermore, collect all the curtailed active powers in the Hnon-negative vector .

A constant model is adopted for the residential loads andand denote the residential-load active and reactive

powers at node H. The corresponding H vectors thatcollect the load active and reactive powers are denoted by and, respectively.

D. PV-Inverter Control Strategies

Themaximum reactive power that can be generated/consumedby the th inverter is limited by its rated apparent power, which isdenoted here as , and the available active power [2], [4].Specifically, given , the inverter operating space in theplane is given by the semicircle

, whose contour is represented by a red-dotted line

in Fig. 2.If the th PV inverter allows RPC, the set of its operating

points is given by

F

which indicates that the active power output is the available PVpower, and the reactive power capability is limited by the inverterrating [2], [4], [29]. The set F is represented by the verticalsegment in Fig. 2(a). When the inverter is not oversized [2], noreactive compensation is possible when ; on the otherhand, the entire inverter rating can be utilized to supply reactivepower when no active power is produced.

Similarly, the set of feasible operating points when an APCstrategy is implemented at inverter [9], [13], [14] is given by

F

and it is depicted in Fig. 2(b).In this paper, an OID strategy whereby PV inverters are

allowed to adjust both active and reactive powers is proposed.Consequently, the set of possible operating points is given by

F

As shown in Fig. 2(c), set F is significantly larger than

F and F , thus offering improved flexibility when

optimizing the performance of the residential distribution sys-tem. Clearly, the control strategy (6) subsumes RPC and APC.A variant of the OID strategy is when the PV inverters arerequired to operate at a sufficiently high PF to limit thecirculation of reactive power throughout the network [29].Thus, if the minimum allowed PF is prescribed to be ,then the amount of reactive power injected or absorbed is

constrained by the two bounds: and

. The resultant set of feasible inverter

operational points is illustrated in Fig. 2(d).It is worth mentioning that a key implementation task relates

to formulating active- and reactive-power set points for PVsystems composed of multiple inverters, e.g., microinvertersystems. Commercially available microinverter systems usuallyinclude a power management unit that communicates viapower-line communication [30], [31]. Consequently, the optimalreal- and reactive-power set points could be relayed from thepower management units to the different inverters via the power-line communication network.

III. OID PROBLEM FORMULATION

Given the operational voltage limits in the residential network,the demanded loads , and the available PV power at thedifferent houses , the objective is to optimally dispatch the PVinverters, which amounts to finding the real and reactive poweroperating points , that optimize the operation of the networkaccording to a well-defined criterion. To this end, pertinent costfunctions are discussed in Section III-A, and the OID problem isformulated in the subsequent Section III-B.

A. Cost Function Formulation

The cost function to be minimized is defined as

where function captures real power losses inthe network, models possible costs of curtailingactive power, and promotes a flat voltage profile. Finally,

, , and denote weighting coefficients. Thethree components in (7) are substantiated as follows.

Fig. 2. Operating regions for the PV inverters are shown by the shaded regionsfor different strategies: (a) RPC, (b) APC at unity PF, (c) proposed OID strategy,and (d) proposed OID strategy with a constraint on the minimum PF.

DALL’ANESE et al.: OPTIMAL DISPATCH OF PHOTOVOLTAIC INVERTERS 489

1) Power losses in the network: The active power losses in thedistribution network are given by [28]

E

R R

where C denotes the current flowing on line .2) Cost associated with active power set points: The cost

incurred by the utility when inverter is required to operate on aset point different from is modeled without loss ofgenerality by the following quadratic function:

H

where the choice of the coefficients is based on specific utility-customer prearrangements [29]. For example, if economic in-dicators are of interest, the coefficients and could be basedon the price at which electricity generated by the PV systemswould be sold back to the utility. Similarly, the amount ofcurtailed power can be minimized by choosing and

, H.3) Voltage deviations from average: Voltage deviations

throughout the network can be minimized by defining

N N

Function encourages flat voltage profiles, since it cap-

tures the distance of the vector collecting N from

the average vector N . It is impor-

tant to note that voltage limits will be enforced in the OIDproblem even if deviations from the average are not penalized(i.e., when ). Nevertheless, as will be demonstratedin the case studies, including this term provides added flexibilityin reducing the deviations of the network voltages from theaverage.

B. OID Problem

Given the cost function in (7), the OID problem is formulatedas follows:

R H

I H

U

N

H

H

H

H

H

H

where is a binary variable indicating whether PV inverter iscontrolled ( ) or not ( ), and it is assumed that

H PV inverters are to be controlled. Clearly, when, it follows from (11f) and (11g) that and, thus implying that PV inverter is delivering the

maximum available PV power at unity PF. Note that, when, the set of admissible set points F for inverter is

specified by constraints (11f)–(11i) [cf., Fig. 2(c) and (d)].Finally, the constraint on is left implicit.

The constraint in (11j) offers the possibility of choosing(or minimizing) the number of controlled inverters in orderto contain the net operational cost of the residential feeder.Additionally, alternating between different subsets of Hfacilitates a more equitable treatment of the inverters [4].Particularly important is constraint (11e), whose goal is toensure that the nodal voltages lie between and (evenduring intervals with peak solar power generation and lowdemand). Finally, (11h)–(11i) follow from the PF constraintexplained in Section II-D.

For a fixed assignment of the binary variables H,boils down to an instance of the OPF problem. Indeed, similar tovarious OPF renditions, is a nonlinear nonconvex problembecause of the balance equations (11b)–(11d), the bilinear termsin (7), and the lower bound in (11e). Furthermore, the presence ofthe binary variables H renders an NP-hard MINP,and finding its global optimum requires solving a number ofsubproblems that increases exponentially ( H ) in the number ofPV inverters. In principle, off-the-shelf MINP solvers can beemployed to find a solution to [25]. However, since thesesolvers are computationally burdensome, they are not suitable forreal-time feeder optimization [25]. Further, they do not generallyguarantee global optimality of their solutions, which translateshere to higher power losses and operational costs.

IV. CONVEX REFORMULATION

In this section, a computationally affordable convex reformu-lation of is derived by leveraging sparsity-promotingregulation approaches to drop the binary selection variables(Section IV-A), and using semidefinite relaxation techniques toaddress the nonconvexity due to bilinear terms and voltage lowerbounds (Section IV-B).

A. Sparsity-Leveraging OID

If PV inverter is not selected, then it provides the maximumactive power at unity PF; consequently, one clearly has that

. On the other hand, and may bedifferent from zero with OID (cf., Fig. 2). Supposing that

H , it follows that the H real-valued vectorT T T isgroup sparse [16];meaning that the subvectors

T (i.e., the “groups” of variables [16]) corresponding toinverters operating at point contain all zeroes, whereas

T for the inverters operated under OID.For example, consider the system in Fig. 1, and suppose thatonly PV inverters in , are to be controlled. Then, it follows

that T T T T T T , withT and T .


One major implication of this group sparsity attribute of thereal-valued vector T T T is that one can discard the binaryvariables H and effect PV inverter selection by leveragingsparsity-promoting regularization techniques. Among possiblecandidates, the following group-Lasso-type function is wellsuited for the problem at hand [16], [17]:

H

where is a tuning parameter. Thus, using (12), the OIDproblem in can be relaxed to

H

Note the absence of the binary selection variables H inthis formulation. The role of is to control the number of

subvectors T that are set to . Particularly, for, all the inverters may curtail active power and provide

reactive power compensation (i.e., H ) and, as is in-creased, the number of inverters participating in OID decreases[16]. Appendix B elaborates further on the PV-inverter selectioncapability offered by the regularization function . Ageneralization of (12) is represented by the weighted counterpart

H , where H

substantiate possible preferences to use (low value of ) ornot (high value of ) specific inverters. With this approach, amore equitable treatment of PV inverters can be facilitated bydiscouraging the use of inverters whose active and reactivepowers are adjusted more frequently.

B. SDP Reformulation

The key to derive an SDP reformulation of is to expressactive and reactive powers injected per node, and voltage magni-tudes as linear functions of the outer-product matrix H.

First, let us consider the constraints in . Let denote

the canonical basis of R and define the admittance-relatedmatrix T per node N . Furthermore, define the

Hermitian matrices H , H ,

and T . Then, the nonconvex constraints (11b)–(11d)and (11e) can be equivalently re-written as (see [19]–[21] forfurther details)

H

H

U

N

which are all linear in the matrix variable .

Now, let us consider the cost function in (7). Define theprojector matrix T , and let

denote the operator that returns a vector collecting thediagonal elements ofmatrix . Then, the function in (7) canbe equivalently expressed in terms of as

. Next, for each distribution line E,define the symmetric matrix R

T , and note that the active power loss on the lineE can be re-expressed as a linear function of the

matrix as R R . Thus,using matrices and E , the cost functionin (7) becomes

Note that for , the surrogate cost is convex inthe matrix variable .

Aiming for an SDP formulation of , Schur’s complementcan be leveraged to convert the nonlinear summands in (12) and(15) to a linear cost over auxiliary optimization variables [24](see Appendix A for details). Specifically, with

H H and serving as auxiliary variables, can beequivalently reformulated as

H H

H

T T

The positive semidefiniteness and rank constraints (16e)–(16f)jointly ensure that for any feasible C , therealways exists a vector of voltages such that H, andthat formulations and solve an equivalentOID problem. For the optima of these two problems,

it holds that T , ,

H , , and

H , where superscript is usedto denote the globally optimal solution.

Unfortunately, is still nonconvex due to the rank-1constraint in (16f). Nevertheless, in the spirit of the semidefinite


relaxation technique [22], one can readily obtain the following(relaxed) SDP reformulation of the OID problem by droppingthe rank constraint:

H H

If the optimal solution of has rank 1, thenit is also a globally optimal solution for the nonconvex pro-blems and [22]. Further, given , the vectorsof voltages and injected currents can be computed as

and , respectively, whereR is the unique nonzero eigenvalue of and the

corresponding eigenvector.A caveat of the semidefinite relaxation technique is that

matrix could have rank greater than 1; in this case, thefeasible rank-1 approximation of obtainable through rankreduction techniques turns out to be generally suboptimal [22].Fortunately though, when the considered radial residentialfeeder is single phase, and the cost (15) is strictly increasingin the line power flows, Theorem 1 of [20] can be convenientlyadapted to the problem at hand to show that a rank-1 solution of

is always attainable provided that is feasible, and thecontrolled inverters are sufficiently oversized. Incidentally,allowing inverters to operate in the region F facilitatesobtaining a rank-1 solution, since higher reactive powers can beabsorbed (a sufficient condition in [20, Thm. 1] requires nodesto be able to absorb a sufficient amount of reactive power).When the house load is not balanced, the OPF formulation forunbalanced distribution systems in [21] can be applied toaccount for discrepancies in nodal voltages and conductorcurrents.

Remark 1: The number of controlled PV inverters dependson the parameter . Provided is feasible, there are (atleast) three viable ways to select so that inverters areselected. 1) First, since both solar irradiation and load typicallymanifest daily/seasonal patterns, can be selected based onhistorical OID results. 2) In the spirit of the so-called cross-validation [16], OID problems featuring different values ofcan be solved in parallel. 3) Finally, note that the optimizedvalues of , are inherently related to the dual variablesassociated with constraints (14a)–(14b) [16], [17]; then, when aprimal–dual algorithm is employed to solve , can beconveniently adjusted during the iterations of this scheme.

Remark 2: A second-order cone program relaxation of theOPF problem for radial systems was recently proposed in [32].However, sufficient conditions for global optimality of theobtained solution require discarding the upper bounds on thereal and reactive power injections, which may not be a realisticsetup for residential distribution systems. In contrast, conditionsderived in [20] for global optimality of the SDP relaxation aretypically satisfied in practice. Furthermore, the virtues of the SDPrelaxation in the case of an unbalanced system operation weredemonstrated in [21].

C. Optimal RPC and APC Strategies

It is immediate to formulate optimal RPC and APC strategiesin the proposed framework. Toward this end, consider thefollowing regularization function:

with and serving as tuning parameters. As intypical (constrained) sparse linear regression problems [33], thesecond and third terms in the right-hand side of (18) promoteentry-wise sparsity in the vectors and , respectively. Thus,by replacing with in , thefollowing inverter dispatch strategies can be readily implemen-ted: 1) : inverters can curtail real power andprovide/consume reactive power, 2)

: APC only is implemented at a subset of PV systems,3) : some of the inverters areoptimally selected to inject/absorb reactive power, and4) : mixed strategy whereby all threestrategies (RPC, APC, and OID) can be implemented.

Compared to the local strategies in [4], [8]–[10], [12], [13],and [27], the resultant RPC and APC tasks are cast here asinstances of the OPF problem, thus ensuring system-leveloptimality guarantees; and, furthermore, through the regulari-zation function (18), they offer the flexibility of selecting thesubset of critical PV inverters that must be controlled in order tofulfill optimization objectives and electrical network con-straints. An OPF-type RPC strategy was previously proposedin [11]; however, inverters providing reactive support werepreselected.

V. NUMERICAL CASE STUDIES

To implement the OID strategy for residential network opti-mization, the utility requires: 1) the network admittance matrix, 2) instantaneous available powers , and 3) the ratings

. Once is solved, the active- and reactive-power setpoints (i.e., , ) are relayed to the inverters.

Consider the distribution network in Fig. 1, which isadopted from [3] and [13]. The pole–pole distance is set to50 m, and the lengths of the drop lines are set to 20 m. Theparameters for the admittance matrix are adopted from [3] and[13], and summarized in Table I. Voltages and in

refer to the minimum and maximum voltage utilizationlimits for the normal operation of residential systems, andthey depend on the specific standard adopted. Since the testsystem in Fig. 1 is adopted from [3] and [13], these limits areset to 0.917 and 1.042 p.u., respectively, in this case study(see p. 11 of the CAN3-C235-83 standard). Notice that thetypical voltages that dictate inverters to shut down according to[5] are higher than .

The optimization packageCVX1 alongwith the interior-point-based solver SeDuMi2 are employed to solve the OID problem inMATLAB. The average computational time required to solve

1Available: http://cvxr.com/cvx/.2Available: http://sedumi.ie.lehigh.edu/.


was 0.27 s on a machine with an Intel Core i7-2600 CPU at3.40GHz. In all the conducted numerical tests, the rank ofmatrix

was always 1, meaning that the globally optimal solution ofwas obtained (cf., Section IV-B).

All 12 houses feature fixed roof-top PV systems, with a dc–acderating coefficient of 0.77 [34]. The dc ratings of the houses areas follows: 5.52 kW for houses , , and (modeled alongthe lines of [35]); 5.70 kW (which corresponds to the averageinstalled rating for residential systems in 2011 [1]) for , ,

, and ; and 9.00 kW for the remaining five houses(modeled along the lines of [36]). As suggested in [4], it isassumed that the PV inverters are oversized by 10% of theresultant ac rating. Further, the minimum PF for the inverters isset to 0.85 [29]. The available active powers during the dayare computed using the System Advisor Model (SAM)3 of theNational Renewable Energy Laboratory (NREL), based on thetypical meteorological year (TMY) data for Minneapolis, MN,USA, during the month of July. The residential load profile isobtained from theOpen Energy Info database, 4 and the base loadexperienced in downtown Saint Paul, MN, USA, during themonth of July is used for this test case. To generate different loadprofiles, the base active power profile is perturbed using aGaussian randomvariablewith zeromean and standard deviation200W and a PF of 0.9 is presumed [3]. Finally, the voltage at thesecondary of the transformer is set at 1.02 p.u., to ensure aminimum voltage magnitude of 0.917 p.u. at whenthe PV inverters do not generate power.

With these data, the peak net active power injection (i.e.,) occurs approximately at solar noon. Standard power-

flow computations yield the voltage profile illustrated in Fig. 3with a black solid line. Note that the voltage magnitude towardthe end of the feeder (nodes 11–19) exceeds the upper limit;furthermore, the voltagemagnitude at houses and (nodes17 and 19) is beyond 1.05 p.u., which is the limit usually set forinverter protection [34].

Consider then implementing the proposed OID strategy,and suppose that the active power losses are to be minimized;i.e., the weighting coefficients in (15) are , , and

. To emphasize the role of the sparsity-promoting regu-larization function, Fig. 4 illustrates the active power curtailedand the reactive power provided by each inverter for[Fig. 4(a) and (b)] and [Fig. 4(c) and (d)]. For , itis clearly seen that all inverters are controlled; in fact, they allcurtail active power from 8:00 A.M. to 6:00 P.M., and injectreactive power during the entire interval. Interestingly, moreactive power is curtailed at houses with higher ac ratings ( ,

, , , and ). When , only seven inverters are

controlled, thus corroborating the ability of the regularizationfunction in (17) in effecting inverter selection. Four remarks arein order: 1) the seven controlled inverters are the ones locatedfar from the transformer; 2) houses at the end of the feedercurtail more active power and absorb more reactive power thanthe others (thus matching the findings in [3] and [13]); 3) all theselected inverters absorb reactive power [whereas they injectpower in Fig. 4(b)]; and 4) it is noted that for , no lessinverters are selected, meaning that at least seven inverters mustbe controlled in order to effect voltage regulation in thisparticular case study (cf. [30]). The resultant voltage profilesare illustrated in Fig. 3, where “OID-d1” refers to the case

, and “OID-d2” to the case . Clearly, although thevoltage limits are enforced in both cases, a flatter voltage profileis obtained in the second case.

Next, the OID strategy is compared with: 1) RPC withoutinverter selection (“RPC-r1”); 2) RPC with inverter selection(“RPC-r2”), where is such that only seven inverters providereactive powers; 3) APC, with (“APC-a1”) [3]; and4) APC with inverter selection (“APC-a2”), where in (18) ischosen such that seven inverters are controlled. It is interestingto note that the minimum PF constraints were not enforced forthe RPC strategy; in fact, is infeasible for a minimum PFconstraint higher than 0.3. Furthermore, the case where ,

(with all other coefficients set to 0), and ,chosen such that seven inverters are selected is also considered.This represents the situation where the utility attempts tominimize the overall power losses in the network, namely thesum of the line losses plus the curtailed power. The resultantOID and APC strategies are marked as “OID-d3” and “APC-a3,” respectively. For a fair comparisonwith RPC, nominimumPF constraints are enforced in OID-d3. As can be seen fromFig. 3, voltage regulation is effected in all the considered cases,with the flattest voltage profile for APC-a1. However, thesestrategies yield different active power losses in the network, aswell as overall active power losses (i.e., the power lost in thelines plus the curtailed power). These two quantities are com-pared in Fig. 5.When there is no price associated with the activepower curtailed, RPC is the one that yields the highest powerlosses in the network (as hinted in [13]); however, APC yieldsthe highest power losses on the lines when . Inter-estingly, strategy “OID-d3” yields the lowest overall active

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 190.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

No OIDRPC−s1RPC−s2APC−a1APC−a2APC−a3OID−d1OID−d2OID−d3

Fig. 3. Voltage profile N at 12:00 P.M. with andwithout inverter control.

TABLE ISINGLE-PHASE -MODEL LINE PARAMETERS

3Available: https://sam.nrel.gov/.4“Commercial andResidentialHourly Load Profiles for all TMY3Locations in

the United States,” accessible from http://en.openei.org/datasets/node/961. De-veloped at the National Renewable Energy Laboratory and made available underthe ODC-BY 1.0 Attribution License.


power losses, thus demonstrating the merits of the proposedOID approach. On the other hand, “OID-d2” yields a goodtradeoff between voltage profile flatness and power loss when

. Table II collects the energy loss in the network (in thecolumn labeled “Network”), the energy curtailed by the in-verters (in the column labeled “Curtailed”), and the total energyloss (in the column labeled “Overall”) for the simulated day.The accumulated energy loss in the business-as-usual approachwith no ancillary services is also reported for comparisonpurposes.

To demonstrate the increased flexibility offered by thefunction in (7), Fig. 6 depicts the voltage profiles ata few houses during the course of the day for different valuesof , for , , from which it is evidentthat the deviations from the average can be minimized byincreasing . Fig. 6(b) illustrates that the lowest overall powerlosses are obtained for (demonstrating that cannot beincreased indiscriminately without considering other optimi-zation objectives).

To better highlight the advantages of the proposed OIDmethod, a long-term impact analysis over the course of a yearis also performed. To this end, the hourly profiles for availablesolar powers are generated using SAM, based on the TMY datafor Minneapolis, MN, USA, and the hourly load profiles avail-able for the whole year in the Open Energy Info database for theTwin Cities, MN, USA. Table III reports the energy loss in thenetwork and the energy curtailed over the whole year. Twosetups are considered: 1) , (i.e., the activepower losses in the network are minimized) and 2) ,

(i.e., the overall power lost is to be minimized). In thefirst setup, it can be clearly seen that the OID strategy yields thelowest losses in the network. The proposed scheme outperformsthe RPC and APC strategies also in the second case, since ityields the lowest overall losses.

To estimate the potential economic savings over the courseof a year, consider using the “average retail price for electricityto ultimate customers by end-use sector” available in themonthly reports of the U.S. Energy Information Administra-tion, for the year 2012, for the state of Minnesota.5 Letdenote the average retail price, and consider solving theOID problem over the whole year in the following twocases: 1) (i.e., only the economiclosses in the network are considered), and 2)

H (i.e., the economic lossemerges from both the losses in the network and the activepower curtailed). Table IV summarizes the overall economiclosses over the year. It can be clearly seen that, by enabling ajoint optimization of the active and reactive power production,OID provides the most economic savings.

VI. CONCLUDING REMARKS AND FUTURE DIRECTIONS

A framework to facilitate high PV penetration in existingresidential distribution networks is proposed in this paper.Through OID, the inverters that have to be controlled in orderto enable voltage regulation are identified, and their optimal

Hou

r of

the

day

8:00

9:00

10:00

11:00

12:00

13:00

14:00

15:00

16:00

17:00

18:000

200

400

600

800

1000

1200

1400

1600

1800

(W)

H1 H2 H4 H6H5H3 H9H8 H10 H11 H12H

(a) (b)7

Hou

r of

the

day

8:00

9:00

10:00

11:00

12:00

13:00

14:00

15:00

16:00

17:00

18:000

20

40

60

80

100

120

140

160

(VAr)

H12H4H1 H2 H3 H5 H7 H8H6 H9 H11H10

Hou

r of

the

day

8:00

9:00

10:00

11:00

12:00

13:00

14:00

15:00

16:00

17:00

18:000

200

400

600

800

1000

1200

1400

1600

1800

H5H3H2H1

(W)

H7 H10H8 H11 H12H4 H6 H9

Hou

r of

the

day

8:00

9:00

10:00

11:00

12:00

13:00

14:00

15:00

16:00

17:00

18:00−200

−150

−100

−50

0(VAr)

H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12

(c) (d)

Fig. 4. Dispatched inverters: (a) curtailed active power, (b) reactive power for , (c) curtailed active power, and (d) reactive power for .

5Available: http://www.eia.gov/totalenergy.


active- and reactive-power set points are obtained. The OIDproblem was cast as an SDP, which is efficiently solvablewith a computational time on the order of the sub-second.Overall, the novel OID framework demonstrates that system-level real-time optimization can facilitate the integration ofPV systems in existing low-voltage distribution systems, andcalls for instituting communication capabilities with residen-tial-scale PV inverters, in order to enable real-time provision-ing of ancillary services. Future efforts include investigatingthe impact of forecast and communication errors on theinverter dispatch task. In addition, the benefits of the proposedOID method can be effectively communicated to practicingengineers by targeting experimental demonstrations in real-world setups.

APPENDIX

A. Derivation of (P3)

An SDP in standard form involves the minimization of alinear function, subject to linear (in)equalities and linear matrixinequalities [24]. The SDP reformulation of relies on the

Schur complement. For the symmetric matrix T ,

TABLE IIIENERGY LOSS IN THE NETWORK AND CURTAILED PV ENERGY FOR THE WHOLE YEAR

( ) ( ) ( )

TABLE IVECONOMIC LOSSES OVER 1 YEAR ($)

8 9 10 11 12 13 14 15 16 17 180

1

2

3

4

5

6

7

8

9

10

Time of the day

(a)

(b)

Act

ive

pow

er lo

ss (

kW)

8 9 10 11 12 13 14 15 16 17 180123456789

1011121314151617

Time of the day

Ove

rall

pow

er lo

st (

kW)

RPC−r1RPC−r2APC−a1APC−a2APC−a3OID−d1OID−d2OID−d3No control

Fig. 5. (a) Active power loss (kW) on the distribution lines and (b) sum of thepower lost on the distribution lines and the one curtailed by the inverters.

8 9 10 11 12 13 14 15 16 17 181

1.01

1.02

1.03

1.04

1.05

1.06

Time of the day

(a)

(b)

Vol

tage

mag

nitu

de (

pu)

r = 0r = 0.5r = 2

8 9 10 11 12 13 14 15 16 17 180123456789

1011121314151617

Time of the day

Ove

rall

pow

er lo

st (

kW)

No controlr = 0r = 0.5r = 2

Fig. 6. Voltagemagnitude at (a) and overall power lost(b) for different values of . In this setup, and .

TABLE IIENERGY LOSS IN THE NETWORK AND CURTAILED PV ENERGY FOR THE SIMULATED DAY


where is invertible, the Schur complement of is given

by T . It follows that is positive definiteif and only if and are positive definite (or, if and only ifwhen , , and are scalars); this result has been leveraged in thederivation of as detailed next.

Aiming for an SDP formulation of , consider introduc-ing the auxiliary variable , replacingwith in the cost of , and adding the constraint

. Then, upon rewriting this constraint as, one can readily obtain (16c) by

using the Schur complement with and. Similarly, (16b) can be obtained by setting

, T , and ; (16c) can be ob-tained by setting , , and ;and (16c) can be obtained by setting ,

T , and .Note finally that the positive semi-definiteness and rank

constraints (16e)–(16f) jointly ensure that there always existsa vector of voltages such that H for any feasible

C [22].

B. Soft-Thresholding on the Inverter Set Point

To rigorously demonstrate the PV-inverter selection capabili-ty offered by the proposed relaxed OID problem, results fromduality theory [37] are leveraged next to derive closed-formexpressions for the optimal inverter set points.

Define the real-valued vector T , and consid-

er rewriting (9) as HT T , with

and T . Note further that

constraint [cf., (13c)] can be re-

expressed in quadratic form as T T ,

with T . Suppose for simplicity that andno PF constraints are imposed, and consider the followingrelaxed OID problem:

H

T T

T H

Problems and are equivalent, and their globally optimalsolutions coincide [37]. Further, it can be shown that Slater’scondition holds [19], and thus has zero duality gap [37,Ch. 6]. Let , denote themultipliers associatedwith (14a)and (14b), the ones with (14d), and , theones with the inverter-related constraints (19a) and (19b), respec-tively. Further, let L de-note the Lagrangian of .

With the optimal dual variables

known, the Lagrangian optimality condition

[37, Prop. 6.2.5] asserts that , can be found as

L .

Thus, exploiting the decomposability of the Lagrangian, it turnsout that the optimal set point for inverter is given as thesolution of the subproblem

T T

where and ,

with T and T . Note thatand . Then, from [17, Th. 1], it follows

that the optimal set points are given by the followingshrinkage and thresholding vector operation:

I

with I if event is true and zero otherwise, and Rthe solution of the scalar optimization problem

T

It can be clearly deduced that when ; i.e.,inverter operates at the unitary-PF set point . On theother hand, when T , the operating point ofinverter is given by . Equation (21) alsoexplains why, with the decreasing of , the number of controlledinverters increases.

REFERENCES

[1] L. Sherwood, (2013, Jul.). U.S. solar market trends 2012 [Online].Available: http://www.irecusa.org.

[2] E. Liu, J. Bebic, andB.Kroposki, “Distribution systemvoltage performanceanalysis for high-penetration photovoltaics,” NREL Technical Monitor,Subcontract Rep. NREL/SR-581-42298, Feb. 2008.

[3] R. Tonkoski, R. Turcotte, and T. H. M. El-Fouly, “Impact of high PVpenetration on voltage profiles in residential neighborhoods,” IEEE Trans.Sustain. Energy, vol. 3, no. 3, pp. 518–527, Jul. 2012.

[4] K. Turitsyn, P. Sulc, S. Backhaus, and M. Chertkov, “Options for controlof reactive power by distributed photovoltaic generators,” Proc. IEEE,vol. 99, no. 6, pp. 1063–1073, Jun. 2011.

[5] IEEE 1547 standard for interconnecting distributed resources with electricpower systems [Online]. Available: http://grouper.ieee.org/groups/scc21/dr_shared/.

[6] California Public Utilities Commission, (2013, Jan.). Advanced invertertechnologies report [Online]. Available: http://www.cpuc.ca.gov.

[7] North American Electric Reliability Corporation, (2012, Mar.). Specialreliability assessment: Interconnection requirements for variable genera-tion [Online]. Available: http://www.nerc.com.

[8] P. Carvalho, P. Correia, and L. Ferreira, “Distributed reactive powergeneration control for voltage rise mitigation in distribution networks,”IEEE Trans. Power Syst., vol. 23, no. 2, pp. 766–772, May 2008.

[9] M. A. Mahmud, M. J. Hossain, H. R. Pota, and A. B. M. Nasiruzzaman,“Voltage control of distribution networks with distributed generationusing reactive power compensation,” in Proc. 37th Annual Conf. IEEEInd. Elec. Soc., Melbourne, Australia, Nov. 2011.

[10] A. Cagnano, E. D. Tuglie, M. Liserre, and R. A. Mastromauro, “Onlineoptimal reactive power control strategy of PV inverters,” IEEE Trans. Ind.Electron., vol. 58, no. 10, pp. 4549–4558, Oct. 2011.

[11] M. Farivar, R. Neal, C. Clarke, and S. Low, “Optimal inverter VAR controlin distribution systems with high PV penetration,” in Proc. IEEE PESGeneral Meeting, San Diego, CA, USA, Jul. 2012.


[12] P. Jahangiri and D. C. Aliprantis, “Distributed volt/VAr control by PVinverters,” IEEE Trans. Power Syst., vol. 28, no. 3, pp. 3429–3439,Aug. 2013.

[13] R. Tonkoski, L. A. C. Lopes, and T. H. M. El-Fouly, “Coordinated activepower curtailment of grid connected PV inverters for overvoltage preven-tion,” IEEE Trans. Sustain. Energy, vol. 2, no. 2, pp. 139–147, Apr. 2011.

[14] R. Tonkoski and L. A. C. Lopes, “Impact of active power curtailment onovervoltage prevention and energy production of PV inverters connected tolow voltage residential feeders,” Renew. Energy, vol. 36, no. 12,pp. 3566–3574, Dec. 2011.

[15] B. A. Robbins, C. N. Hadjicostis, and A. D. Domínguez-García, “A two-stage distributed architecture for voltage control in power distributionsystems,” IEEE Trans. Power Syst., vol. 28, no. 2, pp. 1470–1482,May 2012.

[16] M. Yuan and Y. Lin, “Model selection and estimation in regressionwith grouped variables,” J. Roy. Stat. Soc., vol. 68, no. 1, pp. 49–67,Feb. 2006.

[17] A. T. Puig, A. Wiesel, G. Fleury, and A. O. Hero, “Multidimensionalshrinkage-thresholding operator and group LASSOpenalties,” IEEE SignalProcess. Lett., vol. 18, no. 6, pp. 363–366, Jun. 2011.

[18] X. Bai, H.Wei, K. Fujisawa, and Y.Wang, “Semidefinite programming foroptimal power flow problems,” Int. J. Electr. Power Energy Syst., vol. 30,no. 6–7, pp. 383–392, Jul./Sep. 2008.

[19] J. Lavaei and S. H. Low, “Zero duality gap in optimal power flow problem,”IEEE Trans. Power Syst., vol. 1, no. 1, pp. 92–107, Feb. 2012.

[20] A. Y. Lam, B. Zhang, A. Domínguez-García, and D. Tse, (2012). Optimaldistributed voltage regulation in power distribution networks [Online].Available: http://arxiv.org/abs/1204.5226v1.

[21] E. Dall’Anese, H. Zhu, and G. B. Giannakis, “Distributed optimal powerflow for smart microgrids,” IEEE Trans. Smart Grid, vol. 4, no. 3,pp. 1464–1475, Sep. 2013.

[22] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semidefiniterelaxation of quadratic optimizationproblems,” IEEESignal Process.Mag.,vol. 27, no. 3, pp. 20–34, May 2010.

[23] J. Lavaei, D. Tse, and B. Zhang, “Geometry of power flows and optimiza-tion in distribution networks,” in Proc. IEEE PES General Meeting,San Diego, CA, USA, 2012.

[24] L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev.,vol. 38, no. 1, pp. 49–95, Mar. 1996.

[25] S. Paudyaly, C. A. Canizares, and K. Bhattacharya, “Three-phase distribu-tion OPF in smart grids: Optimality versus computational burden,” in Proc.2nd IEEE PES Int. Conf. Exhib. Innov. Smart Grid Technol., Manchester,U.K., Dec. 2011.

[26] T. Key and B. Seal, “Inverters to provide grid support (DG andstorage),” in Proc. 5th Int. Conf. Integr. Renew. Distrib. Energy Resour.,Berlin, Germany, Dec. 2012.

[27] S. Bolognani and S. Zampieri, “A distributed control strategy for reactivepower compensation in smart microgrids,” IEEE Trans. Automat. Control,vol. 58, no. 11, pp. 2818–2833, Nov. 2013.

[28] J. D. Glover, M. S. Sarma, and T. J. Overbye, Power System Analysis andDesign, Pacific Grove, CA, USA: Thomson Learning, 2008.

[29] M. Braun, J. Künschner, T. Stetz, and B. Engel, “Cost optimal sizing ofphotovoltaic inverters–influence of new grid codes and cost reductions,”in Proc. 25th Eur. PV Solar Energy Conf. Exhib., Valencia, Spain,Sep. 2010.

[30] SolarBridge Technologies, (2013, Oct.). Solarbridge power manager[Online]. Available: http://solarbridgetech.com/products/our-solution/solarbridge-power-manager/.

[31] Enphase Energy, (2013, Oct.). Envoy communications gateway [Online].Available: http://enphase.com/products/envoy/.

[32] M. Farivar and S. H. Low, “Branch flow model: Relaxations and convex-ification (Part I),” IEEE Trans. Power Syst., vol. 28, no. 3, pp. 2554–2564,Aug. 2013.

[33] R. Tibshirani, “Regression shrinkage and selection via the Lasso,” J. Roy.Stat. Soc., vol. 58, no. 1, pp. 267–288, 1996.

[34] D. L. King, S. Gonzalez, G.M. Galbraith, andW. E. Boyson, “Performancemodel for grid-connected photovoltaic inverters,” Sandia National Lab.,Tech. Rep. SAND2007-5036, Sep. 2007.

[35] S. T. Cady, D. Mestas, and C. Cirone, “Engineering systems in the rehome:A net-zero, solar-powered house for the U.S. Department of Energy’s2011Solar Decathlon,” in Proc. IEEE Power Energy Conf. Illinois, Univ.Illinois at Urbana-Champaign, Illinois, IL, USA, 2012.

[36] S. V. Dhople, J. L. Ehlmann, C. J. Murray, S. T. Cady, and P. L. Chapman,“Engineering systems in the gable home:Apassive, net-zero, solar-poweredhouse for the U.S. Department of Energy’s 2009 Solar Decathlon,” in Proc.IEEE Power Energy Conf. Illinois, Univ. Illinois at Urbana-Champaign,Illinois, IL, USA, 2010.

[37] D. P. Bertsekas, A. Nedic, and A. Ozdaglar, Convex Analysis andOptimization, Belmont, MA, USA: Athena Scientific, 2003.

Emiliano Dall’Anese (S’08–M’11) received theLaurea Triennale (B.Sc. degree) and the LaureaSpecialistica (M.Sc. degree) in telecommunicationsengineering from the University of Padova, Italy, in2005 and 2007, respectively, and the Ph.D. degreein information engineering from the Department ofInformation Engineering, University of Padova, Italy,in 2011.From January 2009 to September 2010, he was a

Visiting Scholar at the Department of Electrical andComputer Engineering, University of Minnesota,

Minneapolis, MN, USA. Since January 2011, he has been a PostdoctoralAssociate at the Department of Electrical and Computer Engineering and DigitalTechnology Center, University of Minnesota, Minneapolis, MN, USA.His research interests lie in the areas of power systems, signal processing, and

communications. Current research focuses on energy management in futurepower systems and grid informatics.

Sairaj V. Dhople (S’09–M’13) received the B.S.,M.S., and Ph.D. degrees in electrical engineering,in 2007, 2009, and 2012, respectively, from theUniversity of Illinois at Urbana-Champaign, IL,USA.He is currently an Assistant Professor with the

Department of Electrical and Computer Engineeringat the University of Minnesota (Twin Cities),Minneapolis, MN, USA, where he is affiliated withthe Power and Energy Systems research group. Hisresearch interests include modeling, analysis, andcontrol of power electronics and power systems with

a focus on renewable integration.

Georgios B. Giannakis (F’97) received the diplomain electrical engineering from the National TechnicalUniversity of Athens, Greece, 1981. From 1982 to1986, he was with the University of Southern Califor-nia (USC), LosAngeles, CA, USA,where he receivedtheM.Sc. degree in electrical engineering, in 1983, theM.Sc. degree in mathematics, in 1986, and the Ph.D.degree in electrical engineering, in 1986.Since 1999, he has been a Professor with

the University ofMinnesota,Minneapolis,MN,USA,where he now holds an ADC Chair in Wireless Tele-

communications in the ECE Department, and serves as Director of the DigitalTechnology Center.Hisgeneral interestsspantheareasofcommunications,networking,andstatistical

signal processing—subjects on which he has published more than 360 journalpapers, 600 conference papers, 20 book chapters, two edited books, andtwo research monographs (h-index 105). Current research focuses on sparsityandbigdataanalytics,wireless cognitive radios,mobile adhocnetworks, renewableenergy, power grid, gene-regulatory, and social networks.Dr. Giannakis is the (co-) inventor of 21 patents issued and the (co-) recipient of

eight best paper awards from the IEEESignalProcessing (SP) andCommunicationsSocieties, including the G. Marconi Prize Paper Award in Wireless Communica-tions.He also receivedTechnicalAchievementAwards fromtheSPSociety (2000),from EURASIP (2005), a Young Faculty Teaching Award, and the G. W. TaylorAward forDistinguishedResearch from theUniversity ofMinnesota,Minneapolis,MN,USA.He isaFellowofEURASIPandhasserved the IEEEinanumberofposts,including that of a Distinguished Lecturer for the IEEE-SP Society.


Documents

IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 5, NO. 2, … · 2014-03-24 · Residential Distribution Systems Emiliano Dall’Anese, Member, IEEE, Sairaj V. Dhople, Member, IEEE,