Spin Com 8192

1464 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 3, SEPTEMBER 2013

Distributed Optimal Power Flowfor Smart Microgrids

Emiliano DallAnese, Member, IEEE, Hao Zhu, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE

AbstractOptimal power flow (OPF) is considered for micro-grids, with the objective of minimizing either the power distribu-tion losses, or, the cost of power drawn from the substation and sup-plied by distributed generation (DG) units, while effecting voltageregulation. The microgrid is unbalanced, due to unequal loads ineach phase and non-equilateral conductor spacings on the distribu-tion lines. Similar to OPF formulations for balanced systems, theconsidered OPF problem is nonconvex. Nevertheless, a semidefi-nite programming (SDP) relaxation technique is advocated to ob-tain a convex problem solvable in polynomial-time complexity. En-ticingly, numerical tests demonstrate the ability of the proposedmethod to attain the globally optimal solution of the original non-convex OPF. To ensure scalability with respect to the number ofnodes, robustness to isolated communication outages, and data pri-vacy and integrity, the proposed SDP is solved in a distributedfashion by resorting to the alternating direction method of mul-tipliers. The resulting algorithm entails iterative message-passingamong groups of consumers and guarantees faster convergencecompared to competing alternatives.

Index TermsDistributed optimization, distribution feeders,microgrids, optimal power flow, semidefinite relaxation.

I. INTRODUCTION

M ICROGRIDS are portions of a power distribution net-work located downstream of the distribution substationthat supply a number of industrial and residential loads, andmayinclude distributed generation (DG) and energy storage devices[1]. A microgrid can operate in either grid-connected, islanded,or hybrid modes. Deployment of microgrids promises drasticperformance enhancement of the distribution grid in terms of ef-ficiency and stability, along with increased network scalabilityand resilience to outages.Besides bringing power generation closer to the end user, DG

units offer environment-friendly advantages over conventionalgeneration [1], may provide ancillary services such as reactiveand harmonic compensation [2], [3], and enable DG owners toactively participate in grid operations through supply contracts

Manuscript received October 01, 2012; revised January 12, 2013; acceptedFebruary 16, 2013. Date of publication April 15, 2013; date of current versionAugust 21, 2013. This work was supported by the Institute of Renewable En-ergy and the Environment (IREE) under Grant RL-0010-13, University of Min-nesota. Paper no. TSG-00669-2012.E. DallAnese and G. B. Giannakis are with the Digital Technology Center

and the Department. of Electric and Computer Engineering, University ofMinnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]; [email protected]).H. Zhu is with the Information Trust Institute at the University of Illinois

at Urbana-Champaign, 1308 West Main St, Urbana, IL 61801 USA (e-mail:[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.2013.2248175

and pricing schemes. On the other hand, their operation must becarefully controlled in order to prevent abrupt voltage fluctua-tions, which stem from the well known sensitivity of voltagesto variations of power injections, node over- and under-voltages[4], and drops of the power factor at the distribution substation[5]. To this end, optimal power flow (OPF) approaches are in-creasingly advocated also in this context to ensure efficient op-eration of smart microgrids and effect strict voltage regulation[3], [5][7].OPF problems are deemed challenging because they re-

quire solving nonconvex problems. Nonconvexity stems fromthe nonlinear relationship between voltages and the complexpowers demanded or injected at the nodes. In the context oftransmission networks, the Newton-Raphson method [8] istraditionally employed to obtain a possibly suboptimal solutionof these nonconvex problems. In distribution systems however,its convergence is challenged by the high resistance-to-reac-tance ratio of distribution lines. Alternative approaches includesequential quadratic optimization, steepest descent-basedmethods [5], fuzzy dynamic programming [9], and particleswarm optimization [7]. However, these methods generallyreturn suboptimal load flow solutions, and may be compu-tationally cumbersome [10]. To alleviate these concerns, arelaxed semidefinite programming (SDP) reformulation of theOPF problem for balanced transmission systems was proposedin [11] and [12], where global optimality can be assessed bychecking the rank of the obtained voltage-related matrix. Therelaxed SDP approach was extended to balanced distributionsystems in [13] and [6]. Notably, for networks with a treetopology, [6] and [13] established sufficient conditions underwhich a globally optimal solution is attainable provided theoriginal OPF problem is feasible.Distribution networks are inherently unbalanced because: i)

unequal single-phase loads must be served, and ii) non-equi-lateral conductor spacings of three-phase line segments are in-volved [14]. Further, single-phase DG units may worsen the net-work imbalance. As a consequence, optimization approachescan not rely on single-phase equivalent models as in e.g., [3],[5][7], [13]. For the unbalanced setup, an OPF framework wasproposed in [15], where commercial solvers of nonlinear pro-grams were adopted, and in [16], where quasi-Newton methodswere utilized in conjunction with load flow solvers. However,since these methods are inherently related to gradient descentsolvers of nonconvex programs, they inherit the limitations ofbeing sensitive to initialization, and do not guarantee global op-timality of their solutions.The first contribution of the present paper consists in per-

meating the benefits of SDP relaxation techniques [17] to OPF

U.S. Government work not protected by U.S. copyright.

DALLANESE et al.: DISTRIBUTED OPTIMAL POWER FLOW FOR SMART MICROGRIDS 1465

problems for microgrids operating in an unbalanced setup. Thisoptimization tool has three main advantages: i) it offers the po-tential of finding the globally optimal solution; ii) its worst-casecomputational complexity is quantifiable; and, iii) it can ac-commodate additional thermal and quality-of-power constraintswithout exacerbating the problem complexity [17]. Global opti-mality not only reduces power distribution losses, but also leadsto higher monetary savings compared to suboptimal OPF solu-tions.The OPF problem is solved by a microgrid energy manager

(MEM), which cooperates with local controllers (LCs) locatedthroughout the network. Microgrids can vary in scope, size,and ownership [18]. For those of medium and large size (of adistribution feeder), solving the OPF problem centrally at theMEM may become computationally prohibitive [10]. In fact,interior point SDP solvers do not generally scale well with theproblem size [6], [17], [19]. For a real-time network manage-ment, it is generally required to find a new network operationalsetup rapidly (e.g., in a few seconds or minutes) in order topromptly respond to abrupt load variations and to cope with theintermittent power generation that is typical of renewable-basedDG units. It is then of paramount importance to solve the SDP-based OPF problem in a distributed manner, by decomposingthe main problem into multiple sub-instances that can be solvedefficiently and in parallel. A distributed algorithm is desirablealso when (a group of) customers do not share data with theMEM due to privacy concerns, or because they wish to manageautonomously their DG units in order to pursue individual eco-nomic interests [18]. Finally, a distributed algorithm involvesa modest communication overhead compared to its centralizedcounterpart, as it does not require to pool line, generator, andload data at the MEM, and subsequently disseminate the OPFsolution.Decentralized OPF approaches were first proposed in

[20], [21], where multi-utility transmission systems werepartitioned in autonomously managed areas. Augmented La-grangian methods were employed to decompose the overallOPF problem in per-area instances. A similar approach wasfollowed by e.g., [22], [23] (see also references therein), wherestandard Lagrangian approaches were utilized in conjunctionwith Newton methods. Solving SDP in a distributed fashionis challenging due to the couplings of local voltage-relatedmatrices enforced by the positive semidefinite constraint ofthe global voltage matrix. Results related to positive semidef-inite matrix completion [24] were leveraged in [6], [19] todevelop a distributed OPF algorithms for balanced networksvia dual decomposition. Using the results of [24], and tappinginto the powerful alternating direction method of multipliers(ADMM) [25, Sec. 3.4], a distributed optimization problemfor unbalanced microgrids is formulated here, where each LCsolves an optimization sub-problem, and then exchanges simplemessages with its neighboring LCs. Compared to the dual de-composition schemes of [6], [19], the proposed ADMM-basedapproach offers a markedly improved convergence.In the OPF context, augmented Lagrangian methods (related

to ADMM) were first used in [20], [21] to develop a decentral-ized optimization scheme for (balanced) transmission networks.Off-the-shelf schemes were used to solve the OPF sub-problem

associated with each sub-network. More recently, ADMM wasadvocated in [26] for state estimation, and in [27] for distributedmulti-period OPF in balanced systems. Here, the approaches of[20], [21], [27] are considerably broadened by considering un-balanced distribution networks, and by employing the ADMMto devise a distributed SDP solver.The rest of the paper is organized as follows. Section II re-

capitulates the OPF formulation for microgrids, and Section IIIdevelops its centralized SDP solver. The distributed algorithmis presented in Section IV, while numerical tests are reportedin Section V. Finally, concluding remarks are provided inSection VI1.

II. MODELINGAND PROBLEM FORMULATION

Consider a microgrid comprising nodes collected in theset2 , and overhead or underground linesrepresented by the set of edges .Let node 0 represent the point of common coupling(PCC), taken to be the distribution substation. Define as

and the phases ofline and node , respectively. Letbe the complex line-to-ground voltage at node ofphase , and the current injected at the samenode and phase. As usual, the voltagesare taken as reference for the phasorial representation. Lines

are modeled as -equivalent components [14,Ch. 6], and the phase impedance and shuntadmittance matrices are denoted as and

, respectively. Three- or single-phasetransformers (if any) are modeled as series components withtransmission parameters that depend on the connection type[14, Ch. 8], [15].Per phase , let and denote the active and

reactive powers demanded by a wye-connected load at the bus. Capacitor banks are usually mounted at some nodes to pro-vide reactive power support. Let denote the susceptance ofa capacitor connected at node and phase . Finally, supposethat DG units are located at a subset of nodes , and let

and denote the active and reactive powers suppliedby unit . For conventional DG units, such as diesel gen-erators, the supplied powers can be controlled, and they will bevariables of the OPF problem; on the other hand, andrepresent committed powers for renewable-based sources.Given the demanded loads , the goal is to se-

lect a feasible set of voltages , currents , and powerssupplied by conventional DG units so that the1Notation: Upper (lower) boldface letters will be used for matrices (column

vectors); for transposition; complex-conjugate; and, complex-conjugate transposition; denotes the real part, and the imaginarypart; the imaginary unit. the matrix trace; the matrixrank; denotes the magnitude of a number or the cardinality of a set; andstands for Frobenius norm. Given a vector and a matrix , denotes a

sub-vector containing entries of indexed by the set , andthe sub-matrix with row and column indexes described by and. Finally, and denotes matrices with all zeroes and

ones, respectively.2The symbols defined throughout the paper are recapitulated in Table I.


TABLE INOMENCLATURE

steady-state operation of the microgrid is optimal in a well de-fined sense. To this end, one of the following two objectives isusually pursued.i) Minimization of power losses. The active power flowing on

line is , whereis the complex current from node to node on phase. Thus, the cost to be minimized here is [6]

(1)

where collects all thesteady-state variables.ii) Minimization of the cost of supplied power. Letting the cost

of power drawn from the PCC be denoted by , and theone incurred by the use of DG unit as , one canminimize the cost of supplied power [5]

(2)

Notice that (1) and (2) are equivalent when andfor all .Based on (1)(2), the following OPF problem is considered:

(3a)

(3b)

(3c)

(3d)

(3e)

(3f)

(3g)

where and in (3e) are given minimum and max-imum utilization and service voltages; ;(3f)(3g) are box constraints for the power supplied by conven-tional DG units, and depending on the chosen cost.Similar to OPF variants for transmission networks and

balanced distribution networks, (P1) is a nonlinear nonconvexproblem because of the load flow (3b)(3c) as well as the con-straints (3e). In the next section, an equivalent reformulation of(P1) will be introduced, and its solution will be pursued usingthe SDP relaxation technique.

III. SDP-BASED CENTRALIZED SOLUTION

Consider the complex vectorsand , with and

the complex vectors collecting voltagesand currents per node . Voltages and injectedcurrents abide by Ohms law , where is a symmetricblock matrix of dimensions , whoseentries are given by:i) matrix occupying the off-diagonalblock corresponding to line ; and,

ii) the diagonal block corresponding to nodewith

(4)

where if , otherwiseand

( is formed likewise).The next step consists in expressing the active and reactive

powers injected per node, active powers flowing on the lines,and voltage magnitudes as linear functions of the outer-productmatrix . To this end, define the following admittance-related matrix per node and phase

(5)

whereand denotes the canonical basis of , andlet the matrices andbe defined as

and

,respectively. Then, a linear model in can be establishedusing the following lemma (see also [28] and [12]).Lemma 1: For the Hermitian matrices

(6a)

(6b)

(6c)

(6d)


voltage magnitudes and active as well as reactive powers arelinearly related with as

(7a)

(7b)

(7c)(7d)

with for , and if capacitorbanks are not present at node .

Proof: See the Appendix.Using Lemma 1, problem (P1) is equivalently reformulated

as follows:

(8a)

(8b)

(8c)

(8d)

(8e)(8f)(8g)

where the costs and are re-expressed as

(9a)

(9b)

Problem (P2) is still nonconvex because of the rank-1 con-straint (8g). Nevertheless, the SDP relaxation technique, whichamounts to dropping the rank constraint [17], can be leveragedto obtain the following convex problem:

(10)

Clearly, if the optimal solution of (P3) has rank 1, then it isa globally optimal solution also for the nonconvex problem (P2).Further, since (P1) and (P2) are equivalent, there exists a vector

so that , and the optimal costs of (P1)and (P2) coincide at the optimum. This is formally summarizednext.Proposition 1: Let denote the optimal solution of the

SDP (P3), and assume that . Then, the vector ofline-to-ground voltages where is theunique non-zero eigenvalue of and the correspondingeigenvector, is a globally optimal solution of (P1).

The upshot of the proposed formulation is that a glob-ally optimal solution of (P2) [and hence (P1)] can be ob-tained via standard interior-point solvers in polynomial time.In fact, the worst-case complexity of (P3) is on the order

for gen-eral purpose SDP solvers, with denoting the total numberof constraints and a given solution accuracy [17].Notice however that the sparsity of andthe so-called chordal structure of a matrix can be exploitedto obtain substantial computational savings; see e.g., [29]. Incontrast, gradient descent-based solvers for nonconvex pro-grams, sequential quadratic programming, and particle swarmoptimization, do not guarantee global optimality of the ob-tained solutions and are sensitive to initialization. Here, globaloptimality translates to lower distribution losses and increasedmonetary savings compared to sub-optimal OPF solutions.Since (P3) is a relaxed version of (P2), could have rank

greater than 1. In this case, rank reduction techniques may beemployed to find a feasible rank-1 approximation of pro-vided it exists. For instance, the randomization technique offersa viable way to obtain a rank-1 approximation with quantifiableapproximation error; see e.g., [17] and references therein. Al-beit feasible for (P2), the resultant solution is generally subop-timum [17]. For balanced tree distribution networks, [13] and[6] established conditions under which a rank-1 solution is at-tainable provided the original OPF problem is feasible. Unfor-tunately, when the tree power network is unbalanced, the re-sults of [13] and [6] no longer apply, as explained in the ensuingSection III-B. However, an intuitive argument will be providedin Section III-B to explain why a rank-1 solution is expectedeven in the unbalanced setup. But first, SDP-consistent con-straints on line flows are derived in Section III-A, and a remarkis provided.Remark 1: Step-down or in-line three- or single-phase trans-

former banks (if any) can be accommodated in the formulatedoptimization problems by using their series component models[14, Ch. 8], [15]. If a delta connection is employed on one side ofthe transformer, a small dummy resistance should be includedbetween the primary and the secondary sides (one per phase) inorder to ensure that the matrix obtained by solving (P3)has rank 1; see also [12].

A. Constraints on Line Flows

Constraints on the power dispelled on the distribution lines,or, on the line current magnitudes are generally adopted to pro-tect conductors from overheating (which may eventually triggeran outage event). Using Lemma 1, it turns out that the real powerdissipated on a line can be limited by simply addingthe constraint in(P3), for a given maximum power loss .Consider now the constraint , with a

given upper bound on the magnitude of . Aiming to an SDP-consistent reformulation of this constraint, letdenote the vector collecting the complex currentsflowing through line , and notice that is related


to voltages and as .Next, define the complex matrix

(11)

where is a matrix with elementsand ;

likewise, has dimensions , and its entriesare and .Thus, an SDP-compliant re-formulation of the constraint on thecurrent magnitude is possible as follows.Lemma 2: Consider the Hermitian matrix

(12)

where denotes the canonical basis of .Then, constraint can be equivalently re-ex-pressed as

(13)

Following similar steps, and using (3d), constraints on themagnitude of the injected currents can be derived too.In unbalanced microgrids, it is of prime interest to protect

from overheating also the neutral cable(s) of the distributionlines [14]. Towards this end, let denote the set of groundedneutral cables that are present on the line . Further,let the be the neutral transformation ma-trix, which is obtained from the primitive impedance matrix ofline via Kron reduction [14, Sec. 4.1]. Thus, the neutralcurrents are linearly related to theline currents as . It readily follows fromLemma 2, that the magnitude of the current on the neutral ca-bles can be constrained in (P3) as

(14)

with , thecanonical basis of , and a cap on the magnitudeof .

B. The Rank ConundrumSufficient conditions under which a rank-1 solution is al-

ways obtained provided the original OPF problem is feasiblewere established in [13] and [6] for balanced distributionnetworks with a tree topology. Balanceness implies that equalsingle-phase loads are served,and for each line ,where . To recapitulate the broad out-line of the proofs in [13] and [6], assume for simplicitythat the shunt admittances are all zero. Then, the totalpower flowing from node to is given by

, withthe angle difference between voltages

and . Since the network is balanced, is the same oneach phase . Fixing the voltage magnitudes, theregion of the feasible powers , which isdenoted as , becomes an affine transformation of the unitcircles. Then, if one minimizes a strictly increasing functionof the powers , it follows that the Paretofront of and the one of the convex hull of coincideif [13].Based on this observation, proving that the Pareto re-gions of the feasible power injections at the nodes ofthe nonconvex OPF [the balanced counterpart of (P2)]and the relaxed SDP [the balanced counterpart of (P3)]amounts to showing that: i) the region of feasible powers

is the Cartesian product of the regions; and, ii) the region of the injected powers is

given by an affine transformation of . Specifically, the firstproperty i) follows from the fact that power flows on differentlines are decoupled; that is, it is possible to modify the angle

of a line , while preserving the angle differenceof any other line .Suppose now that the off-diagonal elements of

are not zero; that is, . Thetotal power flowing from node to node is given by

, and it is now a func-tion of , as well as of the angle differences

and .Different from the balanced case, the power flowsare no longer decoupled across lines. In fact, it is impossible toadjust the angles to obtain a new flow on line

, without affecting the angle differencesand for one of the other lines andconnected to the nodes and , respectively. Thus, the resultsof [13] and [6] no longer apply in the unbalanced setup.An analytical characterization of the flow region in the un-

balanced case is challenging because of the number of voltageangles involved and the aforementioned coupling of the linepower flows. Nevertheless, the following simple examples illus-trate why one should expect a rank 1 solution from the relaxedOPF even in an unbalanced setup.Consider a 2-node unbalanced network, and suppose that a

two-phase line connects the two nodes. Let

(as in [30]) and , and assume thatfor and . Fig. 1(a) depicts the feasible re-gion of powers for ,

, , and .Notice that is given by the Minkowski sum of the per-phase regions . It can be seen that the fea-sible flow region (the dark gray area) is a perturbed ellip-soid; specifically, expanding the expression of , it fol-lows that and entail a perturbation of the center andof the axes of the ellipsoid that would be obtained if the linewas balanced. Next, consider minimizing a strictly increasingfunction over . It follows that the set ofPareto optimal points are the ones represented by the red dots inFig. 1(a). Consider now the convex hull of , which amountsto augmenting the dark gray area with the light gray one shown


Fig. 1. Line flow region and its Pareto optimal points. (a) Nonconvex problem.(b) Relaxed problem.

Fig. 2. Feasible power injection region and its Pareto optimal points. (a) Powerinjected at 3 nodes. (b) Injection at 2 and 3 .

in Fig. 1(b). Clearly, points belonging to the light gray area leadto a solution of (P3) with rank higher than 1. Notably, the Paretopoints of and the ones of the relaxed region in Fig. 1(b) coin-cide; since is strictly increasing, the solution of (P3) mustbe on the Pareto boundary; thus, the optimal solution of (P3)has rank 1, and it is an optimal solution also for the nonconvexproblem (P2). Granted that the Pareto points in Figs. 1(a) and1(b) are the same, [13, Lemma 5] can be used to show that thetwo Pareto regions coincide also when constraints on the volt-ages are involved.The region of injected powers for a 3-node network is exam-

ined next. Let, and

assume that for all and .Finally, let , with the power injectedat node and phase . The gray areadepicted in Fig. 2(a) corresponds to the region of feasiblepower injections for

[13], and for theline-line angles ; specifically,

. The angle differ-ence is confined in the set , as higher valuesare not likely to happen in practice [30]. Clearly, the grayregion is nonconvex. If one considers minimizing a strictlyincreasing function of the injected powers over, it follows that the Pareto optimal points are the ones colorcoded red in Fig. 2(a). It can be noticed that the Pareto regiondoes not change if one takes the convex hull of ; therefore,the solution of (P3) has rank 1, and it is an optimal solution for

(P2). Next, fix the power injected (or absorbed) at node 3, andconsider minimizing a strictly increasing function of and. The two-dimensional region of feasible powers and

is depicted in gray in Fig. 2(b). Again, the Pareto front of thegray region and the one of its convex hull coincide.These examples suggest that the nonconvex OPF problem

(P2) and its relaxed counterpart (P3) share their optimal so-lution when

, and the angle differencesbetween conductors are small enough. This further moti-vates efforts toward analytical characterization of the power in-jection region in unbalanced distribution systems.

IV. DISTRIBUTED SOLUTION

Albeit polynomial, the computational complexity incurred bystandard interior-point solvers for SDP does not scale well withthe number of nodes , and the number of constraints [17].Notice that this lack of scalability is incurred also by alternativemethods based on off-the-shelf solvers for nonlinear programs[10]. Furthermore, the communication overhead required to col-lect data from all end users at the MEM, and subsequently dis-seminate the OPF solution back to the LCs, may lead to trafficcongestions and substantial delays in the data delivery. There-fore, a distributed SDP solver, with minimal computational andcommunication costs, is well motivated. A distributed approachis also desirable in order to address possible concerns regardingdata privacy and integrity, and when the microgrid includessingle- or multi-facilities that are managed independently fromthe rest of the network in order to pursue specific economic in-terests [18].Consider partitioning the microgrid into areas

, and suppose that each area is controlled by an LC. In thedistributed SDP solver to be derived, each LC will solve an OPFproblem of reduced dimension for its controlled area. Letbe an extended area defined as

; that is, collects also thenodes belonging to different areas that are connected to bya distribution line. Sets can be interpreted as counterpartsof the regions considered in [20] in the context of transmis-sion systems. Based on the overlaps among , definethe set of neighboring areas for the -th one as

. Finally, let the vector stack the complex line-to-ground voltages of the nodes in (that is, ), andlet and denotethe sub-matrices of and , respec-tively, formed by extracting rows and columns corresponding tonodes in . With these notational conventions, it is possibleto re-write the SDP (P3) as

(15a)

(15b)(15c)


where denotes the set of sub-matrices satisfying thefollowing constraints per area :

(16a)

(16b)

(16c)

(16d)

(16e)

with the additional constraint for the areathat contains the PCC. Further, represents the costassociated with the area ; that is, it collects the terms in(9a) (if ) or (9a) (if ) that pertain to . Forexample, if one wishes to minimize the distribution losses, thecorresponding cost becomes

(17)

with . On the other hand, the ex-pression for is not unique, depending on the specificnetwork topology and operational setup. For instance, if onesassumes that each area is formed by one lateral or one sub-lat-eral [18], then accounts for the cost of power flowinginto the (sub-)lateral from the backbone of the microgrid, andthe cost of power generated within the (sub-)lateral. Instead, ifcustomers are allowed to use DG units only to satisfy their ownneeds, boils down to

. Either way, the equivalent formu-lation (18) effectively expresses the cost as the superpositionof local costs, and divides network constraints on a per-areabasis. However, even with such a decomposition the main chal-lenge lies in the PSD constraint (15a) that couples local ma-trices . Indeed, if all submatrices were non-over-lapping, the PSD constraint on would simplify toper area , and (P4) would be decomposable in sub-problems.However, this equivalence fails to hold here, since submatricesshare entries of . The idea is to identify valid network topolo-gies (that is, valid partitions of the microgrid in smaller areas)for which the PSD constraint decomposition is feasible.To this end, define the following two auxiliary graphs:i) a macro graph , where nodes represent the areas andedges are defined by the neighborhood sets ;and,

ii) a micro graph induced by the sub-matrices; that is, a graph with nodes (one

per phase and node), with an edge connecting the nodesrepresenting the voltages and if the entry of

corresponding to is contained in one of thesub-matrices . Examples of macro graphs areprovided in Figs. 3 and 4.

Based on these auxiliary graphs, results on completing partialHermitian matrices will be leveraged to obtain PSD ones [24].These results rely on the so-termed chordal property of the graph

induced by to establish the equivalence betweenpositive semidefiniteness of the overall matrix and that of allsubmatrices corresponding to the graphs maximal cliques.Towards decomposing the PSD constraint into local ones, the

following assumptions are made, which naturally suggest validpartitions of the microgrid:

(As1) The graph is a tree; and,(As2) for all ,

; that is, no nested extended areas are present.Condition (As1) is quite reasonable in tree distribution net-works; for example, an area can be formed by a pair of nodesthat are connected by a distribution line [6], or by laterals andsub-laterals [18]. (As2) is a technical condition ensuring thatthe subgraph induced by is a maximal clique of , whichallows using the results of [24]. Based on these assumptions,the following can be readily proved.Proposition 2: Under (As1) and (As2), the graph is

chordal, meaning that each of its cycles comprising four ormore nodes has a chord. Furthermore, all its maximal cliquescorrespond to the elements of .As established in [24], the PSD matrix is completable if

and only if is chordal, and all its submatrices correspondingto the maximal cliques of are PSD. Therefore, constrainingto be PSD is tantamount to enforcing the constraint on all

local matrices , . Notice that (As1)requires the macro graph to be a tree, while no conditionsare imposed on the topology of the microgrid. Thus, it may bepossible to find network partitions with an associated chordalgraph also in the case of weakly-meshed microgrids.Next, let collect the indexes

corresponding to the voltages of the nodes thatthe extended areas and share. For example, if areas1 and 2 share nodes and , then indexes thevoltages and . Further, define asthe submatrix of collecting the rows and columns ofcorresponding to the voltages in . With these definitions,and assuming that (As1) and (As2) hold, problem (P4) can bere-written in the following equivalent form:

(18a)

(18b)(18c)

(18d)

where constraint (18c) enforces neighboring areas to consent onthe entries of and that they have in common. Clearly,constraints (18c) couple the optimization problems acrossareas. To enable a fully distributed solution, consider intro-ducing the auxiliary variables and


per area. With these auxiliary variables, (P5) can be equiva-lently re-stated as

(19a)

(19b)(19c)

(19d)

(19e)

(19f)

A similar approach was followed by [19], which utilized ei-ther primal or dual iterations to distribute the OPF in balancedtransmission networks. A distributed OPF for balanced distri-bution feeders was derived in [6], where the dual (sub-)gradientascent was used. Unfortunately, sub-gradient ascent methods donot always lead to a satisfactory solution; when the dual functionis non-differentiable and the step size is fixed, dual and primaliterates converge only on the average. What is more, recoveringthe primal variables from the optimal dual variables is not al-ways guaranteed [31, Sec. 5.5.5]. Besides dealing with unbal-anced power networks, the novelty here consists in solving (P6)distributedly by resorting to the ADMM [25, Sec. 3.4], a pow-erful scheme that has been successfully applied to distributedoptimization and estimation in several contexts. To this end, let

and be the multipliers associated with (19c) and(19d), respectively, and consider the partial quadratically-aug-mented Lagrangian of (19) as

(20)

where is a positive constant [25, Sec. 3.4]. Then, theADMM amounts to iteratively performing the following steps( denotes the iteration index):[S1] Update primal variables:

(21)

[S2] Update auxiliary variables:

(22)

[S3] Update dual variables:

(23)

(24)

In step [S1], the per-area matrices are obtained byminimizing (20), where variables andthe multipliers are kept fixed to their previous iteration values.Likewise, the auxiliary variables are updated in [S2] by fixing

to their up-to-date values. Finally, the dual vari-ables are updated in [S3] via dual sub-gradient ascent.Interestingly, the ADMM iterations [S1][S3] can be sim-

plified by exploiting the favorable decomposability of the La-grangian. To this end, the following lemma is first needed.Lemma 3: If the multipliers are initialized as

, then for every pair of neighboring areasand it holds that for each

. Likewise, for each if.

Proof: See the Appendix.Using Lemma 3, steps [S1][S3] can be simplified as follows.

Furthermore, convergence to the solution of the centralized SDP(P3) is established.Proposition 3: If , then [S1][S3]

boil down to the following primal-dual updates:Update per area as:

(25a)

(25b)

(25c)

(25d)

where the local Lagrangian in (25a) is given by

and the vectors and collect the real and imag-inary parts, respectively, of the entries of the matrix

.Update dual variables locally per area :

(26)


Fig. 3. First Test: IEEE 37-node feeder.

Fig. 4. Second test: (a) 10-node microgrid; (b) corresponding graph .

(27)

Furthermore, for any the iterates ,produced by are convergent, and

for all , withsub-matrices of the optimal solution of (P3).

Proof: See the Appendix.At each iteration, the LC of area receives from the LCs of its

neighboring areas matrices , and updates thelocal multipliers via (26)(27). These multi-pliers are locally stored at area , and they are not exchangedamong LCs (in contrast, multipliers are exchanged per iterationin [19]). Then, LC updates by solving , andtransmits to its neighboring areas .

V. NUMERICAL TESTS

The SDP-based solver is tested here using the following twonetworks, operating in a grid-connected mode: the IEEE 37-node test feeder [30] shown in Fig. 3; and the 10-node 3-phase network depicted in Fig. 4.

The optimization package 3, along with the interior-pointbased solver [32] are employed to implement the cen-tralized and distributed solvers in .The 4.8 kV network of Fig. 3 is an actual portion of

power a distribution network located in California; all thedemanded complex powers are spot loads, and the net-work loading is very unbalanced [30]. Compared to theoriginal scheme however, 7 DG units are placed at nodes

. Specifically, single-phaseconventional DG units supply a maximum real power of 50 kWper phase, and are operated at a unit power factor (PF); that is,

for all . Line impedances and shuntadmittances are computed based on the dataset in [30]. Finally,delta-wye conversions are performed whenever necessary. Asfor the network of Fig. 4(a), the line admittances are all setto

(see [30]), which gives rise to an unbalancedoperation of the network. Shunt admittances are neglected.Single-phase conventional DG units are placed at nodes

, and they can supply a maximum real power of50 kW per phase, at unit PF. All the loads are spot, and theloading is assumed balanced. Specifically, the active and reac-tive loads are set to kWand , respectively, oneach phase. The voltage magnitude at the PCC is 4.16 kV. Withthese two choices, the performance of the proposed approachcan be assessed for different network sizes, line characteristics,loading, and different network partitions shown next.The minimum and maximum utilization and service volt-

ages are set to andfor all nodes. Thus, voltage regulation is enforced withoutrequiring changes in the voltage regulator taps (as ine.g., [15]). Further, the voltage at the PCC is set to

. The average com-putational time required by to solve the centralizedproblem (P3) was 9.0 s and 0.3 s (machine with Intel Corei72600 CPU @ 3.40 GHz), which is significantly lowerthan the time required by commercial solvers for non-linearprograms (see e.g., [10]).Tables II and III list the real power drawn at the PCC

, the total power generated by the DG units, and the overall power losses and

costs of supplied power, when the cost (2) is employed. Thecosts of supplied power are set to /MW and

. Notice that mini-mizing (2) is tantamount to minimizing the power loss(1) when

and for all . In fact, it can be clearlyseen in Tables II and III that the power loss is minimized forthis choice of . Powers , , and remain the samewhen ; however, when (which holds duringthe late-night hours) the DG units reduce the generated activepowers; further, the power loss becomes significantly higherwhen DG units are not used at the maximum extent. Fig. 5 de-picts the active powers generated by DG units as a function of3[Online] Available: http://cvxr.com/cvx/


Fig. 5. Power generated by the DG units [kW].

, for the IEEE 37-node test feeder. It can be noticed thatthe DG units electrically close to the PCC are not utilized when

; on the other hand, DG 6 and 7 still operate at morethan of their maximum capacity.Interestingly, the rank of matrix was always 1. There-

fore, the globally optimal solutions of (P2) [and hence of thenonconvex (P1)] were always attained. In other words, nolower power losses or costs of supplied power can be attainedwith alternative OPF solution approaches. The rank of matrix

was greater than 1 for the IEEE 37-node test feeder when(two non-zero eigenvalues, with and

). Nevertheless, a rank-1 solution was readily ob-tained upon raising the voltage magnitude at the PCC from 1 to1.02 pu. This further prompts an analytical characterization ofthe feasibility region of (P1). Additional tests were performedon the IEEE 13-node feeder [30], and rank-1 matriceswere again always obtained (results are not reported here dueto space limitation).Convergence of the proposed distributed SDP solver is show-

cased in Fig. 6, where the 37-node feeder is partitioned as shownin Fig. 3. This partition resembles the case where laterals includemulti-facilities that are managed independently from the rest ofthe network [18]. The trajectories corresponding to

per iteration , are reported for dif-ferent values of the ADMM parameter , and are comparedwith the ones obtained by using the sub-gradient ascent-baseddistributed algorithm developed in [6] ( denotes the step size,which is assumed to be either constant, or monotonically de-creasing). It can be noticed that the proposed distributed solverexibit a considerably faster convergence than the one based onthe sub-gradient. Considering that the nominal voltage of thefeeder is 4.8 kV, in less then than 50 iterations the average gapbetween the entries of and is on the order of afew volts. Further, notice that the convergence rate is approx-imately the same for , , and . Theaverage computational time required by SeDuMi to solve eachsub-problem was of , and 2.7 seconds. Lower computa-tional times can be obtained by selecting areas of smaller size.Fig. 7 illustrates the convergence of the distributed SDP

solved when the considered 10-node network is partitioned asshown in Fig. 4. The ADMM-based method outperforms theone based on the sub-gradient here too. In this case, the gapsbetween the elements of matrices rapidly vanishwhen and after approximately 100 iterations.

TABLE IIRESULTS FOR THE IEEE 37-NODE FEEDER

TABLE IIIRESULTS FOR THE 10-NODE NETWORK IN FIG. 4

Fig. 6. Convergence of the ADMM, with feeder partitioned as in Fig. 3.

Fig. 7. Convergence of the ADMM, network partitioned as shown in Fig. 4.

In this case, the average time required by to solve eachsub-problem was 0.1 seconds.


VI. CONCLUDING REMARKSThe OPF problem was considered for microgrids operating

in an unbalanced setup. Inspite of the inherent non-convexity,the SDP relaxation technique was advocated to obtain a convexproblem. As corroborated by numerical tests, the main contri-bution of the proposed approach consists in offering the poten-tial to obtain the globally optimal solution of the original non-convex OPF. A distributed SDP solver was also developed byresorting to the ADMM. The distributed algorithm ensures scal-ability with respect to the microgrid size, robustness to commu-nication outages, and preserves data privacy and integrity.

APPENDIXProof of Lemma 1: Equation (7a) can be readily established

by noticing that .To prove (7b), notice first that the injected apparent powerat node and phase is given by

. Next, using , it follows that

, which can be equivalently rewritten as .Thus, the injected real and reactive powers can be obtained byusing, respectively, the real and imaginary parts of . Fi-nally, to prove (7a), define first the vector col-lecting the complex currents flowing from to on each phase,and notice that and

. Then, it follows that.

Proof of Lemma 3: The proof is provided for the multi-pliers ; same steps can be followed for the dual variables

.Suppose that , and notice that the optimization

problem (22) decouples per pair of neighboring areas;specifically, a number of sub-problems is to be solved withrespect to (wrt) the only pair of variablesfor (and clearly ). Neglecting ir-relevant terms, and setting the first-order derivative of

wrtto zero, it readily follows that the minimizer

is given by

(28)

Next, substituting (28) into [S3], and settingyields and

. It then follows by induc-

tion that also for all subsequent iterations.Proof of Proposition 3: Substituting (28) into [S3], and set-

ting , one can readily obtain (26)(27).Consider now . First, substitute (28) into (20), and dis-card irrelevant terms. Next, introduce an auxiliary variableto upper bound the squared norm of , and add the con-straint . Thus, (25a) follows by Schurs comple-ment. Indeed, it is certainly possible to introduce one singlevariable to upper bound the squared norm of the vector

, and derive a linear matrix

inequality similar to (25d) by using Schurs complement. Fi-nally, since the cost in (P6) is convex, and the constraint set isbounded and convex, every limit point of is an op-timal solution to (P6), as established in [25, Prop. 4.2]. Since(P6) and (P3) are equivalent under (As1) and (As2) hold, everylimit point of is also an optimal solution to (P3).

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Emiliano DallAnese (M11) received the LaureaTriennale (B.Sc. degree) and the Laurea Specialistica(M.Sc. degree) in telecommunications engineeringfrom the University of Padova, Italy, in 2005 and2007, respectively, and the Ph.D in informationengineering from the Department of InformationEngineering (DEI), University of Padova, in 2011.From January 2009 to September 2010 he was aVisiting Scholar at the Department of Electrical andComputer Engineering, University of Minnesota,USA. Since January 2011, he has been a Postdoc-

toral Associate at the Department of Electrical and Computer Engineering andDigital Technology Center, University of Minnesota, Minneapolis, MN, USA.

His research interests lie in the areas of statistical signal processing, com-munications, and networking. Current research include optimal energy man-agement in smart power distribution networks; robust, distributed, and spar-sity-aware statistical inference; and, monitoring and optimization of wirelesscognitive radio networks.

Hao Zhu (S07) received her B.S. degree fromTsinghua University, China, in 2006, and the M.Sc.and Ph.D. degrees from the University of Minnesota,Twin Cities, MN, USA, in 2009 and 2012, respec-tively, all in electrical engineering. She is currentlya Postdoctoral Research Associate with the Powerand Energy Systems Group at the University ofIllinois, Urbana-Champaign, IL, USA. Her currentresearch interests include power system monitoringand dynamics, distribution system operations andcontrol, and the smart grid.

Georgios B. Giannakis (F97) received his Diplomadegree in electrical engi neering from the NationalTechnical University of Athens, Greece, in 1981. andthe M.Sc. degree in electrical engineering, the M.Sc.degree in mathematics, and the Ph.D. degree in elec-trical engineering from the University of SouthernCalifornia (USC) in 1983, 1986, and 1986, respec-tively. Since 1999, he has been a Professor with theUni- versity of Minnesota, where he now holds anADC Chair in Wireless Telecommunications in theElectric and Computer Engineering Department and

serves as Director of the Digital Technology Center.His general interests span the areas of communications, networking and sta-

tistical signal processingsubjects on which he has published more than 350journal papers, 580 conference papers, 20 book chapters, two edited books, andtwo research monographs (h-index 103). Current research focuses on sparsityin signals and systems, wireless cognitive radios, mobile ad hoc networks, wire-less sensor, renewable energy, power grid, gene-regulatory, and social networks.He is the (co-)inventor of 21 patents issued, and the (co-) recipient of 8 bestpaper awards from the IEEE Signal Processing (SP) and Communications So-cieties, including the G. Marconi Prize Paper Award in Wireless Communica-tions. He also received Technical Achievement Awards from the SP Society(2000), from EURASIP (2005), a Young Faculty Teaching Award, and the G.W. Taylor Award for Distinguished Research from the University of Minnesota.He is a Fellow of EURASIP, and has served the IEEE in a number of posts, in-cluding that of a Distinguished Lecturer for the IEEE-SP Society.

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