1

Optimal Tap Setting of Voltage RegulationTransformers in Unbalanced Distribution Systems

Brett A. Robbins, Student Member, IEEE, Hao Zhu, Member, IEEE,and Alejandro D. Domínguez-García, Member, IEEE

Abstract—In this paper, we propose a method to optimally setthe tap position of voltage regulation transformers in distributionsystems. We cast the problem as a rank-constrained semidefiniteprogram (SDP), in which the transformer tap ratios are capturedby (i) introducing a secondary-side ‘virtual’ bus per transformer,and (ii) constraining the values that these virtual bus voltages cantake according to the limits on the tap positions. Then, by relaxingthe non-convex rank-1 constraint in the rank-constrained SDPformulation, one obtains a convex SDP problem. The tap positionsare determined as the ratio between the primary-side bus voltageand the secondary-side virtual bus voltage that result from theoptimal solution of the relaxed SDP, and then rounded to thenearest discrete tap values. To efficiently solve the relaxed SDP,we propose a distributed algorithm based on the AlternatingDirection Method of Multipliers (ADMM). We present severalcase studies with single- and three-phase distribution systems todemonstrate the effectiveness of the distributed ADMM-basedalgorithm, and compare its results with centralized solutionmethods.

Index Terms—Transformers, Relaxation Methods, DistributedAlgorithms, Decentralized Control, Power System Analysis Com-puting

I. INTRODUCTION

IN power distribution systems, tap-changing under-load(TCUL) transformers are commonly used for regulating

voltage. Traditionally, automatic voltage regulators (AVRs) areutilized to control the transformer tap position based on localvoltage measurements (see e.g., [1], [2]). While this AVR-based control is effective in achieving local voltage regulation,it is likely not optimal in terms of achieving certain overallsystem operational objectives, e.g., minimize power losses andvoltage regulation from some reference value. Motivated bythis, we propose a framework to determine the transformer tapratios in distribution systems that is optimal in some sense.

To address the problem described above, we formulate anoptimal power flow (OPF), where the transformer tap ratios areincluded as decision variables and the objective is to minimizethe total power losses (although, other objectives can beaccomplished as well). In the context of transmission systems,optimal transformer tap setting under the OPF frameworkhas been investigated for decades. For example, in [3], thetransformer tap positions are included as discrete variables inthe OPF problem, which results in a mixed-integer program

B. A. Robbins, H. Zhu, and A. D. Domínguez-García are with theDepartment of Electrical and Computer Engineering, University of Illi-nois at Urbana-Champaign, Urbana, IL, 61801. E-mail: {robbins3, haozhu,aledan}@ILLINOIS.EDU.

This material is based upon work supported by ABB under project“Distributed and Resilient Voltage Control of Distributed Energy Resourcesin the Smart Grid” (University of Illinois contract UIeRA 2013-2955-00-00).

(MIP) formulation. Unfortunately, the computational complex-ity of this formulation grows exponentially as the numberof transformers increases, and thus becomes intractable forlarge systems. To tackle this complexity issue, several papershave proposed to relax transformer tap positions to continuousoptimization variables, and then the solutions to the closestdiscrete valuables (see e.g., [3]–[5]). This alternative approachcan yield acceptable performance without incurring the addedcomplexity. However, all of these approaches are restrictedto standard OPF formulations, and are known to potentiallysuffer from the same convergence issues present in traditionaliterative solvers.

In this paper, we formulate the OPF problem that arisesin the context of voltage regulation in distribution systemsas a rank-constrained semidefinite program (SDP), and sub-sequently obtain a convex SDP problem from the originalSDP formulation by dropping the only non-convex rank-1constraint (see, e.g., [6]–[9]). In general, this rank relaxation isnot guaranteed to attain the global minimum, in particular formesh networks. Interestingly, it has been shown that undersome mild conditions, the optimal solution for the relaxedSDP-based OPF problem turns out to be of rank 1 for tree-structured networks, which are typical of radial distributionsystems [6]–[8]. In this sense, the rank relaxation scheme isactually guaranteed to attain the global optimum of the originalOPF problem. In addition to handling the OPF problem, theSDP-based approach also constitutes a very promising toolto tackle the non-convexity in other monitoring and controlapplications in power distribution systems.

It is possible to extend the SDP-based OPF approach toinclude the tap ratios of TCUL transformers by introducing avirtual secondary-side bus per transformer, which in turn willresult in additional constraints and decision variables [10]–[12]. However, the TCUL transformer model proposed in [11]is limited due to two issues: (i) the relaxed SDP problem couldfail to yield a rank-1 solution, and thus its global optimality isno longer guaranteed; and (ii) it is only applicable to single-phase systems. The first issue arises since the network isequivalently broken into two disconnected parts by introducingvirtual buses associated to each transformer, and the networkdisconnection would lead to multiple solutions of rank 2 [12].Although an optimal rank-1 solution could be recovered inthis case, the conditions for recovering rank-1 solutions areonly possible for single-phase systems [11], [12]. As forthe second issue, it is well known that distribution systemsare unbalanced; this motivates the formulation of the three-phase OPF problem [9]. As it will become more clear later

2

on, it is impossible to enforce the phase separation betweenthe primary- and secondary-side buses for the transformermodel in [11]. To address this issue, we propose an alternativetransformer model by including a highly resistive line betweenthe primary- and secondary-side buses. The proposed methoddoes not introduce additional complexity as compared to [11],but can successfully resolve the two aforementioned issues.Related to our approach, [13] and [14] discuss adding a verysmall resistance term to the model to handle a similar networkdisconnection issue due to the presence of ideal transformers.It is worth pointing out that such a method will maintainphase angles between primary- and secondary-side buses, butwill return incorrect power transfers; hence it is not deemedeffective for modeling transformers.

In order to solve the relaxed SDP problem described earlier,rather than using iterative solvers traditionally used to solvethe OPF problem, we are interested in fast distributed solversto handle the higher computational complexity introduced bythe SDP formulation. Distributed methods for solving theOPF problem have been proposed in power systems in avariety of contexts (see, e.g., [7], [9], [15] and the referencestherein). In particular, the Alternating-Direction Method ofMultipliers (ADMM) has been widely used as a simple, yetpowerful technique for solving distributed convex optimizationproblems [16]. This method has been successfully applied inpower systems for the dispatch of distributed generation anddeferrable loads [9], [15], as well as state estimation [17]. Inthis paper, we leverage the ADMM to solve the relaxed SDP-based optimal tap problem in a distributed fashion. The abilityto perform the optimization tasks in parallel can dramaticallyreduce computation time and complexity, especially for large-scale systems [15]. Note that the proposed distributed solvercan be implemented in a distributed environment or solvedcentrally and parallelized across several processors.

In this paper, we do not consider the real/reactive powersettings of distributed energy resources (DERs) as optimizationvariables since this is covered extensively in [7]–[9]. However,we could easily incorporate the setting of these devices intoour framework as decision variables and modify the constraintsand objective function as appropriate. In practice, we envisiona two time-scale architecture that categorizes devices as eitherslow or fast time-scale devices with the idea of controllingthem separately. Conventional voltage regulation devices, e.g.,TCUL transformers, would be considered slow time-scaledevices, whereas power electronic interfaced DERs would bethe latter. Then, given that fast (and uncontrolled) changes inDER active generation (consumption) might cause the voltageto deviate from some reference voltage, a second optimization[7] or a feedback control scheme [18], executed at regularintervals (e.g., every minute), could be utilized to determinethe active/reactive power setting of controllable DERs. Thesetwo time-scale separation ideas have been suggested in [7] andwe intend to explore them further.

The remainder of this paper is organized as follows. Sec-tion II introduces the system model and formulates the trans-former tap-setting optimization problem. In Section III, werewrite the OPF as a convex SDP, and introduce a modifiedtransformer model that will allow us to extend this framework

Ipt

Ist

+

Vpt

−

+

Vs′t

−

+

Vst

−

at : 1 yptst

(a) Classical transformer model.

Spts′tSpts′t

+

Vpt

−

+

Vs′t

−

+

Vst

−

yptstat : 1

(b) Equivalent transformer model.

Figure 1: Tap-changing under load transformer models

to a three-phase unbalanced system. The distributed solver isgiven in Section IV. Section V presents the case studies andconcluding remarks are presented in Section VI.

II. PROBLEM FORMULATION

In this section, we first introduce the standard TCUL trans-former model in the literature. Then, we describe the powersystem model adopted in this work and formulate the single-phase optimal power flow (OPF) problem that includes thetransformer tap positions as decision variables.

A. Standard Transformer Model

Figure 1(a) shows the standard model of the tth TCULtransformer located on the distribution line segment (pt, st)of some distribution system. Without loss of generality, weassume that the primary side of the transformer is closest to thefeeder head, and the admittance for the attached distributionline segment(s) and core losses (which are typically ignoredin distribution systems [1], [19]) are referred to the secondaryside as yptst . Given the tap ratio at, this model contains anideal transformer directly connected to bus pt and the virtualsecondary-side bus s′t such that Vpt = atVs′t . The tap ratioat is a discrete variable that typically takes on 33 possiblevalues {a−16, . . . , a0, . . . , a16}, uniformly distributed arounda0 to create a specified range around the rated voltage of thetransformer (which corresponds to a0). For instance, the tapscan move up and down 16 positions from the nominal tap ratioa0 = 1 with each step corresponding to 5/8% p.u. change.With the typical nominal voltage at 1 p.u., the tap ratio isbounded by a = a−16 = 0.9 and a = a16 = 1.1 [2]. Then,the line current and bus voltage relationships for the circuit inFig. 1(a), which depend nonlinearly on the tap ratio, are[

IptIst

]=

[yptst/a

2t −yptst/at

−yptst/at yptst

] [VptVst

]. (1)

Figure 1(b) shows an equivalent model to the one inFig. 1(a) that removes the ideal transformer entirely andaugments the network with the virtual bus s′t. We treat the

3

buses pt and s′t as though they are electrically disconnectedand introduce an additional variable Spts′t to account for thepower transferred across the removed ideal transformer, e.g.,buses pt and s′t will have a net injection of −Spts′t and Spts′t ,respectively. Unlike the model in (1), the primary/secondarypower is independent of the tap ratio. The key advantagesof this alternative model are that: (i) the tap ratio is onlynecessary in order to define the secondary-side bus voltage asVs′t = Vpt/at, and (ii) the admittance matrix for the equivalentcircuit will remain constant. Hence, in the remainder of thepaper we will use the primary/secondary power relationshipof the alternative transformer model in Fig. 1(b). Note thattransformers with fixed turn ratios are easily incorporated bymodifying the admittance matrix Y as described by (1).

B. Power System Model

Consider an n-bus power system that has r TCUL trans-formers. Let T := {1, . . . , r} denote the set of transformers.The set of buses incident to the primary-side of a transformeris defined as Np := {pt | t ∈ T }; similarly, for buses incidentto the secondary side, we have that Ns := {st | t ∈ T }.Additionally, the set of virtual buses (introduced in the equiv-alent transformer model in Fig. 1(b)) is defined as Ns′ :={s′t | t ∈ T }. The remaining m system buses define the setNb := {1, 2, . . . ,m}. Thus, the set of physical buses will be

N = Nb ∪Np ∪Ns, (2)

and the set of buses in the augmented network created byadding the virtual buses is

Na = N ∪Ns′ , (3)

where |Na| = n+ r.The edge-set that represents the set of distribution line

segments (which could contain conductors for single-, two-,or three-phase circuits) is E ⊆ Na ×Na such that (i, k) ∈ Eis the distribution line between buses i and k. The admittancematrix for the single-phase network Y ∈ C(n+r)×(n+r) willreflect the topology of the augmented network. Furthermore,we define the set Sps′ := {Spts′t | t ∈ T } as the powertransferred through the transformers. This balanced single-phase model will be extended to the general unbalanced three-phase case in Section III-D.

Finally, in order to include the equivalent transformer modelin Fig. 1(b), the power flow equations will be formulateddepending on the type of bus as follows:

1) No transformer incident to a bus: Consider the casewhen there are no transformers incident to bus i. Let Hi :={i}∪{k | (i, k) ∈ E} be the set of buses electrically connectedto bus i, which has no transformers incident to it. Then, thepower injected in bus i is

Si = Sgi − Sdi =∑k∈Hi

[Y ∗]ik ViV∗k , ∀i ∈ N\Np, (4)

where the generation Sgi and load Sdi are positive quantities.

2) Transformer incident to a bus: As shown in Fig. 1(b), wetrack the power across the transformer via Spts′t , and capturethe tap ratio with the voltage relationship Vs′t = Vpt/at. Ifbus i is incident to the primary-side of a transformer, then thecorresponding power flow equation becomes

Spt − Spts′t =∑k∈Hpt

[Y ∗]ptk VptV∗k , ∀t ∈ T , (5)

or, for the virtual secondary-side bus, we have that

Spts′t =∑k∈Hs′t

[Y ∗]s′tk Vs′tV ∗k , ∀t ∈ T . (6)

Note that the secondary-side buses in Ns are no longer directlyincident to transformers as in the circuit model in Fig. 1(a).

C. Transformer Tap Ratio Optimization

Next, we formulate an OPF problem, the solution of whichwill provide the tap settings of the TCUL transformer in thenetwork. As mentioned earlier, the discrete tap positions leadto an MIP formulation and its complexity grows exponentiallywith the number of TCUL transformers [3]. To tackle this,we relax the values that the discrete transformer tap ratioat ∈ {a, . . . a} can take, and allow at to take values on thecontinuous interval [a, a]. Once the optimal at is obtained, itwill be rounded to the closest discrete value in {a, . . . , a}.

Given some operational objective function f(V, a), whichwe describe in detail later in Section III-C, defined over thesystem voltages V ∈ Cn+r, and by letting tap ratios a ∈ Rrdenote the vector of transformer tap ratios, the OPF problemof interest can be formulated as follows:

minV, a, Sps′

f (V, a, Sps′) (7a)

such that ∑k∈Hi

[Y ∗]ik ViV∗k − Si = 0, ∀i ∈ N\Np (7b)∑

k∈Hpt

[Y ∗]ptk VptV∗k − Spt + Spts′t = 0, ∀t ∈ T (7c)

∑k∈Hs′t

[Y ∗]s′tk Vs′tV ∗k − Spts′t = 0, ∀t ∈ T (7d)

Vpt − atVs′t = 0, ∀t ∈ T (7e)

and

V ≤∣∣Vi∣∣ ≤ V , ∀i ∈ N (7f)

a ≤ at ≤ a, ∀t ∈ T . (7g)

The high granularity of the available tap positions enables thecontinuous tap-ratio representation approach to yield accept-able results without incurring the added complexity of an MIPformulation [3]–[5]. However, the optimization problem in (7)is still challenging due to the nonlinearity in the power flowequations, as captured by constraints (7b)-(7d). Hence, theensuing section will introduce additional relaxations to handlethese nonlinearities in the power flow model.

4

III. CONVEX RELAXATION

In this section, we first reformulate the non-convex OPFproblem in (7) into matrix form. Then, we review a modifiedtransformer model that we proposed in [12]; this model willallow us to handle three-phase unbalanced OPF. Finally, weuse the modified transformer model to develop the convexrelaxation of the matrix-based OPF formulation in (7).

A. Matrix-Based OPF Formulation

Motivated by the convex relaxation approach in [10]–[12],we will reformulate (7) as an equivalent semidefinite program(SDP). To this end, the complex power injection at bus i ∈ Nis given by

Si = Vi∑k∈Hi

[Y ∗]ik V∗k , (8)

where S, V ∈ Cn+r. We define W ∈ C(n+r)×(n+r) as

W = V V H =

V 21 · · · V1V

∗n+r

.... . .

...V ∗1 Vn+r · · · V 2

n+r

, (9)

where W is a positive semidefinite (PSD) matrix (W � 0)with rank 1. Interestingly, the complex power in (8) is linearlyrelated to the entries of W as follows:

Si = Tr (HiW ) , (10)

with Hi := Y HEi, where Ei := eieTi and ei is a vector with

all entries equal to zero except the ith one that is equal to one.Furthermore, the complex power flowing from bus i to k overline (i, k) ∈ E is given by

Sik = Tr (AikW ) , (11)

where Aik := −eTk Y

HeiEik and Eik := (ei − ek) eTi .

We remove the tap ratio from the voltage relationship in (7e)and (7g) by constraining the voltage on the secondary-side ofthe transformer relative to the primary side, i.e.,

a2 V 2pt ≤ V 2

s′t≤ a2 V 2

pt . (12)

Therefore, the equivalent matrix formulation of the problemin (7) is as follows:

minW�0, Sps′

f (W ) (13a)

such that

Tr (HiW )− Si = 0, ∀i ∈ N\Np (13b)Tr (HptW )− Spt + Spts′t = 0, ∀t ∈ T (13c)

Tr(Hs′tW

)− Spts′t = 0, ∀t ∈ T (13d)

and

V 2 ≤ [W ]ii ≤ V2, ∀i ∈ N (13e)

a2 [W ]ptpt ≤ [W ]s′ts′t≤ a2 [W ]ptpt , ∀t ∈ T (13f)

and[W ]pts′t

= [W ]s′tpt≥ 0, ∀t ∈ T (13g)

rank(W ) = 1. (13h)

The constraint (13g) ensures that Vpt and Vs′t have the samephase angle. Once the solution to (13) is obtained, the tap ratio

Spts′tSpts′t

+

Vpt

−

+

Vs′t

−

+

Vst

−

yptstzt

Figure 2: Non-ideal transformer model.

at of transformer t can be determined using the bus voltageratio as follows:

at =√V 2pt/V

2s′t

=√

[W ]ptpt/[W ]s′ts′t . (14)

B. Non-ideal Transformer ModelAs it will be discussed in detail in Section III-C, the

optimization problem in (13) can be relaxed to a convex oneby dropping the rank-1 constraint [7]–[9]. For distributionnetworks, it has been shown in [7] that this relaxation approachwould yield a rank-1 solution; therefore, achieving the globaloptimum of the original problem. As pointed out in [12],the transformer model in Fig. 1(b) results in two electricallydisconnected networks. Accordingly, it has been shown thatthe solution to the relaxed problem could be of higher-rank, albeit with no loss of optimality as compared to theoriginal problem. However, the higher-rank solution leads toan arbitrary phase angle difference between the primary-sidebus and downstream buses. Such phase angle ambiguity wouldsignificantly complicate the analysis of the three-phase system,since the angle separation among the three phases can nolonger be enforced at the virtual secondary bus.

The issue discussed here can be resolved by introducing themodified transformer model shown in Fig. 2, where we placean impedance zt between pt and s′t of the ideal transformerintroduced earlier to ‘re-connect’ the network. Choosing anappropriate value of zt will ensure that the power flow inthe modified model almost mimics that of an electricallydisconnected network. As detailed later, this modificationwould help maintain the phase angle consistency on both sidesof the transformer; i.e., θpt ≈ θs′t . This is highly attractivesince it allows one to solve an equivalent convex formulationof the original OPF problem in (7), while enforcing the correctphase shift for the transformers. This is especially importantto extend the OPF framework to three-phase systems wherethe phase separation is lost with the disconnected network.

As discussed in detail in Section V-C, we found via numer-ical simulations that there can be a large range of values for ztthat yield solutions that are sufficiently accurate. However, if|zt| is too small, the augmented network admittance matrix Ycould be problematic as the entries corresponding to (pt, s

′t)

become much larger compared to the rest. Although a small|zt| maintains θpt ≈ θs′t , the power flow through zt willbecome comparable to Spts′t , and thus the system power flowwould be different from the original ideal transformer model.On the other hand, if |zt| becomes too large, then the systembehavior begins to mirror the original disconnected case withideal transformers. The latter scenario will result in a solution

5

W that has a rank greater than one. The value of |zt| needsto be chosen within a specific range, which can vary basedon the system and the gains in the cost function. Interestingly,all of our numerical simulations corroborated that a zt witha resistance value of several orders of magnitude (around 2-4 orders for our test systems) larger than the neighboringdistribution line segments yielded the best results.

C. Rank-Relaxed Convex OPF

We apply the modifications to the transformer model asdescribed in Section III.B and relax the rank-1 constraint in(13) to get a rank-relaxed convex OPF of the form:

minW�0, Sps′

f (W ) (15a)

such thatTr(HiW

)− Si = 0, ∀i ∈ N\Np (15b)

Tr(HptW

)− Spt + Spts′t = 0, ∀t ∈ T (15c)

Tr(Hs′tW

)− Spts′t = 0, ∀t ∈ T (15d)

and

V 2 ≤ [W ]ii ≤ V2, ∀i ∈ N (15e)

a2 [W ]ptpt ≤ [W ]s′ts′t≤ a2 [W ]ptpt , ∀t ∈ T , (15f)

where H incorporates the non-ideal transformer model and(13g) is dropped since the network is connected. The rank-relaxed SDP formulation in (15) is guaranteed to achieve theglobal optimality of the non-convex tap setting problem withthe rank constraint.

The objective function f(W ) includes a term that capturessystem losses, which is necessary to ensure that a rank-1solution can be obtained from the relaxed problem [7]–[9],[11]. It may also include additional terms to capture voltagetracking objectives and power factor targets. The discussionabove can be formalized by considering an objective functionof the form

f(W ) = f0(W ) +∑i

fi(W ), (16)

where

f0 (W ) =∑i∈Na

∑k∈Hi

Re{

Tr(AikW

)}, (17)

which captures the total losses of all distribution line segments.The additional penalty terms in (16),

∑i fi(W ), could be

chosen so as to achieve other objectives of interest; next, wediscuss a few possible choices.

If a given a network has considerable distributed generation,minimizing line losses may not reduce the total demand at the(sub)transmission substation. This can be easily addressed byincluding a penalty term in (16) of the form

f1(W ) = αRe{

Tr(HfdrW

)}, (18)

where α is a positive weighting factor.

Additionally, to minimize voltage magnitude deviationsfrom a specified V refi , we can include a penalty term in (16)of the form

f2(W ) =∑i∈N

wi

((V refi )2 − [W ]ii

)2

, (19)

where {wi} are the positive weighting factors per bus i [20].The weights themselves should be chosen based on: (i) thedistance between the bus and the feeder, and (ii) buses thatare prone to voltage violations. Intuitively, buses near theend of the (sub)laterals should be weighted more than thesenear the feeder in traditional radial distribution networks ofunidirectional power flow.

Finally, utilities aim at operating distribution systems witha unity power factor at the feeder head. Motivated by this, thereactive reactive power injection to the feeder head bus can bepenalized by

f3(W ) = β Im{

Tr(HfdrW

)}, (20)

where β is a positive weighting factor.Note that the candidate cost functions introduced are sepa-

rable among all the buses, this will facilitate the developmentof the distributed solver as detailed soon in Section IV.

D. Extension to Three-Phase Unbalanced Systems

So far, we assumed balanced operation, which reduced thesystem model to a per-phase equivalent; however, distributionsystems are inherently unbalanced with untransposed distri-bution lines and have single-, two-, and three-phase radialfeeds; therefore, a three-phase system extension of the ideasdiscussed so far is well motivated. The authors in [9] discussextending the SDP relaxation OPF approach to three-phaseunbalanced systems; our work focuses on incorporating three-phase TCUL transformers into such formulation. For a single-phase system, a rank-1 exact solution can be recovered eventhough the relaxed OPF in (13), the formulation of which isbased on the ideal transformer model in Fig. 1(b), has higher-rank solutions. The additional constraints on the transformer(13g) will ensure a zero phase angle difference between theprimary- and secondary-side bus voltages of that particularphase; however, the phase angles of buses downstream of thesecondary-side bus are not dependent on the primary sideof the transformer. In a multi-phase network, this impliesthat there is no constraint that enforces the angle differencebetween phases, i.e., θa− θb ≈ 120◦. We can maintain thisphase separation by:(i) reconnecting the network by using the non-ideal trans-

former model we propose in Section III-B, or(ii) constraining the off-diagonal entries of the submatrices

Wptpt and Ws′ts′t

associated with the primary and sec-ondary sides of the transformer.

The issue with (ii) is that the constraints turn out to be highlynon-linear in W, which impedes us from incorporating theminto the SDP OPF formulation in (15), which is why we choseto pursue solution (i).

6

zta

ztb

ztc

[yptst]

Vpta

Vptb

Vptc

Vsta

Vstb

Vstc

Sptas′ta

Sptbs′tb

Sptcs′tc

Sptas′ta

Sptbs′tb

Sptcs′tc

Figure 3: Equivalent three-phase transformer model.

Consider the three-phase TCUL model in Fig. 3; we willfollow a configuration similar to the one used for the non-ideal transformer model in Section III-B. The core losses of thetransformer will be neglected [1], and we will assume that eachphase can independently regulate its secondary-side virtual busvoltage. This will be equivalent to a collection of single-phasetransformers that we describe using the model in Fig. 2, whichare coupled by the secondary-side distribution line admittanceyptst ∈ C3×3. Note that there is no mutual impedance betweenthe primary-side bus and the virtual secondary-side virtual bus.Thus, we can optimize every tap individually and maintainthe proper phase separation. We will use Via , Vib , Vic todistinguish the voltages phasors for each phase at bus i. Wealso define a vector V3φ that will include the voltage phasorsfor all the buses in the network that has at most 3× (n + r)elements if every bus has three-phase circuits. The bus voltagesubindex allows one to maintain the notational consistencywith the earlier single-phase (one-line) case. The only differ-ence lies in that the dimension of the counterpart vectors toaccommodate the three phases. For example, the three-phaseline admittance is a block yik ∈ C3×3, as compared to yik ∈ Cfor single-phase cases. This way, all the analysis and problemformulation so far carries over by defining W3φ = V3φV

H3φ .

The power flow equations will remain unchanged wherewe have an (in)equality constraint per phase; however, themutual impedances of the untransposed lines makes it morecomplicated to compute the line losses compared to (17). Tothis end, let ei ∈ N|V3φ| be the vector with all entries equal tozero except the entries corresponding to each phase present atbus i are set to one. Define the matrices G and K as

G = diag(e1)Y Hdiag(e2) + diag(e2)Y Hdiag(e1) (21)

and

[K]ij =

−1, [e1p ]i = [e2p ]j = 1, p = {a, b, c}1, i = j,0, otherwise,

(22)

where eip is the vector ei conditioned on phase p, i.e., thevector contains a single nonzero entry corresponding to theentry for phase p of bus i. Thus, we update the line losscoefficient matrix Aik with

Aik = GK (23)

to capture the total losses across each distribution line segment.

IV. DISTRIBUTED SOLVER

It is well known that centralized algorithms for solving theSDP problem in (13) are not suitable for large systems (see,e.g., [16]). To address this issue, we propose the use of theAlternating Direction Method-of-Multipliers (ADMM), whichallows for an efficient distributed solution to the convex SDPproblem in (15). The ADMM is proven to be a powerfuldistributed optimization method and offers many benefits[16]. In particular, with ADMM, the complexity of the SDPproblem scales with the sub-area size rather than with the fullnetwork size, and the communication architecture is simplerthan that of a centralized scheme. Suppose we partition thesystem into two areas; Fig. 4 shows a topographical viewof the submatrices W (1), W (2) and the boundary conditionsW (1,2) = W (2,1). The computational complexity per iterationusing the popular interior point method for our SDP problem(15) scales with the fourth-order in the size of matrix W ,or equivalently the number of system buses (see, e.g., [21]).For large systems, small partitions will provide significantsavings in the number of optimization variables as depictedby the empty off-diagonal blocks in Fig. 4; however, theincreased number of boundary conditions will require moresuper-iterations for convergence to a solution.

ADMM iteratively minimizes the augmented Lagrangianover three types of variables: (i) the primary variables, i.e., thebus voltages and transformer power transfers; (ii) the auxiliaryvariables that are used to enforce boundary conditions amongneighboring areas; and (iii) the multipliers for dualizing therelaxed problem. The Lagrangian is designed to be separablerelative to each type of variable so that we can cyclicallyminimize with respect to one variable type while fixing theothers. This allows us to solve the problem distributedly andachieve convergence to the same solution obtained with acentralized solver [16].

We begin by partitioning the system into P := {1, 2 . . . , L}areas such that ∪Li=1A(i) = Na, ∩Li=1A(i) = ∅, and|A(i)| ≥ 1 for all i. To include the coupled buses, eacharea A(i) needs to be augmented, and the extended area isA(i) := A(i) ∪ {y | (x, y) ∈ E , x ∈ A(i), y ∈ A(j), i 6= j}.Then, the neighbors of area A(i) are defined as M(i) :=

{j | A(i) ∩ A(j) 6= ∅}. For area A(i), let W (i) ∈ C|A(i)|×|A(i)|

denote the corresponding local matrix for the outer productof the bus voltages, e.g., the |A(i)| × |A(i)| submatrix of Wcorresponding to area i.

To enforce consistency between partitions, boundary con-ditions are required to constrain the submatrices W (i,j) =W (j,i) ∈ C2×2 for single phase (C6×6 for three phase)

W (1)

W (1,2)

W (2)

W =

Figure 4: Partitioned system variables.

7

Lc (·) =∑i∈P

{f (i)

(W (i)

)+

∑k∈M(i)

[Tr((

Γ(i, j))T (

Re{W (i, j)

}−R(i, j)

))+c

2

∥∥∥Re{W (i, j)

}−R(i, j)

∥∥∥2

F

+ Tr((

Λ(i, j))T (

Im{W (i, j)

}−X(i, j)

))+c

2

∥∥∥Re{W (i, j)

}−X(i, j)

∥∥∥2

F

]} (25)

associated with the overlap between neighboring areas. Wedefine the auxiliary variables for the boundary conditions ofthe local optimization as E(i,j), F (i,j) ∈ R2×2 (similarly R6×6

for the three-phase case). These variables E(i,j) (F (i,j)) areused to enforce the real (imaginary) part of the submatrixequality boundary condition on bus voltages. Note that withoutthe auxiliary variables, our problem would not be separablewith respect to each W (i).

For each area we also define B(i), which captures the setof sub-matrices that satisfy the local power flow and voltageconstraints described in (15b)-(15f) for area A(i). Given thatall of the cost functions in Section III-C are separable per area,we can rewrite the global minimization problem in (15) as

minW (i)�0

∑i∈P

f (i)(W (i)

)(24a)

such thatW (i) ∈ B(i), ∀i ∈ P, (24b)

and

Re{W (i,j)

}− E(i,j) = 0, ∀j ∈M(i), (24c)

Im{W (i,j)

}− F (i,j) = 0, ∀j ∈M(i), (24d)

whereE(i,j) − E(j,i) = 0, ∀j ∈M(i), (24e)

F (i,j) − F (j,i) = 0, ∀j ∈M(i). (24f)

We leverage the relations in (24e) and (24f) when formulatingthe update rules of the distributed algorithm, but they are notenforced directly in the optimization problem. Note that theprimary- and virtual secondary-side buses of a transformercannot reside in two different areas. In our system model,we assume that the transformer t is attached at pt, andthat {pt, s′t} are effectively the same physical bus; thus, itsbehavior is completely captured by B(i) where pt ∈ A(i), andthe boundary conditions will be enforced between buses s′tand st.

A. Augmented Lagrangian

Let Γ(i,j),Λ(i,j) ∈ R2×2 denote the Lagrange multipliersassociated with the equality constraints in (24c) and (24d),respectively, where c > 0 is the penalty coefficient. Theaugmented Lagrangian function for (24) is given in (25) atthe top of this page, and is clearly separable amongst the threegroups of variables such that

Lc(·) =∑i∈PL(i)c (·). (26)

Then, we can cyclically optimize the augmented LagrangianLc(·) with respect to one of the groups of variables while

holding the others constant with the following three-stepupdate rule for the kth iteration:[S1.] Primal Variables Update: We take the infimum of Lc(·)with respect to the primal variables, and update them as

W (i)[k] = arg minW (i)∈B(i),W (i)�0

L(i)c (·), (27)

which is dependent of the dual variables Γ(i, j)[k − 1] andΛ(i, j)[k − 1], and the boundary conditions E(i, j)[k − 1] andF (i, j)[k − 1].[S2.] Auxiliary Variables Update: Recall that E(i, j) = E(j,i)

and F (i, j) = F (j,i), also note that Λ(i, j) = −Λ(j,i); then

∇E(i, j)Lc(·) = Re{W (i, j) +W (j,i)

}− 2E(i, j) = 0, (28)

∇F (i, j)Lc(·) = Im{W (i, j) +W (j,i)

}− 2F (i, j) = 0. (29)

We update the auxiliary variables with

E(i, j)[k] =1

2Re{W (i, j)[k] +W (j,i)[k]

}, (30)

F (i, j)[k] =1

2Im{W (i, j)[k] +W (j,i)[k]

}, (31)

for j ∈M(i).[S3.] Multipliers Update: The gradient for the L(·) withrespect to the dual variables is

∇Γ(i, j)L(i)c (·) = Re

{W (i, j)

}− E(i, j), (32)

∇Λ(i, j)L(i)c (·) = Im

{W (i, j)

}− F (i, j). (33)

We initialize all the multipliers to zero; then, we solve the dualvariables using an ascent method and apply (30)–(31). Thus,for j ∈M(i), the update rules for the dual variables are

Γ(i, j)[k]=Γ(i, j)[k−1]+c

2Re{W (i, j)[k]−W (j,i)[k]

}, (34)

Λ(i, j)[k]=Λ(i, j)[k−1]+c

2Im{W (i, j)[k]−W (j,i)[k]

}. (35)

Note that it follows naturally that Γ(i,j) = −Γ(j,i) andΛ(i,j) = −Λ(j,i).

Although Steps S1-S3 are formulated for the single-phaseproblem, they can easily be extended to solve three-phaseunbalanced systems as well by accounting for all phasevoltages per bus. The detailed extension is omitted here forbrevity; however, the numerical results presented in the nextsection include this extension.

8

Feeder

651

650

632

693

671

680

645646 633 634

684611

652

675

A(1)

A(2)

Figure 5: 15-Bus unbalanced distribution system.

V. CASE STUDIES

In this section, we illustrate the ability of the distributed,ADMM-based algorithm proposed in Section IV to optimallyset TCUL tap positions for both single- and three-phase casesof a 15-bus network. We also demonstrate the effectivenessof the algorithm on the IEEE 123-bus test system [19]. In allour studies, the voltage magnitude inequality constraints forall cases are limited to 1 p.u. ± 4.8%, rather than the common± 5% to account for discrepancies associated with roundingto nearest discrete tap position.

We performed the simulations in MATLAB using the CVXpackage [22] with the symmetric cone solver SeDuMi [23].This software package was used to solve the centralizedproblem and to update the primal variables W (1), . . . ,W (L)

in step S1 of the distributed algorithm.While in Section III-C we provided several penalty terms

for different performance objectives, the cost function used inthe case studies in Sections V-A and V-B only considers thedistribution line losses as defined in (17), i.e., f(W ) = f0(W ).

A. 15-Bus Distribution System

We begin with the 15-bus network shown in Fig. 5, whichwe derived from the IEEE 13-bus—a three-phase, unbal-anced distribution system, (see, e.g., [1], [19]). The systemhas a three-phase voltage regulation transformer betweenbuses 650 and 632. The rest of the system contains single-,two-, and three-phase sublaterals. Buses 650 and 651 wereadded between the feeder and the transformer so that thetransformer was not directly connected to the slack bus. Bus693 was added to account for the distributed load along line(632, 671), and bus 692 was removed since it corresponds toa closed switch connected between buses 671 and 675.

In Fig. 5, busses are color coded for areasA(1) andA(2); theextended areas A(1) and A(2) are distinguished by the dashedlines circling the areas. The overlap occurs at buses 632 and693 where W (1,2) ∈ C2×2 for the 14-bus single-phase caseand W (1,2) ∈ C6×6 for the following three-phase case.

1) Single-Phase Results: For the single-phase system, weisolate phase C from Fig. 5 since it is the dominant phaseof the 15-bus network, and create a 14-bus single-phase case

Table I: SINGLE-PHASE 14-BUS NETWORK

Centralized Distributed ExhaustiveOPF OPF Search

f(W ) 0.559 0.466, 0.094 0.554

|Vpt | 1.007 1.007 1.007

|Vs′t| 1.060 1.060 −

|Vst | 1.048 1.048 1.048

Spts′t

11.05 + j4.01 11.04 + j4.01 −

St 0.005 0.005 −

Tap −8 −8 −8

CPU Time 0.8 s − 0.1 s

50 100 150 200 250 3000

0.2

0.4

0.6

Iteration

f(W

)

f(W )

f (1)(W (1))

f (2)(W (2))

(a) Objective Functions

50 100 150 200 250 300

0.95

1

1.05

Iteration

Tap

Rat

ioContinuous

Discrete

(b) Tap position

100 200 300 400 500 600

0.95

1

1.05

Iteration

Tap

Rat

io

c = 10 c = 30

c = 50 c = 70

c = 100

(c) Convergence for Various Penalty Parameters

Figure 6: 14-bus, single-phase results.

that excludes bus 652 from the network topology since phaseC is not present on that bus. The results obtained using: (i) acentralized algorithm, (ii) our distributed algorithm, and (iii)an exhaustive search where we enumerate all of the possibletap ratio combinations are shown in Table I. All three methodsreturn the same optimal tap position. The exhaustive searchuses an ideal transformer model and the difference in thecost function f(W ) compared to the other two methods isdue to the loss St through the non-ideal transformer model.Figures 6(a) and 6(b) show the evolution of the area costfunctions and the corresponding tap ratio for c = 60. Noticethat the optimal position occurs near iteration 250 where theglobal f(W ) remains relatively unchanged after iteration 125,so there are several tap ratios that will result in acceptable

9

200 400 600 800 1,0000

0.5

1

Iteration

f(W

)

f(W )

f (1)(W (1))

f (2)(W (2))

(a) Objective Functions

200 400 600 800 1,000

0.95

1

1.05

Iteration

Tap

Rat

io

tatbtc

(b) Tap position

Figure 7: 15-bus, three-phase results.

solutions. Figure 6(c) shows the effect of changing the penaltyparameter c. For the 14-bus network, the fastest convergenceoccurs when c ≈ 30.

2) Three-Phase Results: In the centralized case, the 15-bus, three-phase unbalanced system problem will have 1763optimization variables since W ∈ C41×41. In contrast, thedistributed case has a 21.8% reduction for a combined total of1379 variables (910 for area 1 and 469 for area 2). The resultsfor the three-phase case are listed in Table II with the progressof the distributed algorithm displayed in Fig. 7. The solutionsto the relaxed problem (incorporating the non-ideal trans-former model) obtained with the centralized and distributedsolvers yielded the same tap positions of {−7,−8,−6}. Theexhaustive search solution yielded a different result for phaseA, with the final tap positions being {−6,−8,−6}. However,if we check the centralized and distributed result by solving thepower flow equations, we found the solution to be acceptable.Similar to the single-phase case, there are several solutionsaround the selected tap positions with nearly the same costsfor the objective function.

From the results in Table I, we can see that the exhaustivesearch method for a single tap resulted in a faster computationthan the relaxed centralized problem, with CPU times of 0.1 sversus 0.8 s, respectively. In the three-phase case, the addedcomputational complexity introduced by the dimensional in-crease associated with the additional phases resulted in anincrease of the CPU time for the exhaustive search solution to107.2 s, while the relaxed centralized case increased slightlyto 2.0 s. The ADMM-based solution that we proposed wastested using a serial implementation on a single-core; thus,we intentionally did not include the CPU time for this case,we refer the reader to [9], [15], [16] for computational benefitsof ADMM.

B. 123-Bus Distribution SystemFigure 8 shows the one-line diagram for the IEEE 123-bus,

three-phase distribution system, which includes four three-

Table II: THREE-PHASE 15-BUS NETWORK

Centralized Distributed Exhaustive PowerOPF OPF Search Flow

f(W ) 1.170 1.047, 0.123 1.170 1.167

1.028 1.028 1.028 1.028|Vpt | 1.017 1.017 1.017 1.017

1.023 1.023 1.023 1.023

1.073 1.073 − −|Vs′t| 1.073 1.073 − −

1.064 1.064 − −

1.048 1.048 1.043 1.050|Vst | 1.044 1.044 1.042 1.042

1.043 1.043 1.041 1.041

8.5 + j4.1 8.5 + j4.1 − −Spts

′t

9.6 + j4.7 9.6 + j4.7 − −9.8 + j3.3 9.8 + j3.3 − −

St 0.004 0.004 − −

Tap -7, -8, -6 -7, -8, -6 -6, -8 , -6 -7, -8, -6

CPU Time 2.0 s − 107.2 s −

Feeder

119

120

63

75

42

91

8 1110

1215

16

13

17

18

19

54 55 56 57 59

14

21

2426

28

32 33 34

2022

2325

2729

31 3035

39 37 40 41

384345

47 50 51 52 53

36

4244

4649

48

62 6061

58

64

65

6667

68

72

104

106

110

113

9395

97

92949698

99

69

72

77

70 7376

75

76 78 79

100101

102103105

107108109

111112

114115 116 117 118

80

8190

91 84

85

86 88

87

898382

#1

#3

#2

#4

Figure 8: IEEE 123-bus distribution system.

phase voltage regulation transformers [19]; we also dividethe system into six areas. This is a comprehensive systemthat is mostly unbalanced and contains overhead/undergrounddistribution line segments with single-, two-, and three-phasebranches. We set the voltage at the feeder to 1.01 p.u. sothat there are 18 buses experiencing an under-voltage with theworst being 0.937 p.u. on bus 118. Note that for this particularcase, we did not include results for an exhaustive search of thetransformer tap settings since the number combinations withthe hardware available is 339. Therefore we can not obtaina solution in a reasonable amount of time. In contrast, thecentralized convex relaxation took 71.5 s of CPU time to reachthe solution.

In this case, we have that W ∈ C259×259 and the centralizedrelaxed OPF has 268,854 optimization variables. The central-ized algorithm yielded a solution such that the tap positionsare in the neutral position for regulators 1 and 2, are set to{−5,−1,−3} for regulator 3, and are set to {−2,−1,−1}

10

for regulator 4. The minimum voltage is raised to 0.985 p.u.and the network losses are 0.8286 p.u. The bus voltage errorbetween the continuous and the rounded discrete tap positionshas an average of 6.63 × 10−4 p.u. and a standard deviationof 0.0017 p.u. with a maximum error of 0.0029 p.u.

In contrast, the distributed algorithm results in 59,436optimization variables, which is a 77.89% reduction from thecentralized scheme. The distributed method returned slightlydifferent results for the tap positions: regulators 1 and 2taps are set to the neutral position, regulator 3 taps are setto {−4,−1,−2}, and regulator 4 taps are set to {0, 1, 0}.The minimum voltage in this configuration is 0.965 p.u. onbus 118. However, the minimum computed network lossesfor this configuration are 0.8303 p.u., which represents a0.3% difference from the centralized result. For this particularnetwork and loading, the tap positions of regulator 3 impactthe network conditions the most since this transformer isconnected to the feeder. As the number of transformers onthe network increases, there could be multiple solutions thatminimize the cost function.

C. Impact of the Choice of ztNext, we explore the impact of the zt on the solution of the

15-bus, unbalanced three-phase optimization; the results arecaptured in Fig. 9. In each subfigure, the thick red vertical linerepresents the resistance of the neighboring distribution linesegments with the vertical lines to the left and right representapproximately an order of magnitude less and every two ordersof magnitude larger, respectively.

The objective function for this particular case is of the form

f(W ) = f0(W ) +∑i∈N

(1− [W ]ii)2, (36)

where the summation term penalizes the voltage deviationsfrom 1 p.u. Figure 9(a) shows the optimization variables Spts′tfor the power transferred through the transformer and thephase difference θpt −θs′t between the primary and secondarysides of the transformer versus the impedance of zt. InFig. 9(b) we plot the power transferred through zt and the rankof the W returned from the convex optimization. In Fig. 9(c)we show cost function f(W ) and the normalized percent error

e (W,V (at)) = 100

√√√√∑i∈N

(√[W ]ii − Vi(at)Vi(at)

)2

, (37)

which is determined by the difference between the voltagemagnitudes recovered from W (zt) and the voltage magnitudesshown in Fig. 9(d) that are computed from the power flow withrounded discrete tap positions.

In Section III-B, we proposed to choose |zt| several ordersof magnitude larger than the impedance of the adjacent distri-bution line segments. For the 15-bus system, the optimizationsolution tends to match the power flow results better for smallvalues of zt, the desired results were captured when zt ischosen such that |zt| is two to four orders of magnitudelarger than the magnitude of the impedance of neighboringdistribution lines. Within this interval, the power transferred

10−2 100 102 104 106

0

5

10

Orders of Magnitude of zt

Re{ S

pts′ t

}[p

.u.]

-3

0

3

(θpt−θ s

′ t)

[deg

]

Re {Sps}∆θ

(a) Power transferred through Spts′t

and phase mismatch betweenpt and s′t versus |zt|.

10−2 100 102 104 106

-1

0

1

Orders of Magnitude of zt

Re{S

t}[

p.u.

]0

1

2

3

4

5

Ran

k(W

)

Re {St}Rank(W )

(b) Power transferred through zt and the rank of W .

10−2 100 102 104 1060

3

6

9

Orders of Magnitude of zt

e(W

,V(a

t))

1.2

1.3

f(W

)

e (W,V (at))

f(W )

(c) Normalized percentage of the voltage error and the objectivefunction in the form f(W ) = f0(W ) + f2(W ).

10−2 100 102 104 106

0.95

1

1.05

Orders of Magnitude of zt

|V|[

p.u.

]

(d) Power flow results for bus voltages from rounded tap positions.

Figure 9: The effects of the choice of zt on the optimizationresults for the 15-bus unbalanced distribution system.

11

through zt converges to zero and the system values in theoptimization converge to a steady-state. We also see that therank of W is still one with a subtle discrepancy in the angledifference across the transformer of less than 3 degrees and anormalized voltage magnitudes error of less than 2% betweenthe power flow results and the resulting W (zt). After fourorders of magnitude difference, the system behaves as thedisconnected case and we are no longer able to accuratelyrecover the solution. Note that these results are for this specificcase. In other scenarios we have found that by choosing|zt| approximately two orders of magnitude larger than theneighboring distribution line segments is a good initial value.

VI. CONCLUDING REMARKS

In this paper, we developed a method to optimally set,via a distributed ADMM-based algorithm, tap positions ofvoltage regulation transformers in distribution systems. Wedemonstrated the applicability of this method via numericalexamples involving single- and three-phase test systems.

Future work will include improving the convergence ofthe distributed ADMM-based algorithm. We also intend toapply this distributed optimization approach to a system-wide voltage tracking control scheme. In this case, we wouldperiodically dispatch the TCUL transformers and then drawupon our previous work in [18] to handle fast-transients thatcan be corrected with reactive power support.

APPENDIX

A. Notation

T Set of transformers indexed {1, . . . , r}a Set of transformer tap ratios indexed {a1, . . . , ar}Np Set of bus incident to the primary side of a trans-

former indexed {p1, . . . , pr}Ns Set of bus incident to the secondary side of a

transformer indexed {s1, . . . , sr}Ns′ Set of bus incident to the virtual secondary side of a

transformer indexed {s′1, . . . , s′r}Nb Set of buses not incident to a transformerN Set of physical busesNa Augmented set of buses that includes Ns′Sps Set of transformer power transfers indexed

{Sp1s′1 , . . . , Sprs′r}zt Impedance added for the non-ideal transformer

modelW Semi positive-definite matrix of bus voltagesW (i,j) Submatrix of W (i) boundary conditions with neigh-

bor jE(i,j) Auxiliary variable associated with Re{W (i,j)}F (i,j) Auxiliary variable associated with Im{W (i,j)}P Set of partitions indexed 1, . . . , LA(i) Set of buses in the area defined by i ∈ PA(i) The extended area containing neighboring busesM(i) The partitions neighboring iB(i) The set of power flow constraints and inequalities for

partition iLc(·) The augmented Lagrangian with penalty parameter cΓ(i,j) Boundary condition dual variable for Re{W (i,j)}Λ(i,j) Boundary condition dual variable for Im{W (i,j)}

REFERENCES

[1] W. H. Kersting, Distribution System Modeling and Analysis. New York,NY: CRC Press, 2001.

[2] P. W. Sauer and M. A. Pai, “A comparison of discrete vs continuousdynamic models of tap-changing-under-load transformers,” in Proc. BulkPower System Voltage Phenomena III: Voltage Stability, Security, &Control, Davos, Switzerland, Aug 1994.

[3] W.-H. E. Liu, A. D. Papalexopoulos, and W. F. Tinney, “Discreteshunt controls in a newton optimal power flow,” IEEE Trans. on PowerSystems, vol. 7, no. 4, pp. 1509–1518, 1992.

[4] E. Acha, H. Ambriz-Perez, and C. R. Fuerte-Esquivel, “Advancedtransformer control modeling in an optimal power flow using newton’smethod,” IEEE Trans on Power Systems, vol. 15, no. 1, pp. 290–298,2000.

[5] M. M. Adibi, R. A. Polyak, I. A. Griva, L. Mili, and S. Ammari,“Optimal transformer tap selection using modified barrier-augmentedlagrangian method,” in Proc. IEEE Power Engineering Society GeneralMeeting, vol. 2, 2003.

[6] S. Bose, D. F. Gayme, S. H. Low, and K. M. Chandy, “Optimal powerflow over tree networks,” in Proc. of Allerton Conf. on Comm., Control,& Computing, 2011, pp. 1342–1348.

[7] B. Zhang, A. Lam, A. Domínguez-García, and D. Tse, “An optimal anddistributed method for voltage regulation in power distribution systems,”IEEE Trans. on Power Systems, 2014.

[8] J. Lavaei, D. Tse, and B. Zhang, “Geometry of power flows in treenetworks,” in Proc. IEEE Power and Energy Society General Meeting,2012, pp. 1–8.

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

[10] R. A. Jabr, “Optimal power flow using an extended conic quadraticformulation,” IEEE Trans. on Power Systems, vol. 23, no. 3, pp. 1000–1008, Aug 2008.

[11] J. Lavaei, “Zero duality gap for classical opf problem convexifiesfundamental nonlinear power problems,” in Proc. American ControlConference (ACC), 2011, pp. 4566–4573.

[12] B. A. Robbins, H. Zhu, and A. D. Dominguez-Garcia, “Optimal tapsettings for voltage regulation transformers in distribution networks,” inProc. 2013 North American Power Symposium (NAPS), 2013, pp. 1–6.

[13] J. Lavaei and S. H. Low, “Convexification of optimal power flowproblem,” in Proc. 48th Allerton Conf. on Communication, Control, &Computing, 2010, pp. 223–232.

[14] J. Lavaei and S. H. Low, “Zero duality gap in optimal power flowproblem,” IEEE Trans on Power Systems, vol. 27, no. 1, pp. 92–107,2012.

[15] M. Kraning, E. Chu, J. Lavaei, and S. Boyd, “Dynamic network energymanagement via proximal message passing,” Foundations and Trends inOptimization, vol. 1, no. 2, pp. 1–54, 2013.

[16] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributedoptimization and statistical learning via the alternating direction methodof multipliers,” Foundations and Trends in Machine Learning, vol. 3,no. 1, pp. 1–122, 2011.

[17] H. Zhu and G. B. Giannakis, “Multi-area state estimation using dis-tributed sdp for nonlinear power systems,” in Proc. of IEEE Converenceon Smart Grid Communications, Nov 2012, pp. 623–628.

[18] B. A. Robbins, C. N. Hadjicostis, and A. D. Dominguez-Garcia, “A two-stage distributed architecture for voltage control in power distributionsystems,” IEEE Trans on Power Systems, vol. 28, no. 2, pp. 1470–1482,May 2013.

[19] IEEE Power and Energy Society. (2010,Sept) Distribution test feeders. [Online]. Available:http://www.ewh.ieee.org/soc/pes/dsacom/testfeeders/index.html

[20] E. Dall’Anese, S. Dhople, and G. B. Giannakis, “Optimal dis-patch of photovoltaic inverters in residential distribution systems,”arXiv:1307.3751, 2014.

[21] Y. Ye, “Linear conic programming,” Dec 2004, Stanford MS&E314/CME336 Course Notes.

[22] S. Boyd, “Cvx: Matlab software for disciplined convex programming,”Mar 2014. [Online]. Available: http://cvxr.com/cvx/

[23] CORAL Lab, “Sedumi,” Mar 2014. [Online]. Available:http://sedumi.ie.lehigh.edu

12

Brett A. Robbins (S’07) received his B.S. and M.S. degrees in electricalengineering from the University of Illinois at Urbana-Champaign in 2008and 2011, respectively. Currently, he is pursuing the Ph.D. degree from theUniversity of Illinois at Urbana-Champaign. His research interests include themodeling, simulation, and control of power systems, and the integration andcontrol of distributed energy resources.

Hao Zhu (M’12) is an Assistant Professor of ECE at the University ofIllinois, Urbana-Champaign (UIUC). She received her B.S. from TsinghuaUniversity, Beijing, China in 2006, and the M.Sc. and Ph.D. degrees fromthe University of Minnesota, Minneapolis in 2009 and 2012, respectively.She has been working as a postdoctoral research associate on power systemvalidation with the Information Trust Institute at UIUC 2012-2013. Her currentresearch interests include power system monitoring and operations, dynamicsand stability, distribution systems, and energy data analytics.

Alejandro D. Domínguez-García (S’02, M’07) received the degree ofElectrical Engineer from the University of Oviedo (Spain) in 2001 andthe Ph.D. degree in electrical engineering and computer science from theMassachusetts Institute of Technology, Cambridge, MA, in 2007.

He is an Associate Professor in the Electrical and Computer EngineeringDepartment at the University of Illinois at Urbana-Champaign, where heis affiliated with the Power and Energy Systems area; he also has been aGrainger Associate since August 2011. He is also an Associate Professor inthe Coordinated Science Laboratory and in the Information Trust Institute,both at the University of Illinois at Urbana-Champaign. His research interestsare in the areas of system reliability theory and control, and their applica-tions to electric power systems, power electronics, and embedded electronicsystems for safety-critical/fault-tolerant aircraft, aerospace, and automotiveapplications.

Dr. Domínguez-García received the NSF CAREER Award in 2010, and theYoung Engineer Award from the IEEE Power and Energy Society in 2012.He is an editor of the IEEE TRANSACTIONS ON POWER SYSTEMS andthe IEEE POWER ENGINEERING LETTERS.