Extended Radial Distribution ACOPF Model: Retrieving ... - SPS

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Extended Radial Distribution ACOPF Model:Retrieving Exactness via Convex Iteration

Ibrahim Alsaleh , Member, IEEE, Lingling Fan , Senior Member, IEEE

Mohammadhafez Bazrafshan , Member, IEEE,

Abstract—Internalizing the distribution system’s fundamentalcomponents into the alternating-current optimal power flow(ACOPF) problem is of essence for solution quality and physicalviability of control setpoints. In this manuscript, we formulatea problem to minimize the voltage deviations from the nominalvalue subject to physical constraints of a comprehensive distribu-tion ACOPF model. The model encompasses a mixture of wye anddelta loads, distributed energy resources (DERs), and step-voltageregulators (SVRs). We expand upon the branch flow model withnon-convex constraints capturing primary-to-secondary SVRvoltage relationships as well as rank-one constraints belongingto power flow and delta-connected net injections. Relaxing theconstraints renders a semidefinite program (SDP) whose ACfeasibility depends on the solution exactness (proximity of allpositive semidefinite (PSD) matrices to being rank 1). For theunderlying model, three sources of inexactness are identified: (i)the non-monotonic voltage-positioning (VP) objective function,(ii) the relaxed delta-connected load and DER constraints, and(iii) the relaxed constraints for SVRs with continuous andnon-uniformly-operated tap positions. We propose to ultimatelycircumvent this rank conundrum via the application of convexiteration whereby the inexact solution initializes a sequence ofrank-constrained problems. The correlation among the previouscomponents allows the convergence to rank-1 solutions. Casestudies on three IEEE distribution feeders evince the merits ofthe proposed problem.

Index Terms—rank-constrained semi-definite programming,convex iteration, unbalanced distribution systems.

I. INTRODUCTION

D ISTRIBUTION systems are increasingly challenged bythe continued emergence of distributed energy resources

(DERs) signaling the need for optimized computational toolsthat coordinate with existing control devices.

The optimal power flow (OPF) problem, first introduced byCarpentier [1], lies at the root of power system optimization,determining a minimum-cost operating point subject to anelectric system’s physical and security constraints. The alter-nating current OPF (ACOPF) problem is non-convex due to thenon-linearity of power flow equations, and can only guaranteelocally-optimal solutions. Various optimization algorithms tosolve the problem have been extensively surveyed in [2]–[4]. Although nonlinear programming (NLP) solvers may findlocally optimal solutions that happen to also be globallyoptimal, they cannot provide certificates of global optimality.

I. Alsaleh is with the Department of Electrical Engineering, University ofHail, Hail 55476, Saudi Arabia (e-mail: [email protected]).

L. Fan is with the Department of Electrical Engineering, University of SouthFlorida, Tampa, FL 33620, USA (e-mail: [email protected]).

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ConvexD-ACOPF

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Fig. 1. Conceptual illustration of the convex relaxation (dashed blue) ofthe non-convex sets (solid green). f1 is the relaxed true minimum andthus feasible to the non-convex problem, whereas f2 is a lower bound andinfeasible to the non-convex problem.

Recent research has focused on convex relaxations in aneffort to find globally-optimal solutions for the ACOPF prob-lem. The breakthrough occurred when Jabr [5] and later Baiet al. [6] led the first convex relaxations of the ACOPF,respectively using second-order conic programming (SOCP)and semidefinite programming (SDP). This paved the wayfor assessing the optimality [7], exploiting sparsity [8]–[10]or cutting planes [11] to enhance computational performance,and developing other relaxations such as SOCP branch flowmodel (BFM) [12], moment relaxation [13], quadratic convex(QC) relaxation [14], and new SDP relaxation (nSDR) [15].The previous models have exclusively targeted single-phase(radial and meshed) systems. Convex relaxations can alsocomplement the NLP solver in providing the distance of theNLP solution to the global optimal.

The distribution ACOPF (D-ACOPF) is typically formulatedas a mixed-integer NLP (MINLP) problem with discretevariables pertaining to control devices and lines. In [16], [17],the primal dual interior point method is used as the local solverwithin an iterative algorithm to optimize the switching statesof those devices. In addition, accounting for the three-phasemodel within the D-ACOPF is of physical importance due tothe presence of unbalanced loads and lines and the mixed wye-delta load connections in most distribution systems. Reference[18] explores to solve an NLP multi-phase D-ACOPF withmixed load connections and relaxed discrete variables.

In recent years, the aforesaid convex programming tech-niques have been exploited to improve the tractability of themulti-phase D-ACOPF. Reference [19] was first to developa multi-phase SDP D-ACOPF model that incorporates wye-connected loads and conductor mutual coupling. Another sem-inal, and more stable, multi-phase SDP model was proposed

https://orcid.org/0000-0002-8228-2421

https://orcid.org/0000-0002-4686-1602

https://orcid.org/0000-0002-1331-1321

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by Gan and Low [20] based on the BFM. For radial D-ACOPF problems, meeting the rank-1 condition is tantamountto the AC feasibility of the solution [21]. Fig. 1 conceptuallyillustrates the difference between an exact and inexact solution.A plethora of applications has been layered on top of theseconvex models to optimize control setpoints communicated toDERs and legacy devices.

One such legacy device is the step-voltage regulator (SVR)which is essentially an autotransformer with a tap-changingmechanism that aids in voltage control during net loadchanges. Conventionally, incorporating adjustable SVRs intothe D-ACOPF problem would require discrete variables fortap ratios as well as bilinear constraints for voltage andpower relationships on the two sides of the SVR. Furthermore,accounting for the delta connections of loads and DERs addsto the non-convexity of the problem due to the non-linearrelationship between phase-to-ground power injections anddelta connection.

Though challenging, developing a unified convex D-ACOPFthat handles broader connections of SVRs, loads, and DERsis critical not only for the voltage control, but also for thephysical viability of the solutions. This paper expands uponthe multi-phase SDP BFM to account for all connections ofSVRs and net loads. The non-linearity is tackled with convexrelaxations, leveraging practical assumptions of phase angledifferences and rank relaxations, whereas solution feasibilityis retrieved by the convex iteration. In what follows, we castlight on the pertinent literature with an emphasis on solutionexactness.

A. Literature Review on SDP Formulations of D-ACOPF

Step-Voltage Regulators: In multi-phase SDP D-ACOPF,the SVR voltage and power transfer are translated to trilinearconstraints, whereas tap positions are assumed to take contin-uous values to bypass the computational difficulty of solvinga mixed-integer SDP problem [22]–[26]. The diagonal of thesecondary-side voltage can be implicitly confined within thetap ratio limit. However, leaving the non-diagonal elementsunconstrained triggers inexactness, for which a non-ideal SVRwith tunable resistance is proposed in [22]. References [23],[24] propound a SDP constraint that outputs a unified tapselection among phases. Most of the studies focused on thewye-connected SVRs. Theoretical basis for relaxing delta-connected SVR constraints using the bus admittance matrix isdeveloped in [25]. A more exact and stable formulation usingthe branch flow model and McCormick envelopes is proposedin [26]. The inexact solutions are then used to retrieve feasiblevoltages using the Z-bus method [27].

Delta-connected Loads and DERs: Because the phase-to-ground power injections have a non-linear relationship withdelta connections, prior studies are limited to wye-connecteddistribution systems. The notion of extending the multiphaseconvex D-ACOPF to account for delta-connected power injec-tions was first introduced by [28], where an additional PSDconstraint describing the power injections across each pair ofphases are enforced into the BFM, referred to as extendedBFM (EBFM). Unfortunately, the experiments reveal that the

relaxation yields inexact solutions with respect to the delta-connected injections and is therefore replaced by a linearapproximation.

Voltage-Positioning Objective: The inexactness of D-ACOPF convex relaxations has been attributable to the choiceof objective functions. In general, convex relaxations of ra-dial D-ACOPF problems are exact for objective functionsrepresenting the generation cost or losses [12], [20]. On theother hand, non-monotonic objective functions such as thevoltage-positioning (VP) often lead to inexact solutions, andare therefore scarcely employed by the literature. In [29]–[31],researchers have studied exactness retrieval for D-ACOPFproblems that optimize the voltage profile. Reference [29]proposes an approach based on the difference of convexprogramming (DCP). Strengthening the relaxation using thistechnique entails an additional concave inequality constraintthat is tightened sequentially. A non-iterative approach basedon the rank-1 approximation is explored in [30] using multipleSDP and linear equality constraints. Though solutions aretighter compared to the SOCP relaxation, exactness is notachieved. Most recently, [31] employs an inner approximationand iteratively recovers exactness via bound tightening. Othermethods are based on penalizing certain variables which isspecific for meshed and single-phase ACOPF [32], [33]. Theaforesaid techniques have solely focused on single-phase D-ACOPF [29]–[33].

B. ContributionSpurred by the preceding modeling and optimization lim-

itations of multi-phase distribution systems, this paper offersthe following contributions:• Formulation of a rank-1 constrained multi-phase D-

ACOPF problem that is amenable to SDP relaxation andaccounts for a mixture of wye and delta-connected loadsand DERs as well as all common connections of SVRswith non-uniform tap operation.

• Introduction of voltage flattening objective to improvevoltage control through SVR tap decisions.

• Utilization of the convex iteration technique for system-atic rank reduction and retrieval of feasible solutions, andcomparisons with convex and non-convex formulations.

The rest of this manuscript is organized as follows. SectionII presents the generic power distribution system model, de-scribes the various connections of SVRs, and formulates theoverall non-convex EBFM. Section III presents various tech-niques to convexify the EBFM with an iterative methodologyto drive PSD matrices to rank-1. Numerical case studies areperformed in Section IV, and conclusions are drawn in SectionV.

II. PROBLEM FORMULATION

A. General NotationsLet R and C respectively denote the sets of real and complex

numbers. Calligraphic letters, e.g., N , denote sets of indices.We use |· | to denote the absolute value or cardinality of aset. Also, superscripts (T), (C), and (H) denote the transpose,element-wise conjugate, and conjugate transpose (Hermitian)of matrices and vectors.

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Node i<latexit sha1_base64="7/tc0iMawM1Ul2tCQpN7v+bSdsg=">AAAB+HicbVDLSgNBEJz1GeMjqx69DCaCp7AbBT0GvXiSCOYByRJmJ51kyOyDmV4xLvkSLx4U8eqnePNvnCR70MSChqKqm+4uP5ZCo+N8Wyura+sbm7mt/PbO7l7B3j9o6ChRHOo8kpFq+UyDFCHUUaCEVqyABb6Epj+6nvrNB1BaROE9jmPwAjYIRV9whkbq2oUOwiOmt1EPaEmUJl276JSdGegycTNSJBlqXfur04t4EkCIXDKt264To5cyhYJLmOQ7iYaY8REbQNvQkAWgvXR2+ISeGKVH+5EyFSKdqb8nUhZoPQ580xkwHOpFbyr+57UT7F96qQjjBCHk80X9RFKM6DQF2hMKOMqxIYwrYW6lfMgU42iyypsQ3MWXl0mjUnbPypW782L1KosjR47IMTklLrkgVXJDaqROOEnIM3klb9aT9WK9Wx/z1hUrmzkkf2B9/gAWRpK1</latexit>

Node k<latexit sha1_base64="pd4e+hdlea3tOB8u8ls3VOETNR0=">AAAB+HicbVDJSgNBEO1xjXHJqEcvjYngKcxEQY9BL54kglkgGUJPTyVp0rPQXSPGIV/ixYMiXv0Ub/6NneWgiQ8KHu9VUVXPT6TQ6Djf1srq2vrGZm4rv72zu1ew9w8aOk4VhzqPZaxaPtMgRQR1FCihlShgoS+h6Q+vJ37zAZQWcXSPowS8kPUj0ROcoZG6dqGD8IjZbRwALQ1L465ddMrOFHSZuHNSJHPUuvZXJ4h5GkKEXDKt266ToJcxhYJLGOc7qYaE8SHrQ9vQiIWgvWx6+JieGCWgvViZipBO1d8TGQu1HoW+6QwZDvSiNxH/89op9i69TERJihDx2aJeKinGdJICDYQCjnJkCONKmFspHzDFOJqs8iYEd/HlZdKolN2zcuXuvFi9mseRI0fkmJwSl1yQKrkhNVInnKTkmbySN+vJerHerY9Z64o1nzkkf2B9/gAZUpK3</latexit>

Fig. 2. Illustration of the wye- and delta-connected DERs and loads.

B. Power Distribution System Model

Let Nb and E denote the node and edge sets of a distributionsystem with a composite of single, two, and three transmissionconductors. For a radially-fed system, each node i ∈ Nbhas a unique predecessor, such that number of ordered pairs,(i, k) ∈ E , is |E| = |Nb \ 1|. G ⊂ N represents a set of DERnodes. Let T = {1, . . . , t} denote the set of SVRs. We useEt ⊂ E to distinguish line segments with SVRs. Additionally,we introduce Nt to denote the set of virtual nodes, i′ ∈ Nt,incident to the SVR’s secondary side, where i′ lies between iand k. Thus, the aggregate set of physical and virtual nodesis

N = Nb ∪Nt

For notational simplicity, we assume N and E have three-phase connection in this section, while adapting the modelto systems with missing phases is discussed later. We definethe phase sets as Φ = {a, b, c}, Φ′ = {b, c, a}, and Φ4 ={ab, bc, ca}. The root node, i = 0, has a fixed three-phasevoltage given by Vref = Vnom

[1, e−j2π/3, ej2π/3

]T. For all

descendent nodes, the voltage vector is defined as Vi ∈ C|Φ|,where i > 0 ∈ N . Similarly, let Iik ∈ C|Φ| define the three-phase currents flowing through the series impedance zik ∈C|Φ|×|Φ|, while Ik,4 ∈ C|Φ4| define those flowing through thedelta-connected net injections as depicted in Fig. 2. For easeof exposition, we consider a joint wye- and delta-connectednet injection at every i ∈ Nb, respectively defined as si,Y =[sai,Y, s

bi,Y, s

ci,Y

]Tand si,4 =

[sabi,4, s

bci,4, s

cai,4

]T. Each entry

of the net-injection vectors is composed of (sφi,g−sφi,d), whereg and d denote the DER and demand complex powers. yi ∈C|Φ| defines the shunt admittance connected at node i.

In what follows, we formulate a comprehensive multi-phasepower system model. We then lift the system variables toformulate an extended branch flow model.

1) Ohm’s Law: The difference between voltage vectorsacross the ends of each (i, k) connected through a seriesimpedance is expressed as:

Vk = Vi − zikIik ∀(i, k) ∈ E (1)

2) Power Balance Equation: the KCL equation is firstwritten for ith node:

Ii =∑

(i,k)∈EIik − Imi ∀i ∈ Nb (2)

Ii ∈ C|Φ| is the net injection current flowing into i. Imi andIik are the branch currents flowing towards and away fromnode i, respectively. Multiplying (2) by the voltage diagonal,diag(V H

i ), and conjugating the resultant renders the powerbalance at node i, which can be written as:

si =∑

(i,k)∈Ediag(ViI

Hik)

−(diag(VmI

Hmi)− zmiImiIHmi

)∀i ∈ Nb (3)

The net injection current, Ii, is also a function of currentsflowing from DER-load ensembles and shunt elements, whichcan be expressed as follows:

Ii = Ii,Y + ΓTIi,4 − yiVi ∀i ∈ Nb (4)

where

Γ =

1 −1 00 1 −1−1 0 1

Again, multiplying (4) by diag(V H

i ) and taking conjugateyields

si = si,Y + diag(ViIHi,4Γ)− diag(ViV

Hi y

Hi ) ∀i ∈ Nb (5)

Derivations of (3) and (5) are provided in Appendix Aand B. While the wye-connected net injection is readilyformulated in (5), the net injections across the phase pairsare diag(ΓViI

Hi,4).

3) Step-Voltage Regulator (SVR): A set of autotransformersmake up the three-phase SVR, which can be either wye,closed-delta, or open-delta connected. The latter requires twoautotransformers across two pairs of phases. Further, discreteSVRs are typically equipped with ±8 or ±16 tap positions.

To formulate a universal SVR model that suits all connec-tions, we use the following assumptions:• Autotransformers have separate tap changing circuits,

enabling a non-uniform tap setting for each phase. Gang-operated SVRs are discussed in [23], [24].

• We rely on the generalized ratio gain and constantsprovided in Table I for type-B SVRs.

• The series impedance is neglected. This assumption isjustifiable for system-level studies given that the SVR’sseries impedance is approximately one tenth of that ofthe two-winding transformer [34].

• For open-delta connected SVRs, the two autotransformersare installed between phase ab and cb.

The three connections of SVRs are illustrated in Fig. 3. Letrt ∈ R3 be the ratio vector of SVR t ∈ T , whose entriesare consistent with the connection type. For example, rt =[rabt , 1, r

cbt ]T for open-delta connection. Equation (6) sets the

SVR’s tap changing limit.

r ≤ rt ≤ r ∀t ∈ T (6)

Let (i, k) ∈ Et, hence the voltage at and the current flowingaway from i′ ∈ Nt can be expressed as follows:

Vi = AtVi′ ∀i′ ∈ Nt (7a)

Iik = (ATt )−1Ii′k ∀(i, k) ∈ Et (7b)

4

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(c)

Fig. 3. Type-B SVR Configuration: (a) wye connection, (b) closed-delta connection, and (c) open-delta connection.

TABLE IRATIO GAINS AND GENERALIZED CONSTANTS OF THE TYPE-B SVR CONFIGURATIONS

SVR Ratio and Constants Wye Closed-Delta Open-Delta

At

rat 0 0

0 rbt 0

0 0 rct

rabt 1− rabt 0

0 rbct 1− rbct1− rcat 0 rcat

rabt 1− rabt 0

0 1 0

0 1− rcbt rcbt

B

1 0 0

0 1 0

0 0 1

1 −1 0

0 1 −1−1 0 1

1 −1 0

0 1 0

0 −1 1

C 0

0 1 0

0 0 1

1 0 0

0 1 0

0 0 0

0 1 0

Utilizing the generalized constants, B and C, in Table I, theratio gain, At, is composed as follows

At = diag(rt)B + C ∀t ∈ T (8)

From (7), we can infer the following power balance rela-tionship:

diag(Vi′I

Hi′k

)= diag

(A−1t ViI

HikAt

)∀(i, k) ∈ Et (9)

C. Rank-Constrained ACOPF

1) Matrix-based Variables: In this section, the previ-ous constraints are reformulated to conform to the rank-constrained EBFM [20], [28]. To this end, the vector variableproducts are lifted to be matrix variables:

Vi = ViVHi , Iik = IikI

Hik, Ii,4 = Ii4I

Hi,4

Sik = ViIHik, Si = ViI

Hi,4

The real and surrogate variables are interrelated through thefollowing PSD matrices, defined as Fik and D4i :

Fik =

[ViIik

] [ViIik

]H

=

[Vi SikSHik Iik

](10a)

D4i =

[ViIi,4

] [ViIi,4

]H

=

[Vi SiSHi Ii,4

](10b)

With the surrogate variables, constraints (1), (3), (5), (7a),and (9) can subsequently be re-written as:

Vk = Vi − (SikzHik + zikS

Hik) + zikIikz

Hik ∀(i, k) ∈ E

(11a)

si =∑

(i,k)∈Ediag(Sik)− (diag(Smi)− zmiImi) ∀i ∈ Nb

(11b)

si = si,Y + diag(SiΓ)− diag(ViyHi ) ∀i ∈ Nb (11c)

si,4 = diag(ΓSi) ∀i ∈ Nb (11d)

Vi = AtVi′ATt ∀i′ ∈ Nt (11e)

diag (Si′k) = diag(A−1t SikAt

)∀(i, k) ∈ Et (11f)

2) Objective Function: Minimizing the nodal voltage devi-ations from a desired value is set as an operational objective ofthe optimization problem. Let κi ∈ R|Φ| denote a non-negativevector variable, defined for every i ∈ N and constrained as:

∀i ∈ N :

κi ≥ diag(Vi)− V des (12a)

κi ≥ V des − diag(Vi) (12b)κi ≥ 0 (12c)

where V des is the desired squared voltage value. Thus, the VPis achieved by minimizing fv =

∑i∈N

∑φ∈Φ κ

φi .

5

3) Non-Convex EBFM Problem:

PNC : min fv (13a)

s. t. (6), (8), (10), (11), (12) (13b)

V0 = VrefVHref (13c)

V ≤ diag(Vi) ≤ V ∀i′ ∈ N (13d)

diag(Ii,4) ≤ I ∀(i, k) ∈ E (13e)

0 ≤ <(sg,i) ≤ P i ∀i ∈ G (13f)|=(si,g)| ≤ ξ<(sg,i) ∀i ∈ G (13g)Fik � 0 ∀(i, k) ∈ E (13h)

D4i � 0 ∀i ∈ N (13i)rank(Fik) = 1 ∀(i, k) ∈ E (13j)

rank(D4i ) = 1 ∀i ∈ N (13k)

(13d) constrains the squared voltages. (13e) sets an upperbound on the currents through the delta connection. (13f)-(13g) are the DER’s active and reactive power constraints,respectively. The DERs are assumed to be constant powerfactor (PF), and thus ξ =

√(1− PF2)/PF. (13j)-(13k) are

the rank 1 constraints related to the power flow and the delta-connected net injection.

The non-convexity of PNC originates from: (i) the trilinearequality constraints in (11e)-(11f), and (ii) the rank-1 con-straints in (13j)-(13k).

III. CONVEX RELAXATIONS AND ITERATION

In this section, we summarize the techniques employed torender PNC a convex problem that solves to lower-boundsolutions. Moreover, we propose the use of convex iterationto retrieve the exactness of the relaxed model (upper-boundsolutions), thereby enabling the recovery of system originalvariables.

A. Convex Relaxations

1) Relaxing Rank-1 Constraints: The non-convex rank-1 constraints, (13j)-(13k), are dropped from the problem.Reference [35] provides a spate of sufficient conditions underwhich the relaxation of (13j) holds exact. For distributionfeeders operated with tree topology and objective functionsminimizing line losses and generation costs, the relaxation of(13j) is numerically exact for IEEE benchmark feeders [20].On the other hand, reference [28] finds that dropping (13k)can only produce lower-bound solutions.

2) Relaxing SVR Voltage Constraint: By defining the fol-lowing new variables:

Vi = BVi′BT, Vi = BVi′CT, Vi = CVi′BT

(14a)

Vi = diag(rt)Vidiag(rt) + diag(rt)Vi + Vidiag(rt) (14b)

the primary-side voltage constraint in (11e) can then beexpanded as follows:

Vi = Vi + CVi′CT (15)

It is evident from (14b) and (15) that Vi is a function of atrilinear term (first term) and bilinear terms (second and third),

whereas Vi is linear in both Vi′ and Vi. Therefore, we relaxthe diagonal and non-diagonal elements of Vi.

a) Diagonal Elements: given that the tap ratios areconfined within the inequality constraint in (6), the relaxationof the diagonal elements of (14b) is obtained by the set ofconstraints in (16):

diag(Vi) ≥ r2diag(Vi) + rdiag(Vi) + rdiag(Vi) (16a)

diag(Vi) ≤ r2diag(Vi) + rdiag(Vi) + rdiag(Vi) (16b)

Note that diag(Vi) = diag(Vi) because they are transposeof each other. Also, for wye-connected SVRs, only the firstterm to the RHS of (16) is non-zero.

b) Non-diagonal Elements: First, we make use of thefact that angle differences among any pair of phases, (φ, φ′)do not radically deviate from their nominal in multi-phaseradial distribution feeders [26, Assumption 3], such that thefollowing inequality holds for any θ > 0:

90◦ ≤ 120◦ − θ ≤ θφi − θφ′

i ≤ 120◦ + θ ≤ 180◦ (17)

Coupled with the fact that constraining elements with apair combination of {(a, b), (b, c), (c, a)} is sufficient giventhe Hermitian symmetry of a 3 × 3 PSD matrix [10, Proof],the non-diagonal real and imaginary elements of (14b) arerelaxed as

∀φ ∈ Φ, φ′ ∈ Φ′ :

<(Vφφ′

i ) ≤ r2<(Vφφ′

i ) + r<(Vφφ′

i ) + r<(Vφφ′

i ) (18a)

<(Vφφ′

i ) ≥ r2<(Vφφ′

i ) + r<(Vφφ′

i ) + r<(Vφφ′

i ) (18b)

=(Vφφ′

i ) ≥ r2=(Vφφ′

i ) + r=(Vφφ′

i ) + r=(Vφφ′

i ) (18c)

=(Vφφ′

i ) ≤ r2=(Vφφ′

i ) + r=(Vφφ′

i ) + r=(Vφφ′

i ) (18d)

A rank-1 primary-side voltage PSD matrix entails thatits recovered voltage vector, Vi, is a result of some AtVi′ .Observe that the previous constraints are based on the premisethat SVR taps are independently-controlled.

Remark 1: The model is only outlined for type-B SVRswhich are more common in distribution systems [34]. The gainmatrices of type-A SVRs are different, e.g., [36]. Therefore,the proposed SVR relaxation cannot be readily extended totype-A delta-connected SVRs.

3) Relaxing SVR Power Constraint: Constraint (11f) isrelaxed as (19) based on observation of the ratio matrixfor wye and delta SVR connections. The diagonal powerelements through wye-connected SVRs are equivalent dueto the diagonal ratio matrix. For delta-connected SVRs, theconstraint is relaxed based on the conservation of power.

Wye SVRs: diag(Si′k) = diag(Sik) (19a)Delta SVRs: tr(Si′k) = tr(Sik) (19b)

4) Overall Relaxed Problem: The convexified version of(13) is written as:

PLB : min fv

s. t. (10), (11a)-(11d), (12), (13c)-(13g), (14a)(15), (16), (18), (19a)

(20)

6

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(Non-Convex)<latexit sha1_base64="AsYA3BVHc1Vs3zVzjTReWPsvgDk=">AAAB/nicbVDLSgMxFM3UV62vUXHlZrAIdWGZqYIui924kgr2Ae1QMultG5pJhiRTWoaCv+LGhSJu/Q53/o1pOwutHggczrmHe3OCiFGlXffLyqysrq1vZDdzW9s7u3v2/kFdiVgSqBHBhGwGWAGjHGqaagbNSAIOAwaNYFiZ+Y0RSEUFf9CTCPwQ9zntUYK1kTr2UaGtYWyCyZ3g5xXBRzCennXsvFt053D+Ei8leZSi2rE/211B4hC4Jgwr1fLcSPsJlpoSBtNcO1YQYTLEfWgZynEIyk/m50+dU6N0nZ6Q5nHtzNWfiQSHSk3CwEyGWA/UsjcT//Nase5d+wnlUayBk8WiXswcLZxZF06XSiCaTQzBRFJzq0MGWGKiTWM5U4K3/OW/pF4qehfF0v1lvnyT1pFFx+gEFZCHrlAZ3aIqqiGCEvSEXtCr9Wg9W2/W+2I0Y6WZQ/QL1sc3GhyVkw==</latexit>

(Non-Convex)<latexit sha1_base64="AsYA3BVHc1Vs3zVzjTReWPsvgDk=">AAAB/nicbVDLSgMxFM3UV62vUXHlZrAIdWGZqYIui924kgr2Ae1QMultG5pJhiRTWoaCv+LGhSJu/Q53/o1pOwutHggczrmHe3OCiFGlXffLyqysrq1vZDdzW9s7u3v2/kFdiVgSqBHBhGwGWAGjHGqaagbNSAIOAwaNYFiZ+Y0RSEUFf9CTCPwQ9zntUYK1kTr2UaGtYWyCyZ3g5xXBRzCennXsvFt053D+Ei8leZSi2rE/211B4hC4Jgwr1fLcSPsJlpoSBtNcO1YQYTLEfWgZynEIyk/m50+dU6N0nZ6Q5nHtzNWfiQSHSk3CwEyGWA/UsjcT//Nase5d+wnlUayBk8WiXswcLZxZF06XSiCaTQzBRFJzq0MGWGKiTWM5U4K3/OW/pF4qehfF0v1lvnyT1pFFx+gEFZCHrlAZ3aIqqiGCEvSEXtCr9Wg9W2/W+2I0Y6WZQ/QL1sc3GhyVkw==</latexit>

PLB<latexit sha1_base64="piKOmSUOKYN7XAUMvo9MpVa9flk=">AAAB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GWpGxcuKtgHNCFMppN26OTBzI0YQv0VNy4UceuHuPNvnLZZaOuBC4dz7uXee/xEcAWW9W2U1tY3NrfK25Wd3b39A/PwqKviVFLWobGIZd8nigkesQ5wEKyfSEZCX7CeP7me+b0HJhWPo3vIEuaGZBTxgFMCWvLMqgPsEfwgb0+93JEhvm1NPbNm1a058CqxC1JDBdqe+eUMY5qGLAIqiFID20rAzYkETgWbVpxUsYTQCRmxgaYRCZly8/nxU3yqlSEOYqkrAjxXf0/kJFQqC33dGRIYq2VvJv7nDVIIrtycR0kKLKKLRUEqMMR4lgQecskoiEwTQiXXt2I6JpJQ0HlVdAj28surpNuo2+f1xt1Frdkq4iijY3SCzpCNLlET3aA26iCKMvSMXtGb8WS8GO/Gx6K1ZBQzVfQHxucP0TmU3g==</latexit>

(Convex)<latexit sha1_base64="UgQCtXp7RTyVxkPA3lEd6Ci1lC8=">AAAB+nicbVDLTgIxFO3gC/E16NLNRGKCGzKDJroksnGJiTwSmJBOuUBDp520HYSMfIobFxrj1i9x599YYBYKnqTJyTn35N6eIGJUadf9tjIbm1vbO9nd3N7+weGRnT9uKBFLAnUimJCtACtglENdU82gFUnAYcCgGYyqc785Bqmo4A96GoEf4gGnfUqwNlLXzhc7GiYmmFQFH8NkdtG1C27JXcBZJ15KCihFrWt/dXqCxCFwTRhWqu25kfYTLDUlDGa5TqwgwmSEB9A2lOMQlJ8sTp8550bpOX0hzePaWai/EwkOlZqGgZkMsR6qVW8u/ue1Y92/8RPKo1gDJ8tF/Zg5WjjzHpwelUA0mxqCiaTmVocMscREm7ZypgRv9cvrpFEueZel8v1VoXKb1pFFp+gMFZGHrlEF3aEaqiOCHtEzekVv1pP1Yr1bH8vRjJVmTtAfWJ8/YH2UEw==</latexit>

PUB<latexit sha1_base64="E8XLFiII7wtoU9iaI4L5jazxWjU=">AAAB/HicbVBNS8NAEN34WetXtEcvi0XwVJIq6LHUi8cKpi20IWy2m3bp5oPdiRhC/CtePCji1R/izX/jts1BWx8MPN6bYWaenwiuwLK+jbX1jc2t7cpOdXdv/+DQPDruqjiVlDk0FrHs+0QxwSPmAAfB+olkJPQF6/nTm5nfe2BS8Ti6hyxhbkjGEQ84JaAlz6wNgT2CH+SdwsuHMsROu/DMutWw5sCrxC5JHZXoeObXcBTTNGQRUEGUGthWAm5OJHAqWFEdpoolhE7JmA00jUjIlJvPjy/wmVZGOIilrgjwXP09kZNQqSz0dWdIYKKWvZn4nzdIIbh2cx4lKbCILhYFqcAQ41kSeMQloyAyTQiVXN+K6YRIQkHnVdUh2Msvr5Jus2FfNJp3l/VWu4yjgk7QKTpHNrpCLXSLOshBFGXoGb2iN+PJeDHejY9F65pRztTQHxifP97vlOc=</latexit>

(Convex)<latexit sha1_base64="UgQCtXp7RTyVxkPA3lEd6Ci1lC8=">AAAB+nicbVDLTgIxFO3gC/E16NLNRGKCGzKDJroksnGJiTwSmJBOuUBDp520HYSMfIobFxrj1i9x599YYBYKnqTJyTn35N6eIGJUadf9tjIbm1vbO9nd3N7+weGRnT9uKBFLAnUimJCtACtglENdU82gFUnAYcCgGYyqc785Bqmo4A96GoEf4gGnfUqwNlLXzhc7GiYmmFQFH8NkdtG1C27JXcBZJ15KCihFrWt/dXqCxCFwTRhWqu25kfYTLDUlDGa5TqwgwmSEB9A2lOMQlJ8sTp8550bpOX0hzePaWai/EwkOlZqGgZkMsR6qVW8u/ue1Y92/8RPKo1gDJ8tF/Zg5WjjzHpwelUA0mxqCiaTmVocMscREm7ZypgRv9cvrpFEueZel8v1VoXKb1pFFp+gMFZGHrlEF3aEaqiOCHtEzekVv1pP1Yr1bH8vRjJVmTtAfWJ8/YH2UEw==</latexit>

Solution<latexit sha1_base64="Tx5qMkqlQaNtMOlCS6ihqdBT8sw=">AAAB+HicbVBNSwMxEM36WetHVz16WSyCp7JbBT0WvXisaD+gXUo2TdvQbLIkE7Eu/SVePCji1Z/izX9jtt2Dtj4YeLw3w8y8KOFMg+9/Oyura+sbm4Wt4vbO7l7J3T9oamkUoQ0iuVTtCGvKmaANYMBpO1EUxxGnrWh8nfmtB6o0k+IeJgkNYzwUbMAIBiv13FIX6COkd5KbTJj23LJf8WfwlkmQkzLKUe+5X92+JCamAgjHWncCP4EwxQoY4XRa7BpNE0zGeEg7lgocUx2ms8On3olV+t5AKlsCvJn6eyLFsdaTOLKdMYaRXvQy8T+vY2BwGaZMJAaoIPNFA8M9kF6WgtdnihLgE0swUcze6pERVpiAzapoQwgWX14mzWolOKtUb8/Ltas8jgI6QsfoFAXoAtXQDaqjBiLIoGf0it6cJ+fFeXc+5q0rTj5ziP7A+fwBpoqTuw==</latexit>

Mat

rix-b

ased

<latexit sha1_base64="cw2XZUf/qI+Izo2xkn1Dq8IYg7w=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gUN5akCrosunEjVLAPaEOZTCbt0MmDmRtpCQV/xY0LRdz6He78GydtFtp64MLhnHtn7j1uLLgCy/o2CkvLK6trxfXSxubW9o65u9dUUSIpa9BIRLLtEsUED1kDOAjWjiUjgStYyx3eZH7rkUnFo/ABxjFzAtIPuc8pAS31zIOTLrARpHcEJB+dZU95k1LPLFsVawq8SOyclFGOes/86noRTQIWAhVEqY5txeCkRAKngk1K3USxmNAh6bOOpiEJmHLS6foTfKwVD/uR1BUCnqq/J1ISKDUOXN0ZEBioeS8T//M6CfhXTsrDOAEW0tlHfiIwRDjLAntcMgpirAmhkutdMR0QSSjoxLIQ7PmTF0mzWrHPK9X7i3LtOo+jiA7RETpFNrpENXSL6qiBKErRM3pFb8aT8WK8Gx+z1oKRz+yjPzA+fwDdnJVr</latexit>

Var

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Ran

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Con

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Exact?<latexit sha1_base64="rQyS81ZShXdj2oftxx12etbb7pA=">AAAB9XicbVDLSgNBEJz1GeMr6tHLYBA8hd0o6M2gCB4jmAcka5id9CZDZh/M9GrCkv/w4kERr/6LN//GSbIHTSxoKKq66e7yYik02va3tbS8srq2ntvIb25t7+wW9vbrOkoUhxqPZKSaHtMgRQg1FCihGStggSeh4Q2uJ37jEZQWUXiPoxjcgPVC4QvO0EgPbYQhpjdDxvFyTDuFol2yp6CLxMlIkWSodgpf7W7EkwBC5JJp3XLsGN2UKRRcwjjfTjTEjA9YD1qGhiwA7abTq8f02Chd6kfKVIh0qv6eSFmg9SjwTGfAsK/nvYn4n9dK0L9wUxHGCULIZ4v8RFKM6CQC2hUKOMqRIYwrYW6lvM+UycAElTchOPMvL5J6ueSclsp3Z8XKVRZHjhySI3JCHHJOKuSWVEmNcKLIM3klb9aT9WK9Wx+z1iUrmzkgf2B9/gBxK5J3</latexit>

Yes<latexit sha1_base64="NqyCXZYgP9yHQ7cDctqLIYQt+cQ=">AAAB8XicbVDLTgJBEJzFF+IL9ehlIjHxRHbRRI9ELx4xkYfChswODUyYnd3M9BrJhr/w4kFjvPo33vwbB9iDgpV0UqnqTndXEEth0HW/ndzK6tr6Rn6zsLW9s7tX3D9omCjRHOo8kpFuBcyAFArqKFBCK9bAwkBCMxhdT/3mI2gjInWH4xj8kA2U6AvO0EoPHYQnTO/BTLrFklt2Z6DLxMtIiWSodYtfnV7EkxAUcsmMaXtujH7KNAouYVLoJAZixkdsAG1LFQvB+Ons4gk9sUqP9iNtSyGdqb8nUhYaMw4D2xkyHJpFbyr+57UT7F/6qVBxgqD4fFE/kRQjOn2f9oQGjnJsCeNa2FspHzLNONqQCjYEb/HlZdKolL2zcuX2vFS9yuLIkyNyTE6JRy5IldyQGqkTThR5Jq/kzTHOi/PufMxbc042c0j+wPn8ARF6kSw=</latexit>

No<latexit sha1_base64="eEdMCZ9as7ttI7h1q2lvWQGf/x8=">AAAB8HicbVDLSgNBEJyNrxhfUY9eBoPgKexGQY9BL54kgnlIsoTZySQZMo9lplcMS77CiwdFvPo53vwbJ8keNLGgoajqprsrigW34PvfXm5ldW19I79Z2Nre2d0r7h80rE4MZXWqhTatiFgmuGJ14CBYKzaMyEiwZjS6nvrNR2Ys1+oexjELJRko3ueUgJMeOsCeIL3Vk26x5Jf9GfAyCTJSQhlq3eJXp6dpIpkCKoi17cCPIUyJAU4FmxQ6iWUxoSMyYG1HFZHMhuns4Ak+cUoP97VxpQDP1N8TKZHWjmXkOiWBoV30puJ/XjuB/mWYchUnwBSdL+onAoPG0+9xjxtGQYwdIdRwdyumQ2IIBZdRwYUQLL68TBqVcnBWrtydl6pXWRx5dISO0SkK0AWqohtUQ3VEkUTP6BW9ecZ78d69j3lrzstmDtEfeJ8/OM6Qrg==</latexit>

Convex Iteration<latexit sha1_base64="w9p7LI/JD5MhlP8mv24KvEqztOA=">AAACAnicbVDLSgNBEJz1GeNr1ZN4GQyCp7AbBT0Gc9FbBBOFJITZSW8yZHZ2mekNCUvw4q948aCIV7/Cm3/j5HHwVdBQVHXT3RUkUhj0vE9nYXFpeWU1t5Zf39jc2nZ3dusmTjWHGo9lrO8CZkAKBTUUKOEu0cCiQMJt0K9M/NsBaCNidYOjBFoR6yoRCs7QSm13v4kwxCDMKrEawJBeIeipNW67Ba/oTUH/En9OCmSOatv9aHZinkagkEtmTMP3EmxlTKPgEsb5ZmogYbzPutCwVLEITCubvjCmR1bp0DDWthTSqfp9ImORMaMosJ0Rw5757U3E/7xGiuF5KxMqSREUny0KU0kxppM8aEdo4ChHljCuhb2V8h7TjNsgTN6G4P9++S+pl4r+SbF0fVooX8zjyJEDckiOiU/OSJlckiqpEU7uySN5Ji/Og/PkvDpvs9YFZz6zR37Aef8C+TWXzA==</latexit>

Exact

Relax

ation

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Variab

leP

rojection

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PNC<latexit sha1_base64="L/E/UDfKOOts/P0flTO6PELtvoU=">AAAB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0GWxG1dSwT6gCWEynbRDJw9mbsQS4q+4caGIWz/EnX/jtM1CWw9cOJxzL/fe4yeCK7Csb6O0tr6xuVXeruzs7u0fmIdHXRWnkrIOjUUs+z5RTPCIdYCDYP1EMhL6gvX8SWvm9x6YVDyO7mGaMDcko4gHnBLQkmdWHWCP4AdZO/cyR4b4tpV7Zs2qW3PgVWIXpIYKtD3zyxnGNA1ZBFQQpQa2lYCbEQmcCpZXnFSxhNAJGbGBphEJmXKz+fE5PtXKEAex1BUBnqu/JzISKjUNfd0ZEhirZW8m/ucNUgiu3IxHSQosootFQSowxHiWBB5yySiIqSaESq5vxXRMJKGg86roEOzll1dJt1G3z+uNu4ta87qIo4yO0Qk6Qza6RE10g9qogyiaomf0it6MJ+PFeDc+Fq0lo5ipoj8wPn8A1cqU4Q==</latexit>

X⇤<latexit sha1_base64="BoEY/cLUC6d52ZnsR4KX6nbmeqM=">AAAB83icbVDLSgNBEJyNrxhfUY9eBoMgHsJuFPQY9OIxgnlAdg2zk95kyOyDmV4xLPkNLx4U8erPePNvnCR70MSChqKqm+4uP5FCo21/W4WV1bX1jeJmaWt7Z3evvH/Q0nGqODR5LGPV8ZkGKSJookAJnUQBC30JbX90M/Xbj6C0iKN7HCfghWwQiUBwhkZyXYQn9IOsM3k465UrdtWegS4TJycVkqPRK3+5/ZinIUTIJdO669gJehlTKLiESclNNSSMj9gAuoZGLATtZbObJ/TEKH0axMpUhHSm/p7IWKj1OPRNZ8hwqBe9qfif100xuPIyESUpQsTni4JUUozpNADaFwo4yrEhjCthbqV8yBTjaGIqmRCcxZeXSatWdc6rtbuLSv06j6NIjsgxOSUOuSR1cksapEk4ScgzeSVvVmq9WO/Wx7y1YOUzh+QPrM8fGHeRtw==</latexit>

W⇤<latexit sha1_base64="verjeBUcf4u+l3Bw2r/F+ZqnH3U=">AAAB83icbVBNS8NAEN3Ur1q/qh69BIsgHkpSBT0WvXisYD+giWWznbRLN5uwOxFL6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzgkRwjY7zbRVWVtfWN4qbpa3tnd298v5BS8epYtBksYhVJ6AaBJfQRI4COokCGgUC2sHoZuq3H0FpHst7HCfgR3QgecgZRSN5HsITBmHWnjyc9coVp+rMYC8TNycVkqPRK395/ZilEUhkgmrddZ0E/Ywq5EzApOSlGhLKRnQAXUMljUD72ezmiX1ilL4dxsqURHum/p7IaKT1OApMZ0RxqBe9qfif100xvPIzLpMUQbL5ojAVNsb2NAC7zxUwFGNDKFPc3GqzIVWUoYmpZEJwF19eJq1a1T2v1u4uKvXrPI4iOSLH5JS45JLUyS1pkCZhJCHP5JW8Wan1Yr1bH/PWgpXPHJI/sD5/ABbwkbY=</latexit>

W⇤<latexit sha1_base64="verjeBUcf4u+l3Bw2r/F+ZqnH3U=">AAAB83icbVBNS8NAEN3Ur1q/qh69BIsgHkpSBT0WvXisYD+giWWznbRLN5uwOxFL6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzgkRwjY7zbRVWVtfWN4qbpa3tnd298v5BS8epYtBksYhVJ6AaBJfQRI4COokCGgUC2sHoZuq3H0FpHst7HCfgR3QgecgZRSN5HsITBmHWnjyc9coVp+rMYC8TNycVkqPRK395/ZilEUhkgmrddZ0E/Ywq5EzApOSlGhLKRnQAXUMljUD72ezmiX1ilL4dxsqURHum/p7IaKT1OApMZ0RxqBe9qfif100xvPIzLpMUQbL5ojAVNsb2NAC7zxUwFGNDKFPc3GqzIVWUoYmpZEJwF19eJq1a1T2v1u4uKvXrPI4iOSLH5JS45JLUyS1pkCZhJCHP5JW8Wan1Yr1bH/PWgpXPHJI/sD5/ABbwkbY=</latexit>

Compute W⇤<latexit sha1_base64="pumnFDFf02VmNWODUokMJoTOa88=">AAACBXicbVC7TsMwFHXKq5RXgBEGiwoJMVRJQYKxogtjkehDakPluE5r1XnIvkFUURYWfoWFAYRY+Qc2/gYn7QAtR7J0fM69ss9xI8EVWNa3UVhaXlldK66XNja3tnfM3b2WCmNJWZOGIpQdlygmeMCawEGwTiQZ8V3B2u64nvnteyYVD4NbmETM8ckw4B6nBLTUNw97wB4gqYd+FAPDKc7vrpe007vTvlm2KlYOvEjsGSmjGRp986s3CGnsswCoIEp1bSsCJyESOBUsLfVixSJCx2TIupoGxGfKSfIUKT7WygB7odQnAJyrvzcS4is18V096RMYqXkvE//zujF4l07CgyxhQKcPebHAEOKsEjzgklEQE00IlVz/FdMRkYSCLq6kS7DnIy+SVrVin1WqN+fl2tWsjiI6QEfoBNnoAtXQNWqgJqLoET2jV/RmPBkvxrvxMR0tGLOdffQHxucPx5aYvQ==</latexit>

Tr (X⇤W⇤) ✏<latexit sha1_base64="iHOVPwey5eBi2ZyFoiQzyU4bdMQ=">AAACJHicbZDLSgMxFIYz9V5vVZdugkVQF2WmCgpuRDcuK/QGnSqZ9EwbmrmYnBHL0Idx46u4ceEFF258FjNtBW8HAl/+c36S83uxFBpt+93KTU3PzM7NL+QXl5ZXVgtr63UdJYpDjUcyUk2PaZAihBoKlNCMFbDAk9Dw+mdZv3EDSosorOIghnbAuqHwBWdopKvCsYtwi2lVDV0JPu6Mrp6fNoeXe1/cyFiJbg93zdA1dSHWQmbuol2yR0X/gjOBIplU5arw4nYingQQIpdM65Zjx9hOmULBJQzzbqIhZrzPutAyGLIAdDsdLTmk20bpUD9S5oRIR+p3R8oCrQeBZyYDhj39u5eJ//VaCfpH7VSEcYIQ8vFDfiIpRjRLjHaEAo5yYIBxJcxfKe8xxTiaXPMmBOf3yn+hXi45+6XyxUHx5HQSxzzZJFtkhzjkkJyQc1IhNcLJHXkgT+TZurcerVfrbTyasyaeDfKjrI9PDP2mQw==</latexit>

Fig. 4. Flowchart of the iterative approach to recover AC feasibility.

The problem in (20) provides a lower bound to the originalproblem, (13), because of the introduced relaxations com-pounded with the voltage-related objective function.

B. Retrieving Exactness via Convex Iteration

The convex iteration is generally understood to refer toconvex problems whose rank relaxation is strengthened itera-tively [37]. The technique has shown potential, in the powersystem area, for the economic dispatch (ED) problem in thedistribution [38] and transmission [10] systems. The formerconsiders the case where DER offer prices differ per phase,which inflicts higher-rank solutions.

1) Convex Iteration Application: We can succinctly referto all N × N PSD matrices in (20) as X, where N = 6 inour case. By arranging the eigenvalues, λ(X∗) ∈ RN , in adescending order, the rank-1 condition is attained if the traceof each X is equal to its first eigenvalue:

Tr(X)− λ1 = Tr(X)−N∑n=1

λn = 0 (21)

Equation (21) establishes that eigenvalues of n > 1 are zeros.This can alternatively be expressed as

N∑n=2

λn = Tr(XW) = 0 (22)

where W is a PSD matrix, referred to as the direction matrix,which corresponds to the zero eigenvalues with a trace equalto N . It is originally an optimal solution to a SDP problemwith known X∗ [37]. For computational efficiency, a closed-form solution to W can be computed upon solving (20) andutilizing the singular value decomposition (SVD) of X∗ as

X∗ = UΛUT (23)

W = U(:, 2 : N)U(:, 2 : N)T (24)

The direction matrix is therefore exploited to re-formulate(20) as (26) with linear regularization terms tr(XW∗) ap-pended to the objective function and updated iteratively todrive X to rank-1. The problem (20) can serve as an initial-ization as shown in Fig. 4. An identity matrix of W can alsobe used to initialize the problem.

References [10] and [37] demonstrate that choice of weight-ing factors, w1 and w2, can affect the quality of the solution,and thus large or small values may be required to mini-mize the regularization terms. Moreover, the problem couldreach a stalling point, where matrices obtained from previousiterations no longer lower the rank of the PSD matrices.Reference [37] found that the problem can be re-initializedby randomizing the direction matrix at the stalling point. Thedirection matrix is randomized as follows:

W = U(:, 2 : N)(U(:, 2 : N)T + rand(N, 1)U(:, 1)T

)(25)

Hence, the algorithm can be extended to detect stallingpoints, either by comparing objective values or regularizationterms of two consecutive iterations. It should be noted thatstalling may persist despite re-initialization, and thus conver-gence may fail.

2) Overall Problem:

PUB : min fv + w1

∑(i,k)∈E

tr(FikW∗ik)

+ w2

∑i∈N4

tr(D4i W∗i )

s. t. (10), (11a)-(11d), (12), (13c)-(13g), (14a)(15), (16), (18), (19a)

(26)

IV. NUMERICAL EXPERIMENTS

The convex iteration problem is implemented in CVX [39],[40], and solved by Mosek [41]. Simulations are performed ona laptop with Intel Core i7 2.7 GHz processor, 16 GB RAM,and MAC OS 10.14.

We conduct multiple case studies on IEEE 13-bus, 37-bus,123-bus and 8500-node feeders, whose data is found in [42],to demonstrate: 1) the effectiveness of the proposed methodin driving inexact solutions to AC feasible and exact ones,2) the importance of internalizing the physical connection ofloads, DERs, and SVRs, 3) the advantage of the proposedmethod over those proposed by the literature, and 4) the gapbetween the solutions to the relaxed problem and the non-convex problem.

The 13-bus and 123-bus feeders have a composite of wye-and delta-connected loads, while the 8500-node feeder is allwye-connected and the 37-bus feeder is all delta-connected.Distributed loads are halved and lumped to adjacent buses.Capacitors and transformers are modeled. Depending on theirstatus, switches are modeled as open or replaced by short lines.The line configurations of the normally-closed switches in the13-bus and 123-bus feeders are assumed to be 601 and 1,respectively. The lengths are set at 10 feet.

Unity voltage is assumed at the substation (Vnom = 1 p.u.).The voltages are confined as per ANSI ±5% limits, i.e. V =

7

TABLE IISIMULATION RESULTS WITH SVRS, DELTA CONNECTIONS AND VP OBJECTIVE

Feeder Con. SVRfv ν θi′ Rounded SVR Taps Eig. Ratio

Iter.Time

PLB PUB % (deg.) a b c λ2/λ1 λ42 /λ41 (s)

IEE

E13

-bus Y

Y 0.1645 0.3283 1.5 0.51 -4 -6 -6 2×10−12 - 2 0.114 0.1580 0.3281 1.5 0.38 -3 -4 -4 2×10−11 - 2 0.13

0.2955 0.8901 3.5 2 .50 -6 - -6 2×10−12 - 2 0.13

Y-Y 0.1164 0.3039 1.7 0.18 -7 -3 -5 2×10−12 5×10−12 2 0.154 0.1086 0.3040 1.7 0.61 -4 -2 -4 3×10−12 1×10−12 2 0.15

0.118 0.6167 2.3 2.63 -6 - -4 3×10−12 2×10−12 3 0.26

IEE

E37

-bus Y

Y 0.5068 0.6222 1.0 1.10 -5 -7 -8 4×10−10 - 2 0.194 0.5708 0.6061 1.0 0.58 -3 -4 -6 1×10−12 - 2 0.19

0.6822 2.0444 2.4 4.37 -3 - -6 1×10−10 - 2 0.18

Y 0.2783 0.3974 0.4 0.21 -6 -5 -5 1×10−08 6×10−08 3 0.474 0.1578 0.3973 0.4 0.26 -3 -3 -3 4×10−08 1×10−07 3 0.46

0.1644 2.3288 3.2 2.42 -4 - -3 2×10−11 3×10−11 3 0.44

IEE

E12

3-bu

s

Y

Y 0.1117 0.6647 0.8 1.19-41,12

-41, 04-31, 03

3×10−12 - 2 2.8003,-14 04

4 0.1044 0.6645 0.8 1.10-31,02

-31, 04-21, 03

6×10−09 - 2 4.1703, 04 04

0.1939 2.4197 2.3 1.98-71,02 - -41, 03

8×10−10 - 2 4.83-13, 04 -14

M 0.1066 0.6764 0.8 1.19-21, 02

-31-21, 03

1×10−10 - 2 3.5603,04 04

Y-

Y 0.1028 0.6776 0.9 1.32-31, 02

-41, 04-31, 03

1×10−11 2×10−11 2 3.3403,-14 04

4 0.0969 0.6776 0.9 1.13-21, 12

-31, 04-21, 03

2×10−08 6×10−08 2 4.3703, 04 04

0.1002 2.9750 2.3 1.96-71, 02 - -51, 03

6×10−10 7×10−10 5 11.27-03, 04 -14

M 0.0983 0.6853 0.9 1.17-21, 02

-31-21, 03

1×10−11 1×10−10 2 3.5303, 04 04

IEE

E85

00-n

ode

Y

Y 2.1689 25.4840 2.2 1.75-81,-32 -81,-32 -71,-21

1×10−10 - 3 16.11-43,-34 33,-24 -13,-24

4 1.8576 25.3059 2.1 1.39-51,-22 -51,-32 -41,-22

1×10−11 - 3 17.16-33,-24 -23,-14 -13,-14

M 1.9997 28.6653 2.2 1.59-51,-32 -51,-32 -41,-22

4×10−11 - 3 14.95-43,-24 -43 -13,-14

Y-4

Y 2.1249 25.2974 2.1 1.83-81,-32 -81,-32 -61, 22

6×10−10 2×10−9 6 31.10-43,-34 -33,-24 -13,-24

4 1.9771 25.2104 2.1 1.41-41,-22 -51,-32 -41,-23

1×1−11 1×1−10 3 17.76-23,-24 -23,-14 -13,-14

M 1.9831 28.6065 2.1 1.36-51,-32 -51,-32 -43,-22

1×10−10 5×10−10 3 18.29-43,-24 -53 -13,-14

1.052 and V = 0.952. The turns ratio limits are r = 1.05 andr = 0.95.

DERs are placed on node 634, 671, 611, and 652 of the 13-bus feeder, node 724, 744, 730, and 732 of the 37-bus feeder,and node 29, 49, 50, 55, and 66 for the 123-bus feeder. TheDER penetration levels are 30% for the 13-bus feeder and60% for each of the 37-bus and 123-bus feeders. Based on thepenetration of the feeder, the per-phase maximum real powerof each DER is calculated as:

Pφ

i = Penetration×∑i∈N

∑φ∈Φi

<(sφL,i)

|Φi| × |G|(27)

The power factor is fixed at 0.9 (ξ = 0.48).The 8500-node system is simulated without DERs or ca-

pacitors to test the SVRs’ capability to flatten the voltage.

A. Performance of the Convex Iteration Problem

We examine the problem performance to retrieve AC feasi-bility considering wye-only and actual bus connections, as wellas all possible SVR connections. We use Y and4 to symbolizewye and delta (closed-delta) connections of net loads (SVRs),while is used for open-delta SVRs.

For the 123-bus and 8500-node feeders, we also assumemixed connections of the four SVRs, denoted as M, withSVR#1 as closed-delta, SVR#2 and SVR#3 as wye, andSVR#4 as open-delta. Tap ratios for all configurations areretrievable as in Appendix B. Moreover, mixed wye-deltaloads are created for the 8500-node feeder, where 30% of allloaded three-phase buses are assumed delta-connected. Thebold symbols indicate the actual connections of loads andSVRs from the feeder data [42].

8

Table II summarizes the results. Columns 4 and 5 list thelower-bound objective values of the inexact solutions, i.e.,PLB, and those obtained at the convergence of the convexiteration. Columns 6-10 are based on the retrieved voltagesand tap ratios, and are computed as follows:

ν = 100×maxi,φ|V φi − 1| (28)

θi′ = maxi′,φ,φ′

|θφi′ − θφ′

i′ − 120◦| (29)

Tapφ = round

(rφt − 1

r

)∀φ ∈ Φ, t ∈ T (30)

Equation (28) measures the percentage of the maximumvoltage deviation from 1 p.u. for voltages downstream ofthe first SVR’s primary side. Equation (29) measures theangle difference from 120◦, respectively. Equation (30) roundsthe tap position to the nearest discrete value, where r =(r − r)/No. of taps.

Exactness is customarily assessed upon obtaining the solu-tion using the eigenvalue-ratio metric, where smaller ratios in-dicate the PSD matrix’s proximity to being rank-1. Therefore,columns 11 and 12 respectively compute the eigenvalue ratiosfor Dik and Fi averaged over all edges and delta-connectednodes.

In what follows, we highlight the key points from Table IIand the performance of the proposed approach:

• A first glance at the objective values of the relaxedproblems demonstrates that PUB achieves an upper-boundsolution.

• The small eigenvalue ratios at convergence affirm theproposed method’s success to drive the relaxed problemtowards exactness (single dominant eigenvalue).

• The maximum angle difference, θi′ , for all cases aresmall. This points to the validity of (17) upon which therelaxation (18) is based.

• Interestingly, accounting for the SVR connections and themixed wye-delta load connections impacts the objectivevalue, voltage deviation, and optimal tap positions, em-phasizing the need for integrating the various connectionsinto convex D-ACOPF to improve the quality and accu-racy of the voltage control.

• In general, SVR wye and closed-delta configurationsachieve the least voltage deviation.

• With middle-sized feeders, the problem converges aftertwo iterations, except for the 37- and 123-bus cases withopen-delta SVRs. For those two cases, larger weightsare required for convergence. Weights w1 and w2 arerespectively set as 13 and 12 for the 37-bus case, andas 7 and 7 for the 123-bus case. On the other hand, theother cases converge with small weights.

• Retrieving the AC feasibility for a comparatively-largefeeder demonstrates the scalability of the algorithm. Forall load and SVR configurations, the problem convergewith both weights being set at 20 or larger.

• The 8500-node case with open-delta SVRs is infeasiblebecause voltages at phase b fall below 0.95 p.u., andtherefore is not included.

TABLE IIICONVEX ITERATION WITH DIFFERENT SVR MODELS

ModelRef. SVR fv

Rounded SVR TapsItr

Timea b c (s)

[22] Y 0.9010-21, 02

-41, 04-31, 02

7 12.94-13, 04 -14

[23],[24] Y 0.7653-51, 02

-51, 04-51, 02

6 11.7003, 04 04

[26]

Y 0.6647-41, 02

-41, 04-31, 02

2 4.9203,-14 04

4 0.6645-31, 02

-31, 04-21, 02

2 5.4303, 04 04

2.4197-61, 02 - -51, 02

3 7.64-13, 04 -14

Fig. 5. The decay of the regularization term with different SVR models. Withthe diagonal relaxation, the problem stalls after iteration 4 (continuous line),and thus randomization is used to achieve convergence (dashed line).

B. Comparison with SVR Models in [22]–[24] and [26]

We explore applying the convex iteration with previouslyavailable relaxations of the SVR model proposed in [22]–[24]and [26] on the 123-bus case. Given that these references arelimited to wye-connected networks, delta connections are notconsidered, and thus the VP objective and SVR model relax-ations become the only sources of inexactness. The maximumiterations and ε are respectively set to 30 and 1× 10−04.

We highlight the following key points from Table III andcomment on the commonalities with and differences from theproposed SVR model:• Despite using various values of w1, Fig. 5 shows that

the convex iteration stalls at iteration 4 for the diagonalrelaxation presented in [22]. The randomization tech-nique is therefore employed to re-initialize the directionmatrices, which results in rank-1 solutions. Comparedto the proposed model, this model only relaxes thediagonal elements of the primary- and secondary-sidevoltage variables, and leaves the non-diagonal ones un-constrained. This lends support to the use of the non-diagonal relaxation in (18).

• The main feature of the wye-connected SVR modelproposed in [23], [24] is that it enforces a gang operationof the tap positions. This explains the increase in the

9

TABLE IVRESULTS OF THE NLP PROBLEM WARM-STARTED BY THE

CONVEX RELAXATION

Feeder fvGap(%)

Rounded SVR Taps Timea b c (s)

13-bus 0.3039 0.008 -7 -3 -5 0.36

37-bus 0.3288 0.005 -4 - -3 0.74

123-bus 0.6776 0.007-41, 02

-41, 02-31, 02

10.6403,-04 04

8500-node 25.4970 0.051

-81,-42 -81,-42 -61,-2262.95-43,-34 -33,-24 -14,-24

objective value, and may be responsible for the slowconvergence.

• While the SVR model in [26] produces the same resultsas the proposed one, the problem convergence is slightlyslower, presumably because it involves: (i) more variablesand inequality constraints for the voltage-ratio terms andthe rank-1 approximation, and (ii) a SDP constraint forthe tap ratio variable.

• The SVR relaxation in [26] relies on tighter pre-definedmaximum angle difference in order to define the boundsof the McCormick constraints. The results reported inTable III are obtained by selecting θi′ = 15◦.

C. Comparison with the Non-convex Problem

In this section, we compare the solutions to the convexproblem with solutions produced by the non-convex NLPformulation equivalent to PNC (13). Considering the originalconnections of loads and SVRs for each feeder with DERsremoved, the NLP formulation is implemented in YALMIP[43] and solved by IPOPT [44]. The (relative) convergencetolerance of IPOPT is set to 1× 10−4.

The NLP solver is supplied by two types of initializations:One initialization is based on the solution to the convexproblem and another is based on running a Z-Bus load flowwith taps fixed. In all instances, both types of initializationsyield similar objective values for the NLP formulation whichare tabulated in the second column of Table IV. The thirdcolumn in Table IV details the percentage gap in differenceof objective values between the NLP formulation and that ofthe convex problem. The obtained SVR tap positions from theNLP solution closely match those retrieved from the convexproblem, with exceptions highlighted in bold in columns fourthrough six of Table IV. Both the convex problem and the NLPformulation turn out to yield almost identical voltage profiles,depicted in Fig. 6. The NLP formulation was observed to havesmaller solve times when initialized by the solution of theconvex problem rather than the load flow solution. Solve timesreported in Table IV refer to NLP solve time when initializedby the solution of the convex problem.

Detailed comparison of solve times between the convexformulation and the NLP formulation with both types of ini-tializations are depicted in Fig. 7. For 123-bus and 8500-nodefeeders, we observed that initializing the NLP formulationby the convex solution provides a three-fold speed-up. Wehighlight here that the load flow solution is obtained with taps

fixed at −5—a value close to the optimal values in Table IV.Fixing tap values so that load-flow voltages are within boundsis generally nontrivial. It must be stressed that it is possiblethat NLP solvers get stuck in local optima [45] and fail toreach global optima. Albeit even locally optimal solutionsobtained by NLP solvers may indeed be satisfactory. In suchcases, specialized convex formulations for the multi-phase D-ACOPF problem with mixed connections of loads and SVRscan complement the NLP solver by quantifying the globaloptimality gap or by improving convergence time of NLPsolvers via supplying high-quality initializations. Particular toour test cases, the proposed algorithm readily yields an exactsolution and certifies global optimality.

V. CONCLUSION

In this paper, we augment the convex D-ACOPF to solveapplications of particular relevance to the distribution systemoptimization. Namely, we present an extended model that ac-counts for the various connections of loads, DERs, and SVRs.Moreover, we set the objective function to minimize voltagedeviations. The relaxations of the introduced constraints in-evitably place the solution outside the feasible region of theoriginal non-convex problem. We then propose to utilize aconvex iteration approach to steer the relaxed solution towardsa physically-meaningful one, in which true voltage variablesare recoverable. Simulations and comparative studies withexisting literature demonstrate the efficacy of the proposedmethod in retrieving AC feasible solutions from the convexrelaxation and thus numerically certifying global optimality.Future work will include incorporating voltage dependency ofloads into the present framework. We also intend to solve amulti-period D-ACOPF to minimize diurnal consumption andSVR switching actions.

APPENDIX ALINEAR ALGEBRA

A. Power Balance

Multiplying the KCL equation in (2) with the diag(V Hi )

results in:

diag(V Hi )Ii =

∑(i,k)∈E

diag(V Hi )Iik − diag(V H

i )Imi (31)

knowing that for any vector U(diag(V H)U

)C= diag(V UH)

then, by taking conjugate of (31) and invoking (1):

diag(ViIHi ) =

∑(i,k)∈E

diag(VmIHik)− diag(VmI

Hmi − zmiImiIH

mi)

(32)

B. Net Injection

Repeating the same manipulation on (4) yields:

diag(ViIHi ) = diag(ViI

Hi,Y) + diag(ViI

Hi,4Γ)− diag(ViV

Hi y

Hi )

(33)

Note that (32) and (33) are (3) and (5).

10

(a) (b) (c)

1

(d)

Fig. 6. Voltage profiles produced by the non-convex problem in comparison to the voltage profile retrieved from the convex problem for the (a) 13-bus case,(b) 37-bus case, (c) 123-bus case, and (d) 8500-node case. Buses are enumerated successively starting from bus 1. The p.u. difference of the two profiles,plotted and scaled using the right axis, indicates that both solutions are comparable.

C. The SVR Power Transfer

The complex power through ideal SVR is obtained bymultiplying both sides of (7a) by (At)

−1, and (7b) by (At)T:

Vi′ = (At)−1Vi (34a)

Ii′k = ATt Iik (34b)

By multiplying (34b) by V ′Hi and taking the conjugate, thepower balance:

Vi′IHi′k = (At)

−1ViIHikAt (35)

In this form, (35) is equivalent to the power constraintin (11f). Note that for wye-connected SVRs, diag(Si′k) =diag(Sik) is satisfied. For open-delta connected SVRs:

diag(Si′k) =

Saaik ± ηabSbaikSbbik ∓ (ηabSbaik + ηcbSbcik)

Sccik ± ηcbSbcik

(36)

where ηφ ≤ rφt − 1 is a small value decided by the respectivetap ratio. With tr(Si′k), the non-diagonal powers elementsare zeroed out. Hence, diag(Sik) is largely determined bydiag(Si′k). Similar conclusions can be drawn about the closed-delta connection. Therefore, downplaying the contributionfrom non-diagonal elements returns a diagonal equality con-straint for all SVR configurations.

11

Fig. 7. Comparison of solve time in seconds between the convex and theNLP formulations. Initialization supplied by the convex formulation reducethe NLP solve time in comparison to that supplied by the load-flow.

APPENDIX BRECOVERING EFFECTIVE TAP RATIOS

Given (14b), the effective per-phase tap ratio is retrieved bysolving the following:

g(rφt ) = V φφi (rφt )2 + 2V φφi rφt − V φφi = 0 (37)

Knowing that V φφi = 0 for wye SVRs, the effective ratio isretrieved as in (38a). For delta SVRs, the intermediate theoremis invoked knowing that rφt ∈ [r, r] to find one solution to (37)as in (38b).

Wye SVRs: rφt =

√V φφi

V φφi(38a)

Delta SVRs: rφt =1

V φφi

(−V φφi +

√(V φφi )2 + V φφi V φφi

)(38b)

REFERENCES

[1] J. Carpentier, “Contribution to the economic dispatch problem,” Bulletinde la Societe Francoise des Electriciens, vol. 3, no. 8, pp. 431–447,1962.

[2] M. Huneault and F. D. Galiana, “A survey of the optimal power flowliterature,” IEEE Transactions on Power Systems, vol. 6, no. 2, pp. 762–770, 1991.

[3] J. A. Momoh, R. Adapa, and M. E. El-Hawary, “A review of selectedoptimal power flow literature to 1993. i. nonlinear and quadratic pro-gramming approaches,” IEEE Transactions on Power Systems, vol. 14,no. 1, pp. 96–104, 1999.

[4] A. Castillo and R. P. O’Neill, “Survey of approaches to solving theacopf,” Federal Energy Regulatory Commission, Tech. Rep, vol. 11,2013.

[5] R. A. Jabr, “Radial distribution load flow using conic programming,”IEEE transactions on power systems, vol. 21, no. 3, pp. 1458–1459,2006.

[6] X. Bai, H. Wei, K. Fujisawa, and Y. Wang, “Semidefinite programmingfor optimal power flow problems,” International Journal of ElectricalPower & Energy Systems, vol. 30, no. 6-7, pp. 383–392, 2008.

[7] J. Lavaei and S. H. Low, “Zero duality gap in optimal power flowproblem,” IEEE Transactions on Power Systems, vol. 27, no. 1, pp.92–107, 2011.

[8] R. A. Jabr, “Exploiting sparsity in sdp relaxations of the opf problem,”IEEE Transactions on Power Systems, vol. 27, no. 2, pp. 1138–1139,2011.

[9] D. K. Molzahn, J. T. Holzer, B. C. Lesieutre, and C. L. DeMarco,“Implementation of a large-scale optimal power flow solver basedon semidefinite programming,” IEEE Transactions on Power Systems,vol. 28, no. 4, pp. 3987–3998, 2013.

[10] M. Ma, L. Fan, Z. Miao, B. Zeng, and H. Ghassempour, “A sparseconvex ac opf solver and convex iteration implementation based on 3-node cycles,” Electric Power Systems Research, vol. 180, p. 106169,2020.

[11] Z. Miao, L. Fan, H. G. Aghamolki, and B. Zeng, “Least squares estima-tion based sdp cuts for socp relaxation of ac opf,” IEEE Transactionson Automatic Control, vol. 63, no. 1, pp. 241–248, 2017.

[12] M. Farivar and S. H. Low, “Branch flow model: Relaxations andconvexification-part i,” IEEE Transactions on Power Systems, vol. 28,no. 3, pp. 2554–2564, 2013.

[13] D. K. Molzahn and I. A. Hiskens, “Sparsity-exploiting moment-basedrelaxations of the optimal power flow problem,” IEEE Transactions onPower Systems, vol. 30, no. 6, pp. 3168–3180, 2014.

[14] C. Coffrin, H. L. Hijazi, and P. Van Hentenryck, “The qc relaxation:A theoretical and computational study on optimal power flow,” IEEETransactions on Power Systems, vol. 31, no. 4, pp. 3008–3018, 2015.

[15] C. Bingane, M. F. Anjos, and S. Le Digabel, “Tight-and-cheap conicrelaxation for the ac optimal power flow problem,” IEEE Transactionson Power Systems, vol. 33, no. 6, pp. 7181–7188, 2018.

[16] L. W. de Oliveira, S. Carneiro Jr, E. J. De Oliveira, J. Pereira, I. C.Silva Jr, and J. S. Costa, “Optimal reconfiguration and capacitor allo-cation in radial distribution systems for energy losses minimization,”International Journal of Electrical Power & Energy Systems, vol. 32,no. 8, pp. 840–848, 2010.

[17] Q. Nguyen, H. V. Padullaparti, K.-W. Lao, S. Santoso, X. Ke, andN. Samaan, “Exact optimal power dispatch in unbalanced distributionsystems with high pv penetration,” IEEE Transactions on Power Systems,vol. 34, no. 1, pp. 718–728, 2018.

[18] S. Paudyal, C. A. Canizares, and K. Bhattacharya, “Optimal operationof distribution feeders in smart grids,” IEEE Transactions on IndustrialElectronics, vol. 58, no. 10, pp. 4495–4503, 2011.

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

[20] L. Gan and S. H. Low, “Convex relaxations and linear approximationfor optimal power flow in multiphase radial networks,” in 2014 PowerSystems Computation Conference. IEEE, 2014, pp. 1–9.

[21] N. Li, L. Chen, and S. H. Low, “Exact convex relaxation of opf for radialnetworks using branch flow model,” in 2012 IEEE Third InternationalConference on Smart Grid Communications (SmartGridComm). IEEE,2012, pp. 7–12.

[22] B. A. Robbins, H. Zhu, and A. D. Domínguez-García, “Optimal tapsetting of voltage regulation transformers in unbalanced distributionsystems,” IEEE Transactions on Power Systems, vol. 31, no. 1, pp. 256–267, 2015.

[23] Y. Liu, J. Li, L. Wu, and T. Ortmeyer, “Chordal relaxation based acopffor unbalanced distribution systems with ders and voltage regulationdevices,” IEEE Transactions on Power Systems, vol. 33, no. 1, pp. 970–984, 2017.

[24] Y. Liu, J. Li, and L. Wu, “Coordinated optimal network reconfigurationand voltage regulator/der control for unbalanced distribution systems,”IEEE Transactions on Smart Grid, vol. 10, no. 3, pp. 2912–2922, 2018.

[25] M. Bazrafshan, N. Gatsis, and H. Zhu, “Optimal tap selection of step-voltage regulators in multi-phase distribution networks,” in 2018 PowerSystems Computation Conference (PSCC). IEEE, 2018, pp. 1–7.

[26] M. Bazrafshan, N. Gatsis, and H. Zhu, “Optimal power flow with step-voltage regulators in multi-phase distribution networks,” IEEE Transac-tions on Power Systems, vol. 34, no. 6, pp. 4228–4239, 2019.

[27] M. Bazrafshan and N. Gatsis, “Convergence of the z-bus method forthree-phase distribution load-flow with zip loads,” IEEE Transactionson Power Systems, vol. 33, no. 1, pp. 153–165, 2017.

[28] C. Zhao, E. Dall-Anese, and S. H. Low, “Optimal power flow inmultiphase radial networks with delta connections,” National RenewableEnergy Lab.(NREL), Golden, CO (United States), Tech. Rep., 2017.

[29] W. Wei, J. Wang, N. Li, and S. Mei, “Optimal power flow of radialnetworks and its variations: A sequential convex optimization approach,”IEEE Transactions on Smart Grid, vol. 8, no. 6, pp. 2974–2987, 2017.

[30] Q. Li and V. Vittal, “Non-iterative enhanced sdp relaxations for optimalscheduling of distributed energy storage in distribution systems,” IEEETransactions on Power Systems, vol. 32, no. 3, pp. 1721–1732, 2016.

[31] N. Nazir and M. Almassalkhi, “Voltage positioning using co-optimization of controllable grid assets in radial networks,”

12

IEEE Transactions on Power Systems, to be published,doi:10.1109/TPWRS.2020.3044206.

[32] R. Madani, S. Sojoudi, and J. Lavaei, “Convex relaxation for optimalpower flow problem: Mesh networks,” IEEE Transactions on PowerSystems, vol. 30, no. 1, pp. 199–211, 2014.

[33] R. Madani, M. Ashraphijuo, and J. Lavaei, “Promises of conic relax-ation for contingency-constrained optimal power flow problem,” IEEETransactions on Power Systems, vol. 31, no. 2, pp. 1297–1307, 2015.

[34] W. H. Kersting, Distribution system modeling and analysis. CRC press,2012.

[35] F. Zhou, Y. Chen, and S. H. Low, “Sufficient conditions for exact semi-definite relaxation of optimal power flow in unbalanced multiphase radialnetworks,” in 2019 IEEE 58th Conference on Decision and Control(CDC), 2019, pp. 6227–6233.

[36] R. R. Mojumdar, P. Arboleya, and C. González-Morán, “Step-voltageregulator model test system,” in 2015 IEEE Power Energy SocietyGeneral Meeting, 2015, pp. 1–5.

[37] J. Dattorro, Convex Optimization and Euclidean Distance Geometry.Palo Alto, California, USA: Meboo Publishing, 2005.

[38] W. Wang and N. Yu, “Chordal conversion based convex iteration algo-rithm for three-phase optimal power flow problems,” IEEE Transactionson Power Systems, vol. 33, no. 2, pp. 1603–1613, 2017.

[39] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convexprogramming, version 2.1.”

[40] M. Grant and S. Boyd, “Graph implementations for nonsmooth convexprograms,” in Recent Advances in Learning and Control, ser. LectureNotes in Control and Information Sciences, V. Blondel, S. Boyd, andH. Kimura, Eds. Springer-Verlag Limited, 2008, pp. 95–110, https://web.stanford.edu/~boyd/papers/pdf/graph_dcp.pdf.

[41] M. ApS, “The mosek optimization toolbox for matlab manual.” 2017.[Online]. Available: https://docs.mosek.com/8.0/toolbox/index.html

[42] IEEE PES Distribution Systems Analysis Subcommittee Radial TestFeeders, https://site.ieee.org/pes-testfeeders/, accessed August 2020.

[43] J. Löfberg, “Yalmip : A toolbox for modeling and optimization inmatlab,” in In Proceedings of the CACSD Conference, Taipei, Taiwan,2004.

[44] A. Wächter and L. T. Biegler, “On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,”Mathematical programming, vol. 106, no. 1, pp. 25–57, 2006.

[45] A. Venzke, S. Chatzivasileiadis, and D. K. Molzahn, “Inexact convexrelaxations for ac optimal power flow: Towards ac feasibility,” ElectricPower Systems Research, vol. 187, p. 106480, 2020. [Online]. Available:https://www.sciencedirect.com/science/article/pii/S0378779620302832

Ibrahim Alsaleh received the B.S. degree in elec-trical engineering from the University of Hail, Hail,Saudi Arabia, in 2012, the M.S. and Ph.D. degreesin electrical engineering from the University ofSouth Florida, Tampa, FL in 2016 and 2020. Heis currently an Assistant Professor with the Depart-ment of Electrical Engineering, University of Hail.His current research interests include power-systemoptimization and applications of renewable energy.

Lingling Fan (SM’08) received the B.S. and M.S.degrees in electrical engineering from SoutheastUniversity, Nanjing, China, in 1994 and 1997, re-spectively, and the Ph.D. degree in electrical engi-neering from West Virginia University, Morgantown,in 2001. Currently, she is a full professor at theUniversity of South Florida, Tampa, where she hasbeen since 2009. She was a Senior Engineer inthe Transmission Asset Management Department,Midwest ISO, St. Paul, MN, form 2001 to 2007,and an Assistant Professor with North Dakota State

University, Fargo, from 2007 to 2009. Her research interests include powersystems and power electronics. Dr. Fan serves as Editor-in-Chief of IEEEElectrification Magazine and Associate Editor for IEEE Trans. Energy Con-version.

Mohammadhafez Bazrafshan received the B.Sc.degree from Iran University of Science and Technol-ogy in 2012 and the M.Sc. and Ph.D. degrees fromthe University of Texas at San Antonio respectivelyin 2014 and 2018, all in electrical engineering. Hisresearch efforts are primarily focused on applicationsof machine learning and optimization in smart powergrids.

https://doi.org/10.1109/TPWRS.2020.3044206

https://web.stanford.edu/~boyd/papers/pdf/graph_dcp.pdf

https://web.stanford.edu/~boyd/papers/pdf/graph_dcp.pdf

https://docs.mosek.com/8.0/toolbox/index.html

https://site.ieee.org/pes-testfeeders/

https://www.sciencedirect.com/science/article/pii/S0378779620302832

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