1

Tightening QC Relaxations of AC Optimal PowerFlow Problems via Complex Per Unit Normalization

Mohammad Rasoul Narimani, Daniel K. Molzahn, and Mariesa L. Crow

Abstract—Optimal power flow (OPF) is a key problem inpower system operations. OPF problems that use the nonlinearAC power flow equations to accurately model the network physicshave inherent challenges associated with non-convexity. To ad-dress these challenges, recent research has applied various convexrelaxation approaches to OPF problems. The QC relaxation isa promising approach that convexifies the trigonometric andproduct terms in the OPF problem by enclosing these termsin convex envelopes. The accuracy of the QC relaxation stronglydepends on the tightness of these envelopes. This paper presentstwo improvements to these envelopes. The first improvementleverages a polar representation of the branch admittances inaddition to the rectangular representation used previously. Thesecond improvement is based on a coordinate transformation viaa complex per unit base power normalization that rotates thepower flow equations. The trigonometric envelopes resulting fromthis rotation can be tighter than the corresponding envelopes inprevious QC relaxation formulations. Using an empirical analysiswith a variety of test cases, this paper suggests an appropriatevalue for the angle of the complex base power. Comparingthe results with a state-of-the-art QC formulation reveals theadvantages of the proposed improvements.

Index Terms—Optimal power flow, Convex relaxation

I. INTRODUCTION

OPTIMAL power flow (OPF) problems are central tomany tasks in power system operations. OPF problems

optimize an objective function, such as generation cost, subjectto both the network physics and engineering limits. The ACpower flow equations introduce non-convexities in OPF prob-lems. Due to these non-convexities, OPF problems may havemultiple local optima [1], [2] and are generally NP-Hard [3].

Many research efforts have focused on algorithms for ob-taining locally optimal or approximate OPF solutions [4].Recent research has also developed convex relaxations of OPFproblems [5]. Convex relaxations bound the optimal objectivevalues, can certify infeasibility, and, in some cases, provablyprovide globally optimal solutions to OPF problems.

The capabilities of convex relaxations are, in many ways,complementary to those of local solution algorithms. Forinstance, relaxations’ objective value bounds can certify howclose a local solution is to being globally optimal. Accordingly,local algorithms and relaxations are used together in spatialbranch-and-bound methods [6]. Solutions from relaxations arealso useful for initializing some local solvers [7]. Relaxationsare also needed for certain solution algorithms for robustOPF problems [8]. Moreover, the objective value boundsprovided by relaxations are directly useful in other contexts,e.g., [9], [10]. The tractability and accuracy of these and otheralgorithms are largely determined by the employed relaxation’stightness. Tightening relaxations is thus an active researchtopic [5].

The quadratic convex (QC) relaxation is a promising ap-proach that encloses the trigonometric and product terms inthe polar representation of power flow equations within convexenvelopes [11]. These envelopes are formed with linear andsecond-order cone programming (SOCP) constraints, result-ing in a convex formulation. The QC relaxation’s tightnessstrongly depends on the quality of these convex envelopes.This paper focuses on improving these envelopes.

Previous work has proposed a variety of approaches fortightening the QC relaxation. These include valid inequalities,such as “Lifted Nonlinear Cuts” [12], [13] and constraintsthat exploit bounds on the differences in the voltage mag-nitudes [14]. Additionally, since the accuracies of the trigono-metric and product envelopes in the QC relaxation rely onthe voltage magnitude and angle difference bounds, boundtightening approaches can significantly strengthen the QCrelaxation [12], [15]–[19]. When bound tightening approachesprovide sign-definite angle difference bounds (i.e., the upperand lower bounds on the angle differences have the same sign),tighter trigonometric envelopes can be applied [12].

This paper proposes two improvements to further tightenQC relaxations of OPF problems. The first improvementleverages a polar representation of the branch admittances inaddition to the rectangular representation used in previous QCformulations. Within certain ranges, portions of the trigono-metric envelopes resulting from the polar admittance repre-sentation are at least as tight (and generally tighter) than thecorresponding portions of the envelopes from the rectangularadmittance representation. In other ranges, the trigonometricenvelopes from the polar admittance representation neithercontain nor are contained within the envelopes from therectangular admittance representation. Thus, combining theseenvelopes tightens the QC relaxation, with empirical resultssuggesting limited impacts on solution times.

The polar admittance representation also enables our secondimprovement. We exploit a degree of freedom in the OPF for-mulation related to the per unit base power normalization. Se-lecting a complex base power (Sbase “ |Sbase| ejψ) results in acoordinate transformation that rotates the power flow equationsrelative to the typical choice of a real-valued base power.We leverage the associated rotational degree of freedom ψto obtain tighter envelopes for the trigonometric functions.While previously proposed power flow algorithms [20] andstate estimation algorithms [21] use similar formulations, thispaper is, to the best of our knowledge, the first to exploit thisrotational degree of freedom to improve convex relaxations.

This paper is organized as follows. Sections II and IIIreview the OPF formulation and the previously proposed QCrelaxation, respectively. Section IV describes the coordinate

2

changes underlying our improved QC relaxation. Section Vthen presents these improvements. Section VI empiricallyevaluates our approach. Section VII concludes the paper.An extended version of this paper in [22] further describesand analyzes the envelopes used in the relaxation and alsoconsiders parallel lines and more general line models.

II. OVERVIEW OF THE OPTIMAL POWER FLOW PROBLEM

This section formulates the OPF problem using a polarvoltage phasor representation. The sets of buses, generators,and lines are N , G, and L, respectively. The set R con-tains the index of the bus that sets the angle reference. LetSdi “ P di ` jQdi and Sgi “ P gi ` jQgi represent the complexload demand and generation, respectively, at bus i P N , wherej “

?´1. Let Vi and θi represent the voltage magnitude

and angle at bus i P N . Let gsh,i ` jbsh,i denote the shuntadmittance at bus i P N . For each generator, define a quadraticcost function with coefficients c2,i ě 0, c1,i, and c0,i. Forsimplicity, we consider a single generator at each bus bysetting the generation limits at buses without generators tozero. Upper and lower bounds for all variables are indicatedby p ¨ q and p ¨ q, respectively.

For ease of exposition, each line pl,mq P L is modeledas a Π circuit with mutual admittance glm ` jblm and shuntadmittance jbc,lm. Extensions to more general line modelsthat allow for off-nominal tap ratios and non-zero phase shiftsare straightforward and available in [22, Appendix A]. Defineθlm “ θl ´ θm for pl,mq P L. The complex power flow intoeach line terminal pl,mq P L is denoted by Plm ` jQlm, andthe apparent power flow limit is Slm. The OPF problem is

minÿ

iPGc2,i pP

gi q

2` c1,i P

gi ` c0,i (1a)

subject to p@i P N , @ pl,mq P LqP gi ´ P

di “ gsh,i V

2i `

ÿ

pl,mqPL,s.t. l“i

Plm `ÿ

pl,mqPL,s.t. m“i

Pml, (1b)

Qgi ´Qdi “ ´bsh,i V

2i `

ÿ

pl,mqPL,s.t. l“i

Qlm `ÿ

pl,mqPL,s.t. m“i

Qml, (1c)

θr “ 0, r P R, (1d)

P gi ď P gi ď Pg

i , Qgiď Qgi ď Q

g

i , (1e)

V i ď Vi ď V i, (1f)

θlm ď θlm ď θlm, (1g)

Plm“glmV2l ´glmVlVm cos pθlmq´blmVlVm sin pθlmq , (1h)

Qlm “ ´pblm ` bc,lm{2qV2l ` blmVlVm cos pθlmq

´ glmVlVm sin pθlmq , (1i)

Pml“glmV2m´glmVlVm cos pθlmq`blmVlVm sin pθlmq , (1j)

Qml “ ´pblm ` bc,lm{2qV2m ` blmVlVm cos pθlmq

` glmVlVm sin pθlmq , (1k)

pPlmq2` pQlmq

2ď`

Slm˘2, pPmlq

2` pQmlq

2ď`

Slm˘2.

(1l)

The objective (1a) minimizes the generation cost. Con-straints (1b) and (1c) enforce power balance at each bus.Constraint (1d) sets the reference bus angle. The constraints

in (1e) bound the active and reactive power generation ateach bus. Constraints (1f)–(1g), respectively, bound the voltagemagnitudes and voltage angle differences. Constraints (1h)–(1k) relate the active and reactive power flows with the voltagephasors at the terminal buses. The constraints in (1l) limit theapparent power flows into both terminals of each line.

III. THE QC RELAXATION OF THE OPF PROBLEM

The QC relaxation convexifies the OPF problem (1) byenclosing the nonconvex terms in convex envelopes [11]. Therelevant nonconvex terms are the square V 2

i , @i P N , and theproducts VlVm cospθlmq and VlVm sinpθlmq, @pl,mq P L. Theenvelope for the generic squared function x2 is xx2yT :

xx2yT “

#

qx :

#

x ě x2,

qx ď px` xqx´ xx.(2)

where qx is a lifted variable representing the squared term.Envelopes for the generic trigonometric functions sinpxq andcospxq are xsinpxqyS and xcospxqy

C :

xsinpxqyS“

$

’

’

’

’

&

’

’

’

’

%

qS :

$

’

’

’

’

&

’

’

’

’

%

qSďcos´

xm

2

¯

x´ xm

2

¯

`sin´

xm

2

¯

if xď 0ď x,

qSěcos´

xm

2

¯

x` xm

2

¯

´sin´

xm

2

¯

if xď 0ď x,

qS ě sinpxq´sinpxqx´x

px´ xq ` sin pxq if x ě 0,qS ď sinpxq´sinpxq

x´xpx´ xq ` sin pxq if x ď 0,

(3)

xcospxqyC “

#

qC :

#

qC ď 1´ 1´cospxmq

pxmq2x2,

qC ě cospxq´cospxqx´x

px´ xq ` cos pxq ,(4)

where xm “ maxp|x| , |x|q. The envelopes xsinpxqyS andxcospxqy

C in (3) and (4) are valid for ´π2 ď x ď π

2 .Similar envelopes for the sine and cosine functions are de-fined analogously for other angle difference ranges. (See [22,Appendix A].) The lifted variables qS and qC are associatedwith the envelopes for the functions sinpθlmq and cospθlmq.The QC relaxation of the OPF problem in (1) is:

minÿ

iPNc2,i pP

gi q

2` c1,i P

gi ` c0,i (5a)

subject to p@i P N , @ pl,mq P LqP gi ´ P

di “ gsh,i wii `

ÿ

pl,mqPL,s.t. l“i

Plm `ÿ

pl,mqPL,s.t. m“i

Pml, (5b)

Qgi ´Qdi “ ´bsh,i wii `

ÿ

pl,mqPL,s.t. l“i

Qlm `ÿ

pl,mqPL,s.t. m“i

Qml, (5c)

θr “ 0, r P R, (5d)

P gi ď P gi ď Pg

i , Qgiď Qgi ď Q

g

i , (5e)

V i ď Vi ď V i, (5f)θlm ď θlm ď θlm, (5g)

pV iq2 ď wii ď pV iq

2, wii P@

V 2i

DT, (5h)

Plm “ glmwll ´ glmclm ´ blmslm, (5i)Qlm “ ´pblm ` bc,lm{2qwll ` blmclm ´ glmslm, (5j)Pml “ glmwmm ´ glmclm ` blmslm, (5k)Qml “ ´pblm ` bc,lm{2qwmm ` blmclm ` glmslm, (5l)

pPlmq2` pQlmq

2ď

`

Slm˘2, pPmlq

2` pQmlq

2ď`

Slm˘2,

(5m)

3

clm “ÿ

k“1,...,8

λk ρpkq1 ρ

pkq2 ρ

pkq3 , qClm P xcospθlmqy

C,

Vl “ÿ

k“1,...,8

λkρpkq1 , Vm “

ÿ

k“1,...,8

λkρpkq2 , qClm “

ÿ

k“1,...,8

λkρpkq3 ,

ÿ

k“1,...,8

λk “ 1, λk ě 0, k “ 1, . . . , 8. (5n)

slm “ÿ

k“1,...,8

γk ζpkq1 ζ

pkq2 ζ

pkq3 , qSlm P xsinpθlmqy

S,

Vl “ÿ

k“1,...,8

γkζpkq1 , Vm “

ÿ

k“1,...,8

γkζpkq2 , qSlm “

ÿ

k“1,...,8

γkζpkq3 ,

ÿ

k“1,...,8

γk “ 1, γk ě 0, k “ 1, . . . , 8. (5o)

P 2lm `Q

2lm ď wll `lm, (5p)

`lm “

˜

Y 2lm ´

b2c,lm4

¸

wll ` Y2lmwmm ´ 2Y 2

lmclm ´ bc,lmQlm,

(5q)»

—

—

–

λ1 ` λ2 ´ γ1 ´ γ2λ3 ` λ4 ´ γ3 ´ γ4λ5 ` λ6 ´ γ5 ´ γ6λ7 ` λ8 ´ γ7 ´ γ8

fi

ffi

ffi

fl

ᵀ»

—

—

–

V lV mV lV mV lV mV lV m

fi

ffi

ffi

fl

“ 0. (5r)

where `lm represents the squared magnitude of the currentflow into terminal l of line pl,mq P L and p¨qᵀ is the transposeoperator. The relationship between `lm and the power flowsPlm and Qlm in (5p) tightens the QC relaxation [11], [23].Appendix A in [22] gives an expression for `lm that considerslines with off-nominal tap ratios and non-zero phase shifts.Also, as shown in (5h), wii is associated with the squaredvoltage magnitude at bus i.

The lifted variables clm and slm represent relaxations of thetrilinear terms VlVm cospθlmq and VlVm sinpθlmq, respectively,with (5n) and (5o) formulating an “extreme point” representa-tion of the convex hulls for the trilinear terms VlVm qClm andVlVm qSlm [6], [24]. The auxiliary variables λk, γk P r0, 1s,k “ 1, . . . , 8, are used in the formulations of these convexhulls. The extreme points of VlVm qClm are ρpkq P rVl, Vls ˆ

rVm, Vms ˆ r qClm,qClms, k “ 1, . . . , 8 and the extreme points

of VlVm qSlm are ζpkq P rVl, Vls ˆ rVm, Vms ˆ rqSlm,qSlms,

k “ 1, . . . , 8. Since sine and cosine are odd and evenfunctions, respectively, clm “ cml and slm “ ´sml.

A “linking constraint” from [19] is also enforced in (5r).This linking constraint is associated with the bilinear termsVlVm that are shared in VlVm cospθlmq and VlVm sinpθlmq.

IV. COORDINATE TRANSFORMATIONS

The improvements to the QC relaxation’s envelopes thatare our main contributions are based on certain coordinatetransformations. This section describes these transformations.We first form the power flow equations using polar represen-tations of the lines’ mutual admittances. We then introduce acomplex base power in the per unit normalization that providesa rotational degree of freedom in the power flow equations.

While this section uses a Π circuit line model for simplicity,extensions to more general line models are straightforward.These extensions are presented in [22, Appendix A].

A. Power Flow Equations with Admittance in Polar Form

Equations (1h)–(1k) model the power flows through aline pl,mq P L via a rectangular representation of the line’smutual admittance, glm`jblm. In (5i)–(5l), the QC relaxationfrom [11] uses this rectangular admittance representation.

The line flows can be equivalently modeled using a polarrepresentation of the mutual admittance, Ylmejδlm , whereYlm “

a

g2lm ` b2lm and δlm “ arctan pblm{glmq are the mag-

nitude and angle of the mutual admittance for line pl,mq P L,respectively. Using polar admittance coordinates, the complexpower flows Slm and Sml into each line terminal are:

Slm “ Vlejθl

ˆˆ

Ylmejδlm ` j

bc,lm2

˙

Vlejθl ´ Ylme

jδlmVmejθm

˙˚

,

(6a)

Sml “ Vmejθm

ˆ

´YlmejδlmVle

jθl`

ˆ

Ylmejδlm`j

bc,lm2

˙

Vmejθm

˙˚

,

(6b)

where p¨q˚ is the complex conjugate. Taking the real andimaginary parts of (6) yields the active and reactive line flows:

Plm “ RepSlmq “ Ylm cospδlmqV2l ´ YlmVlVm cospθlm ´ δlmq,

(7a)Qlm “ ImpSlmq “ ´ pYlm sinpδlmq ` bc,lm{2qV

2l

´ YlmVlVm sinpθlm ´ δlmq, (7b)

Pml “ RepSmlq “ Ylm cospδlmqV2m ´ YlmVlVm cospθlm ` δlmq,

(7c)Qml “ ImpSmlq “ ´ pYlm sinpδlmq ` bc,lm{2qV

2m

` YlmVlVm sinpθlm ` δlmq. (7d)

With the rectangular admittance representation, the activeand reactive power flow equations (1h)–(1i) each have twotrigonometric terms (i.e., cospθlmq and sinpθlmq). Conversely,there is only one trigonometric term in each of the power flowequations that use the polar admittance representation (7) (e.g.,cospθlm ´ δlmq for Plm and sinpθlm ´ δlmq for Qlm). Whilethese formulations are equivalent, the differing representationsof the trigonometric terms suggest the possibility of usingdifferent trigonometric envelopes. The QC formulation we willpropose in Section V-C exploits these differences.

B. Rotated Power Flow Formulation

The base power used in the per unit normalization is tradi-tionally chosen to be a real-valued quantity. More generally,complex-valued choices for the base power are also acceptableand can provide benefits for some algorithms. For instance,certain power flow [20] and state estimation algorithms [21],[25] leverage formulations with a complex-valued base power.

To improve the QC relaxation’s trigonometric envelopes,this section reformulates the OPF problem with a complexbase power. Let Sorigbase and Snewbasee

jψ denote the original and thenew base power, respectively, where Sorigbase, S

newbase, and ψ are

real-valued. Thus, the original base Sorigbase is real-valued, whilethe new base Snewbasee

jψ is complex-valued with magnitudeSnewbase and angle ψ. Quantities associated with the new basepower will be accented with a tilde, p˜q. Complex power flowsin the original base and the new base are related as:

Slm “ Slm ¨Sorigbase

Snewbaseejψ, Sml “ Sml ¨

Sorigbase

Snewbaseejψ.

4

Since changing the magnitude of the base power does not af-fect the arguments of the trigonometric functions in the powerflow equations, we choose Snewbase “ Soldbase. With this choice,

Slm “ Slm{ejψ, Sml “ Sml{e

jψ.

The angle of the base power, ψ, affects the arguments of thetrigonometric functions, as shown in the following derivation:

Slm “ Slm{ejψ “

´

Ylme´jpδlm`ψq ` pbc,lm{2qe

´jpπ2`ψq¯

V 2l

´ YlmVlVmejp´δlm`θlm´ψq, (8a)

Sml “ Sml{ejψ “

´

Ylme´jpδlm`ψq ` pbc,lm{2qe

´jpπ2`ψq¯

V 2m

´ YlmVmVle´jpδlm`θlm`ψq. (8b)

Taking the real and imaginary parts of (8) yields:

Plm“RepSlmq“pYlm cospδlm ` ψq ´ pbc,lm{2q sinpψqqV 2l

´ YlmVlVm cospθlm ´ δlm ´ ψq, (9a)

Qlm“ ImpSlmq“´ pYlm sinpδlm ` ψq`pbc,lm{2q cospψqqV 2l

´ YlmVlVm sinpθlm ´ δlm ´ ψq, (9b)

Pml“RepSmlq“pYlm cospδlm ` ψq´pbc,lm{2q sinpψqqV 2m

´ YlmVmVl cospθlm ` δlm ` ψq, (9c)

Qml“ ImpSmlq“´ pYlm sinpδlm ` ψq`pbc,lm{2q cospψqqV 2m

` YlmVmVl sinpθlm ` δlm ` ψq. (9d)

The arguments of the trigonometric functions cospθlm´δlm´ψq, sinpθlm´δlm´ψq, cospθlm`δlm`ψq, and sinpθlm`δlm`ψq in (9) are linear in ψ. For a given ψ, all other trigonometricterms in (9) are constants that do not require special handling.

C. Rotated OPF Problem

We next represent the complex power generation and loaddemands using the new base power:

Sgi “ Sgi ¨Sorigbase

Snewbaseejψ“

Sgiejψ

“P gi ` jQ

gi

ejψ.

Define Sgi “ P gi `jQgi , @i P N . Taking the real and imaginary

parts of Sgi yields the following relationship between thepower generation in the new and original bases:

„

P giQgi

“

„

cospψq sinpψq´ sinpψq cospψq

„

P giQgi

. (10)

The inverse relationship is well defined for any choice of ψsince the matrix in (10) is invertible.

The analogous relationship for the power demands is:„

P diQdi

“

„

cospψq sinpψq´ sinpψq cospψq

„

P diQdi

. (11)

Applying (9)–(11) to (1) yields a “rotated” OPF problem:

minř

iPG c2,i

´

P gi cospψq ´ Qgi sinpψq¯2

` c1,i

´

P gi cospψq ´ Qgi sinpψq¯

` c0,i (12a)

subject to p@i P N , @ pl,mq P LqP gi ´ P

di “ pgsh,i cospψq ´ bsh,i sinpψqqV 2

i

`ÿ

pl,mqPL,s.t. l“i

Plm `ÿ

pl,mqPL,s.t. m“i

Pml, (12b)

Qgi ´ Qdi “ ´pgsh,i sinpψq ` bsh,i cospψqqV 2

i (12c)

`ÿ

pl,mqPL,s.t. l“i

Qlm `ÿ

pl,mqPL,s.t. m“i

Qml,

θr “ 0, r P R, (12d)

P gi ď P gi cospψq ´ Qgi sinpψq ď Pg

i , (12e)

Qgiď Qgi cospψq ` P gi sinpψq ď Q

g

i , (12f)

V i ď Vi ď V i, θlm ď θlm ď θlm, (12g)

pPlmq2 ` pQlmq

2 ď pSlmq2, pPmlq

2 ` pQmlq2 ď pSlmq

2,(12h)

Eq. (9). (12i)

The rotated OPF problem (12) is equivalent to (1) in thatany solution tV ˚, θ˚, P g‹, Qg‹u to (12) can be mapped toa solution tV ˚, θ˚, P g‹, Qg‹u to (1) using (10). Solutionsto both formulations have the same voltage magnitudes andangles, V ˚ and θ˚. Thus, (12) can be interpreted as revealinga degree of freedom associated with choosing the base power’sphase angle ψ. The next section exploits this degree of freedomto tighten the QC relaxation’s trigonometric envelopes.

V. ROTATED QC RELAXATION

This section leverages the coordinate transformations pre-sented in Section IV to tighten the QC relaxation. We firstpropose and analyze new envelopes for the trigonometricfunctions and trilinear terms. We then describe an empiricalanalysis that informs the choice of the base power angle ψ inorder to tighten the relaxation for typical OPF problems.

A. Convex Envelopes for the Trigonometric Terms

A key determinant of the QC relaxation’s tightness isthe quality of the convex envelopes for the trigonometricterms in the power flow equations. The rotated OPF formula-tion (12) has four relevant trigonometric terms for each line:cospθlm´ δlm´ψq, sinpθlm´ δlm´ψq, cospθlm` δlm`ψq,and sinpθlm ` δlm ` ψq, @pl,mq P L. This contrasts with thetwo unique trigonometric terms (cospθlmq and sinpθlmq) perpair of connected buses in the OPF formulation (1).

This would seem to suggest that at least twice as manylifted variables would be required in order to relax the rotatedOPF formulation (12) compared to the original OPF formula-tion (1), i.e., Cpsqlm , Spsqlm and Cprqlm , Sprqlm for relaxing the sendingend quantities cospθlm´δlm´ψq, sinpθlm´δlm´ψq and thereceiving end quantities cospθlm`δlm`ψq, sinpθlm`δlm`ψqin (12) versus qClm and qSlm for relaxing cospθlmq and sinpθlmqin (1). However, this is not the case since the arguments of thetrigonometric terms in the rotated OPF formulation are not in-dependent. For notational convenience, define δlm “ δlm`ψ.The angle sum and difference identities imply the followingrelationships:»

—

—

–

sinpδlm ` θlmq

cospδlm ` θlmq

sinpδlm ´ θlmq

cospδlm ´ θlmq

fi

ffi

ffi

fl

“

»

—

—

–

sinpδlmq cospδlmq

cospδlmq ´ sinpδlmq

sinpδlmq ´ cospδlmq

cospδlmq sinpδlmq

fi

ffi

ffi

fl

„

cospθlmqsinpθlmq

. (13)

5

Rearranging these relationships yields:„

sinpθlm ` δlmq

cospθlm ` δlmq

“

„

αlm βlm´βlm αlm

„

sinpθlm ´ δlmq

cospθlm ´ δlmq

.

(14)where, for notational convenience, αlm “ pcospδlmqq

2 ´

psinpδlmqq2 and βlm “ 2 cospδlmq sinpδlmq. The implication

of (14) is that only two (rather than four) lifted variables aredefined per line (chosen to be the sending end quantities Cpsqlmand Spsqlm for relaxing the trigonometric terms cospθlm ´ δlmq

and sinpθlm ´ δlmq). The remaining trigonometric functions,sinpθlm ` δlmq and cospθlm ` δlmq, are representable interms of sinpθlm ´ δlmq and cospθlm ´ δlmq via the lineartransformation (14). Since the matrix in (14) is invertible forall δlm, the transformation (14) is always well-defined.

While not explicitly including the lifted variables Cprqlm andSprqlm for the receiving end quantities, we tighten the relaxation

of (12) by enforcing the trigonometric envelopes associatedwith both the sending and receiving end quantities using (14):

Cpsqlm P xcospθlm ´ δlm ´ ψqy

C, (15a)

Spsqlm P xsinpθlm ´ δlm ´ ψqy

S, (15b)

αlmSpsqlm ` βlmC

psqlm P xsinpθlm ` δlm ` ψqy

S, (16a)

´βlmSpsqlm ` αlmC

psqlm P xcospθlm ` δlm ` ψqy

C, (16b)

Related special consideration is needed for parallel lines.While the rest of this section considers systems withoutparallel lines for simplicity, [22, Appendix B] discusses thisissue in detail. Using the linear relationships in (14), all rele-vant trigonometric terms in (12) can be represented as linearcombinations of sinpθlm ´ δlm ´ ψq and cospθlm ´ δlm ´ ψqfor each unique pair of connected buses pl,mq P L. The QCrelaxations of (1) and (12) hence have the same number oflifted variables (two per pair of connected buses).

There are two characteristics that distinguish the trigono-metric expressions in (1) and (12): First, the power flowequations (1h)–(1k) contain weighted sums of two trigono-metric functions of θlm, while (9a)–(9d) each contain a singletrigonometric function of θlm. (The trigonometric expressionscospδlm ` ψq, sinpδlm ` ψq, cospψq, and sinpψq in (9a)–(9d) are constants that do not require special consideration.)Second, the base power angle ψ used to formulate (12)provides a degree of freedom that shifts the arguments ofthe trigonometric functions in (9a)–(9d). We next discuss howboth of these characteristics can be exploited to tighten theQC relaxation.

Regarding the first distinguishing characteristic, factoringout ´VlVm to focus on the trigonometric functions showsthat the relaxation of (1h) depends on the quality of aweighted sum of trigonometric envelopes: glm xcos pθlmqy

C`

blm xsin pθlmqyS . The relaxation of (9a) depends on the quality

of the envelope Ylm xcos pθlm ´ δlm ´ ψqyC . (The relaxations

of (1h)–(1k) and (9a)–(9d) are analogous.) To focus on the firstcharacteristic, consider the latter envelope with ψ “ 0.

Fig. 1 on the following page illustrates examples of theseenvelopes for a line with the same mutual admittance (glm `

jblm “ 0.6´ j0.8) for different intervals of angle differences(θlm ď θlm ď θlm). While transmission lines with such largeresistances are atypical, we choose this admittance to bettervisualize our approach in Fig. 1. Our approach is valid for allvalues of line admittances.

To compare these envelopes, we consider their boundaries.As shown in [22, Appendix C], either the upper or lowerboundary of the envelope Ylm xcos pθlm ´ δlmqy

C is at least astight (and sometimes tighter) compared to the correspondingboundary of the envelope glm xcos pθlmqy

C`blm xsin pθlmqy

S

for certain values of δlm, θminlm , and θmaxlm . In this case,there is no general dominance relationship for the otherboundary. For other values of δlm, θminlm , and θmaxlm , noneof the boundaries of the envelope Ylm xcos pθlm ´ δlmqy

C

dominate or are dominated by a boundary of the envelopeglm xcos pθlmqy

C` blm xsin pθlmqy

S . Thus, a QC relaxationthat enforces the intersection of these envelopes is generallytighter than relaxations constructed using either of these en-velopes individually. Section V-D discusses this further.

The second characteristic distinguishing between the en-velopes for (1) and (12) is the ability to choose ψ in the latterenvelopes. As shown in Fig. 2, changing ψ rotates the argu-ments of these envelopes. We also visualize the impacts thatdifferent values of ψ have on the sine and cosine envelopes inan animation available at https://arxiv.org/src/1912.05061v2/anc/rotated_envelope_animation.gif.

Analytically comparing the impacts of different values forψ is not straightforward. Accordingly, this section will laterdescribe an empirical study that suggests a good choice for ψfor typical OPF problems.

B. Envelopes for Trilinear Terms

In addition to the trigonometric functions considered thusfar, the products between the voltage magnitudes and thetrigonometric functions in (9) are another source of non-convexity in the rotated OPF problem (12). We next exploit therelationship between the sending and receiving end trigono-metric functions (14) in order to relax these products using alimited number of additional lifted variables.

Similar to (5n)–(5o), we relax the trilinear products by con-structing linear envelopes using the upper and lower bounds onVl, Vm, cospθlm´δlm´ψq, sinpθlm´δlm´ψq, cospθlm`δlm`ψq, and sinpθlm`δlm`ψq. We use the linear relationship (14)to represent the upper and lower bounds on the receivingend quantities cospθlm ` δlm ` ψq (denoted C

prqlm , Cprqlm ) and

sinpθlm`δlm`ψq (denoted Sprqlm , Sprqlm ) in terms of the boundson the sending end quantities cospθlm´δlm´ψq (denoted Cpsqlm ,Cpsqlm) and sinpθlm ´ δlm ´ ψq (denoted Spsqlm , Spsqlm). We then

enforce constraints on the sending end quantities derived fromthe intersection of the transformed bounds associated with thereceiving end quantities along with the bounds on the sendingend quantities. Intersecting these bounds forms a polytopein terms of the sending end quantities C

psqlm (representing

cospθlm´δlm´ψq) and Spsqlm (representing sinpθlm´δlm´ψq),expressible as a convex combination of its extreme points.

Fig. 3 on the following page shows the bounds on both thesending and receiving end quantities in terms of the sending

6

(a) ´90˝ ď θlm ď 90˝

(b) ´60˝ ď θlm ď 0˝

(c) ´30˝ ď θlm ď 30˝

(d) ´30˝ ď θlm ď 0˝

Figure 1. Comparison of envelopes for the trigonometric terms in (1) and (12).The yellow and magenta regions (with dotted and dashed borders, respec-tively) in (a)–(d) show the envelopes glm xcos pθlmqy

C` blm xsin pθlmqy

S

and Ylm xcos pθlm ´ δlmqyC , respectively. The black solid lines correspond

to the function glm cos pθlmq ` blm sin pθlmq “ Ylm cos pθlm ´ δlmq.

(a) xcos pθlm ´ δlm ´ ψqyC

(b) xsin pθlm ´ δlm ´ ψqyS

Figure 2. Comparison of envelopes for the sine and cosine functions for differ-ent values of ψ. The yellow and red regions (with dashed and dotted borders,respectively) in (a) and (b) show the envelopes xcos pθlm ´ δlm ´ ψqy

C andxsin pθlm ´ δlm ´ ψqy

S , for ψ1 “ ´15˝ and ψ2 “ 45˝, respectively. Theangle difference θlm varies within 0˝ ď θlm ď 72˝, and δlm “ ´53˝.

αlmCpsqlm ´ βlmS

psqlm P rC

prqlm , C

prqlm s and

βlmCpsqlm ` αlmS

psqlm P rS

prqlm , S

prqlm s

Cpsqlm P rC

psqlm , C

psqlms and S

psqlm P rS

psqlm , S

psqlms

Cpsqlm

Spsqlm

Figure 3. A projection of the four-dimensional polytope associated with thetrilinear products between voltage magnitudes and trigonometric functions, interms of the sending end variables Spsqlm and Cpsqlm representing cospθlm ´δlm´ψq and sinpθlm´ δlm´ψq. The polytope formed by intersecting thesending end polytope (yellow) and receiving end polytope (red) is outlinedwith the dashed black lines and has vertices shown by the black dots.

7

end quantities. The yellow region is the polytope formed bythe bounds on cospθlm´δlm´ψq and sinpθlm´δlm´ψq. Thered region is the polytope formed by using (14) to representthe bounds on the receiving end quantities cospθlm`δlm`ψqand sinpθlm` δlm`ψq in terms of the sending end quantitiescospθlm´δlm´ψq and sinpθlm´δlm´ψq. The black dots arethe vertices of the polytope shown by the dashed black linesformed from the intersection of the yellow and red polytopes.Appendix D in [22] shows how to compute these vertices.

Enforcing the constraints associated with both the yellowand red polytopes adds an unnecessary computational burden.We instead restrict the sending end quantities cospθlm´δlm´ψq and sinpθlm ´ δlm ´ ψq to lie within the polytope shownby the black dashed line in Fig. 3. This implicitly ensuressatisfaction of the bounds on the receiving end quantities.

To relax the product terms VlVm cospθlm ´ δlm ´ ψq andVlVm sinpθlm ´ δlm ´ ψq, we first represent the quantitiescospθlm ´ δlm ´ ψq and sinpθlm ´ δlm ´ ψq using liftedvariables C

psqlm and S

psqlm , respectively. We then extend the

polytope shown by the black dashed lines in Fig. 3 usingthe upper and lower bounds on Vl and Vm. The resultingfour-dimensional polytope is the convex hull of the quadrilin-ear polynomial VlVmC

psqlm S

psqlm , which we represent using an

extreme point formulation similar to (5n)–(5o). Let Tlm “

tpCint,1lm , Sint,1lm q, pCint,2lm , Sint,2lm q, . . . , pCint,Nlm , Sint,Nlm qu de-note the coordinates of the intersection points (black dots)in Fig. 3, where N is the number of intersection pointswhich ranges from 4 to 8 depending on the value of ψ.The extreme points of VlVmC

psqlm S

psqlm are then denoted as

ηpkq P rVl, VlsˆrVm, VmsˆTlm, k “ 1, . . . , 4N . The auxiliaryvariables λk P r0, 1s, k “ 1, . . . , 4N , are used to form theconvex hull of the quadrilinear term VlVmC

psqlm S

psqlm .

The envelopes for the trilinear terms are:

clm “ÿ

k“1,...,4N

λk ηpkq1 η

pkq2 η

pkq3 , slm “

ÿ

k“1,...,4N

λk ηpkq1 η

pkq2 η

pkq4 ,

Vl “ÿ

k“1,...,4N

λkηpkq1 , Vm “

ÿ

k“1,...,4N

λkηpkq2 , S

psqlm “

ÿ

k“1,...,4N

λkηpkq4 ,

Cpsqlm “

ÿ

k“1,...,4N

λkηpkq3 ,

ÿ

k“1,...,4N

λk “ 1, λk ě 0, k “ 1, . . . , 4N ,

Cpsqlm P xcospθlm ´ δlm ´ ψqy

C,

Spsqlm P xsinpθlm ´ δlm ´ ψqy

S,

αlmSpsqlm ` βlmC

psqlm P xsinpθlm ` δlm ` ψqy

S,

´ βlmSpsqlm ` αlmC

psqlm P xcospθlm ` δlm ` ψqy

C, (17)

where the last four trigonometric envelope constraints corre-spond to (15)–(16).

Note that (17) precludes the need for the linking con-straint (5r) that relates the common term VlVm in the productsVlVm sinpθlmq and VlVm cospθlmq.

C. QC Relaxation of the Rotated OPF ProblemReplacing the squared and trilinear terms with the corre-

sponding lifted variables in the rotated OPF formulation (12)results in our proposed “Rotated QC” (RQC) relaxation:

min (12a) (18a)subject to p@i P N ,@ pl,mq P LqP gi ´ P

di “ pgsh,i cospψq ´ bsh,i sinpψqqwii

`ÿ

pl,mqPL,s.t. l“i

Plm `ÿ

pl,mqPL,s.t. m“i

Pml, (18b)

Qgi ´ Qdi “ ´pgsh,i sinpψq ` bsh,i cospψqqwii

`ÿ

pl,mqPL,s.t. l“i

Qlm `ÿ

pl,mqPL,s.t. m“i

Qml, (18c)

Plm “ pYlm cospδlm ` ψq ´ bc,lm{2 sinpψqqwll ´ Ylmclm, (18d)Qlm “ ´pYlm sinpδlm ` ψq ` bc,lm{2 cospψqqwll ´ Ylmslm,

(18e)Pml “ ´Ylmclm ` pYlm cospδlm ` ψq ´ bc,lm{2 sinpψqqwmm,

(18f)Qml “ Ylmslm ´ pYlm sinpδlm ` ψq ` bc,lm{2 cospψqqwmm,

(18g)

P 2lm ` Q

2lm ď wll ˜lm, (18h)

˜lm “

ˆ

b2c,lm{4` Y2lm ´ Ylmbc,lm cospδlm ` ψq sinpψq

` Ylmbc,lm sinpδlm ` ψq cospψq

˙

wll ` Y2lmwmm

``

´2Y 2lm cospδlm ` ψq ` Ylmbc,lm sinpψq

˘

clm

``

2Y 2lm sinpδlm ` ψq ` Ylmbc,lm cospψq

˘

slm, (18i)Equations (5h), (12d)–(12h), (17). (18j)

Note that trilinear terms are relaxed via the extreme pointapproach in (17) that yields the convex hulls for these terms.The variables clm and slm are relaxations of the trilinearterms VlVm cospθlm´ δlm´ψq and VlVm sinpθlm´ δlm´ψq,respectively. Appendix A in [22] gives an expression for ˜

lm

that considers off-nominal tap ratios and non-zero phase shifts.

D. Tightened QC Relaxation of the Rotated OPF ProblemApplying the angle sum and difference identities in com-

bination with (14) reveals a linear relationship between thetrigonometric functions used in the original QC relaxation (5),cospθlmq and sinpθlmq, and those in the RQC relaxation (18),cospθlm ´ δlm ´ ψq and sinpθlm ´ δlm ´ ψq:

„

cospθlmqsinpθlmq

“Mlm

„

sinpθlm ´ δlm ´ ψqcospθlm ´ δlm ´ ψq

, (19)

where the constant matrix Mlm is defined as

Mlm “1

2

ˆ„

sinpδlm ` ψq cospδlm ` ψqcospδlm ` ψq ´ sinpδlm ` ψq

„

αlm βlm´βlm αlm

`

„

´ sinpδlm ` ψq cospδlm ` ψqcospδlm ` ψq sinpδlm ` ψq

˙

with αlm and βlm defined as in (14). As mentioned inSection V-A, the RQC relaxation (18) can be further tightenedby additionally enforcing the envelopes xcos pθlmqy

C andxsin pθlmqy

S used in the original QC relaxation (5). Thisresults in the “Tightened Rotated QC” (TRQC) relaxation:

min (12a) (20a)subject to p@i P N ,@ pl,mq P Lq

Mlm

„

Cpsqlm

Spsqlm

P

„

xcospθlmqyC

xsinpθlmqyS

(20b)

Equations (5h), (12d)–(12h), (17), (18b)–(18i). (20c)

8

E. An Empirical Analysis for Determining the Rotation ψ

The key parameter in our proposed QC formulation is therotation ψ. Choosing an appropriate value for ψ using ananalytical method is challenging because, as shown in Fig. 1,ψ simultaneously affects the envelopes for both the sine andcosine functions such that values of ψ which lead to tighterenvelopes for the cosine function can result in looser envelopesfor the sine function (and vice-versa). Moreover, the singlevalue for ψ applied to the entire system requires balancing theimpacts of ψ among all lines simultaneously. Thus, choosingan appropriate value for ψ is not straightforward. We thereforeuse the following empirical analysis to choose a value for ψthat works well for a range of test cases. In our results, wedenote the best value of ψ for each case as ψ˚.

Fig. 4 shows the optimality gaps for the PGLib-OPF testcases as a function of ψ, each normalized by the maximumgap for that case over all values for ψ. The results in the figurewere generated by sweeping ψ from ´90˝ to 90˝ in steps of0.5˝. (The figure is exactly symmetric for values of ψ from90˝ to -90˝.) The shaded red bands around the median line(in black) show every fifth percentile of the results.

The results in Fig. 4 indicate that good values of ψ areconsistent across the test systems. Thus, we suggest using ψ “80˝, which is where the median of the optimality gaps over allthe test cases was smallest. Moreover, the symmetry in Fig. 4implies that selecting ψ within the intervals [´90˝, ´80˝],[´15˝, ´5˝], and [80˝, 90˝] results in nearly the smallestoptimality gaps for almost all of the test cases compared tothe optimality gaps from the RQC relaxation using ψ˚.

Figure 4. Normalized optimality gap as a function of ψ for PGLib-OPF cases.

Our ongoing work is generalizing the rotated QC for-mulation proposed in this paper by allowing for differentvalues of ψ for each line in the system in order to constructtighter envelopes (i.e., independent choices of ψlm for eachpl,mq P L rather than one value of ψ for the entire system).While complicating the overall formulation, this generalizationsimplifies the impacts from choosing each ψlm and thus hasthe potential to enable analytical methods for identifying thebest values for ψlm for each pl,mq P L. One possible approachis to compute the value of ψlm that minimizes the areas ofthe envelopes (i.e., the yellow and red regions in Fig. 2).

VI. NUMERICAL RESULTS

This section demonstrates the effectiveness of the proposedapproach using selected test cases from the PGLib-OPF v18.08benchmark library [26]. These test cases were selected sinceexisting relaxations fail to provide tight bounds on the bestknown objective values. Our implementations use Julia 0.6.4,JuMP v0.18 [27], PowerModels.jl [28], and Gurobi 8.0 asmodeling tools and the solver. The results are computed usinga laptop with an i7 1.80 GHz processor and 16 GB of RAM.

Table I summarizes the results from applying the QC (5),RQC (18), and TRQC (20) relaxations to selected test cases.The first column lists the test cases. The next group of columnsrepresents optimality gaps, defined as

Optimality Gap “ˆ

Local Solution´ QC BoundLocal Solution

˙

. (21)

The optimality gaps are defined using the local solutions tothe non-convex problem (1) from PowerModels.jl. The finalgroup of columns shows the solver times.

Comparing the second and third columns in Table I revealsthat using admittances in polar form without rotation (i.e., theRQC relaxation (18) with ψ “ 0) can improve the optimalitygaps of some test cases (e.g., improvements of 3.76% and3.19% for “case30_ieee” and “case24_ieee_rts__api”, respec-tively, relative to the original QC relaxation (5)) . However, theRQC relaxation with ψ “ 0 has worse performance in othercases, such as “case300_ieee” and “case14_ieee__sad”, whichhave 0.02% and 2.29% larger optimality gaps, respectively.

Using a non-zero value for ψ can improve the optimalitygaps. Solving the RQC relaxation (18) with the suggestedψ “ 80˝ obtained from the empirical analysis in Section V-Eresults in 1.08% better optimality gaps, on average, comparedto the original QC relaxation. The RQC relaxation (18) withψ˚ (the best value of ψ for each case) provides optimalitygaps that are not worse than those obtained by the original QCrelaxation (5) for all test cases, yielding an improvement of1.36% on average compared to the original QC relaxation. Asone specific example, the gap from the original QC relaxationfor “case162_ieee_dtc__sad” is 6.22% compared to 6.30% forthe RQC relaxation (18) with ψ “ 0 relaxation (18). Use ofthe suggested ψ “ 80˝ reduces the gap to 5.65%, which issuperior to the gap obtained from the QC relaxation (5). Usingψ˚ further reduces the optimality gap to 5.59%.

Enforcing the envelopes from both the original QC relax-ation and the RQC relaxation, i.e., the TRQC relaxation (20),further improves the optimality gaps. Solving the TRQC relax-ation (20) with the suggested ψ “ 80˝ results in 1.29% bettergaps, on average, compared to the original QC relaxation.The TRQC relaxation with ψ˚ yields optimality gaps thatare 1.57% and 0.21% better, on average, compared to theoriginal QC relaxation and the RQC relaxation with ψ˚. Theadditional envelopes xsin pθlmqy

S and xcos pθlmqyC in the

TRQC relaxation increase the average solver time by 22%.Fig. 5 visualizes the optimality gaps for variants of the

QC relaxation over a range of test cases. Positive valuesindicate an improvement in the optimality gap of the associatedvariant relative to the original QC relaxation (5). The testcases are sorted in order of increasing optimality gaps obtained

9

Table IRESULTS FROM APPLYING THE QC AND RQC RELAXATIONS WITH VARIOUS OPTIONS TO SELECTED PGLIB TEST CASES

Test Cases QCGap (%)

RQC(ψ “ 0)

Gap (%)

RQC(ψ “ 80˝)Gap (%)

RQC pψ˚q TRQC(ψ “ 80˝)Gap (%)

TRQC (ψ˚) QCTime(sec)

RQCTime(sec)

TRQCTime(sec)Gap (%) ψ˚ Gap (%) ψ˚

case3_lmbd 0.97 0.97 0.89 0.79 ´81˝ 0.84 0.63 11˝ 0.34 0.01 0.01case30_ieee 18.67 14.91 13.14 12.11 65˝ 13.14 11.82 ´25˝ 0.33 0.03 0.03case118_ieee 0.77 0.90 0.65 0.64 70˝ 0.64 0.62 70˝ 0.55 0.19 0.23case300_ieee 2.56 2.58 2.43 2.26 ´13˝ 2.32 2.24 ´13˝ 1.54 1.50 3.15case9241_pegase 1.71 1.70 1.70 1.69 ´10˝ 1.70 1.69 ´10˝ 265.39 190.80 297.56case3_lmbd__api 4.57 4.31 4.42 4.28 2˝ 4.17 3.93 ´71˝ 0.51 0.01 0.01case24_ieee_rts__api 11.02 7.83 7.51 7.24 79˝ 7.31 6.98 ´11˝ 0.71 0.03 0.04case39_epri__api 1.71 1.38 1.33 1.33 ´11˝ 1.32 1.32 79˝ 0.39 0.05 0.05case73_ieee_rts__api 9.54 8.12 7.36 7.36 ´10˝ 7.24 7.24 ´10˝ 1.00 0.29 0.37case118_ieee__api 28.67 28.03 26.82 26.52 ´8˝ 27.11 26.38 ´8˝ 0.53 0.20 0.97case179_goc__api 5.86 6.01 5.57 4.90 ´81˝ 4.90 4.06 ´78˝ 0.82 0.61 0.64case14_ieee__sad 19.16 21.45 17.89 16.30 77˝ 15.82 15.39 ´12˝ 0.35 0.03 0.03case24_ieee_rts__sad 2.74 2.55 2.31 2.19 78˝ 2.26 2.12 ´12˝ 0.40 0.05 0.06case30_ieee__sad 5.66 5.95 4.81 4.59 ´13˝ 4.56 4.45 66˝ 0.32 0.05 0.06case73_ieee_rts__sad 2.37 2.24 1.98 1.90 79˝ 1.84 1.82 78˝ 0.41 0.28 0.41case118_ieee__sad 6.67 8.10 5.45 5.39 81˝ 5.45 5.07 69˝ 0.58 0.25 0.39case162_ieee_dtc__sad 6.22 6.30 5.65 5.59 ´14˝ 5.65 5.54 76˝ 0.86 0.55 0.84case300_ieee__sad 2.34 2.59 1.80 1.61 83˝ 1.78 1.59 83˝ 1.94 1.29 2.06

AC: AC local solution from (1), QC Gap: Optimality gap for the QC relaxation from (5), RQC Gap: Optimality gap for the Rotated QC relaxation from (18),TRQC Gap: Optimality gap for the Tightened Rotated QC Relaxation from (20), ψ˚: Use of the best ψ for this case.

Opti

mal

ity

gap

dif

fere

nce

s (%

)

RQC: Eq (16) with =0°

RQC: Eq (16) with =80°

RQC: Eq (16) with *

TRQC: Eq (18) with =80°

TRQC: Eq (18) with *

Figure 5. Comparison of optimality gap differences with respect to theoriginal QC relaxation (5) for different QC relaxation variants.

from the original QC relaxation. The TRQC relaxation withψ˚ achieves the smallest optimality gaps. While the RQCrelaxation with ψ “ 0 obtains a worse optimality gap forsome test cases compared to the original QC relaxation, boththe RQC and TRQC relaxations with ψ˚ outperform the QCrelaxation for all test cases. As expected from the analysis inSection V-E, applying the suggested ψ “ 80˝ results in goodperformance across a variety of test cases.

VII. CONCLUSION

This paper proposes and empirically tests two improve-ments for strengthening QC relaxations of OPF problems bytightening the envelopes used for the trigonometric terms.The first improvement represents the line admittances in polarform. The second improvement applies a complex base powernormalization with angle ψ in order to rotate the argumentsof the trigonometric terms. An empirical analysis is used tosuggest a good value for ψ. Comparison to the state-of-the-art QC relaxation reveals the effectiveness of the proposed

improvements. Our ongoing work is extending the RQC re-laxation to allow for distinct values of ψ for each line.

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Mohammad Rasoul Narimani (S’14-M’20) is apostdoc at Texas A&M University, College Sta-tion. He received the B.S. and M.S. degrees inelectrical engineering from the Razi University andShiraz University of Technology, respectively. Healso received the Ph.D. electrical engineering fromMissouri University of Science & Technology. Hisresearch interests are in the application of optimiza-tion techniques to electric power systems.

Daniel K. Molzahn (S’09-M’13-SM’19) is an As-sistant Professor in the School of Electrical andComputer Engineering at the Georgia Institute ofTechnology. He also holds an appointment as a com-putational engineer in the Energy Systems Divisionat Argonne National Laboratory and is a fellow ofthe Strategic Energy Institute at Georgia Tech. Hewas a Dow Postdoctoral Fellow in Sustainability atthe University of Michigan, Ann Arbor. He receivedthe B.S., M.S., and Ph.D. degrees in electrical en-gineering and the Masters of Public Affairs degree

from the University of Wisconsin–Madison, where he was a National ScienceFoundation Graduate Research Fellow. His research focuses on optimizationand control of electric power systems.

Mariesa L. Crow (S’83–M’90–SM’94–F’10) isthe Fred Finley Distinguished Professor Emeritusof Electrical Engineering at Missouri S&T. Shereceived her BSE in Electrical Engineering fromthe University of Michigan and her Ph.D. in Elec-trical Engineering from the University of Illinois-Urbana/Champaign. She joined Missouri Universityof Science & Technology in 1991. Her area ofprofessional interest is computational methods andpower electronics applications to renewable energysystems, microgrids, and energy storage. She is a

Registered Professional Engineer in the State of Missouri and a Fellow of theIEEE.