1

Implementation of a Large-Scale Optimal PowerFlow Solver Based on Semidefinite ProgrammingDaniel K. Molzahn,Student Member, IEEE, Jesse T. Holzer, Bernard C. Lesieutre,Senior Member, IEEE,

and Christopher L. DeMarco,Member, IEEE

Abstract—The application of semidefinite programming tothe optimal power flow (OPF) problem has recently attractedsignificant research interest. This paper provides advances inmodeling and computation required for solving the OPF problemfor large-scale, general power system models. Specifically, asemidefinite programming relaxation of the OPF problem ispresented that incorporates multiple generators at the samebus and parallel lines. Recent research in matrix completiontechniques that decompose a single large matrix constrainedto be positive semidefinite into many smaller matrices hasmade solution of OPF problems using semidefinite programmingcomputationally tractable for large system models. We providethree advances to existing decomposition techniques: a matrixcombination algorithm that further decreases solver time, amodification to an existing decomposition technique that extendsits applicability to general power system networks, and a methodfor obtaining the optimal voltage profile from the solution to adecomposed semidefinite program.

Index Terms—Optimal power flow, Semidefinite optimization

I. I NTRODUCTION

T HE optimal power flow (OPF) problem determines anoptimal operating point for an electric power system in

terms of a specified objective function, subject to both networkequality constraints (i.e., the power flow equations, whichmodel the relationship between voltages and power injections)and engineering limits (e.g., inequality constraints on voltagemagnitudes, active and reactive power generations, and flowson transmission lines and transformers). Total variable gener-ation cost per unit time is typically chosen as the objectivefunction.

The OPF problem is typically nonconvex due to the non-linear power flow equations [1]. Nonconvexity of the OPFproblem has made solution techniques an ongoing researchtopic. Many OPF solution techniques have been proposed, in-cluding successive quadratic programs, Lagrangian relaxation,genetic algorithms, particle swarm optimization, and interiorpoint methods [2], [3].

Recently, significant attention has focused on the applicationof semidefinite programming to the OPF problem [4], [5].Using a rank relaxation, the OPF problem is reformulated as aconvex semidefinite program. If the relaxed problem satisfies arank condition (i.e., the semidefinite program has zero dualitygap), the globally optimal solution to the original OPF problemcan be determined in polynomial time. No prior OPF solutionmethod offers a guarantee of finding a global solution in

University of Wisconsin-Madison Department of ElectricalandComputer Engineering: molzahn@wisc.edu, lesieutre@engr.wisc.edu,demarco@engr.wisc.edu Department of Mathematics: holzer@math.wisc.edu

polynomial time; semidefinite programming approaches thushave a substantial advantage over traditional solution tech-niques. Note, however, that the rank condition is not alwayssatisfied, which means that semidefinite relaxations do not givephysically meaningful solutions for all realistic power systemmodels [6], [7]. The solution to the semidefinite relaxationforsuch cases may provide a good starting point for a local searchalgorithm. Further, the solution to the semidefinite relaxationalways yields a lower bound to the unknown global solution,which provides a measure of sub-optimality for any localsolution. It is important to note, however, that a solutionto the semidefinite relaxation with non-zero duality gap isnot necessarily a better approximation to the true solutionthan other solution techniques. For instance, when appliedtothe three-bus system in [6], the DC OPF, a common linearapproximation to the OPF problem [3], yields more accurateactive power generation and active power Lagrange multipliers(LMPs), as compared to a traditional AC OPF calculation, thanthe solution with non-zero duality gap from the semidefiniterelaxation.

Recent research has investigated conditions under whichthe rank condition is satisfied; to date, sufficient conditionsfor rank condition satisfaction include highly limiting re-quirements on power injection and voltage magnitude limitsand either radial networks (typical of distribution systemmodels) or unrealistically dense placement of controllablephase shifting transformers [8]–[11]. Additional work includesusing semidefinite programming to create voltage stabilitymargins [12].

This paper first focuses on modeling aspects that mustbe addressed in order to apply the semidefinite program togeneral power system models. The first issue addressed isthat of allowing multiple generators at the same bus. Byequating bus power injections with power generation, existingformulations only allow a single generator to exist at a bus.Weuse the concept of equal marginal generation cost to produceaformulation allowing for multiple generators at the same bus,each with separate cost functions and generation limits. Thispaper considers both quadratic and convex piecewise-lineargenerator cost functions.

A method for incorporating flow limits on parallel linesis then presented. Existing formulations limit the total flowbetween two buses, which cannot properly account for parallellines with different electrical properties and flow limits.In con-trast, the proposed method limits the flow on each individualline. Lines in this formulation can have off-nominal voltageratios and/or non-zero phase shifts.

2

This paper next advances research in the computationaltractability of applying semidefinite programming to largepower system models. Semidefinite programming relaxationsof the OPF problem constrain a2n × 2n symmetric matrixto be positive semidefinite, wheren is the number buses inthe system. The semidefinite program size thus grows as thesquare of the number of buses, which makes solution of theOPF problem by semidefinite programming computationallychallenging for large systems. Recent work using matrix com-pletion [13]–[15] reduces the computational burden inherent insolving large systems by taking advantage of the sparse matrixstructure created by realistic power system models. Sojoudiand Lavaei [10], Jabr [16], and Bai and Wei [17] presentformulations that decompose the single large2n × 2n pos-itive semidefinite matrix constraint into positive semidefiniteconstraints on many smaller matrices. If the matrices fromthese decompositions satisfy a rank condition, the2n × 2nmatrix also satisfies the rank condition and the optimal solutioncan be obtained. Sojoudi and Lavaei’s decomposition [10]uses a cycle basis of the network. The matrix decompositionapproach used by both Jabr [16] and Bai and Wei [17] is basedon the maximal cliques of a chordal extension of the network.

We provide several enhancements to the existing decom-positions. Specifically, we present a heuristic algorithm forcombining some of the small matrices resulting from thedecomposition. Since linking constraints are required betweenelements of the decomposed matrices that refer to the sameelement in the2n× 2n matrix, it is not always advantageousto create the smallest possible matrices. Combining matriceseliminates some of these linking constraints, which can resultin significant computational speed increases. We justify theclaim that the proposed algorithm can substantially increasecomputational speed using both theoretical arguments andseveral test cases.

Note that this paper considers a centralized application ofthese decompositions in the sense of creating one semidefiniteprogram that is solved on a single computer as opposed tocreating many subproblems that are solved using decentralizedtechniques as in [18]. Centralized application allows for solu-tion with existing generic semidefinite programming solvers.

A further enhancement presented in this paper is a techniquefor recovering the optimal voltage profile from the decom-posed matrices. While the steps are relatively straightfor-ward, existing literature does not detail a method for actuallyobtaining the optimal voltage profile from a solution to adecomposed formulation.

Although we focus on the maximal clique decompositionproposed by both Jabr [16] and Bai and Wei [17] due tothe voluminous literature on matrix completion with chordalextensions (e.g., [13]–[15]), both of these enhancements couldbe applied to Sojoudi and Lavaei’s decomposition [10] as well.

We also describe a modification to the maximal cliquedecomposition as formulated by Jabr [16] that allows forapplication to general power systems. This formulation cre-ates a chordal extension of the network using a Choleskyfactorization of the absolute value of the imaginary part ofthe bus admittance matrix. However, this matrix may fail tobe positive definite (for instance, in networks with significant

shunt capacitive compensation), thus preventing calculation ofa Cholesky factorization. We describe an alternative matrixthat is always positive definite and gives an equivalent chordalextension, thus broadening the applicability of this decompo-sition to general networks.

This paper is organized as follows. Section II provides boththe classical formulation of the OPF problem and a proposedsemidefinite programming relaxation that incorporates multi-ple generators at the same bus and parallel lines, includinglines with off-nominal voltage ratios and/or non-zero phase-shifts. Section III first gives an overview of the maximalclique decomposition and then presents three advances indecompositions for large-scale system models: an algorithmthat improves computation speed by combining matrices, amodification to Jabr’s maximal clique decomposition thatextends its applicability to general power system networks,and a technique for recovering the optimal voltage profilefrom a solution to a decomposed formulation. Section III alsodiscusses rank condition satisfaction for large system models.

II. T HE OPF PROBLEM AND MODELING ISSUES

We first present the OPF problem as it is classically formu-lated. Specifically, this formulation is in terms of rectangularvoltage coordinates, active and reactive power generation, andapparent power line-flow limits. Each bus may have multi-ple generators, and parallel lines are allowed. This classicalOPF formulation is generally nonconvex. We then describea semidefinite programming relaxation of the OPF problemadopted from [5] that handles the modeling issues of multiplegenerators at the same bus and parallel lines.

A. Classical OPF Formulation

Consider ann-bus power system, whereN = {1, 2, . . . , n}represents the set of all buses. DefineG as the set of allgenerators, withGi the set of generators at busi. Let Gq

represent the set of all generators with quadratic cost functions,with Gq

i those such generators at busi. Let Gpw represent theset of generators with piecewise-linear cost functions, withGpwi those such generators at busi. (Some of these sets may

be empty.)PGg + jQGg represents the active and reactivepower output of generatorg ∈ G. PDi + jQDi representsthe active and reactive load demand at each busi ∈ N .Vi = Vdi + jVqi represents the voltage phasors in rectangularcoordinates at each busi ∈ N . Superscripts “max” and “min”denote specified upper and lower limits. LetY = G + jBdenote the network admittance matrix.L represents the set of all lines, where linek ∈ L has

terminals at buseslk andmk, with parallel lines allowed (i.e.,more than one line between the same terminal buses). LetSk

represent the apparent power flow on the linek ∈ L.Define a cost function associated with each generator,

typically representing a dollar/hour variable operating cost.This paper considers quadratic and convex piecewise-linearcost functions in (1a) and (1b), respectively. In (1a), the termscg2, cg1, andcg0 represent the quadratic cost coefficients forgeneratorg ∈ Gq. In (1b), generatorg ∈ Gpw has a piecewise-linear cost function composed ofrg line segments specified

3

by slopesmg1, . . . , mgrg and breakpoints(agt, bgt) , t =1, . . . , rg, whereagt is the power generation coordinate andbgt is the cost coordinate for the breakpoint.

Cqg (PGg) = cg2P

2Gg + cg1PGg + cg0 (1a)

Cpwg (PGg) =

mg1 (PGg − ag1) + bg1, PGg ≤ ag1

mg2 (PGg − ag2) + bg2, ag1 < PGg ≤ ag2

......

mgr (PGg − agr) + bgr, agr < PGg

(1b)

The classical OPF problem is then

min∑

g∈Gq

Cqg (PGg) +

∑

g∈Gpw

Cpwg (PGg) (2a)

subject to

PminGg ≤ PGg ≤ Pmax

Gg ∀g ∈ G (2b)

QminGg ≤ QGg ≤ Qmax

Gg ∀g ∈ G (2c)(

V mini

)2≤ V 2

di + V 2qi ≤ (V max

i )2 ∀i ∈ N (2d)

|Sk| ≤ Smaxk ∀k ∈ L (2e)

∑

g∈Gi

(PGg)− PDi = Vdi

n∑

h=1

(GihVdh −BihVqh) (2f)

+ Vqi

n∑

h=1

(BihVdh +GihVqh) ∀i ∈ N

∑

g∈Gi

(QGg)−QDi = Vdi

n∑

h=1

(−BihVdh −GihVqh) (2g)

+ Vqi

n∑

h=1

(GihVdh −BihVqh) ∀i ∈ N

Note that this formulation limits the apparent power flowmeasured at each end of a given line, recognizing that activeand reactive line losses can cause these quantities to differ.

B. Semidefinite Programming Relaxation of the OPF Problem

This section first describes the semidefinite relaxation of theOPF problem, including the capability to incorporate parallellines and multiple generators at the same bus. Letei denote theith standard basis vector inRn. Define the matrixYi = eie

Ti Y,

where the superscriptT indicates the transpose operator.Matrices employed in the bus power injection and voltage

magnitude constraints are

Yi =1

2

[

Re(

Yi + Y Ti

)

Im(

Y Ti − Yi

)

Im(

Yi − Y Ti

)

Re(

Yi + Y Ti

)

]

(3)

Yi = −1

2

[

Im(

Yi + Y Ti

)

Re(

Yi − Y Ti

)

Re(

Y Ti − Yi

)

Im(

Yi + Y Ti

)

]

(4)

Mi =

[

eieTi 0

0 eieTi

]

(5)

A “line” in this formulation includes both transmission linesand transformers, where transformers may have both a phase

shift and/or an off-nominal voltage ratio. That is, linek ismodeled as aΠ circuit (with series admittancegk + jbk andshunt capacitancesbsh, k

2 ) in series with an ideal transformer(with turns ratio 1 : τke

jθk ) as in [3]. Note that a smallminimum resistance is enforced on all lines in accordancewith [5]. Define fi as theith standard basis vector inR2n.Matrices employed in the line-flow constraints are

Zkl=gkτ2k

(

flkfTlk+ flk+nf

Tlk+n

)

− cl(

flkfTmk

+ fmkfTlk+ flk+nf

Tmk+n + fmk+nf

Tlk+n

)

+ sl(

flkfTmk+n + fmk+nf

Tlk− flk+nf

Tmk

− fmkfTlk+n

)

(6)

Zkm= gk

(

fmkfTmk

+ fmk+nfTmk+n

)

− cm(

flkfTmk

+ fmkfTlk+ flk+nf

Tmk+n + fmk+nf

Tlk+n

)

+ sm(

flk+nfTmk

+ fmkfTlk+n − flkf

Tmk+n − fmk+nf

Tlk

)

(7)

Zkl= −

(

2bk + bsh,k2τ2k

)

(

flkfTlk+ flk+nf

Tlk+n

)

+ cl(

flkfTmk+n + fmk+nf

Tlk− flk+nf

Tmk

− fmkfTlk+n

)

+ sl(

flkfTmk

+ fmkfTlk+ flk+nf

Tmk+n + fmk+nf

Tlk+n

)

(8)

Zkm= −

(

bk +bsh,k2

)

(

fmkfTmk

+ fmk+nfTmk+n

)

+ cm(

flk+nfTmk

+ fmkfTlk+n − flkf

Tmk+n − fmk+nf

Tlk

)

+ sm(

flkfTmk

+ fmkfTlk+ flk+nf

Tmk+n + fmk+nf

Tlk+n

)

(9)

where, for notational convenience,

cl =(

gk cos (θk) + bk cos(

θk +π

2

))

/ (2τk) (10)

cm =(

gk cos (−θk) + bk cos(

−θk +π

2

))

/ (2τk) (11)

sl =(

gk sin (θk) + bk sin(

θk +π

2

))

/ (2τk) (12)

sm =(

gk sin (−θk) + bk sin(

−θk +π

2

))

/ (2τk) (13)

To write the semidefinite relaxation, first define the vectorof voltage coordinates

x =[

Vd1 Vd2 . . . Vdn Vq1 Vq2 . . . Vqn]

(14)

Then define the rank one matrix

W = xxT (15)

The active and reactive power injections at busi are thengiven by tr (YiW) and tr

(

YiW)

, respectively, wheretrindicates the matrix trace operator (i.e., sum of the diagonalelements). The square of the voltage magnitude at busi isgiven by tr (MiW).

Similarly, the active and reactive line flows for linek ∈ L atterminal busl are given bytr (Zkl

W) and tr(

ZklW

)

. Dueto the asymmetry introduced by allowing transformers withoff-nominal voltage ratios and non-zero phase shifts, we also

4

require separate matrices to represent active and reactivepowerflows from the other terminal of linek at busm: tr (Zkm

W)and tr

(

ZkmW

)

, respectively.Replacement of the rank one constraint (15) by the less

stringent constraintW � 0, where� 0 indicates the corre-sponding matrix is positive semidefinite, yields the semidefi-nite relaxation. The semidefinite relaxation is “tight” when thesolution has zero duality gap. The duality gap for a solutionto the semidefinite relaxation refers to the difference betweenthe optimal objective value for the semidefinite relaxationand the objective value for a global solution to the classi-cal formulation of the OPF problem (2). A solution to thesemidefinite relaxation has zero duality gap if and only if therank conditionrank (W) ≤ 2 is satisfied. For a solution withzero duality gap, a unique rank one matrixW can be recoveredby enforcing the known voltage angle at the reference bus [5].

The semidefinite relaxation of the OPF problem is

min∑

g∈Gq

αg +∑

g∈Gpw

βg (16a)

subject to

PminGg ≤ PGg ≤ P

maxGg ∀g ∈ G (16b)

− PDi +∑

g∈Gi

PGg = tr (YiW) ∀i ∈ N (16c)

Qmini ≤ tr

(

YiW)

≤ Qmaxi ∀i ∈ N (16d)

(

Vmini

)2

≤ tr (MiW) ≤ (V maxi )2 ∀i ∈ N (16e)

− (Smaxk )2 tr (Zkl

W) tr(

ZklW

)

tr (ZklW) −1 0

tr(

ZklW

)

0 −1

� 0 ∀k ∈ L (16f)

− (Smaxk )2 tr (ZkmW) tr

(

ZkmW)

tr (ZkmW) −1 0

tr(

ZkmW)

0 −1

� 0 ∀k ∈ L (16g)

[

cg1PGg + cg0 − αg√cg2PGg√

cg2PGg −1

]

� 0 ∀g ∈ Gq (16h)

{βg ≥ mgt (PGg − agt) + bgt ∀t = 1, . . . , rg} ∀g ∈ Gpw

(16i)

W � 0 (16j)

where apparent power line-flow limits and quadratic generatorcost functions are implemented using Schur’s complementformula in (16f)–(16g) and (16h), respectively; in (16i), thepiecewise-linear generator cost functions are implementedusing the “constrained cost variable” method as in [3]; and,fornotational convenience, the maximum and minimum reactivepower injections at each bus are defined as

Qmaxi = −QDi +

∑

g∈Gi

QmaxGg (17)

Qmini = −QDi +

∑

g∈Gi

QminGg (18)

The dual form of the semidefinite relaxation (i.e., the dual of(16)) requires definition of Lagrange multipliers correspond-ing to each constraint in (16). Define vectors of Lagrangemultipliers associated with lower inequality bounds on activepower, reactive power, and voltage magnitude asψ

k, γ

i, and

µi, and those associated with upper bounds asψk, γi, and

µi, respectively. Define a scalar variableλi as the aggregatedLagrange multiplier (i.e., the locational marginal price (LMP))of active power at each busi. Note thatλi is not constrained tobe non-negative. Define two3×3 symmetric matrices per linefor the line-flow limits measured from each line terminal:Hkl

andHkm, with H

cdkl

andHcdkm

indicating the(c, d) element ofthe corresponding matrix. Define2×2 symmetric matrices foreach generator with a quadratic cost function:Rg, with R

cdg

the (c, d) element ofRg. Define a Lagrange multiplierζgt foreach line segmentt of each generatorg with a piecewise-linearcost function.

Define a matrix-valued functionA.

A =∑

i∈N

{

λiYi +(

γi − γi

)

Yi +(

µi − µi

)

Mi

}

+ 2∑

k∈L

{

H12klZkl

+H12km

Zkm+H

13klZkl

+H13km

Zkm

}

(19)

Define a scalar real-valued functionρ.

ρ =∑

i∈N

λiPDi + γiQmin

i − γiQmaxi + µ

i

(

V mini

)2− µi (V

maxi )2

+∑

g∈Gqi

(

ψgPminGg − ψgP

maxGg + cg0 −R

22g

)

−

∑

g∈Gpwi

rg∑

t=1

(ζgt (mgtagt − bgt))

−

∑

k∈L

{

(Smaxk )2

(

H11kl

+H11km

)

+H22kl

+H22km

+H33kl

+H33km

}

(20)

The dual form of the semidefinite programing relaxation ofthe OPF problem is then written as

max ρ (21a)

subject to

A � 0 (21b)

Hkl� 0, Hkm � 0 ∀k ∈ L (21c)

Rg � 0, R11g = 1 ∀g ∈ Gq (21d)

rg∑

t=1

ζgt = 1 ∀g ∈ Gpw (21e)

{

λi = cg1 + 2√cg2R

12g + ψg − ψ

g∀g ∈ Gq

i

}

∀i ∈ N (21f){

λi =

rg∑

t=1

ζgtmgt ∀g ∈ Gpwi

}

∀i ∈ N (21g)

ψg≥ 0, ψg ≥ 0, γ

i≥ 0, γi ≥ 0, µ

i≥ 0, µi ≥ 0, ζgt ≥ 0

(21h)The semidefinite relaxation has zero duality gap and yields

a physically meaningful solution if and only if the solutionto(21) satisfies the rank conditiondim (null (A)) ≤ 2.

C. Discussion

Several aspects of the semidefinite programming formu-lation deserve special attention. We focus on those aspectsthat differ from previous formulations (e.g., [5]) due to the

5

proposed formulation’s allowing of multiple generators atthesame bus and the possibility of parallel lines.

The semidefinite relaxation includes the possibility of mul-tiple generators at the same bus. As shown in (21f) and (21g),all generators at the same busi must have the same aggregateactive power Lagrange multiplierλi. This is related to theprinciple of equal marginal costs in the economic dispatchproblem [19]. Since generator reactive power injections donotappear in the cost function of (2), reactive power Lagrangemultipliers are only needed for each generator bus ratherthan for each generator. This is seen in (17) and (18), whichdetermine the allowed range of busi reactive power injection.

The semidefinite relaxation also includes the possibilityof parallel lines (i.e., multiple lines with the same terminalbuses) and the ability to represent transformers with off-nominal voltage ratios and/or non-zero phase-shifts. Previousformulations limited line flows by constraining the total powerflow between two buses, precluding the ability to separatelylimit line flows on parallel lines. This modeling flexibilitycomes at the price of additional complexity. Incorporatingparallel lines removes the ability to form the line-flow matricesdirectly from the bus admittance matrix, instead requiringthemore complicated expressions in (6)–(9). Incorporating off-nominal voltage ratios and non-zero phase shifts breaks thesymmetry of theΠ model such that different line-flow matricesare required for each line terminal (i.e.,Zkl

in (6) and Zkl

in (8) for active and reactive power flows measured fromthe sending terminal andZkm

in (7) andZkmin (9) for the

receiving terminal).

For large system models, numerical difficulties in thesemidefinite programming solver may prevent convergenceto acceptable precision. We have found several pragmatictechniques that reduce numerical difficulties with large sys-tems. First, ignore engineering limits that will clearly not bebinding at the solution. Many power system data sets specifylarge values for limits that are intended to be unconstrained,particularly for reactive power generation and line-flow limits.We do not incorporate terms corresponding to very largelimits. Similarly, some generators specified with quadratic costfunctions actually have linear cost functions (i.e.,cg2 = 0).The correspondingRg matrix is eliminated. These techniquesdo not affect the optimality of the resulting solution.

Numerical difficulties often occur when the system modelhas very “tight” limits. For instance, the active power gener-ation of a synchronous condenser is constrained to be zero.A second technique for reducing numerical difficulties is touse equality constraints rather than inequality constraints tomodel these limits. When the power output of a generator isconstrained to a very small range, fix the generator at the mid-point of this range and directly add the associated generationcost to the objective function. The degree of suboptimalityof the resulting solution can be estimated by multiplying theLagrange multiplier corresponding to the equality constraintby the half of the difference between the maximum andminimum limits.

III. A DVANCES IN MATRIX COMPLETION

DECOMPOSITIONS

In this section, we describe several advances in the decom-position techniques used to reduce the computational burdenof semidefinite relaxations of large OPF problems. First, wereview the maximal clique decomposition as introduced byJabr [16]. Next, we present a matrix combination algorithmthat significantly reduces the required computation time ofthis decomposition. We then discuss the rank condition prop-erties of solutions to large system models. We next presenta modification to Jabr’s formulation of the maximal cliquedecomposition that extends this decomposition to generalnetworks rather than only networks with admittance matricesthat satisfy a definiteness requirement. Finally, we describe atechnique for obtaining the optimal voltage profile from thedecomposed matrices.

A. Overview of Jabr’s Maximal Clique Decomposition

Jabr’s formulation of the maximal clique decompositionuses a matrix completion theorem [13]. Several graph theoreticdefinitions aid understanding of this theorem. A “clique” isasubset of the graph vertices for which each vertex in the cliqueis connected to all other vertices in the clique. A “maximalclique” is a clique that is not a proper subset of another clique.A graph is “chordal” if each cycle of length four or more nodeshas a chord, which is an edge connecting two nodes that arenot adjacent in the cycle. The maximal cliques of a chordalgraph can be determined in linear time [20]. See [16] and [21]for more details on these definitions.

The matrix completion theorem can now be stated. LetA

be a symmetric matrix with partial information (i.e., not allentries ofA have known values) with an associated undirectedgraph. Note that the graph of interest has the power systembuses as vertices and the branch susceptances as edge weights.The matrix A can be completed to a positive semidefinitematrix (i.e., the unknown entries ofA can be chosen suchthat A � 0) if and only if the submatrices associated witheach of the maximal cliques of the graph associated withA

are all positive semidefinite.The matrix completion theorem allows replacing the single

large2n× 2n positive semidefinite constraint (21b) by manysmaller matrices that are each constrained to be positivesemidefinite. This significantly reduces the problem size forlarge, sparse power networks.

There are two important aspects of this decomposition thatare relevant to our advances. First, since the maximal cliquescan have non-empty intersection (i.e., contain some of thesame buses), different matrices may contain elements that referto the same element in the2n× 2n matrix. Therefore, linkingconstraints are required to force equality between elementsthat are shared between maximal cliques. To specify theselinking constraints, Jabr recommends forming a “clique tree”:a maximum weight spanning tree of a graph with nodescorresponding to the maximal cliques and the edge weightsbetween each node pair given by the number of shared busesin each clique pair. A maximal weight spanning tree of thisgraph, which can be calculated using Prim’s algorithm [22],

6

is used to reduce the number of linking constraints: equalityconstraints are only enforced between the appropriate elementsin maximal cliques that are adjacent in the maximal weightspanning tree.

Second, graphs corresponding to realistic power networksare not chordal. A chordal extension of the graph is thusrequired in order to use the matrix completion theorem. Achordal extension adds edges between non-physically con-nected nodes (i.e., edges in the chordal extension of the graphmay exist between buses that are not connected by a line inthe power system) to obtain a chordal graph. Jabr recommendsobtaining a chordal extension using a Cholesky factorizationof the absolute value of the imaginary part of the network’sadmittance matrix. To minimize the total number of edges, aCholesky factorization with minimum fill-in is obtained froma minimum degree ordering of the row/column indices [23].

B. Matrix Combination Algorithm

We first describe a modification to the maximal cliquedecomposition that yields a significant computational speedimprovement. This modification accounts for the trade-offbetween the size of maximal cliques and the number of linkingconstraints. Smaller maximal cliques generally reduce thetotalsize of the positive semidefinite constrained matrices. Theoverlap between maximal cliques, as determined by the cliquetree approach, establishes the number of linking constraints.

The maximal clique decomposition uses a Cholesky factor-ization with minimum fill-in to obtain small maximal cliquesas a heuristic for minimizing the number of variables inthe positive semidefinite matrix constraints. This approachdoes not account for the computational burden of the linkingconstraints. The optimization literature provides theoreticalsupport for the concept of reducing computational burden bycombining matrices (see [13] and Section 4 of [14]). Specifi-cally, this literature discusses the potential trade off insolvertime between the sizes of the semidefinite-constrained matricesand the number of linking constraints, which require solutionof a system of linear equations in the semidefinite programalgorithm. Combining matrices eliminates the need for linkingconstraints between the matrices at the computational price ofa larger matrix. A heuristic for combining matrices to gain thebenefits of small matrices while reducing linking constraintsthus has the potential for computational speed improvements.

Many common semidefinite program solvers, such asSeDuMi [24] and SDPT3 [25], use primal–dual methods thatsolve both the primal and dual problems simultaneously. More-over, since a primal constraint corresponds to a dual variable,the “size” of the semidefinite program can be estimated byadding the total number of scalar variables required to formthe matrices with the number of linking constraints. Thisapproximation for the “size” of a semidefinite program formsthe basis of our matrix combination heuristic. In a greedymanner, we repeatedly combine the pair of matrices that mostreduces the “size” of the semidefinite program as measuredby the total number of variables plus the number of linkingconstraints.

We next detail our matrix combination heuristic. LetL bea parameter specifying the maximum number of matrices.

Consider a semidefinite program formed from the chordalextension of a power system network, with maximal cliquei containingdi buses. Since the matrices corresponding tothe maximal cliques are symmetric and contain both realand imaginary voltage components, matrixi (correspondingto maximal cliquei) has di (2di + 1) variables. If maximalcliques i and k, adjacent in the clique tree, sharesik buses,then sik (2sik + 1) linking constraints are required betweenthe corresponding matrices. For each pair of adjacent maximalcliques in the clique tree, the change in the optimizationproblem “size” ∆ik if the cliques i and k were combinedis given by

∆ik = dik (2dik + 1)− di (2di + 1)

− dk (2dk + 1)− sik (2sik + 1) (22)

where dik = di + dk − sik is the number of buses in thecombined clique.

While the number of matrices is greater thanL, combinea pair of adjacent maximal cliques with smallest∆ik. Thenrecalculate the value of∆ik for all maximal cliques adjacentto the newly combined clique. Repeat until the number ofmatrices is equal toL. Constrain the resulting set of matricesto be positive semidefinite in the OPF formulation.

We test this heuristic using two large system models: theIEEE 300-bus system [26] and a 3012-bus model of the Polishsystem for evening peak demand in winter 2007-2008 [3].These systems were chosen to examine how the heuristicscales with system size. Matrix decomposition techniques donot result in a notable speed improvement for small systems;no matrix decomposition significantly reduced the computa-tional time for the IEEE 14, 30, and 57-bus systems [26].

The OPF formulation in (21) was implemented usingYALMIP version 3 [27], SeDuMi version 1.3 [24], andMATLAB R2011a. One computer with an 64-bit Intel i7-2600Quad Core CPU at 3.40 GHz with 16 GB of RAM was used torun the formulation. By using the matrix completion decom-positions to exploit the inherent sparsity, these computationalresources were sufficient for the 300 and 3012-bus systemmodels. A tolerance of1× 10−9 for SeDuMi’seps was usedin the calculation of the results.

Figures 1a and 1b show variation in total solver time(i.e., the time used by the semidefinite programming solverSeDuMi), with the parameterL. These figures do not includethe set up time required to initialize the formulation. However,the set up time is typically only 15% to 20% of the solver time.

The solver times for Jabr’s formulation of the maximalclique decomposition are the rightmost points (no matrixcombinations) in Figures 1a and 1b. AsL decreases fromthe rightmost point, the solver times decrease by, at most,approximately a factor of 2.5 for the 300-bus system and afactor of 3.6 for the 3012-bus system as compared to the solvertime without combining matrices.

The solution time graphs in Figures 1a and 1b appear tobe “noisy,” which we attribute to differences in the numberof iterations needed to reach the specified tolerance. That is,solver times can vary among choices ofL if it requires one

7

0 50 100 150 200 2500

5

10

15300−Bus System: Solver Time (sec) vs. L

L (Maximum Number of Matrices)

Sol

ver

Tim

e (s

ec)

No Matrix CombiningMatrix Combining

(a) Solver time vs.L for IEEE 300-Bus System. Note that the time withL = 1 of 69.5 seconds is not shown on the plot.

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

3012−Bus System: Solver Time (sec) vs. L

L (Maximum Number of Matrices)

Sol

ver

Tim

e (s

ec)

No Matrix CombiningMatrix Combining

(b) Solver time vs.L for 3012-Bus System. Note that the time withL = 2of approximately1 × 105 seconds is not shown on the plot and the systemcould not be solved withL = 1.

Fig. 1. Solver Time vs.L

more solver iteration to reach the desired solution tolerance.Despite this noise, there is a clear trend. The plots show thatreducingL results in significant improvements in solver timeas compared to the case without matrix combining. However,as L continues to decrease, the speed improvements fromremoving linking constraints are overcome by the additionalvariables required for the larger matrices (in the extreme,returning to a single2n × 2n matrix for L = 1). Thus, thesolver times exhibit a steep increase for smallL. (Solutiontimes for very smallL are not shown on Figures 1a and 1b.For the 300-bus system, the full2n × 2n matrix constraintcorresponding toL = 1 solved in 69.5 seconds. The 3012-bussystem could not be solved withL = 1; the system requiredapproximately1.0× 105 seconds forL = 2.)

Rather than combining matrices until below a specifiedparameter value, we also tried combining matrices until nopair of adjacent maximal cliques had a negative value of∆ik (i.e., stopping combining matrices once the heuristicindicated no further advantage to doing so). In our numericalexperience, however, this approach typically did not identifya set of matrices that minimized the solver time. The numberof matrices for which no remaining adjacent pairs of maximalcliques had negative∆ik wasL = 150 for the 300-bus systemand L = 1376 for the 3012-bus system. In both systems,faster solver times were obtained for smaller values ofL. Thisreinforces the fact that our measure of semidefinite programsize is a heuristic approximation of the computational burden.

Based on these empirical results, choosingL equal to 10%of the initial number of matrices appears to give near minimumsolver times. (ExpressingL as a percentage of the originalnumber of matrices allows for easy comparison betweensystems.) A minimum computational time consistently appearsat approximately this value ofL for the available power systemmodels; however, limited diversity of available large systemmodels precludes more general comments on this value. Wespeculate that system models that are strongly interconnected(i.e., have a relatively low amount of sparsity) will inherently

have large maximal cliques and thus not benefit from as manymatrix combinations as compared to more sparsely connectedsystem models. We therefore anticipate that a larger value ofL would be appropriate for strongly interconnected models.

Note that solution times are not greatly affected by the ad-dition of parallel lines or multiple generators at the same bus;models with these features have comparable computationalburden relative to other models with the same total numbersof lines and generators.

Table I summarizes these results by providing the solvertimes for each system / decomposition pair along with a“speed up factor” (SUF) for the improvement of the matrixcombination approach withL = 10% of the original numberof matrices as compared to not combining matrices. Resultsusing the full2n×2nmatrix for the 3012-bus system could notbe computed. Note that solver times withL = 10% for othermodels of the Polish system that is represented in the 3012-bus system model are available in Table II. Also note that theproposed heuristic yields improvements for the intermediate-sized 118-bus system.

System 2n× 2n No Combining SUFCombining (L = 10%)

IEEE 118-bus 6.63 4.84 2.06 2.349IEEE 300-bus 69.45 13.18 5.71 2.309

Polish 3012-bus – 3578.5 1197.4 2.989

TABLE ISOLVER T IMES (SEC) FOR VARIOUS ALGORITHMS

We attempted several alternatives to the proposed heuristic:a variant of the proposed algorithm that, at each step, randomly(weighted by ∆ik) selects a pair of maximal cliques tocombine; the heuristic proposed in [14]; and a “top-down”approach that groups maximal cliques using a normalized cutalgorithm on the clique tree. As compared to the proposedheuristic, these alternatives sometimes had comparable, butnot faster, solution times.

8

0 5 10 15 20 25 30 3510

−2

100

102

104

106

108

Eigenvalues of Typical A Matrix

Eigenvalue Index

Eig

enva

lue

(a) Eigenvalues for a TypicalA Matrix

0 10 20 30 40 50 6010

−2

100

102

104

106

108

Eigenvalues of Indeterminate A Matrix

Eigenvalue Index

Eig

enva

lue

(b) Eigenvalues for an IndeterminateA Matrix

Fig. 2. Eigenvalues for SelectedA Matrices of the Solution to the 3012-Bus System

C. Analysis of Duality Gap Properties of Solution to OPFProblems for Large System Models

While solutions to many OPF problems satisfy the rankcondition and thus have zero duality gaps [5], it is knownthat some small system models yield solutions with non-zeroduality gaps [6], [7]. Until the recent exploitation of powersystem sparsity, computational challenges have precludedin-vestigation of the rank properties of the semidefinite relaxationfor large system models. Harnessing the computational meth-ods described in this paper, we conduct further investigationof rank condition satisfaction for large system models. Notethat, as in the previous section, the results in this sectionarecalculated with a minimum line resistance of1×10−4 per unitin accordance with [5] and with SeDuMi’s tolerance parametereps set to1× 10−9.

When using a matrix completion decomposition, solutionsto the dual formulation of the semidefinite relaxation consistof a set ofA matrices. For a solution that satisfies the rankcondition, the nullspaces of allA matrices have dimensionless than or equal to two. However, for numerical reasons,solvers do not yield a “hard zero” value for the eigenvaluescorresponding to the nullspaces of these matrices. Therefore, itcan be difficult to determine when anA matrix has nullspacewith dimension two. For illustration of this challenge, Figure 2shows the eigenvalues, sorted in order of ascending magnitude,for selectedA matrices from theL = 10% decompositionof the Polish 3012-bus system model. With two smallesteigenvalues that are four orders of magnitude below the nextsmallest eigenvalues, Figure 2a shows a typical matrix thathas nullspace with dimension two. Conversely, the smallesteigenvalues in Figure 2b are only two orders of magnitudebelow the next smallest eigenvalues; the nullspace dimensionfor this matrix is more difficult to determine. Characterizingthe overall satisfaction of the rank condition for the Polish3012-bus system is correspondingly difficult.

To evaluate the satisfaction of the rank condition formethods that exploit power system sparsity by using matrixcompletion decompositions, we propose the following metricto measure closeness to a nullspace with dimension two.

The metric is based on the ratio between the third andsecond smallest magnitude eigenvalues. The minimum suchratio among all theA matrices is termed the “minimumeigenvalue ratio.” If the solution did yield “hard zeros” for zeroeigenvalues, the second smallest eigenvalue would be zero andthe third smallest eigenvalue would be non-zero, resultingina minimum eigenvalue ratio of infinity. In practice, numericalissues result in minimum eigenvalue ratios that are large(typical values are greater than1× 107 for small systems thatare known to satisfy the rank condition). Further, if the solutiondoes not satisfy the rank condition, both the second and thirdsmallest eigenvalues will have similar magnitudes near zero,therefore yielding a small value for the minimum eigenvalueratio. Thus, a large value for the minimum eigenvalue ratioindicates a solution with zero duality gap while a small valueindicates a non-zero duality gap solution. Note that morecomplex metrics than the proposed minimum eigenvalue ratioare possible; the proposed metric is intended to be a simplebut meaningful measure.

Table II shows the minimum eigenvalue ratios for severaltest systems. The solution times withL = 10% are also given.The systems with more than 300 buses are representationsof the Polish grid with various levels of modeling detail anddifferent loading scenarios (winter peak (wp), winter off peak

System Min Eigenvalue Max Mismatch Solver TimeModel Ratio (L = 10%)

IEEE 118-bus 2.86× 109 3.9× 10−5 MVAr 2.1 secIEEE 300-bus 2.25× 102 4.7× 100 MVAr 5.7 sec2383-bus (wp) 7.90× 102 2.9× 102 MVAr 730 sec2736-bus (sp) 3.07× 104 2.7× 10−2 MVAr 622 sec2737-bus (sop) 4.11× 104 3.7× 10−1 MVAr 607 sec2746-bus (wp) 8.65× 104 5.5× 10−2 MW 752 sec2746-bus (wop) 1.95× 104 1.4× 10−1 MW 738 sec3012-bus (wp) 1.72× 102 4.1× 102 MVAr 1197 sec3120-bus (sp) 5.84× 102 4.6× 101 MVAr 1619 sec3375-bus (wp) 1.64× 102 5.2× 102 MVAr 1457 sec

TABLE IIMEASURES OFRANK CONDITION SATISFACTION AND SOLVER T IMES

FOR VARIOUS SYSTEM MODELS

9

50 100 150 200 2500

1

2

3

4

5

PQ Bus Index

PQ

Bus

Mis

mat

ch (

MW

, MV

Ar)

Mismatch at PQ Buses (IEEE 300−Bus System)

P mismatch (MW)Q mismatch (MVAr)

(a) Power Mismatch for IEEE 300-Bus System

500 1000 1500 2000 2500 30000

50

100

150

200

250

300

350

400

450

PQ Bus Index

Mis

mat

ch (

MW

, MV

Ar)

Power Mismatch at PQ Buses (Polish 3012−Bus System)

P mismatch (MW)Q mismatch (MVAr)

(b) Power Mismatch for Polish 3012-Bus System

Fig. 3. Active and Reactive Power Mismatch at PQ Buses

(wop), summer peak (sp), and summer off peak (sop)). Theseresults indicate that, according to the proposed metric, thelarge system models do not satisfy the rank condition aswell as many smaller system models, which generally haveminimum eigenvalue ratios greater than1 × 107 when theysatisfy the rank condition. Since, other than the IEEE 118 and300-bus systems, the large system models all represent thesame Polish system, the lack of more diverse system modelslimits the ability to make more general statements concerningsatisfaction of the rank condition for large system models.

An alternative test for satisfaction of the rank condition isbased on the mismatch between the calculated and specifiedactive and reactive power injections at PQ buses. To recoveracandidate voltage profile, we form the closest rank one matrixto the solution’sW matrix using the eigenvector associatedwith the largest eigenvalue ofW. If the solution has zeroduality gap, the matrixW is rank one and the resulting voltageprofile will satisfy the power injection equality constraints atthe PQ buses. Conversely, the closest rank one matrix to asolution with non-zero duality gap will typically not yielda voltage profile that satisfies the power injection equalityconstraints at PQ buses. Thus, the mismatch between thecalculated and specified power injections at PQ buses providesan alternative measure for satisfaction of the rank condition.

Figures 3a and 3b show the mismatch between the specifiedand calculated active and reactive power injections at PQ busesfor the 300-bus and 3012-bus systems, respectively, sortedinorder of increasing active power mismatch. The voltage profileyields small mismatches for the majority of buses, but a fewbuses display large mismatches in both active and reactivepower. The large power mismatches indicate solutions havingnon-zero duality gap. With small mismatch at the majorityof PQ buses, such solutions with non-zero duality gap mayprovide good starting points for a local search algorithm.Table II shows the maximum mismatch, considering bothactive and reactive powers, for a variety of test systems.Solutions to several of these system models have relativelylarge power mismatches; for instance, mismatches for all testsystems in Table II except for the 118, 2736, and 2746 (wp)

bus systems are greater than the default Newton solutiontolerance of 0.1 MW/MVAr used by the power flow solutionpackage PSS/E [28]. Large power mismatches indicate that thecorresponding solutions do not satisfy the rank condition.Notethe correlation between the minimum eigenvalue ratio andthe maximum power mismatch, which supports the validityof these measures of rank condition satisfaction.

D. Extending Jabr’s Formulation of the Maximal Clique De-composition to All Systems

The first step in Jabr’s formulation creates a chordal ex-tension of the network using a Cholesky factorization of theabsolute value of the imaginary part of the bus admittance ma-trix (i.e., chol (|Im (Y)|)). Only positive definite matrices haveCholesky factorizations. Since not all power system networkshave admittance matrices that satisfy|Im (Y)| ≻ 0 (e.g.,networks with sufficiently large shunt capacitances), Jabr’sformulation cannot be universally applied to such networks.

Jabr’s formulation only uses thesparsity pattern(i.e., loca-tion of the non-zero elements) of the Cholesky factorization.Thus, an alternative, positive definite matrix whose Choleskyfactorization exhibits the same sparsity pattern would extendJabr’s formulation to general power systems. We next presentsuch an alternative matrix.

Let D represent the incidence matrix associated with thenetwork (i.e., each row ofD corresponds to a line and hastwo non-zero elements:+1 in the column corresponding tothe line’s “from” bus and−1 in the column correspondingto the line’s “to” bus). The matrixE in (23) has a Choleskyfactorization with the same sparsity pattern aschol (|Im (Y)|).

E = DTD+ In×n (23)

whereIn×n is then× n identity matrix.SinceDT

D has a Laplacian structure, it is positive semidef-inite. Adding an identity matrix increases all eigenvaluesbyone, and thusE is positive definite. Note that the commonmodification for making a Laplacian matrix positive definitevia adding the matrix1 ·1T , where1 is the vector of all ones,

10

is not appropriate due to the fact that this modification makesthe Cholesky factorization ofE dense.

The bus admittance matrixY has generalized Laplacianstructure, with weightings from the line admittances, plusdiag-onal terms corresponding to shunt admittances. TheE matrix’ssimilar construction implies that its Cholesky factorizationhas the same sparsity pattern as the Cholesky factorizationof |Im (Y)|. Using the Cholesky factorization ofE thereforeextends Jabr’s method to general power networks.

E. Obtaining the Optimal Voltage Profile

The solution to a decomposed problem is a set of positivesemidefinite matrices. If all the matrices have nullspaces withappropriate dimension, the optimal voltage profile can berecovered [5], [16]. (For formulations that separate real andimaginary voltage components, like (21), the nullspace of allmatrices must have dimension less than or equal to two.)However, existing literature does not give a detailed methodfor recovering the optimal voltage profile. We next describeatechnique for obtaining the optimal voltage profile.

An overview of this technique follows. First obtain vectorsin the nullspaces (hereafter referred to as nullvectors) ofeachpositive semidefinite constrained matrix. Note that calculationof these nullvectors can be carried out in parallel since thenullspace computation for each matrix can be performedindependently. These nullvectors, when rearranged such thatthey correspond to complex “phasor” voltages, can each bemultiplied by a different complex scalar and remain in theirrespective nullspaces. Since a bus can be in multiple maximalcliques, elements in different vectors may correspond to thesame bus voltage phasor. The complex scalars are chosen suchthat elements of different vectors that correspond to the samebus voltage are equal. A centrally computed linear nullspacecalculation of a specified matrix gives an appropriate choice ofthe scalar values. This allows for specification of a vector thatis a real scalar multiple of the optimal voltage profile. Usinga single binding constraint, the resulting vector is scaledtoobtain the optimal voltage profile.

We next present the details of this technique. Consider anoptimal solution to (21) consisting ofd positive semidefinitematricesAi with dim

(

null(

Ai

))

≤ 2, ∀i ∈ {1, . . . , d}. Letu(i) be a nullvector ofAi. Let ri be the number of buses in themaximal clique corresponding to matrixi. Convert each vectoru(i) to complex “phasor” form:u(i) = u

(i)1:ri

+ ju(i)ri+1:2ri

,where subscript1 : ri indicates the first throughrthi elementsof the corresponding vector.

Vectorsu(i) remain in their corresponding nullspace aftermultiplication by complex scalarsαi. This property is usedto enforce consistency between elements of different vectorsthat correspond to the same bus voltage phasor. Obtaining theoptimal voltage profile requires determining values ofαi thatcreate agreement between all elements representing the samevoltage from the nullvectors of different matrices. This can bevisualized by forming a table with rows corresponding to busindices and columns corresponding to maximal clique indices.If maximal cliquej contains busi, the(i, j) entry of the tableis αj multiplied by the element ofu(j) corresponding to bus

1

2 3

4

5

Fig. 4. Illustrative Voltage Profile Recovery Example Network

``````````

BusClique Index

1 2 3

1 α1u(1)1 α2u

(2)1

2 α1u(1)2

3 α1u(1)3 α2u

(2)2 α3u

(3)1

4 α2u(2)3 α3u

(3)2

5 α5u(3)3

TABLE IIIILLUSTRATIVE VOLTAGE PROFILE RECOVERY EXAMPLE TABLE

i. If maximal cliquej does not contain busi, the (i, j) entryof the table is empty.

Since each row of the table represents a voltage phasor at thecorresponding bus, values ofαi ∀i = 1, . . . , d are chosen suchthat all entries in each row are equal. Appropriate values ofαi

are obtained using a nullvector of an appropriately specifiedmatrix. Specifically, use the following procedure to createamatrix C with d columns that enforces equality of all entriesof each row of the table. For each rowi of the table, findthe first non-empty entry and store the corresponding columnindexj. (All rows of the table will have at least one non-emptyentry because each bus is contained in at least one maximalclique.) While there exists a non-empty entry in rowi withcolumn index greater thanj (let the non-empty entry exist incolumnk), add a row to the matrixC that enforces equalityof the (i, j) and (i, k) entries. Setj = k and repeat until noother non-empty entries exist in rowi with column indicesgreater thanj. Then proceed to rowi+ 1.

Consider the illustrative example system network in Fig-ure 4 and corresponding Table III. This system has threemaximal cliques composed of buses{1, 2, 3}, {1, 3, 4}, and{3, 4, 5}. The corresponding equation for the example is

Cα =

u(1)1 −u

(2)1 0

u(1)3 −u

(2)2 0

0 u(2)2 −u

(3)1

0 u(2)3 −u

(3)2

α1

α2

α3

=

0

0

0

0

(24)

The nullspace calculation has a non-trivial solution if allAi matrices of the solution have nullspaces with dimensionless than or equal to two. (For a solution to the semidefiniterelaxation where some of theA matrices have nullspacedimension greater than two, the nullspace calculation may only

11

have the trivial solutionα = 0, indicating that a consistentvoltage profile cannot be obtained.) A nullvectorα yields atable where all entries of each row have the same value. Createa vectorη of lengthn whereηi is equal to the value of an entryin the ith row of the table. The vectorη is a scalar multipleof the optimal voltage vector.

Sinceα has one degree of freedom in its length, the optimalvoltage profile is a real scalar multipleχ of η. To determinethe value ofχ, one additional piece of information is requiredfrom a binding constraint. Reference [5] suggests the useof a binding voltage magnitude constraint. However, not allsolutions have a binding voltage magnitude constraint (e.g., thethree-bus system in [6]). Optimal solutions to OPF problemshave at least one binding constraint, but not necessarily abinding voltage magnitude constraint.

A binding constraint is identified by a non-zero value of thecorresponding Lagrange multiplier. Consider a solution with abinding voltage magnitude constraint. LetVk be the value ofa binding voltage magnitude constraint at busk. The value ofχ is chosen using this voltage magnitude:

χ =Vk|ηk|

(25)

For solutions without a binding voltage magnitude constraint,use an alternative binding constraint to determineχ.

The optimal voltage profile is then constructed by scalingηby χ and rotating the resulting vector to obtain zero referenceangle.

V opt = χηe−jθref (26)

whereθref is the angle of the element ofη corresponding tothe reference bus.

IV. CONCLUSION AND FUTURE WORK

This paper has addressed two categories of practical issuesassociated with implementing a large-scale optimal power flowsolver based on semidefinite programming: modeling issuesassociated with general power systems and computationalissues. Specific modeling issues addressed include multiplegenerators at the same bus and limiting flows on parallellines. Both quadratic and convex piecewise-linear generatorcost functions are considered.

The paper provides three computational advances for ex-ploiting power system sparsity using matrix completion de-compositions. First, a proposed matrix combination algorithmconsiders the impact of “linking constraints” between elementsin certain decomposed matrices that refer to the same elementin the original 2n × 2n matrix. Since combining matriceseliminates linking constraints, matrix combination can reducecomputation time. Calculations using test systems show theefficacy of the matrix combination approach: the IEEE 300-bus system shows a factor of approximately 2.3 decreasein solver time and a 3012-bus model of the Polish systemshows a factor of 3.0 decrease in solver time compared to notcombining matrices. The rank characteristics of solutionstoOPF problems for large system models are also examined.

Next, Jabr’s formulation of the maximal clique decomposi-tion [16] was extended to general power system networks. Thisformulation uses a Cholesky factorization of the absolute valueof the imaginary part of the bus admittance matrix. Since aCholesky factorization requires a positive definite matrix, thisapproach cannot be used for some networks (e.g., networkswith large shunt capacitive compensation). Jabr’s formulationonly uses the sparsity pattern of the Cholesky factorization.We propose an alternative positive definite matrix with thesame sparsity pattern to extend Jabr’s formulation to generalpower system networks.

A final computational advance is a method for constructingthe optimal voltage profile from a solution consisting ofdecomposed matrices. Although existing literature discussesthe use of matrix decompositions [10], [16], [17], it doesnot give a detailed method for obtaining the optimal voltageprofile.

Future work includes investigation of alternative load mod-els. Currently, the formulation includes the capability forconstant power and constant impedance load models. Anothercommon load model is constant current, which is not triviallyincorporated into the semidefinite programming formulation.Investigation of whether a constant current model can beincluded in a semidefinite programming-based OPF solver isthus valuable.

ACKNOWLEDGMENT

The authors acknowledge support of this work by U.S.Department of Energy under award #DE-SC0002319, as wellas by the National Science Foundation under IUCRC award#0968833. Daniel Molzahn acknowledges the support of theNational Science Foundation Graduate Research Fellowship.The authors additionally thank the anonymous reviewers fortheir comments and suggestions.

REFERENCES

[1] B. Lesieutre and I. Hiskens, “Convexity of the Set of Feasible Injectionsand Revenue Adequacy in FTR Markets,”IEEE Transactions on PowerSystems, vol. 20, no. 4, pp. 1790 – 1798, November 2005.

[2] Z. Qiu, G. Deconinck, and R. Belmans, “A Literature Survey of OptimalPower Flow Problems in the Electricity Market Context,” inPowerSystems Conference and Exposition, 2009. PSCE ’09. IEEE/PES, March2009, pp. 1–6.

[3] R. Zimmerman, C. Murillo-Sanchez, and R. Thomas, “MATPOWER:Steady-State Operations, Planning, and Analysis Tools forPower SystemsResearch and Education,”IEEE Transactions on Power Systems, no. 99,pp. 1–8, 2011.

[4] 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.

[5] J. Lavaei and S. Low, “Zero Duality Gap in Optimal Power FlowProblem,” IEEE Transactions on Power Systems, vol. 27, no. 1, pp. 92–107, Feb. 2012.

[6] B. C. Lesieutre, D. K. Molzahn, A. R. Borden, and C. L. DeMarco,“Examining the Limits of the Application of Semidefinite Programmingto Power Flow Problems,” in49th Annual Allerton Conference onCommunication, Control, and Computing, 2011, Sept. 28-30 2011.

[7] W. A. Bukhsh, A. Grothey, K. I. McKinnon, and P. A. Trodden, “LocalSolutions of Optimal Power Flow,” University of Edinburgh School ofMathematics, Tech. Rep. ERGO 11-017, 2011, [Online]. Available: http://www.maths.ed.ac.uk/ERGO/pubs/ERGO-11-017.html.

[8] B. Zhang and D. Tse, “Geometry of Feasible Injection Region of PowerNetworks,” in 49th Annual Allerton Conference on Communication,Control, and Computing, 2011, Sept. 28-30 2011.

12

[9] S. Bose, D. Gayme, S. Low, and K. Chandy, “Optimal Power Flow OverTree Networks,” in49th Annual Allerton Conference on Communication,Control, and Computing, 2011, Sept. 28-30 2011.

[10] S. Sojoudi and J. Lavaei, “Physics of Power Networks Makes HardOptimization Problems Easy To Solve,” in2012 IEEE Power & EnergySociety General Meeting, July 22-27 2012.

[11] J. Lavaei, D. Tse, and B. Zhang, “Geometry of Power Flowsin TreeNetworks,” in2012 IEEE Power & Energy Society General Meeting, July22-27 2012.

[12] D. K. Molzahn, B. C. Lesieutre, and C. L. DeMarco, “A SufficientCondition for Power Flow Insolvability with Applications to VoltageStability Margins,” To appear inIEEE Transactions on Power Systems.

[13] M. Fukuda, M. Kojima, K. Murota, K. Nakataet al., “ExploitingSparsity in Semidefinite Programming via Matrix CompletionI: GeneralFramework,”SIAM Journal on Optimization, vol. 11, no. 3, pp. 647–674,2001.

[14] K. Nakata, K. Fujisawa, M. Fukuda, M. Kojima, and K. Murota,“Exploiting Sparsity in Semidefinite Programming via Matrix CompletionII: Implementation and Numerical Results,”Mathematical Programming,vol. 95, no. 2, pp. 303–327, 2003.

[15] S. Kim, M. Kojima, M. Mevissen, and M. Yamashita, “Exploiting Spar-sity in Linear and Nonlinear Matrix Inequalities via Positive SemidefiniteMatrix Completion,”Mathematical Programming, vol. 129, no. 1, pp. 33–68, 2011.

[16] R. A. Jabr, “Exploiting Sparsity in SDP Relaxations of the OPFProblem,” IEEE Transactions on Power Systems, vol. PP, no. 99, p. 1,2011.

[17] X. Bai and H. Wei, “A Semidefinite Programming Method with GraphPartitioning Technique for Optimal Power Flow Problems,”InternationalJournal of Electrical Power & Energy Systems, 2011.

[18] A. Lam, B. Zhang, and D. Tse, “Distributed algorithms for optimalpower flow problem,”arXiv preprint arXiv:1109.5229, 2011.

[19] J. Glover, M. Sarma, and T. Overbye,Power System Analysis andDesign. Thompson Learning, 2008.

[20] R. Tarjan and M. Yannakakis, “Simple Linear-Time Algorithms to TestChordality of Graphs, Test Acyclicity of Hypergraphs, and SelectivelyReduce Acyclic Hypergraphs,”SIAM Journal on Computing, vol. 13, p.566, 1984.

[21] G. Valiente,Algorithms on Trees and Graphs. Springer Verlag, 2002.[22] J. Gross and J. Yellen,Graph Theory and its Applications. CRC press,

2006.[23] P. Amestoy, T. Davis, and I. Duff, “Algorithm 837: AMD, An Ap-

proximate Minimum Degree Ordering Algorithm,”ACM Transactions onMathematical Software (TOMS), vol. 30, no. 3, pp. 381–388, 2004.

[24] J. Sturm, “Using SeDuMi 1.02, A MATLAB Toolbox for OptimizationOver Symmetric Cones,”Optimization Methods and Software, vol. 11,no. 1, pp. 625–653, 1999.

[25] R. Tutuncu, K. Toh, and M. Todd, “Solving Semidefinite-Quadratic-Linear Programs using SDPT3,”Mathematical Programming, vol. 95,no. 2, pp. 189–217, 2003.

[26] Power Systems Test Case Archive. University of Washington De-partment of Electrical Engineering. [Online]. Available:http://www.ee.washington.edu/research/pstca/

[27] J. Lofberg, “YALMIP: A Toolbox for Modeling and Optimizationin MATLAB,” in IEEE International Symposium on Computer AidedControl Systems Design, 2004. IEEE, 2004, pp. 284–289.

[28] Siemens PTI, “OPF Manual,”Power System Simulation for Engineering(PSS/E), vol. 31.0, December 2007.

Daniel K. Molzahn (S’09) received the B.S. andthe M.S. degrees in electrical engineering from theUniversity of Wisconsin-Madison in 2008 and 2010,respectively, where he is currently pursuing thePh.D. degree.

Jesse T. Holzer is a Ph.D. student in the MathDepartment at the University of Wisconsin-Madison.His academic interests are variational inequalities forlarge scale equilibrium problems and optimizationmodeling for energy systems. In his free time hecan be found running for distance or biking the hillswest of Madison.

Bernard C. Lesieutre (S’86-M’93-SM’06) receivedthe B.S., M.S., and Ph.D. degrees in electrical en-gineering from the University of Illinois at Urbana-Champaign.

He is currently an Associate Professor of Electri-cal and Computer Engineering at the University ofWisconsin-Madison. His research interests includethe modeling, monitoring, and analysis of electricpower systems and electric energy markets.

Christopher L. DeMarco (M’85) is the GraingerProfessor of Electrical and Computer Engineeringat the University of Wisconsin-Madison, where hehas been a member of the faculty since 1985. Atthe University of Wisconsin-Madison, he has servedas Electrical and Computer Engineering DepartmentChair, and currently serves as Site Director forthe Power Systems Engineering Research Center(PSERC). His research and teaching interests centeron dynamics, optimization and control of electricalenergy systems. He received the B.S. degree at the

Massachusetts Institute of Technology, and the Ph.D. degree at the Universityof California, Berkeley, both in electrical engineering and computer sciences.