Convex Relaxation of Optimal Power Flow - Netlab:...

IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15–27, MARCH 2014 1

Convex Relaxation of Optimal Power FlowPart I: Formulations and Equivalence

Steven H. Low EAS, Caltech [email protected]

Abstract—This tutorial summarizes recent advances in theconvex relaxation of the optimal power flow (OPF) problem,focusing on structural properties rather than algorithms. PartI presents two power flow models, formulates OPF and theirrelaxations in each model, and proves equivalence relations amongthem. Part II presents sufficient conditions under which the convexrelaxations are exact.

I. INTRODUCTION

For our purposes an optimal power flow (OPF) problemis a mathematical program that seeks to minimize a certainfunction, such as total power loss, generation cost or userdisutility, subject to the Kirchhoff’s laws as well as capacity,stability and security constraints. OPF is fundamental in powersystem operations as it underlies many applications such aseconomic dispatch, unit commitment, state estimation, stabilityand reliability assessment, volt/var control, demand response,etc. There has been a great deal of research on OPF sinceCarpentier’s first formulation in 1962 [2]. An early solutionappears in [3] and extensive surveys can be found in e.g. [4],[5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].

Power flow equations are quadratic and hence OPF can beformulated as a quadratically constrained quadratic program(QCQP). It is generally nonconvex and hence NP-hard. Alarge number of optimization algorithms and relaxations havebeen proposed. A popular approximation is a linear program,called DC OPF, obtained through the linearization of the powerflow equations e.g. [16], [17], [18], [19], [20]. See also [21]for a more accurate linear approximation. To the best of ourknowledge solving OPF through semidefinite relaxation is firstproposed in [22] as a second-order cone program (SOCP) forradial (tree) networks and in [23] as a semidefinite program(SDP) for general networks in a bus injection model. It is firstproposed in [51], [57] as an SOCP for radial networks in thebranch flow model of [45]. See Remark 6 below for moredetails. While these convex relaxations have been illustratednumerically in [22] and [23], whether or when they will turnout to be exact is first studied in [24]. Exploiting graph sparsityto simplify the SDP relaxation of OPF is first proposed in [25],[26] and analyzed in [27], [28].

Convex relaxation of quadratic programs has been appliedto many engineering problems; see e.g. [29]. There is a richtheory and extensive empirical experiences. Compared withother approaches, solving OPF through convex relaxation offersseveral advantages. First, while DC OPF is useful in a widevariety of applications, it is not applicable in other applications;

A preliminary and abridged version has appeared in [1].

see Remark 10. Second a solution of DC OPF may not befeasible (may not satisfy the nonlinear power flow equations).In this case an operator may tighten some constraints in DCOPF and solve again. This may not only reduce efficiency butalso relies on heuristics that are hard to scale to larger systemsor faster control in the future. Third, when they converge,most nonlinear algorithms compute a local optimal usuallywithout assurance on the quality of the solution. In contrasta convex relaxation provides for the first time the ability tocheck if a solution is globally optimal. If it is not, the solutionprovides a lower bound on the minimum cost and hence abound on how far any feasible solution is from optimality.Unlike approximations, if a relaxed problem is infeasible, itis a certificate that the original OPF is infeasible.

This two-part tutorial explains the main theoretical resultson semidefinite relaxations of OPF developed in the last fewyears. Part I presents two power flow models that are useful indifferent situations, formulates OPF and its convex relaxationsin each model, and clarifies their relationship. Part II [30]presents sufficient conditions that guarantee the relaxations areexact, i.e. when one can recover a globally optimal solution ofOPF from an optimal solution of its relaxations. We focus onbasic results using the simplest OPF formulation and does notcover many relevant works in the literature, such as stochasticOPF e.g. [31], [32], [33], distributed OPF e.g. [34], [35], [36],[37], [38], [39], new applications e.g. [40], [41], or what to dowhen relaxation fails e.g. [42], [43], [44], to name just a few.

Outline of paper

Many mathematical models have been used to model powernetworks. In Part I of this two-part paper we present two suchmodels, we call the bus injection model (BIM) and the branchflow model (BFM). Each model consists of a set of power flowequations. Each models a power network in that the solutions ofeach set of equations, called the power flow solutions, describethe steady state of the network. We prove that these two modelsare equivalent in the sense that there is a bijection between theirsolution sets (Section II). We formulate OPF within each modelwhere the power flow solutions define the feasible set of OPF(Section III). Even though BIM and BFM are equivalent someresults are much easier to formulate or prove in one model thanthe other; see Remark 2 in Section II.

The complexity of OPF formulated here lies in the noncon-vexity of power flow equations that gives rise to a nonconvexfeasible set of OPF. We develop various characterizations ofthe feasible set and design convex supersets based on thesecharacterizations. Different designs lead to different convex


relaxations and we prove their relationship (Sections IV and V).When a relaxation is exact an optimal solution of the originalnonconvex OPF can be recovered from any optimal solution ofthe relaxation. In Part II [30] we present sufficient conditionsthat guarantee the exactness of convex relaxations.

Branch flow models are originally proposed for networkswith a tree topology, called radial networks, e.g. [45], [46],[47], [48], [49], [50], [51], [52], [53]. They take a recursivestructure that simples the computation of power flow solutions,e.g. [54], [55], [47]. The model of [45], [46] also has alinearization that offers several advantages over DC OPF inBIM; see Remark 10. The linear approximation provides simplebounds on the branch powers and voltage magnitudes in thenonlinear BFM (Section VI). These bounds are used in [56] toprove a sufficient condition for exact relaxation.

Finally we make algorithmic recommendations in SectionVII based on the key results presented here. Most proofs canbe found in the original papers (or the arXiv version of thistutorial) and are omitted.

Notations

Let C denote the set of complex numbers and R the set ofreal numbers. For a 2 C, Re a and Im a denote the real andimaginary parts of a respectively. For any set A ✓ Cn, convAdenotes the convex hull of A. For a 2R, [a]+ := max{a,0}. Fora,b 2C, a b means Re a Re b and Im a Im b. We abusenotation to use the same symbol a to denote either a complexnumber Rea+ i Ima or a 2-dimensional real vector a =(Rea,Ima) depending on the context.

In general scalar or vector variables are in small letters, e.g.u,w,x,y,z. Most power system quantities however are in capitalletters, e.g. S jk,Pjk,Q jk, I j,Vj. A variable without a subscriptdenotes a vector with appropriate components, e.g. s := (s j, j =0, . . . ,n), S := (S jk,( j,k) 2 E). For vectors x,y, x y denotescomponentwise inequality.

Matrices are usually in capital letters. The transpose ofa matrix A is denoted by AT and its Hermitian (complexconjugate) transpose by AH . A matrix A is Hermitian if A=AH .A is positive semidefinite (or psd), denoted by A ⌫ 0, if A isHermitian and xHAx � 0 for all x 2 Cn; in particular if A ⌫ 0then by definition A = AH . For matrices A,B, A ⌫ B meansA�B is psd. Let Sn be the set of all n⇥n Hermitian matricesand Sn

+ the set of n⇥n psd matrices.A graph G=(N,E) consists of a set N of nodes and a set E ✓

N⇥N of edges. If G is undirected then ( j,k)2 E if and only if(k, j) 2 E. If G is directed then ( j,k) 2 E only if (k, j) 62 E; inthis case we will use ( j,k) and j ! k interchangeably to denotean edge pointing from j to k. We sometimes use G = (N, E)to denote a directed graph. By “ j ⇠ k” we mean an edge ( j,k)if G is undirected and either j ! k or k ! j if G is directed.Sometimes we write j 2 G or ( j,k) 2 G to mean j 2 N or( j,k) 2 E respectively. A cycle c := ( j1, . . . , jK) is an orderedset of nodes jk 2 N so that ( jk, jk+1) 2 E for k = 1, . . . ,K withthe understanding that jK+1 := j1. In that case we refer to a linkor a node in the cycle by ( jk, jk+1) 2 c or jk 2 c respectively.

II. POWER FLOW MODELS

In this section we describe two mathematical models ofpower networks and prove their equivalence. By a “mathemat-ical model” we mean a set of variables and a set of equationsrelating these variables. These equations are motivated by thephysical system, but mathematically, they are the starting pointfrom which all claims are derived.

A. Bus injection modelConsider a power network modeled by a connected undi-

rected graph G(N+,E) where N+ := {0}[N, N := {1,2, . . . ,n},and E ✓ N+⇥N+. Each node in N+ represents a bus and eachedge in E represents a transmission or distribution line. Weuse “bus” and “node” interchangeably and “line” and “edge”interchangeably. For each edge (i, j) 2 E let yi j 2 C be itsadmittance. A bus j 2 N+ can have a generator, a load, both orneither. Let Vj be the complex voltage at bus j 2 N+ and |Vj|denote its magnitude. Bus 0 is the slack bus. Its voltage is fixedand we assume without loss of generality that V0 = 1\0� perunit (pu). Let s j be the net complex power injection (generationminus load) at bus j 2 N+.

The bus injection model (BIM) is defined by the followingpower flow equations that describe the Kirchhoff’s laws:

s j = Âk: j⇠k

yHjk Vj(V H

j �V Hk ), j 2 N+ (1)

Let the set of power flow solutions V for each s be:

V(s) := {V 2 Cn+1 | V satisfies (1)}

For convenience we include V0 in the vector variable V :=(Vj, j 2 N+) with the understanding that V0 := 1\0� is fixed.

Remark 1: Bus types. Each bus j is characterized by twocomplex variables Vj and s j, or equivalently, four real variables.The buses are usually classified into three types, dependingon which two of the four real variables are specified. For theslack bus 0, V0 is given and s0 is variable. For a generator bus(also called PV -bus), Re(s j) = p j and |Vj| are specified andIm(s j) = q j and \Vj are variable. For a load bus (also calledPQ-bus), s j is specified and Vj is variable. The power flowor load flow problem is: given two of the four real variablesspecified for each bus, solve the n+1 complex equations in (1)for the remaining 2(n+ 1) real variables. For instance whenall n buses j 6= 0 are all load buses, the power flow problemsolves (1) for the n complex voltages Vj, j 6= 0, and the powerinjection s0 at the slack bus 0. This can model a distributionsystem with a substation at bus 0 and n constant-power loads atthe other buses. For optimal power flow problems p j and |Vj| ongenerator buses or s j on load buses can be variables as well. Forinstance economic dispatch optimizes real power generations p jat generator buses; demand response optimizes demands s j atload buses; and volt/var control optimizes reactive powers q j atcapacitor banks, tap changers, or inverters. These remarks alsoapply to the branch flow model presented next.

B. Branch flow modelIn the branch flow model we adopt a connected directed

graph G = (N+, E) where each node in N+ := {0,1, . . . ,n}


represents a bus and each edge in E ✓ N+ ⇥N+ representsa transmission or distribution line. Fix an arbitrary orientationfor G and let m := |E| be the number of directed edges inG. Denote an edge by ( j,k) or j ! k if it points from nodej to node k. For each edge ( j,k) 2 E let z jk := 1/y jk be thecomplex impedance on the line; let I jk be the complex currentand S jk = Pjk + iQ jk be the sending-end complex power frombuses j to k. For each bus j 2 N+ let Vj be the complex voltageat bus j. Assume without loss of generality that V0 = 1\0� pu.Let s j be the net complex power injection at bus j.

The branch flow model (BFM) in [57] is defined by thefollowing set of power flow equations:

Âk: j!k

S jk = Âi:i! j

�Si j � zi j|Ii j|2

�+ s j, j 2 N+ (2a)

I jk = y jk(Vj �Vk), j ! k 2 E (2b)S jk = Vj IH

jk, j ! k 2 E (2c)

where (2b) is the Ohm’s law, (2c) defines branch power, and(2a) imposes power balance at each bus. The quantity zi j|Ii j|2represents line loss so that Si j � zi j|Ii j|2 is the receiving-endcomplex power at bus j from bus i.

Let the set of solutions x := (S, I,V ) of BFM for each s be:

X(s) := {x 2 C2m+n+1 | x satisfies (2)}

For convenience we include V0 in the vector variable V :=(Vj, j 2 N+) with the understanding that V0 := 1\0� is fixed.

C. EquivalenceEven though the bus injection model (1) and the branch flow

model (2) are defined by different sets of equations in termsof their own variables, both are models of the Kirchhoff’s lawsand therefore must be related. We now clarify the precise sensein which these two mathematical models are equivalent. We saytwo sets A and B are equivalent, denoted by A ⌘ B, if there isa bijection between them [58].

Theorem 1: V(s)⌘ X(s) for any power injections s.

Remark 2: Two models. Given the bijection between the so-lution sets V(s) and X(s) any result in one model is in principlederivable in the other. Some results however are much easier tostate or derive in one model than the other. For instance BIMallows a much cleaner formulation of the semidefinite program(SDP) relaxation. BFM for radial networks has a convenientrecursive structure that allows a more efficient computation ofpower flows and leads to a useful linear approximation of BFM;see Section VI. The sufficient condition for exact relaxation in[56] provides intricate insights on power flows that are hardto formulate or prove in BIM. Finally, since BFM directlymodels branch flows S jk and currents I jk, it is easier to use forsome applications. We will therefore freely use either modeldepending on which is more convenient for the problem athand.

III. OPTIMAL POWER FLOW

A. Bus injection modelAs mentioned in Remark 1 an optimal power flow problem

optimizes both variables V and s over the solution set of the

BIM (1). In addition all voltage magnitudes must satisfy:

v j |Vj|2 v j, j 2 N+ (3)

where v j and v j are given lower and upper bounds on voltagemagnitudes. Throughout this paper we assume v j > 0 to avoidtriviality. The power injections are also constrained:

s j s j s j, j 2 N+ (4)

where s j and s j are given bounds on the injections at buses j.Remark 3: OPF constraints. If there is no bound on the load

or on the generation at bus j then s j =�•� i• or s j = •+ i•respectively. On the other hand (4) also allows the case wheres j is fixed (e.g. a constant-power load), by setting s j = s j to thespecified value. For the slack bus 0, unless otherwise specified,we always assume v0 = v0 = 1 and s0 =�•� i•, s0 = •+ i•.Therefore we sometimes replace j 2N+ in (3) and (4) by j 2N.

We can eliminate the variables s j from the OPF formulationby combining (1) and (4) into

s j Âk:( j,k)2E

yHjk Vj(V H

j �V Hk ) s j, j 2 N+ (5)

Then OPF in the bus injection model can be defined just interms of the complex voltage vector V . Define

V := {V 2 Cn+1 | V satisfies (3), (5)} (6)

V is the feasible set of optimal power flow problems in BIM.Let the cost function be C(V ). Typical costs include the cost

of generating real power at each generator bus or line loss overthe network. All these costs can be expressed as functions ofV . Then the problem of interest is:OPF:

minV

C(V ) subject to V 2 V (7)

Since (5) is quadratic, V is generally a nonconvex set. OPF isthus a nonconvex problem and NP-hard to solve in general.

B. Branch flow model

Denote the variables in the branch flow model (2) by x :=(S, I,V,s) 2 C2(m+n+1). We can also eliminate the variables s jas for the bus injection model by combining (2a) and (4) but itwill prove convenient to retain s := (s j, j 2 N+) as part of thevariables. Define the feasible set in the branch flow model:

X := {x 2 C2(m+n+1) | x satisfies (2), (3), (4)} (8)

Let the cost function in the branch flow model be C(x). Thenthe optimal power flow problem in the branch flow model is:OPF:

minx

C(x) subject to x 2 X (9)

Since (2) is quadratic, X is generally a nonconvex set. OPF isthus a nonconvex problem and NP-hard to solve in general.

Remark 4: OPF equivalence. By Theorem 1 there is abijection between V and X. Throughout this paper we assumethat the cost functions in BIM and BFM are equivalent underthis bijection and we abuse notation to denote them by the


same symbol C(·). Then OPF (7) in BIM and (9) in BFM areequivalent.

Remark 5: OPF variants. OPF as defined in (7) and (9) isa simplified version that ignores other important constraintssuch as line limits, security constraints, stability constraints,and chance constraints; see extensive surveys in [4], [5], [6],[7], [8], [9], [10], [11], [12], [13], [14], [15], [59], [60] anda recent discussion in [61] on real-life OPF problems. Someof these can be incorporated without any change to the resultsin this paper (e.g. see [57], [62] for models that include shuntelements and line limits). Indeed a shunt element y j at bus jcan be easily included in BIM by modifying (1) into:

s j = Âk: j⇠k

yHjk Vj(V H

j �V Hk )+ yH

j |Vj|2

or included in BFM by modifying (2a) into:

Âk: j!k

S jk + yHj |Vj|2 = Â

i:i! j


�+ s j

C. OPF as QCQP

Before we describe convex relaxations of OPF we first showthat, when C(V ) :=V HCV is quadratic in V for some Hermitianmatrix C, OPF is indeed a quadratically constrained quadraticprogram (QCQP) by converting it into the standard form. Wewill use the derivation in [62] for OPF (7) in BIM. OPF (9) inBFM can similarly be converted into a standard form QCQP.

Define the (n+1)⇥ (n+1) admittance matrix Y by

Yi j =

8>><

>>:

Âk:k⇠i

yik, if i = j

�yi j, if i 6= j and i ⇠ j0 otherwise

Y is symmetric but not necessarily Hermitian. Let I j be the netinjection current from bus j to the rest of the network. Thenthe current vector I and the voltage vector V are related by theOhm’s law I = YV . BIM (1) is equivalent to:

s j = VjIHj = (eH

j V )(IHe j)

where e j is the (n+ 1)-dimensional vector with 1 in the jthentry and 0 elsewhere. Hence, since I = YV , we have

s j = tr�eH

j VV HY He j�

= tr�Y He jeH

j�

VV H

= V HY Hj V

where Yj := e jeHj Y is an (n+1)⇥(n+1) matrix with its jth row

equal to the jth row of the admittance matrix Y and all otherrows equal to the zero vector. Yj is in general not Hermitianso that V HY H

j V is in general a complex number. Its real andimaginary parts can be expressed in terms of the Hermitian andskew Hermitian components of Y H

j defined as:

F j :=12�Y H

j +Yj�

and Y j :=12i

�Y H

j �Yj�

Then

Re s j =V HF jV and Im s j =V HY jV

Let their upper and lower bounds be denoted by

p j := Re s j and p j := Re s j

q j := Re s j and q j := Re s j

Let Jj := e jeHj denote the Hermitian matrix with a single 1 in

the ( j, j)th entry and 0 everywhere else. Then OPF (7) can bewritten as a standard form QCQP:

minV2Cn+1

V HCV (10a)

s.t. V HF jV p j, V H(�F j)V �p j (10b)

V HY jV q j, V H(�Y j)V �q j (10c)

V HJjV v j, V H(�Jj)V �v j (10d)

where j 2 N+ in (10).

IV. FEASIBLE SETS AND RELAXATIONS: BIMIn this and the next section we derive semidefinite relaxations

of OPF and clarify their relations. The cost function C of OPFis usually assumed to be convex in its variables. The difficultyof OPF formulated here thus arises from the nonconvex feasiblesets V for BIM and X for BFM. The basic approach to derivingconvex relaxations of OPF is to design convex supersets of(equivalent sets of) V or X and minimize the same cost functionover these supersets. Different choices of convex supersets leadto different relaxations, but they all provide a lower bound toOPF. If every optimal solution of a convex relaxation happensto lie in V or X then it is also feasible and hence optimal forthe original OPF. In this case we say the realxation is exact.

In this section we present three characterizations of thefeasible set V in BIM. These characterizations naturally suggestconvex supersets and semidefinite relaxations of OPF, and weprove equivalence relations among them. In the next sectionwe treat BFM. In Part II of the paper we discuss sufficientconditions that guaranteed exact relaxations.

A. PreliminariesSince OPF is a nonconvex QCQP there is a standard

semidefinite relaxation through the equivalence relation: for anyHermitian matrix M, V HMV = tr MVV H = tr MW for a psdrank-1 matrix W . Applying this transformation to the QCQPformulation (10) leads to an equivalent problem of the form:

minW2Sn+1

tr CW

s.t. tr ClW bl , W ⌫ 0, rank W = 1

for appropriate Hermitian matrices Cl and real numbers bl .This problem is equivalent to (10) because given a psd rank-1 solution W , a unique solution V of (10) can be recoveredthrough rank-1 factorization W = VV H . Unlike (10) whichis quadratic in V this problem is convex in W except thenonconvex rank-1 constraint. Removing the rank-1 constraintyields the standard SDP relaxation.

We now generalize this intuition to characterize the feasibleset V in (6) in terms of partial matrices. These characterizationslead naturally to SDP, chordal, and second-order cone program(SOCP) relaxations of OPF in BIM, as shown in [58], [28].


We start with some basic definitions on partial matrices andtheir completions; see e.g. [63], [64], [65] for more details. Fixany connected undirected graph F with n vertices and m edgesconnecting distinct vertices.1 A partial matrix WF is a set of2m+n complex numbers defined on F :

WF := {[WF ] j j, [WF ] jk, [WF ]k j |nodes j and edges ( j,k) of F}

WF can be interpreted as a matrix with entries partially specifiedby these complex numbers. If F is a complete graph (in whichthere is an edge between every pair of vertices) then WF is afully specified n⇥n matrix. A completion W of WF is any fullyspecified n⇥n matrix that agrees with WF on graph F , i.e.,

[W ] j j = [WF ] j j, [W ] jk = [WF ] jk for j,( j,k) 2 F

Given an n⇥n matrix W we use WF to denote the submatrix ofW on F, i.e., the partial matrix consisting of the entries of W de-fined on graph F . If q is a clique (a fully connected subgraph) ofF then let WF(q) denote the fully-specified principal submatrixof WF defined on q. We extend the definitions of Hermitian,psd, and rank-1 for matrices to partial matrices, as follows.A partial matrix WF is Hermitian, denoted by WF = W H

F , if[WF ] jk = [WF ]Hk j for all ( j,k) 2 F ; it is psd, denoted by WF ⌫ 0,if WF is Hermitian and the principal submatrices WF(q) are psdfor all cliques q of F ; it is rank-1, denoted by rank WF = 1, if theprincipal submatrices WF(q) are rank-1 for all cliques q of F .We say WF is 2⇥2 psd (rank-1), denoted by WF( j,k)⌫ 0 (rankWF( j,k) = 1) if, for all edges ( j,k) 2 F , the 2⇥ 2 principalsubmatrices

[WF ]( j,k) :=[WF ] j j [WF ] jk[WF ]k j [WF ]kk

�

are psd (rank-1). F is a chordal graph if either F has no cycle orall its minimal cycles (ones without chords) are of length three.A chordal extension c(F) of F is a chordal graph that containsF , i.e., c(F) has the same vertex set as F but an edge set that isa superset of F’s edge set. In that case we call the partial matrixWc(F) a chordal extension of the partial matrix WF . Every graphF has a chordal extension, generally nonunique. In particular acomplete supergraph of F is a trivial chordal extension of F .

For our purposes chordal graphs are important because ofthe result [63, Theorem 7] that every psd partial matrix has apsd completion if and only if the underlying graph is chordal.When a positive definite completion exists, there is a uniquepositive definite completion, in the class of all positive definitecompletions, whose determinant is maximal. Theorem 2 belowextends this to rank-1 partial matrices.

B. Feasible setsWe can now characterize the feasible set V of OPF defined

in (6). Recall the undirected connected graph G = (N+,E) thatmodels a power network. Given a voltage vector V 2 V definea partial matrix WG :=WG(V ): for j 2 N+ and ( j,k) 2 E,

[WG] j j := |Vj|2 (11a)[WG] jk := VjV H

k =: [WG]Hk j (11b)

1In this subsection we abuse notation and use n,m to denote generalintegers unrelated to the number of buses or lines in a power network.

Then the constraints (5) and (3) imply that the partial matrixWG satisfies 2

s j Âk:( j,k)2E

yHjk�[WG] j j � [WG] jk

� s j, j 2 N+ (12a)

v j [WG] j j v j, j 2 N+ (12b)

Following Section III-C these constraints can also be writtenin a (partial) matrix form as:

p j tr F jWG p j

q j tr Y jWG q j

v j tr JjWG v j

The converse is not always true: given a partial matrix WGthat satisfies (12) it is not always possible to recover a voltagevector V in V. Indeed this is possible if and only if WG hasa completion W that is psd rank-1, because in that case Wsatisfies (12) since y jk = 0 if ( j,k) 62 E and it can be uniquelyfactored as W = VV H with V 2 V. We hence seek conditionsadditional to (12) on the partial matrix WG that guarantee thatit has a psd rank-1 completion W from which V 2 V can berecovered. Our first key result provides such a characterization.

We say that a partial matrix WG satisfies the cycle conditionif for every cycle c in G

Â( j,k)2c

\Wjk = 0 mod 2p (13)

When \Wjk represent voltage phase differences across each linethen the cycle condition imposes that they sum to zero (mod 2p)around any cycle. The next theorem, proved in [58, Theorem3] and [28], implies that WG has a psd rank-1 completion W ifand only if WG is 2⇥2 psd rank-1 on G and satisfies the cyclecondition (13), if and only if it has a chordal extension Wc(G)

that is psd rank-1. 3

Consider the following conditions on (n+ 1)⇥ (n+ 1) ma-trices W and partial matrices Wc(G) and WG:

W ⌫ 0, rank W = 1 (14)Wc(G) ⌫ 0, rank Wc(G) = 1 (15)

WG( j,k)⌫ 0, rank WG( j,k) = 1, ( j,k) 2 E, (16)

Theorem 2: Fix a graph G on n+1 nodes and any chordalextension c(G) of G. Assuming Wj j > 0,

⇥Wc(G)

⇤j j > 0 and

[WG] j j > 0, j 2 N+, we have:(1) Given an (n+ 1)⇥ (n+ 1) matrix W that satisfies (14),

its submatrix Wc(G) satisfies (15).(2) Given a partial matrix Wc(G) that satisfies (15), its sub-

matrix WG satisfies (16) and the cycle condition (13).(3) Given a partial matrix WG that satisfies (16) and the

cycle condition (13), there is a completion W of WG thatsatisfies (14).

2The constraint (12a) can also be written compactly in terms of theadmittance matrix Y as in [66]:

s diag�WY H� s

3The theorem also holds with psd replaced by negative semidefinite.


Informally Theorem 2 says that (14) is equivalent to (15) isequivalent to (16)+(13). It characterizes a property of the fullmatrix W (rank W = 1) in terms of its submatrices Wc(G) andWG. This is important because the submatrices are typicallymuch smaller than W for large sparse networks and mucheasier to compute. The theorem thus allows us to solve simplerproblems in terms of partial matrices as we now explain.

Define the set of Hermitian matrices:

W := {W 2 Sn+1 | W satisfies (12), (14)} (17)

Fix any chordal extension c(G) of G and define the set ofHermitian partial matrices Wc(G):

Wc(G) := {Wc(G) | Wc(G) satisfies (12), (15)} (18)

Finally define the set of Hermitian partial matrices WG:

WG := {WG | WG satisfies (12), (13), (16)} (19)

Note that the definition of psd for partial matrices implies thatWc(G) and WG are Hermitian. The assumption v j > 0, j 2 N+

implies that all matrices or partial matrices have strictly positivediagonal entries.

Theorem 2 implies that given a partial matrix Wc(G) 2Wc(G)

or a partial matrix WG 2WG there is a psd rank-1 completionW 2W from which a solution V 2V of OPF can be recovered.In fact we know more: given any Hermitian partial matrix WG(not necessarily in WG), the set of all completions of WG thatsatisfies the condition in Theorem 2(3) consists of a singlepsd rank-1 matrix and infinitely many indefinite non-rank-1matrices; see [58, Theorems 5 and 8] and discussions therein.Hence the psd rank-1 completion W of a WG 2WG is unique.

Corollary 3: Given a partial matrix Wc(G) 2Wc(G) or WG 2WG there is a unique psd rank-1 completion W 2W.

Recall that two sets A and B are equivalent (A ⌘ B) if thereis a bijection between them. Even though W,Wc(G),WG aredifferent kinds of spaces Theorem 2 and Corollary 3 implythat they are all equivalent to the feasible set of OPF.

Theorem 4: V⌘W⌘Wc(G) ⌘WG.

Theorem 4 suggests three equivalent problems to OPF. Weassume the cost function C(V ) in OPF depends on V onlythrough the partial matrix WG defined in (11). For example if thecost is total real line loss in the network then C(V )=Â j Re s j =Â j Âk:( j,k)2E Re

�[WG] j j � [WG] jk

�yH

jk. If the cost is a weighted

sum of real generation power then C(V ) = Â j

⇣c j Re s j + pd

j

⌘

where pdj are the given real power demands at buses j; again

C(V ) is a function of the partial matrix WG. Then Theorem 4implies that OPF (7) is equivalent to

minW

C(WG) s.t. W 2 W (20)

where W is any one of the sets W,Wc(G),WG. Specifically,given an optimal solution W opt in W, it can be uniquelydecomposed into W opt = V opt(V opt)H . Then V opt is in V andan optimal solution of OPF (7). Alternatively given an optimalsolution W opt

F in Wc(G) or WG, Corollary 3 guarantees that W optF

has a unique psd rank-1 completion W opt in W from which anoptimal V opt 2V can be recovered. In fact given a partial matrix

WG 2WG (or Wc(G) 2Wc(G)) there is a more direct constructionof a feasible solution V 2V of OPF than through its completion;see Section IV-D.

C. Semidefinite relaxations

Hence solving OPF (7) is equivalent to solving (20) overany of W,Wc(G),WG for an appropriate matrix variable. Thedifficulty with solving (20) is that the feasible sets W, Wc(G),and WG are still nonconvex due to the rank-1 constraints andthe cycle condition (13). Their removal leads to SDP, chordal,and SOCP relaxations of OPF respectively.

Relax W, Wc(G) and WG to the following convex supersets:

W+ := {W 2 Sn+1 |WG satisfies (12),W ⌫ 0}W+

c(G) := {Wc(G) |WG satisfies (12),Wc(G) ⌫ 0}W+

G := {WG |WG satisfies (12),WG( j,k)⌫ 0, ( j,k) 2 E}

Define the problems:OPF-sdp:

minW

C(WG) s.t. W 2W+ (21)

OPF-ch:

minWc(G)

C(WG) s.t. Wc(G) 2W+c(G) (22)

OPF-socp:

minWG

C(WG) s.t. WG 2W+G (23)

The condition WG( j,k)⌫ 0 in the definition of W+G is equivalent

to [WG] jk = [WG]Hk j and (recall the assumption v j > 0, j 2 N+)

[WG] j j > 0, [WG]kk > 0, [WG] j j[WG]kk ��[WG] jk

��2

This is a second-order cone and hence OPF-socp is indeed anSOCP in the rotated form.

Remark 6: Literature. SOCP relaxation for OPF seems to befirst proposed in [22] for the bus injection model (1), and in[51], [57] for the branch flow model (2) as explained in the nextsection. By defining a new set of variables v j := |Vj|2, R jk :=|Vj||Vk|cos(q j �qk), and I jk := |Vj||Vk|sin(q j �qk) where q j :=\Vj, [22] rewrites the bus injection model (1) in the complexdomain as a set of linear equations in these new variables inthe real domain and the following quadratic equations:

v jvk = R2jk + I2

jk

Relaxing these equalities to v jvk � R2jk + I2

jk enlarges the so-lution set to a second-order cone that is equivalent to W+

G inthis paper. SDP relaxation is first proposed in [23] for the businjection model and analyzed in [24]. Chordal relaxation forOPF is first proposed in [25], [26] and analyzed in [27], [28].

D. Solution recovery

When the convex relaxations OPF-sdp, OPF-ch, OPF-socpare exact, i.e., if their optimal solutions W sdp, W ch

ch , W socpG

happen to lie in W, Wc(G), WG respectively, then an optimalsolution V opt of the original OPF can be recovered from these


solutions. Indeed the recovery method works not just for anoptimal solution, but any feasible solution that lies in W, Wc(G)

or WG. Moreover, given a W 2 W or a Wc(G) 2 Wc(G), theconstruction of V depends on W or Wc(G) only through theirsubmatrix WG. We hence describe the method for recoveringthe unique V from a WG, which may be a partial matrix in WGor the submatrix of a (partial) matrix in W or Wc(G).

Let T be an arbitrary spanning tree of G rooted at bus 0. LetP j denote the unique path from node 0 to node j in T . Recallthat V0 = 1\0� without loss of generality. For i = 1, . . . ,n, let

|Vi| :=q[WG]ii

\Vi := � Â( j,k)2Pi

\ [WG] jk

Then it can be checked that V is in (6) and feasible for OPF.

E. Tightness of relaxationsSince W✓W+, Wc(G) ✓W+

c(G), WG ✓W+G , the relaxations

OPF-sdp, OPF-ch, OPF-socp all provide lower bounds on OPF(7) in light of Theorem 4. How do these relaxations comparein terms of computational efficiency and tightness?

OPF-socp is the simplest computationally. OPF-ch usuallyrequires more computation than OPF-socp but much less thanOPF-sdp for large sparse networks (even though OPF-ch canbe as complex as OPF-sdp in the worse case [64], [65]). Therelative tightness of the relaxations depends on the networktopology. For a general mesh network OPF-sdp is as tight arelaxation as OPF-ch and they are strictly tighter than OPF-socp. For a tree (radial) network the hierarchy collapses andall three are equally tight. We now make this precise.

Consider their feasible sets W+, W+c(G) and W+

G . We say thata set A is an effective subset of a set B, denoted by A v B, if,given a (partial) matrix a 2 A, there is a (partial) matrix b 2 Bthat has the same cost C(a) =C(b). We say A is similar to B,denoted by A ' B, if A v B and B v A. Note that A ⌘ B impliesA'B but the converse may not be true. The feasible set of OPF(7) is an effective subset of the feasible sets of the relaxations;moreover these relaxations have similar feasible sets when thenetwork is radial. This is a slightly different formulation of thesame results in [58], [28].

Theorem 5: V v W+ ' W+c(G) v W+

G . If G is a tree thenVvW+ 'W+

c(G) 'W+G .

Let Copt,Csdp,Cch,Csocp be the optimal values of OPF (7),OPF-sdp (21), OPF-ch (22), OPF-socp (23) respectively. The-orem 4 and Theorem 5 directly imply

Corollary 6: Copt � Csdp = Cch � Csocp. If G is a tree thenCopt �Csdp =Cch =Csocp.

Remark 7: Tightness. Theorem 5 and Corollary 6 imply thatfor radial networks one should always solve OPF-socp sinceit is the tightest and the simplest relaxation of the three.For mesh networks there is a tradeoff between OPF-socpand OPF-ch/OPF-sdp: the latter is tighter but requires heaviercomputation. Between OPF-ch and OPF-sdp, OPF-ch is usuallypreferable as they are equally tight but OPF-ch is usually muchfaster to solve for large sparse networks. See [25], [26], [28],[27], [67] for numerical studies that compare these relaxations.

F. Chordal relaxationTheorem 2 through Corollary 6 apply to any chordal exten-

sion c(G) of G. The choice of c(G) does not affect the optimalvalue of the chordal relaxation but determines its complexity.Unfortunately the optimal choice that minimizes the complexityof OPF-ch is NP-hard to compute.

This difficulty is due to two conflicting factors in choosinga c(G). Recall that the constraint Wc(G) ⌫ 0 in the definitionof W+

c(G) consists of multiple constraints that the principalsubmatrices Wc(G)(q)⌫ 0, one for each (maximal) clique q ofc(G). When two cliques q and q0 share a node their submatricesWc(G)(q) and Wc(G)(q0) share entries that must be decoupledby introducing auxiliary variables and equality constraints onthese variables. The choice of c(G) determines the number andsizes of these submatrices Wc(G)(q) as well as the numbers ofauxiliary variables and additional decoupling constraints. Onthe one hand if c(G) contains few cliques q then the submatricesWc(G)(q) tend to be large and expensive to compute (e.g. ifc(G) is the complete graph then there is a single clique, butWc(G) =W and OPF-ch is identical to OPF-sdp). On the otherhand if c(G) contains many small cliques q then there tendsto be more overlap and chordal relaxation tends to requiremore decoupling constraints. Hence choosing a good chordalextension c(G) of G is important but nontrivial. See [64], [65]and references therein for methods to compute efficient chordalrelaxations of general QCQP. For OPF [27] proposes effectivetechniques to reduce the number of cliques in its chordalrelaxation. To further reduce the problem size [67] proposesto carefully drop some of the decoupling constraints, thoughthe resulting relaxation can be weaker.

V. FEASIBLE SETS AND RELAXATIONS: BFMWe now present an SOCP relaxation of OPF in BFM pro-

posed in [51], [57] in two steps. We first relax the phase anglesof V and I in (2) and then we relax a set of quadratic equalitiesto inequalities. This derivation pinpoints the difference betweenradial and mesh topologies. It motivates a recursive version ofBFM for radial networks (Section VI) and the use of phaseshifters for convexification of mesh networks (Part II [30]).

A. Feasible setsConsider the following set of equations in the variables x :=

(S,`,v,s) in R3(m+n+1):4

Âk: j!k

S jk = Âi:i! j

(Si j � zi jì j)+ s j, j 2 N+ (24a)

v j � vk = 2Re�zH

jkS jk�� |z jk|2` jk, j ! k 2 E

(24b)v j` jk = |S jk|2, j ! k 2 E (24c)

4The use of complex variables is only a shorthand and should be interpretedas operations in real variables. For instance (24a) is a shorthand for

Âk: j!k

Pjk = Âi:i! j

�Pi j � ri jì j

�+ p j

Âk: j!k

Q jk = Âi:i! j

�Qi j � xi jì j

�+q j


and define the solution set as:

Xnc := {x 2 R3(m+n+1) | x satisfies (3), (4), (24)}

Note that the vector v includes v0 and s includes s0. The model(24) is first proposed in [45], [46]. 5 It can be derived as arelaxation of BFM (2) as follows. Taking the squared magnitudeof (2c) and replacing |Vj|2 and |I jk|2 by v j and ` jk respectivelyyield (24c). To obtain (24b), use (2b)–(2c) to write Vk = Vj �z jkS jkV�1

j and take the squared magnitude on both sides toeliminate the phase angles of V and I. These operations definea mapping h :C2(m+n+1) !R3(m+n+1) by: for any x=(S, I,V,s),h(x) := (S,`,v,s) with ` jk = |I jk|2 and v j = |Vj|2.

Throughout this paper we assume the cost function C(x) inOPF (9) depends on x only through x := h(x). For examplefor total real line loss C(x) = Â( j,k)2E Rez jk` jk. If the cost is aweighted sum of real generation power then C(x) = Â j(c j p j +pd

j ) where p j are the real parts of s j and pdj are the given real

power demands at buses j; again C(x) depends only on x.Then the model (24) is a relaxation of BFM (2) in the sense

that the feasible set X of OPF in (9) is an effective subsetof Xnc, X v Xnc, since h(X) ✓ Xnc. We now characterize thesubset of Xnc that is equivalent to X.

Given an x := (S,`,v,s) 2 R3(m+n+1) define b (x) 2 Rm by

b jk(x) := \�v j � zH

jkS jk�, j ! k 2 E (25)

Even though x does not include phase angles of V , x impliesa phase difference across each line j ! k 2 E given by b jk(x).The subset of Xnc that is equivalent to X are those x for whichthere exists q such that q j �qk = b jk(x). To state this preciselylet B be the m⇥n (transposed) reduced incidence matrix of G:

Bl j =

8><

>:

1 if edge l 2 E leaves node j�1 if edge l 2 E enters node j0 otherwise

where j 2 N. Consider the set of x 2 Xnc such that

9q that solves Bq = b (x) mod 2p (26)

i.e., b (x) is in the range space of B (mod 2p). A solution q(x),if exists, is unique in (�p,p]n. Define the set

X := {x 2 R3(m+n+1) | x satisfies (3), (4), (24), (26)}

The following result characterizes the feasible set X of OPF inBFM and follows from [57, Theorems 2, 4].

Theorem 7: X⌘ X✓ Xnc.The bijection between X and X is given by h defined aboverestricted to X. Its inverse h�1(S,`,v,s) = (S, I,V,s) is definedon X in terms of q(x) by:

Vj := pv j eiq j(x), j 2 N (27a)

I jk :=p` jk ei(q j(x)�\S jk), j ! k 2 E (27b)

The condition (26) is equivalent to the cycle condition (13)in the bus injection model. To see this fix any spanning tree

5The original model, called the DistFlow equations, in [45], [46] is forradial (distribution) networks, but its extension here to mesh networks is trivial.

T = (N,ET ) of the (directed) graph G. We can assume withoutloss of generality (possibly after re-labeling the links) that ETconsists of links l = 1, . . . ,n. Then B can be partitioned into

B =

BTB?

�

where the n⇥n submatrix BT corresponds to links in T and the(m�n)⇥n submatrix B? corresponds to links in T? := G\T .Similarly partition b (x) into

b (x) =

bT (x)b?(x)

�

The next result, proved in [57, Theorems 2 and 4], provides amore explicit characterization of (26) in terms of b (x). Whenit holds this characterization has the same interpretation of thecycle condition in (13): the voltage angle differences impliedby x sum to zero (mod 2p) around any cycle. Formally let b bethe extension of b from directed to undirected links: for eachj ! k 2 E let b jk(x) := b jk(x) and bk j(x) := �b jk(x). We sayc := ( j1, . . . , jK) is an undirected cycle if, for each k = 1, . . . ,K,either jk ! jk+1 2 E or jk+1 ! jk 2 E with the interpretationthat jK+1 := j1; ( jk, jk+1) 2 c denotes one of these links.

Theorem 8: An x 2 Xnc satisfies (26) if and only if aroundeach undirected cycle c we have

Â( j,k)2c

b jk(x) = 0 mod 2p (28)

In that case q(x) = P�B�1

T bT (x)�

is the unique solution of(26) in (�p,p]n, where P(f) projects f to (�p,p]n.

Theorem 8 determines when the voltage magnitudes v of agiven x can be assigned phase angles q(x) so that the resultingx := h�1(x) is a power flow solution in X.

B. SOCP relaxation

The set Xnc that contains the (equivalent) feasible set X ofOPF is still nonconvex because of the quadratic equalities in(24c). Relax them to inequalities:

v j ` jk � |S jk|2, ( j,k) 2 E (29)

and define the set:

X+ :={x 2 R3(m+n+1) | x satisfies (3), (4), (24a), (24b), (29)}

Clearly X h⌘ X ✓ Xnc ✓ X+; see Figure 1. Moreover X+ is asecond-order cone in the rotated form.

The three sets X, Xnc, X+ define the following problems:OPF:

minx

C(x) subject to x 2 X (30)

OPF-nc:

minx

C(x) subject to x 2 Xnc (31)

OPF-socp:

minx

C(x) subject to x 2 X+ (32)

The next theorem follows from the results in [57] and implies


h

h−1

C2(m+n+1) R3(m+n+1)

Xnc

X X

X+

Fig. 1: Feasible sets X of OPF (9) in BFM, its equivalent setX (defined by h) and its relaxations Xnc and X+. If G is a treethen X= Xnc.

that OPF (9) is equivalent to minimization over X and OPF-socp is its SOCP relaxation. Moreover for radial networksvoltage and current angles can be ignored and OPF (9) isequivalent to OPF-nc.

Theorem 9: X⌘X✓Xnc ✓X+. If G is a tree then X⌘X=Xnc ✓ X+.

Let Copt be the optimal cost of OPF (9) in the branchflow model. Let Copf, Cnc, Csocp be the optimal costs of OPF(30), OPF-nc (31), OPF-socp (32) respectively defined above.Theorem 9 implies

Corollary 10: Copt =Copf �Cnc �Csocp. If G is a tree thenCopt =Copf =Cnc �Csocp.

Remark 8: SOCP relaxation. Suppose one solves OPF-socpand obtains an optimal solution xsocp := (S,`,v,s) 2 X+. Forradial networks if xsocp attains equality in (29) then xsocp 2Xnc and Theorem 9 implies that an optimal solution xopt :=(S, I,V,s) 2 X of OPF (9) can be recovered from xsocp. Indeedxopt = h�1(xsocp) where h�1 is defined in (27). Alternativelyone can use the angle recovery algorithms in [57, Part I] torecover xopt. For mesh networks xsocp needs to both attainequality in (29) and satisfy the cycle condition (26) in orderfor an optimal solution xopt to be recoverable. Our experiencewith various practical test networks suggests that xsocp usuallyattains equality in (29) but, for mesh networks, rarely satisfes(26) [51], [57], [56], [28]. Hence OPF-socp is effective forradial networks but not for mesh networks (in both BIM andBFM).

C. EquivalenceTheorem 9 establishes a bijection between X and the feasible

set X of OPF (9) in BFM. Theorem 4 establishes a bijectionbetween WG and the feasible set V of OPF (7) in BIM.Theorem 1 hence implies that X ⌘ X ⌘ V ⌘ WG. Moreovertheir SOCP relaxations are equivalent in these two models [58],[28]. Define the set of partial matrices defined on G that are2⇥2 psd rank-1 but do not satisfy the cycle condition (13):

Wnc := {Wnc | WG satisfies (12),WG( j,k)⌫ 0,rank WG( j,k) = 1 for all ( j,k) 2 E}

Clearly WG ✓Wnc ✓W+G in general and WG =Wnc ✓W+

G forradial networks.

Theorem 11: X⌘WG, Xnc ⌘Wnc and X+ ⌘W+G .

The bijection between X+ and W+G is defined as follows. Let

WG ✓C2m+n+1 denote the set of Hermitian partial matrices (in-cluding [WG]00 = v0 which is given). Let x := (S,`,v,s) denotevectors in R3(m+n+1). Define the linear mapping g : WG ! X+

by x = g(WG) where

S jk := yHjk�[WG] j j � [WG] jk

�, j ! k

` jk := |y jk|2�[WG] j j +[WG]kk � [WG] jk � [WG]k j

�, j ! k

v j := [WG] j j, j 2 N+

s j := Âk: j⇠k


�, j 2 N+

Its inverse g�1 : X+ !WG is WG = g�1(x) where [WG] j j := v jfor j 2 N+ and [WG] jk := v j � zH

jkS jk =: [WG]Hk j for j ! k. Themapping g and its inverse g�1 restricted to WG (Wnc) and X(Xnc) define the bijection between them.

VI. BFM FOR RADIAL NETWORKS

Theorem 9 implies that for radial networks the model (24) isexact. This is because the reduced incident matrix B in (26) isn⇥n and invertible, so the cycle condition is always satisfied[57, Theorem 4]. Hence a solution in Xnc can be mapped to abranch flow solution in X by the mapping h�1 defined in (27).For radial networks this model has two advantages: (i) it hasa recursive structure that simplifies computation, and (ii) it hasa linear approximation that provides simple bounds on branchpowers S jk and voltage magnitudes v j, as we now show.

A. Recursive equations and graph orientationThe model (24) holds for any graph orientation of G. It has

a recursive structure when G is a tree. In that case differentorientations have different boundary conditions that initializethe recursion and may be convenient for different applications.Without loss of generality we take bus 0 as the root of thetree. We discuss two different orientations: one where everylink points away from bus 0 and the other where every linkpoints towards bus 0. 6

Case I: Links point away from bus 0. Model (24) reduces to:

Âk: j!k

S jk = Si j � zi j`i j + s j, j 2 N+ (33a)


jkS jk�� |z jk|2` jk, j ! k 2 E (33b)

v j` jk = |S jk|2, j ! k 2 E (33c)

where bus i in (33a) denotes the unique parent of node j (onthe unique path from node 0 to node j), with the understandingthat if j = 0 then Si0 := 0 and `i0 := 0. Similarly when j is aleaf node7 all S jk = 0 in (33a). The model (33) is called theDistFlow equations and first proposed in [45], [46].

Its recursive structure is exploited in [47] to analyze thepower flow solutions given an (s j, j 2 N), as we now explainusing the special case of a linear network with n+ 1 busesthat represents a main feeder. To simplify notation denote

6An alternative model is to use an undirected graph and, for each link ( j,k),the variables (S jk,` jk) and (Sk j,`k j) are defined for both directions, with theadditional equations S jk +Sk j = z jk` jk and `k j = ` jk .

7A node j is a leaf node if there exists no i such that i ! j 2 E.


�S j( j+1),` j( j+1)

�and z j( j+1) by (S j,` j) and z j respectively.

Then the DistFlow equations (33) reduce to (v0 is given):

S j+1 = S j � z j` j + s j+1, j = 0, . . . ,n�1 (34a)v j+1 = v j �2Re(zH

j S j)� |z j|2` j, j = 0, . . . ,n�1 (34b)

v j` j = |S j|2, j = 0, . . . ,n�1 (34c)S0 = s0, Sn = 0 (34d)

Let x j := (S j,` j,v j), j 2 N+. If s0 were known then one canstart with (v0,s0) and use the recursion (34a)–(34c) to computex j in terms of s0 = S0, i.e., (34) can be collapsed into functionsof the scalar variable s0 (recall that (s j, j 2 N) are given):

x j = f j(s0), j 2 N+ (35)

Use the boundary condition (34d) Sn = fn(s0) = 0 to solvefor the scalar variable s0. The other variables x j can thenbe computed from (35). This method can be extended toa general radial network with laterals [47]. See also [68],[69] for techniques for solving the nonlinear equations (35),and [54], [55] for a different recursive approach called theforward/backward sweep for radial networks.Case II: Links point towards bus 0. Model (24) reduces to:

S ji = Âk:k! j

�Sk j � zk j ˆk j

�+ s j, j 2 N+ (36a)

vk � v j = 2Re�zH

k jSk j�� |zk j|2 ˆk j, k ! j 2 E (36b)

vk ˆk j = |Sk j|2, k ! j 2 E (36c)

where i in (36a) denotes the node on the unique path betweennode 0 and node j. The boundary condition is defined by S ji = 0in (36a) when j = 0 and Sk j = 0,`k j = 0 in (36a) when j is aleaf node. An advantage of this orientation is illustrated in thenext subsection in proving a simple bound on v j.

B. Linear approximation and bounds

By setting ` jk = 0 in (33) we obtain a linear approximationof the the branch flow model, with the graph orientation whereall links point away from bus 0:

Âk: j!k

Slinjk = Slin

i j + s j, j 2 N+ (37a)

vlinj � vlin

k = 2Re⇣

zHjkSlin

jk

⌘, j ! k 2 E (37b)

where bus i in (37a) denotes the unique parent of bus j. Theboundary condition is: Slin

i0 := 0 in (37a) when j = 0, and Slinjk = 0

in (37a) when j is a leaf node. This is called the simplifiedDistFlow equations in [46], [70]. It is a good approximation of(33) because the loss z jk` jk is typically much smaller than thebranch power flow S jk.

The next result provides simple bounds on (S,v) in terms oftheir linear approximations (Slin,vlin). Denote by T j the subtreerooted at bus j, including j. We write “k 2 T j” to mean nodek of T j and “(k, l) 2 T j” to mean edge (k, l) of T j. Denote byPk the set of links on the unique path from bus 0 to bus k.

Lemma 12: Fix any v0 and s 2 R2(n+1). Let (S,`,v) and(Slin,vlin) be solutions of (33) and (37) respectively with thegiven v0 and s. Then

(1) For i ! j 2 E

Slini j = � Â

k2T j

sk

Si j = � Âk2T j

sk +

0

@zi jì j + Â(k,l)2T j

zkl`kl

1

A

(2) For i ! j 2 E, Si j � Slini j with equality if only if ì j and

all `kl in T j are zero.(3) For j 2 N+

vlinj = v0 � Â

(i,k)2P j

2Re⇣

zHikSlin

ik

⌘

v j = v0 � Â(i,k)2P j

�2Re

�zH

ikSik�� |zik|2ìk

�

(4) For j 2 N+, v j vlinj .

Lemma 12 says that the power flow Si j on line (i, j) equalsthe total load �Âk2T j sk in the subtree rooted at node jplus the total line loss in supplying these loads. The linearapproximation Slin

i j neglects the line loss and underestimatesthe required power to supply these loads.

Lemma 12(1)–(3) can be easily proved by recursing on(33a)–(33b) and (37). Since Si j � Slin

i j but |zik|2ìk � 0, a directproof of Lemma 12(4) is not obvious. Instead, one can makeuse of Lemma 13 below and define a bijection between thesolutions (S,`,v) of (33) and the solutions (S, ˆ, v) of (36) inwhich v = v. It can be checked that the solutions of (37) andthose of (38) are related by Slin = �Slin and vlin = vlin. ThenLemma 13(4) implies Lemma 12(4).

A linear approximation of (36) is (setting ˆk j = 0):

Slinji = Â

k:k! jSlin

k j + s j, j 2 N+ (38a)

vlink � vlin

j = 2Re⇣

zHk jS

link j

⌘, k ! j 2 E (38b)

Lemma 13: Fix any v0 and s 2 R2(n+1). Let (S, ˆ, v) and(Slin, vlin) be solutions of (36) and (38) respectively with thegiven v0 and s. Then

(1) For all j ! i 2 E

Slinji = Â

k2T j

sk

S ji = Âk2T j

sk � Â(k,l)2T j

zkl ˆkl

(2) For all j ! i 2 E, S ji Slinji with equality if and only if

all `kl in T j are zero.(3) For j 2 N+

vlinj = v0 + Â

(i,k)2P j

2Re⇣

zHik Slin

ik

⌘

v j = v0 + Â(i,k)2P j

�2Re

�zH

ik Sik�� |zik|2 îk

�

(4) For j 2 N+, v j vlinj .

Lemma 13 says that the branch power S ji (towards bus 0)equals the total injection Âk2T j sk in the subtree rooted at bus


j minus the line loss in that subtree. The linear approximationSlin

ji neglects the loss and hence overestimates the branch powerflow. It can be proved by recursing on (36a)–(36b) and (38).

Remark 9: Bounds for SOCP relaxation. Lemmas 12 and 13do not depend on the quadratic equalities (33c) and (36c) aslong as ` jk � 0. In particular the lemmas hold if the equalitieshave been relaxed to inequalities v j` jk � |S jk|2. These boundsare used in [56] to prove a sufficient condition for exact SOCPrelaxation for radial networks.

Remark 10: Linear approximations. For radial networks thelinear approximations (37) and (38) of BFM have two ad-vantages over the (linear) DC approximation of BIM. Firstthey have a simple recursive structure that leads to simplebounds on power flow quantities. Second DC approximationassumes a lossless network (r jk = 0), fixes voltage magnitudes,and ignores reactive power, whereas (37) and (38) do not.This is important for distribution systems where loss is muchhigher than in transmission systems, voltages can fluctuatesignificantly and reactive powers are used to regulate them.On the other hand (37) and (38) are applicable only for radialnetworks whereas DC approximation applies to mesh networksas well. See also [21] for a more accurate linearization of BIMthat addresses the shortcomings of DC OPF.

VII. CONCLUSION

We have presented a bus injection model and a branch flowmodel, formulated several relaxations of OPF, and proved theirrelations. These results suggest a new approach to solving OPFsummarized in Figure 2. For radial networks we recommend

OPFBsocp)

OPF)solu2on)

Recover)or

cycle)condi2on)

Y)

rankB1)

OPFBch) OPFBsdp)

Y)

WGsocp

Y,)mesh)

2x2)rankB1)

Y)radial)

OPFBsocp)

cycle)condi2on)Y)

equality)

Y)radial)

Y,)mesh)

Wc(G )ch W sdp xsocp

V opt xopt

Fig. 2: Solving OPF through semidefinite relaxations.

solving OPF-socp in either BIM or BFM though there ispreliminary evidence that BFM can be more stable numerically.For mesh networks we recommend solving OPF-ch for smallnetworks and OPF-socp followed by a heuristic search for afeasible point for large networks.

The key for this solution strategy is that the relaxations areexact so that an optimal solution of the original OPF can berecovered. In Part II of this paper [30] we summarize sufficientconditions that guarantee exact relaxation.

Acknowledgment. We thank the support of NSF throughNetSE CNS 0911041, ARPA-E through GENI DE-AR0000226,the National Science Council of Taiwan through NSC 103-3113-P-008-001, Southern California Edison, the Los AlamosNational Lab and Caltech’s Resnick Institute. We thank theanonymous reviewers for their helpful suggestions.

REFERENCES

[1] S. H. Low. Convex relaxation of optimal power flow: a tutorial. InIREP Symposium – Bulk Power System Dynamics and Control (IREP),Rethymnon, Greece, August 2013.

[2] J. Carpentier. Contribution to the economic dispatch problem. Bulletinde la Societe Francoise des Electriciens, 3(8):431–447, 1962. In French.

[3] H.W. Dommel and W.F. Tinney. Optimal power flow solutions. PowerApparatus and Systems, IEEE Transactions on, PAS-87(10):1866–1876,Oct. 1968.

[4] J. A. Momoh. Electric Power System Applications of Optimization. PowerEngineering. Markel Dekker Inc.: New York, USA, 2001.

[5] M. Huneault and F. D. Galiana. A survey of the optimal power flowliterature. IEEE Trans. on Power Systems, 6(2):762–770, 1991.

[6] J. A. Momoh, M. E. El-Hawary, and R. Adapa. A review of selectedoptimal power flow literature to 1993. Part I: Nonlinear and quadraticprogramming approaches. IEEE Trans. on Power Systems, 14(1):96–104,1999.

[7] J. A. Momoh, M. E. El-Hawary, and R. Adapa. A review of selected op-timal power flow literature to 1993. Part II: Newton, linear programmingand interior point methods. IEEE Trans. on Power Systems, 14(1):105 –111, 1999.

[8] K. S. Pandya and S. K. Joshi. A survey of optimal power flow methods. J.of Theoretical and Applied Information Technology, 4(5):450–458, 2008.

[9] Stephen Frank, Ingrida Steponavice, and Steffen Rebennack. Optimalpower flow: a bibliographic survey, I: formulations and deterministicmethods. Energy Systems, 3:221–258, September 2012.

[10] Stephen Frank, Ingrida Steponavice, and Steffen Rebennack. Optimalpower flow: a bibliographic survey, II: nondeterministic and hybridmethods. Energy Systems, 3:259–289, September 2013.

[11] Mary B. Cain, Richard P. O’Neill, and Anya Castillo. History of optimalpower flow and formulations (OPF Paper 1). Technical report, US FERC,December 2012.

[12] Richard P. O’Neill, Anya Castillo, and Mary B. Cain. The IV formulationand linear approximations of the AC optimal power flow problem (OPFPaper 2). Technical report, US FERC, December 2012.

[13] Richard P. O’Neill, Anya Castillo, and Mary B. Cain. The computationaltesting of AC optimal power flow using the current voltage formulations(OPF Paper 3). Technical report, US FERC, December 2012.

[14] Anya Castillo and Richard P. O’Neill. Survey of approaches to solvingthe ACOPF (OPF Paper 4). Technical report, US FERC, March 2013.

[15] Anya Castillo and Richard P. O’Neill. Computational performance ofsolution techniques applied to the ACOPF (OPF Paper 5). Technicalreport, US FERC, March 2013.

[16] B Stott and O. Alsac. Fast decoupled load flow. IEEE Trans. on PowerApparatus and Systems, PAS-93(3):859–869, 1974.

[17] O. Alsac, J Bright, M Prais, and B Stott. Further developments in LP-based optimal power flow. IEEE Trans. on Power Systems, 5(3):697–711,1990.

[18] Thomas J. Overbye, Xu Cheng, and Yan Sun. A comparison of the ACand DC power flow models for LMP calculations. In Proceedings of the37th Hawaii International Conference on System Sciences, 2004.

[19] K. Purchala, L. Meeus, D. Van Dommelen, and R. Belmans. Usefulnessof DC power flow for active power flow analysis. In Proc. of IEEE PESGeneral Meeting, pages 2457–2462. IEEE, 2005.

[20] B. Stott, J. Jardim, and O. Alsac. DC Power Flow Revisited. IEEE Trans.on Power Systems, 24(3):1290–1300, Aug 2009.

[21] Carleton Coffrin and Pascal Van Hentenryck. A linear-programmingapproximation of AC power flows. CoRR, abs/1206.3614, 2012.

[22] R.A. Jabr. Radial Distribution Load Flow Using Conic Programming.IEEE Trans. on Power Systems, 21(3):1458–1459, Aug 2006.

[23] X. Bai, H. Wei, K. Fujisawa, and Y. Wang. Semidefinite programmingfor optimal power flow problems. Int’l J. of Electrical Power & EnergySystems, 30(6-7):383–392, 2008.

[24] J. Lavaei and S. H. Low. Zero duality gap in optimal power flow problem.IEEE Trans. on Power Systems, 27(1):92–107, February 2012.

[25] X. Bai and H. Wei. A semidefinite programming method with graphpartitioning technique for optimal power flow problems. Int’l J. ofElectrical Power & Energy Systems, 33(7):1309–1314, 2011.

[26] R. A. Jabr. Exploiting sparsity in sdp relaxations of the opf problem.Power Systems, IEEE Transactions on, 27(2):1138–1139, 2012.

[27] D. Molzahn, J. Holzer, B. Lesieutre, and C. DeMarco. Implementation ofa large-scale optimal power flow solver based on semidefinite program-ming. IEEE Transactions on Power Systems, 28(4):3987–3998, November2013.


[28] Subhonmesh Bose, Steven H. Low, Mani Chandy, Thanchanok Teer-aratkul, and Babak Hassibi. Equivalent relaxations of optimal powerflow. Submitted for publication, 2013.

[29] Z. Luo, W. Ma, A.M.C. So, Y Ye, and S. Zhang. Semidefinite relaxationof quadratic optimization problems. Signal Processing Magazine, IEEE,27(3):20 –34, May 2010.

[30] S. H. Low. Convex relaxation of optimal power flow, II: exactness. IEEETrans. on Control of Network Systems, 2014. to appear.

[31] P. P. Varaiya, F. F. Wu, and J. W. Bialek. Smart operation of smart grid:Risk-limiting dispatch. Proceedings of the IEEE, 99(1):40–57, January2011.

[32] M. Vrakopoulou, M. Katsampani, K. Margellos, J. Lygeros, and G. An-dersson. Probabilistic security-constrained ac optimal power flow. InPowerTech (POWERTECH), 2013 IEEE Grenoble, pages 1–6, June 2013.

[33] Russell Bent, Daniel Bienstock, and Michael Chertkov. Synchronization-aware and algorithm-efficient chance constrained optimal power flow. InIREP Symposium – Bulk Power System Dynamics and Control (IREP),Rethymnon, Greece, August 2013.

[34] B. H. Kim and R. Baldick. Coarse-grained distributed optimal powerflow. IEEE Trans. Power Syst., 12(2):932–939, May 1997.

[35] R. Baldick, B. H. Kim, C. Chase, and Y. Luo. A fast distributedimplementation of optimal power flow. IEEE Trans. Power Syst.,14(3):858–864, August 1999.

[36] G. Hug-Glanzmann and G. Andersson. Decentralized optimal power flowcontrol for overlapping areas in power systems. IEEE Trans. Power Syst.,24(1):327?336, February 2009.

[37] F. J. Nogales, F. J. Prieto, and A. J. Conejo. A decomposition method-ology applied to the multi-area optimal power flow problem. Ann. Oper.Res., (120):99–116, 2003.

[38] Albert Lam, Baosen Zhang, and David N Tse. Distributed algorithms foroptimal power flow problem. In IEEE CDC, pages 430–437, 2012.

[39] M. Kraning, E. Chu, J. Lavaei, and S. Boyd. Dynamic network energymanagement via proximal message passing. Foundations and Trends inOptimization, 1(2):70–122, 2013.

[40] K. Turitsyn, P. Sulc, S. Backhaus, and M. Chertkov. Options for controlof reactive power by distributed photovoltaic generators. Proc. of theIEEE, 99(6):1063 –1073, June 2011.

[41] Albert Y.S. Lam, Baosen Zhang, Alejandro Domınguez-Garcıa, andDavid Tse. Optimal distributed voltage regulation in power distributionnetworks. arXiv, April 2012.

[42] Dzung T. Phan. Lagrangian duality and branch-and-bound algorithms foroptimal power flow. Operations Research, 60(2):275–285, March/April2012.

[43] A. Gopalakrishnan, A.U. Raghunathan, D. Nikovski, and L.T. Biegler.Global optimization of optimal power flow using a branch amp; boundalgorithm. In Proc. of the 50th Annual Allerton Conference on Commu-nication, Control, and Computing (Allerton), pages 609–616, Oct 2012.

[44] C. Josz, J. Maeght, P. Panciatici, and J. Ch. Gilbert. Application of themoment-SOS approach to global optimization of the OPF problem. arXiv,November 2013.

[45] M. E. Baran and F. F Wu. Optimal Capacitor Placement on radialdistribution systems. IEEE Trans. Power Delivery, 4(1):725–734, 1989.

[46] M. E Baran and F. F Wu. Optimal Sizing of Capacitors Placed on ARadial Distribution System. IEEE Trans. Power Delivery, 4(1):735–743,1989.

[47] H-D. Chiang and M. E. Baran. On the existence and uniqueness of loadflow solution for radial distribution power networks. IEEE Trans. Circuitsand Systems, 37(3):410–416, March 1990.

[48] Hsiao-Dong Chiang. A decoupled load flow method for distribution powernetworks: algorithms, analysis and convergence study. InternationalJournal Electrical Power Energy Systems, 13(3):130–138, June 1991.

[49] R. Cespedes. New method for the analysis of distribution networks. IEEETrans. Power Del., 5(1):391–396, January 1990.

[50] A. G. Exposito and E. R. Ramos. Reliable load flow technique forradial distribution networks. IEEE Trans. Power Syst., 14(13):1063–1069,August 1999.

[51] Masoud Farivar, Christopher R. Clarke, Steven H. Low, and K. ManiChandy. Inverter VAR control for distribution systems with renewables.In Proceedings of IEEE SmartGridComm Conference, October 2011.

[52] Joshua A. Taylor and Franz S. Hover. Convex models of distributionsystem reconfiguration. IEEE Trans. Power Systems, 2012.

[53] J. Lavaei, D. Tse, and B. Zhang. Geometry of power flows andoptimization in distribution networks. Power Systems, IEEE Transactionson, PP(99):1–12, 2013.

[54] W. H. Kersting. Distribution systems modeling and analysis. CRC, 2002.

[55] D. Shirmohammadi, H. W. Hong, A. Semlyen, and G. X. Luo. Acompensation-based power flow method for weakly meshed distributionand transmission networks. IEEE Transactions on Power Systems,3(2):753–762, May 1988.

[56] Lingwen Gan, Na Li, Ufuk Topcu, and Steven H. Low. Exact convexrelaxation of optimal power flow in radial networks. IEEE Trans.Automatic Control, 2014.

[57] Masoud Farivar and Steven H. Low. Branch flow model: relaxations andconvexification (parts I, II). IEEE Trans. on Power Systems, 28(3):2554–2572, August 2013.

[58] Subhonmesh Bose, Steven H. Low, and Mani Chandy. Equivalence ofbranch flow and bus injection models. In 50th Annual Allerton Conferenceon Communication, Control, and Computing, October 2012.

[59] Deqiang Gan, Robert J. Thomas, and Ray D. Zimmerman. Stability-constrained optimal power flow. IEEE Trans. Power Systems, 15(2):535–540, May 2000.

[60] F. Capitanescua, J.L. Martinez Ramosb, P. Panciaticic, D. Kirschend,A. Marano Marcolinie, L. Platbroodf, and L. Wehenkelg. State-of-the-art,challenges, and future trends in security constrained optimal power flow.Electric Power Systems Research, 81(8):1731–1741, August 2011.

[61] Brian Stott and Ongun Alsac. Optimal power flow: Basic requirementsfor real-life problems and their solutions. White paper, July 2012.

[62] S. Bose, D. Gayme, K. M. Chandy, and S. H. Low. Quadraticallyconstrained quadratic programs on acyclic graphs with application topower flow. arXiv:1203.5599v1, March 2012.

[63] R. Grone, C. R. Johnson, E. M. Sa, and H. Wolkowicz. Positivedefinite completions of partial Hermitian matrices. Linear Algebra andits Applications, 58:109–124, 1984.

[64] Mituhiro Fukuda, Masakazu Kojima, Kazuo Murota, and KazuhideNakata. Exploiting sparsity in semidefinite programming via matrixcompletion I: General framework. SIAM Journal on Optimization,11:647–674, 2001.

[65] 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,95(2):303–327, 2003.

[66] Baosen Zhang and David Tse. Geometry of the injection region of powernetworks. IEEE Trans. Power Systems, 28(2):788–797, 2013.

[67] Martin S. Andersen, Anders Hansson, and Lieven Vandenberghe.Reduced-complexity semidefinite relaxations of optimal power flow prob-lems. arXiv:1308.6718v1, August 2013.

[68] R.D. Zimmerman and Hsiao-Dong Chiang. Fast decoupled powerflow for unbalanced radial distribution systems. Power Systems, IEEETransactions on, 10(4):2045–2052, 1995.

[69] M.S. Srinivas. Distribution load flows: a brief review. In PowerEngineering Society Winter Meeting, 2000. IEEE, volume 2, pages 942–945 vol.2, 2000.

[70] M.E. Baran and F.F. Wu. Network reconfiguration in distribution systemsfor loss reduction and load balancing. IEEE Trans. on Power Delivery,4(2):1401–1407, Apr 1989.

Steven H. Low (F’08) is a Professor of the Comput-ing+Mathematical Sciences and Electrical Engineer-ing at Caltech, and a Changjiang Chair Professor ofZhejiang University. Before that, he was with AT&TBell Labs, Murray Hill, NJ, and the University ofMelbourne, Australia. He was a co-recipient of IEEEbest paper awards, the R&D 100 Award, and anOkawa Foundation Research Grant. He is on the Tech-nical Advisory Board of Southern California Edisonand was a member of the Networking and InformationTechnology Technical Advisory Group for the US

President’s Council of Advisors on Science and Technology (PCAST). He isa Senior Editor of the IEEE Journal on Selected Areas in Communications,the IEEE Trans. Control of Network Systems, and the IEEE Trans. NetworkScience & Engineering, and is on the editorial board of NOW Foundations andTrends in Networking, and in Electric Energy Systems. He is an IEEE Fellowand received his BS from Cornell and PhD from Berkeley both in EE. Hisresearch interests are in power systems and communication networks.

IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE 2014 1

Convex Relaxation of Optimal Power FlowPart II: Exactness

Steven H. Low EAS, Caltech [email protected]

Abstract—This tutorial summarizes recent advances in theconvex relaxation of the optimal power flow (OPF) problem,focusing on structural properties rather than algorithms. PartI presents two power flow models, formulates OPF and theirrelaxations in each model, and proves equivalence relationsamong them. Part II presents sufficient conditions under whichthe convex relaxations are exact.

I. INTRODUCTION

The optimal power flow (OPF) problem is fundamentalin power systems as it underlies many applications suchas economic dispatch, unit commitment, state estimation,stability and reliability assessment, volt/var control, demandresponse, etc. OPF seeks to optimize a certain objectivefunction, such as power loss, generation cost and/or userutilities, subject to Kirchhoff’s laws as well as capacity,stability and security constraints on the voltages and powerflows. There has been a great deal of research on OPF sinceCarpentier’s first formulation in 1962 [1]. Recent surveys canbe found in, e.g., [2], [3], [4], [5], [6], [7], [8], [9], [10], [11],[12], [13].

OPF is generally nonconvex and NP-hard, and a largenumber of optimization algorithms and relaxations have beenproposed. To the best of our knowledge solving OPF throughsemidefinite relaxation is first proposed in [14] as a second-order cone program (SOCP) for radial (tree) networks and in[15] as a semidefinite program (SDP) for general networksin a bus injection model. It is first proposed in [16], [17]as an SOCP for radial networks in the branch flow modelof [18], [19]. While these convex relaxations have beenillustrated numerically in [14] and [15], whether or when theywill turn out to be exact is first studied in [20]. Exploitinggraph sparsity to simplify the SDP relaxation of OPF is firstproposed in [21], [22] and analyzed in [23], [24].

Solving OPF through convex relaxation offers severaladvantages, as discussed in Part I of this tutorial [25, SectionI]. In particular it provides the ability to check if a solutionis globally optimal. If it is not, the solution provides alower bound on the minimum cost and hence a bound on

Acknowledgment: We thank the support of NSF through NetSECNS 0911041, ARPA-E through GENI DE-AR0000226, Southern CaliforniaEdison, the National Science Council of Taiwan through NSC 103-3113-P-008-001, the Los Alamos National Lab (DoE), and Caltech’s ResnickInstitute. A preliminary and abridged version has appeared in Proceedingsof the IREP Symposium - Bulk Power System Dynamics and Control - IX,Rethymnon, Greece, August 25-30, 2013.

how far any feasible solution is from optimality. Unlikeapproximations, if a relaxed problem is infeasible, it is acertificate that the original OPF is infeasible.

This tutorial presents main results on convex relaxationsof OPF developed in the last few years. In Part I [25], wepresent the bus injection model (BIM) and the branch flowmodel (BFM), formulate OPF within each model, and provetheir equivalence. The complexity of OPF formulated herelies in the quadratic nature of power flows, i.e., the nonconvexquadratic constraints on the feasible set of OPF. We character-ize these feasible sets and design convex supersets that leadto three different convex relaxations based on semidefiniteprogramming (SDP), chordal extension, and second-ordercone programming (SOCP). When a convex relaxation isexact, an optimal solution of the original nonconvex OPF canbe recovered from every optimal solution of the relaxation.In Part II we summarize main sufficient conditions thatguarantee the exactness of these relaxations.

Network topology turns out to play a critical role indetermining whether a relaxation is exact. In Section II wereview the definitions of OPF and their convex relaxationsdeveloped in [25]. We also define the notion of exactnessadopted in this paper. In Section III we present three typesof sufficient conditions for these relaxations to be exact forradial networks. These conditions are generally not necessaryand they have implications on allowable power injections,voltage magnitudes, or voltage angles:

A Power injections: These conditions require that notboth constraints on real and reactive power injectionsbe binding at both ends of a line.

B Voltages magnitudes: These conditions require that theupper bounds on voltage magnitudes not be binding.They can be enforced through affine constraints onpower injections.

C Voltage angles: These conditions require that the volt-age angles across each line be sufficiently close. Thisis needed also for stability reasons.

These conditions and their references are summarized inTables I and II. Some of these sufficient conditions are provedusing BIM and others using BFM. Since these two models areequivalent (in the sense that there is a linear bijection betweentheir solution sets [24], [25]), these sufficient conditionsapply to both models. The proofs of these conditions typicallydo not require that the cost function be convex (they focus


type condition model reference remarkA power injections BIM, BFM [26], [27], [28], [29], [30]

[31], [16], [17]B voltage magnitudes BFM [32], [33], [34], [35] allows general injection regionC voltage angles BIM [36], [37] makes use of branch power flows

TABLE I: Sufficient conditions for radial (tree) networks.

network condition reference remarkwith phase shifters type A, B, C [17, Part II], [38] equivalent to radial networks

direct current type A [17, Part I], [20], [39] assumes nonnegative voltagestype B [40], [41] assumes nonnegative voltages

TABLE II: Sufficient conditions for mesh networks

on the feasible sets and usually only need the cost functionto be monotonic). Convexity is required however for efficientcomputation. Moreover it is proved in [35] using BFM thatwhen the cost function is convex then exactness of the SOCPrelaxation implies uniqueness of the optimal solution forradial networks. Hence the equivalence of BIM and BFMimplies that any of the three types of sufficient conditionsguarantees that, for a radial network with a convex costfunction, there is a unique optimal solution and it can becomputed by solving an SOCP. Since the SDP and chordalrelaxations are equivalent to the SOCP relaxation for radialnetworks [24], [25], these results apply to all three typesof relaxations. Empirical evidences suggest some of theseconditions are likely satisfied in practice. This is importantas most power distribution systems are radial.

These conditions are insufficient for general mesh net-works because they cannot guarantee that an optimal solutionof a relaxation satisfies the cycle condition discussed in [25].In Section IV we show that these conditions are howeversufficient for mesh networks that have tunable phase shiftersat strategic locations. The phase shifters effectively make amesh network behave like a radial network as far as convexrelaxation is concerned. The result can help determine if anetwork with a given set of phase shifters can be convexi-fied and, if not, where additional phase shifters are neededfor convexification. These conditions are also sufficient fordirect current (dc) mesh networks where all variables arein the real rather than complex domain. Counterexamplesare known where SDP relaxation is not exact, especially forAC mesh networks without tunable phase shifters [42], [43].We discuss three recent approaches for global optimizationof OPF when the semidefinite relaxations discussed in thistutorial fail.

We conclude in Section V. All proofs are omitted and canbe found in the original papers or the arXiv version of thistutorial.

II. OPF AND ITS RELAXATIONS

We use the notations and definitions from Part I of thispaper. In this section we summarize the OPF problems andtheir relaxations developed there; see [25] for details.

We adopt in this paper a strong sense of “exactness” wherewe require the optimal solution set of the OPF problemand that of its relaxation be equivalent. This implies thatan optimal solution of the nonconvex OPF problem canbe recovered from every optimal solution of its relaxation.This is important because it ensures any algorithm thatsolves an exact relaxation always produces a globally optimalsolution to the OPF problem. Indeed interior point methodsfor solving SDPs tend to produce a solution matrix with amaximum rank [44], so can miss a rank-1 solution if therelaxation has non-rank-1 solutions as well. It can be difficultto recover an optimal solution of OPF from such a non-rank-1 solution, and our definition of exactness avoids thiscomplication. See Section II-C for detailed justifications.

A. Bus injection modelThe BIM adopts an undirected graph G 1 and can be

formulated in terms of just the complex voltage vectorV 2 Cn+1. The feasible set is described by the followingconstraints:

s j Âk:( j,k)2E

yHjk Vj(V H

j �V Hk ) s j, j 2 N+ (1a)

v j |Vj|2 v j, j 2 N+ (1b)

where s j,s j,v j,v j, possibly ±•± i•, are given bounds onpower injections and voltage magnitudes. Note that the vectorV includes V0 which is assumed given (v0 = v0 and \V0 = 0�)unless otherwise specified. The problem of interest is:OPF:

minV2Cn+1

C(V ) subject to V satisfies (1) (2)

1We will use “bus” and “node” interchangeably and “line” and “link”interchangeably.


For relaxations consider the partial matrix WG defined onthe network graph G that satisfies

s j Âk:( j,k)2E


� s j, j 2 N+ (3a)

v j [WG] j j v j, j 2 N+ (3b)

We say that WG satisfies the cycle condition if for everycycle c in G

Â( j,k)2c

\[WG] jk = 0 mod 2p (4)

We assume the cost function C depends on V only throughVV H and use the same symbol C to denote the cost in termsof a full or partial matrix. Moreover we assume C dependson the matrix only through the submatrix WG defined onthe network graph G. See [25, Section IV] for more detailsincluding the definitions of Wc(G) ⌫ 0 and WG( j,k) ⌫ 0.Define the convex relaxations:OPF-sdp:

minW2Sn+1

C(WG) subject to WG satisfies (3),W ⌫ 0 (5)

OPF-ch:

minWc(G)

C(WG) subject to WG satisfies (3),Wc(G) ⌫ 0 (6)

OPF-socp:

minWG

C(WG) subject to WG satisfies (3),

WG( j,k)⌫ 0, ( j,k) 2 E (7)

For BIM, we say that OPF-sdp (5) is exact if every optimalsolution W sdp of OPF-sdp is psd rank-1; OPF-ch (6) is exactif every optimal solution W ch

c(G) of OPF-ch is psd rank-1 (i.e.,the principal submatrices W ch

c(G)(q) of W chc(G) are psd rank-1

for all maximal cliques q of the chordal extension c(G) ofgraph G); OPF-socp (7) is exact if every optimal solutionW socp

G of OPF-socp is 2⇥2 psd rank-1 and satisfies the cyclecondition (4). To recover an optimal solution V opt of OPF(2) from W sdp or W ch

c(G) or W socpG , see [25, Section IV-D].

B. Branch flow modelThe BFM adopts a directed graph G and is defined by the

following set of equations:

Âk: j!k

S jk = Âi:i! j


�+ s j, j 2 N+ (8a)

I jk = y jk(Vj �Vk), j ! k 2 E (8b)S jk = Vj IH

jk, j ! k 2 E (8c)

Denote the variables in BFM (8) by x := (S, I,V,s) 2C2(m+n+1). Note that the vectors V and s include V0 (given)and s0 respectively. Recall from [25] the variables x :=(S,`,v,s) 2 R3(m+n+1) that is related to x by the mappingx = h(x) with ` jk := |I jk|2 and v j := |Vj|2. The operational

constraints are:

v j v j v j, j 2 N+ (9a)s j s j s j, j 2 N+ (9b)

We assume the cost function depends on x only throughx = h(x). Then the problem in BFM is:OPF:

minx

C(x) subject to x satisfies (8), (9) (10)

For SOCP relaxation consider:

Âk: j!k

S jk = Âi:i! j

(Si j � zi j`i j)+ s j, j 2 N+ (11a)


jkS jk�� |z jk|2` jk, j ! k 2 E (11b)

v j` jk � |S jk|2, j ! k 2 E (11c)

We say that x satisfies the cycle condition if

9q 2 Rn such that Bq = b (x) mod 2p (12)

where B is the m⇥ n reduced incidence matrix and, givenx := (S,`,v,s), b jk(x) := \(v j � zH

jkS jk) can be interpreted asthe voltage angle difference across line j ! k implied by x(See [25, Section V]). The SOCP relaxation in BFM isOPF-socp:

minx

C(x) subject to x satisfies (11), (9) (13)

For BFM, OPF-socp (13) in BFM is exact if every optimalsolution xsocp attains equality in (11c) and satisfies the cyclecondition (12). See [25, Section V-A] for how to recover anoptimal solution xopt of OPF (10) from any optimal solutionxsocp of its SOCP relaxation.

C. ExactnessThe definition of exactness adopted in this paper is more

stringent than needed. Consider SOCP relaxation in BIMas an illustration (the same applies to the other relaxationsin BIM and BFM). For any sets A and B, we say thatA is equivalent to B, denoted by A ⌘ B, if there is abijection between these two sets. Let M(A) denote the setof minimizers when a certain function is minimized over A.

Let V and W+G denote the feasible sets of OPF (2) and

OPF-socp (7) respectively:

V := {V 2 Cn+1 | V satisfies (1)}W+

G := {WG | WG satisfies (3),WG( j,k)⌫ 0,( j,k) 2 E}

Consider the following subset of W+G :

WG := {WG | WG satisfies (3), (4),WG( j,k)⌫ 0,rank WG( j,k) = 1,( j,k) 2 E}

Our definition of exact SOCP relaxation is that M(W+G) ✓

WG. In particular, all optimal solutions of OPF-socp mustbe 2⇥2 psd rank-1 and satisfy the cycle condition (4). Since


WG ⌘V (see [25]), exactness requires that the set of optimalsolutions of OPF-socp (7) be equivalent to that of OPF (2),i.e., M(W+

G) =M(WG)⌘M(V).If M(W+

G) ) M(WG) ⌘ M(V) then OPF-socp (7) is notexact according to our definition. Even in this case, however,every sufficient condition in this paper guarantees that anoptimal solution of OPF can be easily recovered from anoptimal solution of the relaxation that is outside WG. The dif-ference between M(W+

G) =M(WG) and M(W+G))M(WG)

is often minor, depending on the objective function; seeRemarks 1 and 2 and comments after Theorems 5 and 8in Section III. Hence we adopt the more stringent definitionof exactness for simplicity.

III. RADIAL NETWORKS

In this section we summarize the three types of sufficientconditions listed in Table I for semidefinite relaxations ofOPF to be exact for radial (tree) networks. These results areimportant as most distribution systems are radial.

For radial networks, if SOCP relaxation is exact then SDPand chordal relaxations are also exact (see [25, Theorems5, 9]). We hence focus in this section on the exactness ofOPF-socp in both BIM and BFM. Since the cycle conditions(4) and (12) are vacuous for radial networks, OPF-socp (7)is exact if all of its optimal solutions are 2⇥ 2 rank-1 andOPF-socp (13) is exact if all of its optimal solutions attainequalities in (11c). We will freely use either BIM or BFMin discussing these results. To avoid triviality we make thefollowing assumption throughout the paper:

The voltage lower bounds satisfy v j > 0, j 2 N+. Theoriginal problems OPF (2) and (10) are feasible.

A. Linear separabilityWe will first present a general result on the exactness of

the SOCP relaxation of general QCQP and then apply it toOPF. This result is first formulated and proved using a dualityargument in [27], generalizing the result of [26]. It is provedusing a simpler argument in [31].

Fix an undirected graph G = (N+,E) where N+ :={0,1, . . . ,n} and E ✓ N+⇥N+. Fix Hermitian matrices Cl 2Sn+1, l = 0, . . . ,L, defined on G, i.e., [Cl ] jk = 0 if ( j,k) 62 E.Consider QCQP:

minx2Cn+1

xHC0x (14a)

subject to xHClx bl , l = 1, . . . ,L (14b)

where C0,Cl 2 C(n+1)⇥(n+1), bl 2 R, l = 1, . . . ,L, and itsSOCP relaxation where the optimization variable ranges overHermitian partial matrices WG:

minWG

tr C0WG (15a)

subject to tr ClWG bl , l = 1, . . . ,L (15b)WG( j,k)⌫ 0, ( j,k) 2 E (15c)

The following result is proved in [27], [31]. It can beregarded as an extension of [45] on the SOCP relaxationof QCQP from the real domain to the complex domain.Consider: 2

A1: The cost matrix C0 is positive definite.A2: For each link ( j,k) 2 E there exists an a jk such that

\ [Cl ] jk 2 [ai j,ai j +p] for all l = 0, . . . ,L.Let Copt and Csocp denote the optimal values of QCQP (14)and SOCP (15) respectively.

Theorem 1: Suppose G is a tree and A2 holds. ThenCopt = Csocp and an optimal solution of QCQP (14) can berecovered from every optimal solution of SOCP (15).

Remark 1: The proof of Theorem 1 prescribes a simpleprocedure to recover an optimal solution of QCQP (14)from any optimal solution of its SOCP relaxation (15). Theconstruction does not need the optimal solution of SOCP(15) to be 2⇥2 rank-1. Hence the SOCP relaxation may notbe exact according to our definition of exactness, i.e., someoptimal solutions of (15) may be 2⇥ 2 psd but not 2⇥ 2rank-1. If the objective function is strictly convex howeverthen the optimal solution sets of QCQP (14) and SOCP (15)are indeed equivalent.

Corollary 2: Suppose G is a tree and A1–A2 hold. ThenSOCP (15) is exact.

We now apply Theorem 1 to our OPF problem. Recall thatOPF (2) in BIM can be written as a standard form QCQP[27]:

minx2Cn

V HC0V

s.t. V HF jV p j, V H(�F j)V �p j (16a)

V HY jV q j, V H(�Y j)V �q j (16b)

V HJjV �v j, V H(�Jj)V �v j

for some Hermitian matrices C0,F j,Y j,Jj where j 2 N+.A2 depends only on the off-diagonal entries of C0, F j, Y j(Jj are diagonal matrices). It implies a simple pattern on thepower injection constraints (16a)–(16b). Let y jk = g jk � ib jkwith g jk > 0,b jk > 0. Then we have (from [27]):

[Fk]i j =

8><

>:

12Yi j = � 1

2 (gi j � ibi j) if k = i12Y H

i j = � 12 (gi j + ibi j) if k = j

0 if k 62 {i, j}

[Yk]i j =

8><

>:

�12i Yi j = � 1

2 (bi j + igi j) if k = i12iY

Hi j = � 1

2 (bi j � igi j) if k = j0 if k 62 {i, j}

Hence for each line ( j,k) 2 E the relevant angles for A2 are

2All angles should be interpreted as “mod 2p”, i.e., projected onto(�p,p].


those of [C0] jk and

[F j] jk = �12(g jk � ib jk)

[Fk] jk = �12(g jk + ib jk)

[Y j] jk = �12(b jk + ig jk)

[Yk] jk = �12(b jk � ig jk)

as well as the angles of �[F j] jk,�[Fk] jk and�[Y j] jk,�[Yk] jk. These quantities are shown in Figure1 with their magnitudes normalized to a common value andexplained in the caption of the figure.

Φ j"# $% jk

Re

Im

− Φ j#$ %& jk

Φk[ ] jk

− Φk[ ] jk

Ψ j"# $% jk − Ψ k[ ] jk

Ψ k[ ] jk − Ψ j#$ %& jk

lower)bounds)on))pj,qj, pk,qk

α jk

[C0 ] jk

upper)bounds)on))pj,qj, pk,qk

Fig. 1: Condition A2’ on a line ( j,k) 2 E. The quantities([F j] jk, [Fk] jk, [Y j] jk, [Yk] jk) on the left-half plane corre-spond to finite upper bounds on (p j, pk,q j,qk) in (16a)–(16b); (�[F j] jk,�[Fk] jk,�[Y j] jk,�[Yk] jk) on the right-halfplane correspond to finite lower bounds on (p j, pk,q j,qk).A2’ is satisfied if there is a line through the origin, specifiedby the angle a jk, so that the quantities corresponding tofinite upper or lower bounds on (p j, pk,q j,qk) lie on oneside of the line, possibly on the line itself. The load over-satisfaction condition in [26], [30] corresponds to the Im-axis that excludes all quantities on the right-half plane. Thesufficient condition in [29, Theorem 2] corresponds to thered line in the figure that allows a finite lower bound on thereal power at one end of the line, i.e., p j or pk but not both,and no finite lower bounds on reactive powers q j and qk.

Condition A2 applied to OPF (16) takes the following form(see Figure 1):

A2’: For each link ( j,k) 2 E there is a line in the complexplane through the origin such that [C0] jk as well asthose ±[Fi] jk and ±[Yi] jk corresponding to finite loweror upper bounds on (pi,qi), for i = j,k, are all on oneside of the line, possibly on the line itself.

Let Copt and Csocp denote the optimal values of OPF (2) andOPF-socp (7) respectively.

Corollary 3: Suppose G is a tree and A2’ holds.1) Copt =Csocp. Moreover an optimal solution V opt of OPF

(2) can be recovered from every optimal solution W socpG

of OPF-socp (7).2) If, in addition, A1 holds then OPF-socp (7) is exact.

It is clear from Figure 1 that condition A2’ cannot be satis-fied if there is a line where both the real and reactive powerinjections at both ends are both lower and upper bounded(8 combinations as shown in the figure). A2’ requires thatsome of them be unconstrained even though in practice theyare always bounded. It should be interpreted as requiringthat the optimal solutions obtained by ignoring these boundsturn out to satisfy these bounds. This is generally differentfrom solving the optimization with these constraints butrequiring that they be inactive (strictly within these bounds)at optimality, unless the cost function is strictly convex. Theresult proved in [27] also includes constraints on real branchpower flows and line losses. Corollary 3 includes severalsufficient conditions in the literature for exact relaxation asspecial cases; see the caption of Figure 1.

Corollary 3 also implies a result first proved in [16], usinga different technique, that SOCP relaxation is exact in BFMfor radial networks when there are no lower bounds on powerinjections s j. The argument in [16] is generalized in [17, PartI] to allow convex objective functions, shunt elements, andline limits in terms of upper bounds on ` jk. AssumeA3: The cost function C(x) is convex, strictly increasing

in `, nondecreasing in s = (p,q), and independent ofbranch flows S = (P,Q).

A4: For j 2 N+, s j =�•� i•.Popular cost functions in the literature include active powerloss over the network or active power generations, both ofwhich satisfy A3. The next result is proved in [16], [17].

Theorem 4: Suppose G is a tree and A3–A4 hold. ThenOPF-socp (13) is exact.

Remark 2: If the cost function C(x) in A3 is only nonde-creasing, rather than strictly increasing, in `, then A3–A4still guarantee that all optimal solutions of OPF (10) are(i.e., can be mapped to) optimal solutions of OPF-socp (13),but OPF-socp may have an optimal solution that maintainsstrict inequalities in (11c) and hence is infeasible for OPF.Even though OPF-socp is not exact in this case, the proofof Theorem 4 constructs from it an optimal solution of OPF(See the arXiv version of this paper).

B. Voltage upper bounds

While type A conditions (A2’ and A4 in the last sub-section) require that some power injection constraints not bebinding, type B conditions require non-binding voltage upperbounds. They are proved in [32], [33], [34], [35] using BFM.


For radial networks the model originally proposed in [18],[19], which is (11) with the inequalities in (11c) replaced byequalities, is exact. This is because the cycle condition (12)is always satisfied as the reduced incidence matrix B is n⇥nand invertible for radial networks. Following [35] we adoptthe graph orientation where every link points towards node0. Then (11) for a radial network reduces to:

S jk = Âi:i! j

(Si j � zi j`i j)+ s j, j 2 N+ (17a)


jkS jk�� |z jk|2` jk, j ! k 2 E (17b)

v j` jk � |S jk|2, j ! k 2 E (17c)

where v0 is given and in (17a), k denotes the node on theunique path from node j to node 0. The boundary conditionis: S jk := 0 when j = 0 in (17a) and Si j = 0, `i j = 0 when jis a leaf node.3

As before the voltage magnitudes must satisfy:

v j v j v j, j 2 N (18a)

We allow more general constraints on the power injections:for j 2 N, s j can be in an arbitrary set S j that is boundedabove:

s j 2 S j ✓ {s j 2 C |s j s j}, j 2 N (18b)

for some given s j, j 2 N.4 Then the SOCP relaxation isOPF-socp:

minx

C(x) subject to (17), (18) (19)

As defined in Section II-C, OPF-socp (19) is exact if everyoptimal solution xsocp attains equality in (17c). In that casean optimal solution of BFM (10) can be uniquely recoveredfrom xsocp.

We make two comments on the constraint sets S j in (18b).First S j need not be convex nor even connected for convexrelaxations to be exact. They (only) need to be convex tobe efficiently computable. Second such a general constrainton s is useful in many applications. It includes the casewhere s j are subject to simple box constraints, but alsoallows constraints of the form |s j|2 a, |\s j| f j that isuseful for volt/var control [46], or q j 2 {0,a} for capacitorconfigurations.Geometric insight. To motivate our sufficient condition, wefirst explain a simple geometric intuition using a two-busnetwork on why relaxing voltage upper bounds guaranteesexact SOCP relaxation. Consider bus 0 and bus 1 connectedby a line with impedance z := r+ ix. Suppose without lossof generality that v0 = 1 pu. Eliminating S01 = s0 from (17),the model reduces to (dropping the subscript on `01):

p0 � r` = �p1, q0 � x` = �q1, p20 +q2

0 = ` (20)

3A node j 2 N is a leaf node if there is no i such that i ! j 2 E.4We assume here that s0 is unconstrained, and since V0 := 1\0� pu, the

constraints (18) involve only j in N, not N+.

and

v1 � v0 = 2(rp0 + xq0)� |z|2` (21)

Suppose s1 is given (e.g., a constant power load). Then thevariables are (`,v1, p0,q0) and the feasible set consists ofsolutions of (20) and (21) subject to additional constraints on(`,v1, p0,q0). The case without any constraint is instructive

= p02 + q0

2

p0 − r = −p1q0 − x = −q1

c

p0

q0

high v1

low v1

Fig. 2: Feasible set of OPF for a two-bus network without anyconstraint. It consists of the (two) points of intersection ofthe line with the convex surface (without the interior), andhence is nonconvex. SOCP relaxation includes the interiorof the convex surface and enlarges the feasible set to theline segment joining these two points. If the cost functionC is increasing in ` or (p0,q0) then the optimal point overthe SOCP feasible set (line segment) is the lower feasiblepoint c, and hence the relaxation is exact. No constraint on` or (p0,q0) will destroy exactness as long as the resultingfeasible set contains c.

and shown in Figure 2. The point c in the figure correspondsto a power flow solution with a large v1 (normal operation)whereas the other intersection corresponds to a solution witha small v1 (fault condition). As explained in the caption,SOCP relaxation is exact if there is no voltage constraintand as long as constraints on (`, p0,q0) does not remove thehigh-voltage (normal) power flow solution c. Only when thesystem is stressed to a point where the high-voltage solutionbecomes infeasible will relaxation lose exactness. This agreeswith conventional wisdom that power systems under normaloperations are well behaved.

Consider now the voltage constraint v1 v1 v1. Substi-tuting (20) into (21) we obtain

v1 = (1+ rp1 + xq1)� |z|2`

translating the constraint on v1 into a box constraint on `:1|z|2 (rp1 + xq1 +1� v1) ` 1

|z|2 (rp1 + xq1 +1� v1)

Figure 2 shows that the lower bound v1 (corresponding toan upper bound on `) does not affect the exactness of SOCPrelaxation. The effect of upper bound v1 (corresponding toa lower bound on `) is illustrated in Figure 3. As explainedin the caption of the figure SOCP relaxation is exact if the


upper bound v1 does not exclude the high-voltage power flowsolution c and is not exact otherwise.

p0

q0

c

(a) Voltage constraint not binding

op2mal)solu2on)of)SOCP))(infeasible)for)OPF))

p0

q0

c

(b) Voltage constraint binding

Fig. 3: Impact of voltage upper bound v1 on exactness. (a)When v1 (corresponding to a lower bound on `) is notbinding, the power flow solution c is in the feasible setof SOCP and hence the relaxation is exact. (b) When v1excludes c from the feasible set of SOCP, the optimal solutionis infeasible for OPF and the relaxation is not exact.

To state the sufficient condition for a general radial net-work, recall from [25, Section VI] the linear approximationof BFM for radial networks obtained by setting ` jk = 0 in(17): for each s

Slinjk (s) = Â

i2T j

si (22a)

vlinj (s) = v0 +2 Â

(i,k)2P j

Re⇣

zHikSlin

ik (s)⌘

(22b)

where T j denotes the subtree at node j, including j, andP j denotes the set of links on the unique path from j to 0.The key property we will use is, from [25, Lemma 13 andRemark 9]:

S jk Slinjk (s) and v j vlin

j (s) (23)

Define the 2⇥2 matrix function

A jk(S jk,v j) := I � 2v j

z jk�S jk

�T (24)

where z jk := [r jk x jk]T is the line impedance and S jk :=[Pjk Q jk]T is the branch power flows, both taken as 2-dimensional real vectors so that z jk

�S jk

�T is a 2⇥2 matrixwith rank less or equal to 1. The matrices A jk(S jk,v j)describe how changes in the real and reactive power flowspropagate towards the root node 0; see comments below.Evaluate the Jacobian matrix A jk(S jk,v j) at the boundaryvalues:

A jk := A jk

✓hSlin

jk (s)i+, v j

◆

:= I � 2v j

z jk

✓hSlin

jk (s)i+◆T

(25)

Here�[a]+

�T is the row vector [[a1]+ [a2]+] with [a j]+ :=

max{0,a j}.For a radial network, for j 6= 0, every link j ! k identifies

a unique node k and therefore, to simplify notation, we referto a link interchangeably by ( j,k) or j and use A j, A j, z jetc. in place of A jk, A jk, z jk etc. respectively.

AssumeB1: The cost function is C(x) := Ân

j=0 Cj (Res j) with C0strictly increasing. There is no constraint on s0.

B2: The set S j of injections satisfies vlinj (s) v j, j 2 N,

where vlinj (s) is given by (22).

B3: For each leaf node j 2 N let the unique path fromj to 0 have k links and be denoted by P j :=((ik, ik�1), . . . ,(i1, i0)) with ik = j and i0 = 0. ThenAit · · ·Ait0

zit0+1> 0 for all 1 t t 0 < k.

The following result is proved in [35].Theorem 5: Suppose G is a tree and B1–B3 hold. Then

OPF-socp (19) is exact.

We now comment on the conditions B1–B3. B1 requiresthat the cost functions Cj depend only on the injections s j.For instance, if Cj (Res j) = p j, then the cost is total activepower loss over the network. It also requires that C0 bestrictly increasing but makes no assumption on Cj, j > 0.Common cost functions such as line loss or generation costusually satisfy B1. If C0 is only nondecreasing, rather thanstrictly increasing, in p0 then B1–B3 still guarantee that alloptimal solutions of OPF (10) are (effectively) optimal forOPF-socp (19), but OPF-socp may not be exact, i.e., it mayhave an optimal solution that maintains strict inequalities in(17c). In this case the proof of Theorem 5 can be used torecursively construct from it another optimal solution thatattains equalities in (17c).

B2 is affine in the injections s := (p,q). It enforces theupper bounds on voltage magnitudes because of (23).

B3 is a technical assumption and has a simple interpre-tation: the branch power flow S jk on all branches shouldmove in the same direction. Specifically, given a marginalchange in the complex power on line j ! k, the 2⇥2 matrixA jk is (a lower bound on) the Jacobian and describes theeffect of this marginal change on the complex power on theline immediately upstream from line j ! k. The product ofAi in B3 propagates this effect upstream towards the root.B3 requires that a small change, positive or negative, in thepower flow on a line affects all upstream branch powers inthe same direction. This seems to hold with a significantmargin in practice; see [35] for examples from real systems.

Theorem 5 unifies and generalizes some earlier results in[32], [33], [34]. The sufficient conditions in these papers havethe following simple and practical interpretation: OPF-socpis exact provided either

• there are no reverse power flows in the network, or• if the r/x ratios on all lines are equal, or• if the r/x ratios increase in the downstream direction

from the substation (node 0) to the leaves then there are


no reverse real power flows, or• if the r/x ratios decrease in the downstream direction

then there are no reverse reactive power flows.The exactness of SOCP relaxation does not require con-

vexity, i.e., the cost C(x) = Ânj=0 Cj(Res j) need not be a

convex function and the injection regions S j need not beconvex sets. Convexity allows polynomial-time computation.Moreover when it is convex the exactness of SOCP relaxationalso implies the uniqueness of the optimal solution, as thefollowing result from [35] shows.

Theorem 6: Suppose the costs Cj, j = 0, . . . ,n, are convexfunctions and the injection regions S j, j = 1, . . . ,n, are convexsets. If the relaxation OPF-socp (19) is exact then its optimalsolution is unique.

Consider the model of [18] for radial networks, which is(17) with the inequalities in (17c) replaced by equalities. LetX denote an equivalent feasible set of OPF,5 i.e., those x 2R3(m+n+1) that satisfy (17), (18) and attain equalities in (17c).The proof of Theorem 6 reveals that, for radial networks, thefeasible set X has a “hollow” interior.

Corollary 7: Suppose G is a tree. If x and x are distinctsolutions in X then no convex combination of x and x canbe in X. In particular X is nonconvex.

This property is illustrated vividly in several numericalexamples for mesh networks in [47], [48], [49], [50].

C. Angle differencesThe sufficient conditions in [29], [36], [37] require that

the voltage angle difference across each line be small. Weexplain the intuition using a result in [36] for an OPF problemwhere |Vj| are fixed for all j 2 N+ and reactive powers areignored. Under these assumptions, as long as the voltageangle difference is small, the power flow solutions form alocally convex surface that is the Pareto front of its relaxation.This implies that the relaxation is exact. This geometricpicture is apparent in earlier work on the geometry of powerflow solutions, see e.g. [47], and underlies the intuition thatthe dynamics of a power system is usually benign until itis pushed towards the boundary of its stability region. Thegeometric insight in Figures 2 and 3 for BFM and later in thissubsection for BIM says that, when it is far away from theboundary, the local convexity structure also facilitates exactrelaxation. Reactive power is considered in [37, Theorem 1]with fixed |Vj| where, with an additional constraint on thelower bounds of reactive power injections that ensure theselower bounds are not tight, it is proved that if the originalOPF problem is feasible then its SDP relaxation is exact. Thecase of variable |Vj| without reactive power is considered in[36, Theorem 7] but the simple geometric structure is lost.

5There is a bijection between X and the feasible set of OPF (10) (when(18b) are placed by (9b)) [17], [25].

Recall that y jk = g jk � ib jk with g jk > 0,b jk > 0. Let Vj =|Vj|eiq j and suppose |Vj| are given. Consider:

minp,P,q

C(p) (26a)

s.t. p j p j p j, j 2 N+ (26b)

q jk q jk q jk, ( j,k) 2 E (26c)p j = Â

k:k⇠ jPjk, j 2 N+ (26d)

Pjk = |Vj|2g jk � |Vj||Vk|g jk cosq jk + |Vj||Vk|b jk sinq jk

( j,k) 2 E (26e)

where q jk := q j �qk are the voltage angle differences acrosslines ( j,k).

We comment on the constraints on angles q jk in (26).When the voltage magnitudes |Vi| are fixed, constraints onreal power flows, branch currents, line losses, as well asstability constraints can all be represented in terms of q jk.Indeed a line flow constraint of the form |Pjk| P jk becomesa constraint on q jk using the expression for Pjk in (26e). Acurrent constraint of the form |I jk| I jk is also a constrainton q jk since |I jk|2 = |y jk|(|Vj|2 + |Vk|2 �2|VjVk|cosq jk). Theline loss over ( j,k) 2 E is equal to Pjk + Pk j which isagain a function of q jk. Stability typically requires |q jk| tostay within a small threshold. Therefore given constraintson branch power or current flows, losses, and stability,appropriate bounds q jk,q jk can be determined in terms ofthese constraints, assuming |Vj| are fixed.

We can eliminate the branch flows Pjk and angles q jk from(26). Since |Vj|, j 2 N+, are fixed we assume without loss ofgenerality that |Vj|= 1 pu. Define the injection region

Pq :=

(p 2 Rn | p j = Â

k:k⇠ j

�g jk �g jk cosq jk +b jk sinq jk

�,

q jk q jk q jk, j 2 N+,( j,k) 2 E

)(27)

Let Pp := {p 2 Rn | p j p j p j, j 2 N}. Then (26) is:OPF:

minp

C(p) subject to p 2 Pq \Pp (28)

This problem is hard because the set Pq is nonconvex. Toavoid triviality we assume OPF (28) is feasible. For a set A letconvA denote the convex hull of A. Consider the followingproblem that relaxes the nonconvex feasible set Pq \Pp of(28) to a convex superset:OPF-socp:

minp

C(p) subject to p 2 conv(Pq ) \ Pp (29)

We will show below that (29) is indeed an SOCP. It is saidto be exact if every optimal solution of (29) lies in Pq \Ppand is therefore also optimal for (28).

We say that a point x 2 A ✓ Rn is a Pareto optimal point


in A if there does not exist another x0 2 A such that x0 xwith at least one strictly smaller component x0j < x j. ThePareto front of A, denoted by O(A), is the set of all Paretooptimal points in A. The significance of O(A) is that, for anyincreasing function, its minimizer, if exists, is necessarily inO(A) whether A is convex or not. If A is convex then xopt is aPareto optimal point in O(A) if and only if there is a nonzerovector c := (c1, . . . ,cn) � 0 such that xopt is a minimizer ofcT x over A [51, pp.179–180].

AssumeC1: C(p) is strictly increasing in each p j.C2: For all ( j,k) 2 E, � tan�1 b jk

g jk< q jk q jk < tan�1 b jk

g jk.

The following result, proved in [36], [37], says that (29) isexact provided q jk are suitably bounded.

Theorem 8: Suppose G is a tree and C1–C2 hold.1) Pq \Pp = O(conv(Pq ) \ Pp).2) The problem (29) is indeed an SOCP. Moreover it is

exact.

C1 is needed to ensure every optimal solution of OPF-socp(29) is optimal for OPF (28). If C(p) is nondecreasing but notstrictly increasing in all p j, then Pq \Pp ✓O(conv(Pq )\ Pp)and OPF-socp may not be exact according to our definition.Even in that case it is possible to recover an optimal solutionof OPF from any optimal solution of OPF-socp.

Theorem 8 is illustrated in Figures 4 and 5. As explained

pk

2gjk

2bjk

pj

pj, p

k( )

Fig. 4: Feasible set of OPF (28) for a two-bus networkwithout any constraint when |Vj| are fixed and reactivepowers are ignored. It is an ellipse without the interior,hence nonconvex. OPF-socp (29) includes the interior ofthe ellipse and is hence convex. If the cost function C isstrictly increasing in (p j, pk) then the Pareto front of theSOCP feasible set will lie on the lower part of the ellipse,O(Pq ) = Pq , and hence OPF-socp is exact.

in the caption of Figure 4, if there are no constraints thenSOCP relaxation (29) is exact under condition C1. It is clearfrom the figure that upper bounds on power injections do notaffect exactness whereas lower bounds do. The purpose ofcondition C2 is to restrict the angle q jk in order to eliminatethe upper half of the ellipse from Pq . As explained in thecaption of Figure 5, under C2, Pq \Pp =O(conv(Pq ) \ Pp)and hence the relaxation is exact. Otherwise it may not.

pj, p

k( )

Pareto)front)

(a) Exact relaxation with constraint

pj, p

k( )

Pareto)front)

(b) Inexact relaxation with constraint

Fig. 5: With lower bounds p on power injections, the feasibleset of OPF-socp (29) is the shaded region. (a) When thefeasible set of OPF (28) is restricted to the lower half of theellipse (small |q jk|), the Pareto front remains on the ellipseitself, Pq \Pp =O(conv(Pq ) \ Pp), and hence the relaxationis exact. (b) When the feasible set of OPF includes upper halfof the ellipse (large |q jk|), the Pareto front may not lie onthe ellipse if p is large, making the relaxation not exact.

When the network is not radial or |Vj| are not constants,then the feasible set can be much more complicated thanellipsoids [48], [49], [50]. Even in such settings the Paretofronts might still coincide, though the simple geometricpicture is lost. See [47] for a numerical example on anAustralian system or [24] on a three-bus mesh network.

D. EquivalenceSince BIM and BFM are equivalent, the results on exact

SOCP relaxation and uniqueness of optimal solution applyin both models. Recall the linear bijection g from BIM toBFM defined in [25, end of Section V] by x = g(WG) where

S jk := yHjk�[WG] j j � [WG] jk

�, j ! k

` jk := |y jk|2�[WG] j j +[WG]kk � [WG] jk � [WG]k j

�, j ! k

v j := [WG] j j, j 2 N+

s j := Âk: j⇠k


�, j 2 N+

The mapping g allows us to directly apply Theorem 6 toBIM. We summarize all the results for type A and type Bconditions for radial networks. 6

Theorem 9: Suppose G and G are trees. Suppose condi-tions A1–A2’, or A3–A4, or B1–B3 hold. Then

1) BIM: SOCP relaxation (7) is exact. Moreover if C(WG)is convex in ([WG] j j, [WG] jk) then the optimal solutionis unique.

2) BFM: SOCP relaxation (13) is exact. Moreover ifC(x) := Â j Cj(p j) is convex in p then the optimalsolution is unique.

6To apply type C conditions to BFM, one needs to translate theangles q jk to the BFM variables x := (S,`,v,s) through b jk(x), thoughthis will introduce additional nonconvex constraints into OPF of the formq jk b jk(x) q jk .


Since both the SDP and the chordal relaxations are equivalentto the SOCP relaxation for radial networks, these resultsapply to SDP and chordal relaxations as well.

IV. MESH NETWORKS

In this section we summarize a result of [17, Part II] onmesh networks with phase shifters and of [17, Part I], [39],[41] on dc networks when all voltages are nonnegative.

To be able to recover an optimal solution of OPF from anoptimal solution W socp

G /xsocp of SOCP relaxation, W socpG /xsocp

must satisfy both a local condition and a global cyclecondition ((4) for BIM and (12) for BFM); see the definitionof exactness in Section II. The conditions of Section IIIguarantee that every SOCP optimal solution will satisfy thelocal condition (i.e., W socp

G is 2 ⇥ 2 psd rank-1 and xsocp

attains equalities in (11c)), whether the network is radialor mesh, but do not guarantee that it satisfies the cyclecondition. For radial networks, the cycle condition is vacuousand therefore the conditions of Section III are sufficient forSOCP relaxation to be exact. The result of [17, Part II]implies that these conditions are sufficient also for a meshnetwork that has tunable phase shifters at strategic locations.

Similar conditions also extend to dc networks where allvariables are real and the voltages are assumed nonnegative.

A. AC networks with phase shiftersFor BFM the conditions of Section III guarantee that every

optimal solution of OPF-socp (13) attains equalities in (11c)but may or may not satisfy the cycle condition (12). If itdoes then it can be uniquely mapped to an optimal solutionof OPF (10), according to [17, Theorem 2]. If it does notthen the solution is not physically implementable because itdoes not satisfy the power flow equations (Kirchhoff’s laws).For a radial network the reduced incidence matrix B in (12)is n⇥ n and invertible and hence every optimal solution ofthe SOCP relaxation that attains equalities in (11c) alwayssatisfies the cycle condition [17, Theorem 4]. This is not thecase for a mesh network where B is m⇥n with m > n.

It is proved in [17, Part II] however that if the networkhas tunable phase shifters then any SOCP solution thatattains equalities in (11c) becomes implementable even if thesolution does not satisfy the cycle condition. This extends thesufficient conditions A1–A2’, or A3–A4, or B1–B3, or C0–C1 from radial networks to this type of mesh networks.

For BIM the effect of phase shifter is equivalent tointroducing a free variable fc in (4) for each basis cyclec so that the cycle condition can always be satisfied for anyWG. The results presented here however start with a simplepower flow model (30) for networks with phase shifters. Thismodel makes transparent the effect of the spatial distributionof phase shifters and how they impact the exactness of SOCPrelaxation and can be useful in other contexts, such as thedesign of a network of FACTS (Flexible AC TransmissionSystems) devices.

BFM with phase shifters. We consider an idealized phaseshifter that only shifts the phase angles of the sending-endvoltage and current across a line, and has no impedance norlimits on the shifted angles. Specifically consider an idealizedphase shifter parametrized by f jk across line j ! k as shownin Figure 6. As before let Vj denote the sending-end voltage at

kijφ jk

z jk

Fig. 6: Model of a phase shifter in line j ! k.

node j. Define I jk to be the sending-end current leaving nodej towards node k. Let i be the point between the phase shifterf jk and line impedance z jk. Let Vi and Ii be the voltage at iand the current from i to k respectively. Then the effect of anidealized phase shifter, parametrized by f jk, is summarizedby the following modeling assumptions:

Vi = Vj eif jk and Ii = I jk eif jk

The power transferred from nodes j to k is still (defined to be)S jk :=VjIH

jk, which is equal to the power ViIHi from nodes i to

k since the phase shifter is assumed to be lossless. ApplyingOhm’s law across z jk, we define the branch flow model withphase shifters as the following set of equations:

Âk: j!k

S jk = Âi:i! j


�+ s j, j 2 N+ (30a)

I jk = y jk

⇣Vj �Vk e�if jk

⌘, j ! k 2 E (30b)

S jk = VjIHjk, j ! k 2 E (30c)

Without phase shifters (f jk = 0), (30) reduces to BFM(8). Let x := (S, I,V,s) 2 C2(m+n+1) denote the variables in(30). Let x := (S,`,v,s) 2 R3(m+n+1) denote the variables inSOCP relaxation (13). These variables are related throughthe mapping x = h(x) where ` jk = |I jk|2 and v j = |Vj|2. Inparticular, given any solution x of (30), x := h(x) satisfies(11) with equalities in (11c).

Cycle condition. If every line has a phase shifter then thecycle condition changes from (12) to: given any x thatsatisfies (11) with equalities in (11c),

9(q ,f) 2 Rn+m such that Bq = b (x)�f mod 2p (31)

It is proved in [17, Part II] that, given any x that attainsequalities in (11c), there always exists a q in (�p,p]n anda f in (�p,p]m that solve (31). Moreover phase shifters areneeded only on lines not in a spanning tree.

Exact SOCP relaxation. Recall the OPF problem (10) wherethe feasible set X without phase shifters is:

X := {x | x satisfies (30) with f = 0 and (9)}


Phase shifters on every line enlarge the feasible set to:

X := {x | x satisfies (30) for some f and (9)}

Given any spanning tree T of G, let “f 2 T?” be theshorthand for “f jk = 0 for all ( j,k) 2 T ”, i.e., f involvesonly phase shifters in lines not in the spanning tree T . Fixany T . Define the feasible set when there are phase shiftersonly on lines outside T :

XT := {x | x satisfies (30) for some f 2 T? and (9)}

Clearly X ✓ XT ✓ X. Define the (modified) OPF problemwhere there is a phase shifter on every line:OPF-ps:

minx,f

C(x) subject to x 2 X, f 2 Rm (32)

and that where there are phase shifters only outside T :OPF-T :

minx,f

C(x) subject to x 2 XT , f 2 T? (33)

Let Copt, Cps, and CT denote respectively the optimal val-ues of OPF (10), OPF-ps (32), and OPF-T (33). ClearlyCopt �CT �Cps since X✓XT ✓X. Solving OPF (10), OPF-ps (32), or OPF-T (33) is difficult because their feasible setsare nonconvex.

Recall the following sets defined in [25] for networkswithout phase shifters:

X+ := {x | x satisfies (9) and (11)}Xnc := {x | x satisfies (9) and (11) with equalities in (11c)}X := {x | x 2 Xnc and satisfies the cycle condition (12)}

Note that X is defined by the cycle condition without phaseshifters (f = 0 in (31)). As explained in [25, Theorem 9],X is equivalent to the feasible set X of OPF (10). HenceX⌘ X✓ Xnc ✓ X+. A key result of [17, Part II] is

Theorem 10: Fix any spanning tree T of G. Then XT =X ⌘ Xnc.

The implication of Theorem 10 is that, for a mesh network,when a solution of SOCP relaxation (13) attains equalities in(11c) (i.e., it is in Xnc), then it can be implemented with anappropriate setting of phase shifters even when the solutiondoes not satisfy the cycle condition (12). Define the problem:OPF-nc:

minx

C(x) subject to x 2 Xnc (34)

Let Cnc and Csocp denote respectively the optimal values ofOPF-nc (34) and OPF-socp (13). Theorem 10 then implies

Corollary 11: Fix any spanning tree T of G. Then1) X✓ XT = X ⌘ Xnc ✓ X+.2) Copt �CT =Cps =Cnc �Csocp.Hence if an optimal solution xsocp of OPF-socp (13) attains

equalities in (11c) then xsocp solves the problem OPF-nc (34).

If it also satisfies the cycle condition (12) then xsocp 2 Xand it can be mapped to a unique optimal of OPF (10).Otherwise, xsocp can be implemented through an appropriatephase shifter setting f and it attains a cost that lower boundsthe optimal cost of the original OPF without tunable phaseshifters. Moreover this benefit can be attained with phaseshifters only outside an arbitrary spanning tree T of G. Theresult can help determine if a network with a given set ofphase shifters can be convexified and, if not, where additionalphase shifters are needed for convexification [17, Part II].

Corollary 11 also implies that, if SOCP is exact, thenphase shifters cannot further reduce the cost. This can helpdetermine when phase shifters provide benefit to systemoperations.

Hence phase shifters in strategic locations make a meshnetwork behave like a radial network as far as convexrelaxation is concerned. The results of Section III then imply

Corollary 12: Suppose conditions A1–A2’, or A3–A4, orB1–B3, or C1–C2 hold. Then any optimal solution of OPF-socp (13) solves OPF-ps (32) and OPF-T (33).

B. DC networksIn this subsection we consider purely resistive dc networks,

i.e., the impedance z jk = r jk = y�1jk , the power injections

s j = p j, and the voltages Vj are real. We assume all voltagemagnitudes are strictly positive. Formally:D0: Replace (1b) and (11b) by 0 <V j Vj V j, j 2 N+,

and replace (3b) by 0 <V 2j [WG] j j V 2

j , j 2 N+.Type A conditions. Condition D0 immediately implies that thecycle condition (12) in BFM is satisfied by every feasible xof OPF-socp (13), for

b jk(x) := \�v j � zH

jkS jk�

= \⇣

v j � r jk

⇣r�1

jk Vj(Vj �Vk)⌘⌘

= 0

A3–A4 guarantee that any optimal solution of OPF-socpattains equality in (11c) for general mesh networks. Hence[25, Theorem 7] and Theorem 4 imply

Corollary 13: Suppose A3–A4 and D0 hold. Then OPF-socp (13) is exact.

For BIM, consider an OPF as a QCQP (16) where all thematrices are real and symmetric. Even though all the QCQPmatrices in (16) satisfy condition A2’, Corollary 3 is notdirectly applicable as its proof constructs a complex (ratherthan real) V from an optimal solution of OPF-socp. Howeverif there are no lower bounds on the power injections, thenonly F j are involved in the QCQP so all their off-diagonalentries are negative. It is then observed in [39] that [45,Theorem 3.1] directly implies (without needing D0)

Corollary 14: Suppose A1 and A4 hold. Then OPF-sdp(5) and OPF-socp (7) are exact.

Type B conditions. The following result is proved in [41].Consider:


B1’: The cost function is C(x) := Ânj=0 Cj (Res j) with Cj

strictly increasing for all j 2N+. There is no constrainton s0.

B2’: V 1 =V 2 = · · ·=V n; S j = [p j, p j] with p j < 0, j 2 N.B2”: V j = • for j 2 N.

Theorem 15: Suppose at least one of the following holds:• B1, B2” and D0; or• B1’, B2’ and D0.

Then OPF-socp (7) with the additional constraints Wjk � 0,( j,k)2E, is exact. If, in addition, the problem is convex thenits optimal solution is unique.

It is possible to enforce B2” by an affine constraint on thepower injections, similar to (but different from) condition B2for radial networks; see [41] for details. See also [52] for aresult on the uniqueness of SOCP relaxation.

C. General AC networks

Unfortunately no sufficient conditions for exact semidef-inite relaxation for general mesh networks are yet known.There are type A conditions on power injections for exactrelaxation only for special cases: a lossless cycle or losslesscycle with one chord [29], or a weakly cyclic network (whereevery line belongs to at most one cycle) of size 3 [53].

We close by mentioning three recent approaches for globaloptimization of OPF when the relaxations in this tutorial fail.First, higher-order semidefinite relaxations on the Lesserrehierarchy for polynomial optimization [54] have been appliedto solving OPF when SDP relaxation fails [55], [56], [57],[58]. By going up the hierarchy, the relaxations becometighter and their solutions approach a global optimal of theoriginal polynomial optimization [54], [59]. This howevercomes at the cost of significantly higher runtime. Techniquesare proposed in [57], [58] to reduce the problem sizes, e.g., byexploiting sparsity or adding redundant constraints [60], [61],[58] or applying higher-order relaxations only on (typicallysmall) subnetworks where constraints are violated [57].

Second, a branch-and bound algorithm is proposed in[62] where a lower bound is computed from the Lagrangiandual of OPF and the feasible set subdivision is based onrectangular or ellipsoidal bisection. The dual problem issolved using a subgradient algorithm. Each iteration of thesubgradient algorithm requires minimizing the Lagrangianover the primal variables. This minimization is separableinto two subproblems, one being a convex subproblem andthe other having a nonconvex quadratic objective. The lattersubproblem turns out to be a trust-region problem that has aclosed-form solution. It is proved in [62] that the proposedalgorithm converges to a global optimal. This method isextended in [63] to include more constraints and alternativelyuse SDP relaxation for lower bounding the cost.

Finally a new approach is proposed in [64] based onconvex quadratic relaxation of OPF in polar coordinates.

V. CONCLUSION

We have summarized the main sufficient conditions forexact semidefintie relaxations of OPF as listed in TablesI and II. For radial networks these conditions suggest thatSOCP relaxation (and hence SDP and chordal relaxations)will likely be exact in practice. This is corroborated bysignificant numerical experience. For mesh networks they areapplicable only for special cases: networks that have tunablephase shifters or dc networks where all variables are realand voltages are nonnegative. Even though counterexamplesexist where SDP/chordal relaxation is not exact for ACmesh networks numerical experience seems to suggest thatSDP/chordal relaxation tends to be exact in many cases.Sufficient conditions that guarantee exact relaxation for ACmesh networks however remain elusive. The main difficultyis in designing relaxations of the cycle condition (4) or (12).

REFERENCES

[1] J. Carpentier. Contribution to the economic dispatch problem. Bulletinde la Societe Francoise des Electriciens, 3(8):431–447, 1962.

[2] J. A. Momoh. Electric Power System Applications of Optimization.Power Engineering. Markel Dekker Inc.: New York, USA, 2001.

[3] M. Huneault and F. D. Galiana. A survey of the optimal power flowliterature. IEEE Trans. on Power Systems, 6(2):762–770, 1991.

[4] J. A. Momoh, M. E. El-Hawary, and R. Adapa. A review of selectedoptimal power flow literature to 1993. Part I: Nonlinear and quadraticprogramming approaches. IEEE Trans. on Power Systems, 14(1):96–104, 1999.

[5] J. A. Momoh, M. E. El-Hawary, and R. Adapa. A review ofselected optimal power flow literature to 1993. Part II: Newton, linearprogramming and interior point methods. IEEE Trans. on PowerSystems, 14(1):105 – 111, 1999.

[6] K. S. Pandya and S. K. Joshi. A survey of optimal power flow methods.J. of Theoretical and Applied Information Technology, 4(5):450–458,2008.

[7] Stephen Frank, Ingrida Steponavice, and Steffen Rebennack. Optimalpower flow: a bibliographic survey, I: formulations and deterministicmethods. Energy Systems, 3:221–258, September 2012.

[8] Stephen Frank, Ingrida Steponavice, and Steffen Rebennack. Optimalpower flow: a bibliographic survey, II: nondeterministic and hybridmethods. Energy Systems, 3:259–289, September 2013.

[9] Mary B. Cain, Richard P. O’Neill, and Anya Castillo. History ofoptimal power flow and formulations (OPF Paper 1). Technical report,US FERC, December 2012.

[10] Richard P. O’Neill, Anya Castillo, and Mary B. Cain. The IVformulation and linear approximations of the AC optimal power flowproblem (OPF Paper 2). Technical report, US FERC, December 2012.

[11] Richard P. O’Neill, Anya Castillo, and Mary B. Cain. The compu-tational testing of AC optimal power flow using the current voltageformulations (OPF Paper 3). Technical report, US FERC, December2012.

[12] Anya Castillo and Richard P. O’Neill. Survey of approaches to solvingthe ACOPF (OPF Paper 4). Technical report, US FERC, March 2013.

[13] Anya Castillo and Richard P. O’Neill. Computational performance ofsolution techniques applied to the ACOPF (OPF Paper 5). Technicalreport, US FERC, March 2013.

[14] R.A. Jabr. Radial Distribution Load Flow Using Conic Programming.IEEE Trans. on Power Systems, 21(3):1458–1459, Aug 2006.

[15] X. Bai, H. Wei, K. Fujisawa, and Y. Wang. Semidefinite programmingfor optimal power flow problems. Int’l J. of Electrical Power & EnergySystems, 30(6-7):383–392, 2008.

[16] Masoud Farivar, Christopher R. Clarke, Steven H. Low, and K. ManiChandy. Inverter VAR control for distribution systems with renewables.In Proceedings of IEEE SmartGridComm Conference, October 2011.


[17] Masoud Farivar and Steven H. Low. Branch flow model: relaxationsand convexification (parts I, II). IEEE Trans. on Power Systems,28(3):2554–2572, August 2013.

[18] M. E. Baran and F. F Wu. Optimal Capacitor Placement on radialdistribution systems. IEEE Trans. Power Delivery, 4(1):725–734, 1989.

[19] M. E Baran and F. F Wu. Optimal Sizing of Capacitors Placed on ARadial Distribution System. IEEE Trans. Power Delivery, 4(1):735–743, 1989.

[20] J. Lavaei and S. H. Low. Zero duality gap in optimal power flowproblem. IEEE Trans. on Power Systems, 27(1):92–107, February2012.

[21] X. Bai and H. Wei. A semidefinite programming method with graphpartitioning technique for optimal power flow problems. Int’l J. ofElectrical Power & Energy Systems, 33(7):1309–1314, 2011.

[22] R. A. Jabr. Exploiting sparsity in sdp relaxations of the opf problem.Power Systems, IEEE Transactions on, 27(2):1138–1139, 2012.

[23] D. Molzahn, J. Holzer, B. Lesieutre, and C. DeMarco. Implementationof a large-scale optimal power flow solver based on semidefiniteprogramming. IEEE Transactions on Power Systems, 28(4):3987–3998,November 2013.

[24] Subhonmesh Bose, Steven H. Low, Thanchanok Teeraratkul, andBabak Hassibi. Equivalent relaxations of optimal power flow. IEEETrans. Automatic Control, 2014.

[25] S. H. Low. Convex relaxation of optimal power flow, I: formulationsand relaxations. IEEE Trans. on Control of Network Systems, 1(1):15–27, March 2014.

[26] S. Bose, D. Gayme, S. H. Low, and K. M. Chandy. Optimal powerflow over tree networks. In Proc. Allerton Conf. on Comm., Ctrl. andComputing, October 2011.

[27] S. Bose, D. Gayme, K. M. Chandy, and S. H. Low. Quadraticallyconstrained quadratic programs on acyclic graphs with application topower flow. arXiv:1203.5599v1, March 2012.

[28] B. Zhang and D. Tse. Geometry of feasible injection region of powernetworks. In Proc. Allerton Conf. on Comm., Ctrl. and Computing,October 2011.

[29] Baosen Zhang and David Tse. Geometry of the injection region ofpower networks. IEEE Trans. Power Systems, 28(2):788–797, 2013.

[30] S. Sojoudi and J. Lavaei. Physics of power networks makes hardoptimization problems easy to solve. In IEEE Power & Energy Society(PES) General Meeting, July 2012.

[31] Somayeh Sojoudi and Javad Lavaei. Semidefinite relaxation fornonlinear optimization over graphs with application to power systems.Preprint, 2013.

[32] Na Li, Lijun Chen, and Steven Low. Exact convex relaxation of OPFfor radial networks using branch flow models. In IEEE InternationalConference on Smart Grid Communications, November 2012.

[33] Lingwen Gan, Na Li, Ufuk Topcu, and Steven H. Low. On theexactness of convex relaxation for optimal power flow in tree networks.In Prof. 51st IEEE Conference on Decision and Control, December2012.

[34] Lingwen Gan, Na Li, Ufuk Topcu, and Steven H. Low. Optimal powerflow in distribution networks. In Proc. 52nd IEEE Conference onDecision and Control, December 2013.

[35] Lingwen Gan, Na Li, Ufuk Topcu, and Steven H. Low. Exact convexrelaxation of optimal power flow in radial networks. IEEE Trans.Automatic Control, 2014.

[36] Javad Lavaei, David Tse, and Baosen Zhang. Geometry of power flowsand optimization in distribution networks. arXiv, November 2012.

[37] Albert Y.S. Lam, Baosen Zhang, Alejandro Domınguez-Garcıa, andDavid Tse. Optimal distributed voltage regulation in power distributionnetworks. arXiv, April 2012.

[38] Somayeh Sojoudi and Javad Lavaei. Convexification of optimal powerflow problem by means of phase shifters. In Proc. IEEE SmartGridComm, 2013.

[39] Javad Lavaei, Anders Rantzer, and Steven H. Low. Power flowoptimization using positive quadratic programming. In Proceedingsof IFAC World Congress, 2011.

[40] Lingwen Gan and Steven H. Low. Optimal power flow in DC networks.In 52nd IEEE Conference on Decision and Control, December 2013.

[41] Lingwen Gan and Steven H. Low. Optimal power flow in direct currentnetworks. IEEE Trans. Power Systems, 2014. To appear.

[42] B. Lesieutre, D. Molzahn, A. Borden, and C. L. DeMarco. Examiningthe limits of the application of semidefinite programming to powerflow problems. In Proc. Allerton Conference, 2011.

[43] Waqquas A. Bukhsh, Andreas Grothey, Ken McKinnon, and PaulTrodden. Local solutions of optimal power flow. IEEE Trans. PowerSystems, 28(4):4780–4788, November 2013.

[44] Raphael Louca, Peter Seiler, and Eilyan Bitar. A rank minimizationalgorithm to enhance semidefinite relaxations of optimal power flow.In Proc. Allerton Conf. on Communication, Control and Computing,2013.

[45] S. Kim and M. Kojima. Exact solutions of some nonconvex quadraticoptimization problems via SDP and SOCP relaxations. ComputationalOptimization and Applications, 26(2):143–154, 2003.

[46] K. Turitsyn, P. Sulc, S. Backhaus, and M. Chertkov. Options for controlof reactive power by distributed photovoltaic generators. Proc. of theIEEE, 99(6):1063 –1073, June 2011.

[47] I. A. Hiskens. Analysis tools for power systems – contending withnonlinearities. Proc. IEEE, 83(11):1573–1587, November 1995.

[48] I. A. Hiskens and R. Davy. Exploring the power flow solution spaceboundary. IEEE Trans. Power Systems, 16(3):389–395, 2001.

[49] B. C. Lesieutre and I. A. Hiskens. Convexity of the set of feasibleinjections and revenue adequacy in FTR markets. IEEE Trans. PowerSystems, 20(4):1790–1798, 2005.

[50] Yuri V. Makarov, Zhao Yang Dong, and David J. Hill. On convexityof power flow feasibility boundary. IEEE Trans. Power Systems,23(2):811–813, May 2008.

[51] S. P. Boyd and L. Vandenberghe. Convex optimization. CambridgeUniversity Press, 2004.

[52] Chee Wei Tan, Desmond W. H. Cai, and Xin Lou. Resistive networkoptimal power flow: uniqueness and algorithms. Submitted for publi-cation, 2014.

[53] Ramtin Madani, Somayeh Sojoudi, and Javad Lavaei. Convex re-laxation for optimal power flow problem: Mesh networks. In Proc.Asilomar Conference on Signals, Systems and Computers, November2013.

[54] J. B. Lasserre. Global optimization with polynomials and the problemof moments. SIAM Journal on Optimization, 11:796–817, 2001.

[55] C. Josz, J. Maeght, P. Panciatici, and J. Ch. Gilbert. Application ofthe moment-SOS approach to global optimization of the OPF problem.arXiv, November 2013.

[56] D.K. Molzahn and I.A. Hiskens. Moment-based relaxation of the op-timal power flow problem. In Proc. 18th Power Systems ComputationConference (PSCC), August 2014.

[57] D.K. Molzahn and I.A. Hiskens. Sparsity-exploiting moment-basedrelaxations of the optimal power flow problem. arXiv, 1404.5071,2014.

[58] Bissan Ghaddar, Jakub Marecek, and Martin Mevissen. Optimal powerflow as a polynomial optimization problem. arXiv:1404.3626v1, April2014.

[59] P. A. Parrilo. Semidefinite programming relaxations for semialgebraicproblems. Mathematical Programming, 96:293–320, 2003.

[60] H. Waki, S.Kim, M. Kojima, and M. Muramatsu. Sums of squaresand semidefinite programming relaxations for polynomial optimizationproblems with structured sparsity. SIAM Journal on Optimization,17(1):218–242, 2006.

[61] J. B. Lasserre. Convexity in semialgebraic geometry and polynomialoptimization. SIAM Journal on Optimization, 19:1995–2014, 2009.

[62] Dzung T. Phan. Lagrangian duality and branch-and-bound algo-rithms for optimal power flow. Operations Research, 60(2):275–285,March/April 2012.

[63] A. Gopalakrishnan, A.U. Raghunathan, D. Nikovski, and L.T. Biegler.Global optimization of optimal power flow using a branch amp; boundalgorithm. In Proc. of the 50th Annual Allerton Conference onCommunication, Control, and Computing (Allerton), pages 609–616,Oct 2012.

[64] H. L. Hijazi, C. Coffrin, and P. Van Hentenryck. Convex quadraticrelaxations of mixed-integer nonlinear programs in power systems.Technical report, NICTA, Canberra, ACT Australia, September 2013.

For author biography, see Part I of this tutorial [25].

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