Convex Relaxation of Optimal Power Flow

Transcription

1 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH Convex Relaxation of Optimal Power Flow Part I: Formulations and Equivalence Steven H. Low EAS, Caltech slow@caltech.edu Abstract This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact. I. INTRODUCTION For our purposes an optimal power flow (OPF) problem is a mathematical program that seeks to imize a certain function, such as total power loss, generation cost or user disutility, subject to the Kirchhoff s laws as well as capacity, stability and security constraints. OPF is fundamental in power system operations as it underlies many applications such as economic dispatch, unit commitment, state estimation, stability and reliability assessment, volt/var control, demand response, etc. There has been a great deal of research on OPF since Carpentier s first formulation in 1962 [2]. An early solution appears 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 be formulated as a quadratically constrained quadratic program (QCQP). It is generally nonconvex and hence NP-hard. A large number of optimization algorithms and relaxations have been proposed. A popular approximation is a linear program, called DC OPF, obtained through the linearization of the power flow equations e.g. [16], [17], [18], [19], [20]. See also [21] for a more accurate linear approximation. To the best of our knowledge solving OPF through semidefinite relaxation is first proposed in [22] as a second-order cone program (SOCP) for radial (tree) networks and in [23] as a semidefinite program (SDP) for general networks in a bus injection model. It is first proposed in [51], [57] as an SOCP for radial networks in the branch flow model of [45]. See Remark 6 below for more details. While these convex relaxations have been illustrated numerically in [22] and [23], whether or when they will turn out to be exact is first studied in [24]. Exploiting graph sparsity to simplify the SDP relaxation of OPF is first proposed in [25], [26] and analyzed in [27], [28]. Convex relaxation of quadratic programs has been applied to many engineering problems; see e.g. [29]. There is a rich theory and extensive empirical experiences. Compared with other approaches, solving OPF through convex relaxation offers several advantages. First, while DC OPF is useful in a wide variety of applications, it is not applicable in other applications; A preliary and abridged version has appeared in [1]. see Remark 10. Second a solution of DC OPF may not be feasible (may not satisfy the nonlinear power flow equations). In this case an operator may tighten some constraints in DC OPF and solve again. This may not only reduce efficiency but also relies on heuristics that are hard to scale to larger systems or faster control in the future. Third, when they converge, most nonlinear algorithms compute a local optimal usually without assurance on the quality of the solution. In contrast a convex relaxation provides for the first time the ability to check if a solution is globally optimal. If it is not, the solution provides a lower bound on the imum cost and hence a bound on how far any feasible solution is from optimality. Unlike approximations, if a relaxed problem is infeasible, it is a certificate that the original OPF is infeasible. This two-part tutorial explains the main theoretical results on semidefinite relaxations of OPF developed in the last few years. Part I presents two power flow models that are useful in different situations, formulates OPF and its convex relaxations in each model, and clarifies their relationship. Part II [30] presents sufficient conditions that guarantee the relaxations are exact, i.e. when one can recover a globally optimal solution of OPF from an optimal solution of its relaxations. We focus on basic results using the simplest OPF formulation and does not cover many relevant works in the literature, such as stochastic OPF e.g. [31], [32], [33], distributed OPF e.g. [34], [35], [36], [37], [38], [39], new applications e.g. [40], [41], or what to do when relaxation fails e.g. [42], [43], [44], to name just a few. Outline of paper Many mathematical models have been used to model power networks. In Part I of this two-part paper we present two such models, we call the bus injection model (BIM) and the branch flow model (BFM). Each model consists of a set of power flow equations. Each models a power network in that the solutions of each set of equations, called the power flow solutions, describe the steady state of the network. We prove that these two models are equivalent in the sense that there is a bijection between their solution sets (Section II). We formulate OPF within each model where the power flow solutions define the feasible set of OPF (Section III). Even though BIM and BFM are equivalent some results are much easier to formulate or prove in one model than the other; see Remark 2 in Section II. The complexity of OPF formulated here lies in the nonconvexity of power flow equations that gives rise to a nonconvex feasible set of OPF. We develop various characterizations of the feasible set and design convex supersets based on these characterizations. Different designs lead to different convex

2 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH relaxations and we prove their relationship (Sections IV and V). When a relaxation is exact an optimal solution of the original nonconvex OPF can be recovered from any optimal solution of the relaxation. In Part II [30] we present sufficient conditions that guarantee the exactness of convex relaxations. Branch flow models are originally proposed for networks with a tree topology, called radial networks, e.g. [45], [46], [47], [48], [49], [50], [51], [52], [53]. They take a recursive structure that simples the computation of power flow solutions, e.g. [54], [55], [47]. The model of [45], [46] also has a linearization that offers several advantages over DC OPF in BIM; see Remark 10. The linear approximation provides simple bounds on the branch powers and voltage magnitudes in the nonlinear BFM (Section VI). These bounds are used in [56] to prove a sufficient condition for exact relaxation. Finally we make algorithmic recommendations in Section VII based on the key results presented here. Most proofs can be found in the original papers (or the arxiv version of this tutorial) and are omitted. Notations Let C denote the set of complex numbers and R the set of real numbers. For a 2 C, Re a and Im a denote the real and imaginary parts of a respectively. For any set A C n, conva denotes the convex hull of A. For a 2 R, [a] + := max{a,0}. For a,b 2 C, a apple b means Re a apple Re b and Im a apple Im b. We abuse notation to use the same symbol a to denote either a complex number Rea + iima or a 2-dimensional real vector a =(Rea, Im a) 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 capital letters, e.g. S jk,p jk,q jk,i j,v j. A variable without a subscript denotes 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 apple y denotes componentwise inequality. Matrices are usually in capital letters. The transpose of a matrix A is denoted by A T and its Hermitian (complex conjugate) transpose by A H. A matrix A is Hermitian if A = A H. A is positive semidefinite (or psd), denoted by A 0, if A is Hermitian and x H Ax 0 for all x 2 C n ; in particular if A 0 then by definition A = A H. For matrices A,B, A B means A B is psd. Let S n be the set of all n n Hermitian matrices and S n + 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; in this case we will use ( j,k) and j! k interchangeably to denote an edge pointing from j to k. We sometimes use G =(N,Ẽ) 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 :=(j 1,..., j K ) is an ordered set of nodes j k 2 N so that ( j k, j k+1 ) 2 E for k = 1,...,K with the understanding that j K+1 := j 1. In that case we refer to a link or a node in the cycle by ( j k, j k+1 ) 2 c or j k 2 c respectively. II. POWER FLOW MODELS In this section we describe two mathematical models of power networks and prove their equivalence. By a mathematical model we mean a set of variables and a set of equations relating these variables. These equations are motivated by the physical system, but mathematically, they are the starting point from which all claims are derived. A. Bus injection model Consider a power network modeled by a connected undirected graph G(N +,E) where N + := {0}[N, N := {1,2,...,n}, and E N + N +. Each node in N + represents a bus and each edge in E represents a transmission or distribution line. We use bus and node interchangeably and line and edge interchangeably. For each edge (i, j) 2 E let y ij 2 C be its admittance. A bus j 2 N + can have a generator, a load, both or neither. Let V j be the complex voltage at bus j 2 N + and V j denote its magnitude. Bus 0 is the slack bus. Its voltage is fixed and we assume without loss of generality that V 0 = 1\0 per unit (pu). Let s j be the net complex power injection (generation us load) at bus j 2 N +. The bus injection model (BIM) is defined by the following power flow equations that describe the Kirchhoff s laws: s j = Â y H jk V j(v j H k: j k V H k ), j 2 N+ (1) Let the set of power flow solutions V for each s be: V(s) := {V 2 C n+1 V satisfies (1)} For convenience we include V 0 in the vector variable V := (V j, j 2 N + ) with the understanding that V 0 := 1\0 is fixed. Remark 1: Bus types. Each bus j is characterized by two complex variables V j and s j, or equivalently, four real variables. The buses are usually classified into three types, depending on which two of the four real variables are specified. For the slack bus 0, V 0 is given and s 0 is variable. For a generator bus (also called PV -bus), Re(s j )=p j and V j are specified and Im(s j )=q j and \V j are variable. For a load bus (also called PQ-bus), s j is specified and V j is variable. The power flow or load flow problem is: given two of the four real variables specified for each bus, solve the n+1 complex equations in (1) for the remaining 2(n + 1) real variables. For instance when all n buses j 6= 0 are all load buses, the power flow problem solves (1) for the n complex voltages V j, j 6= 0, and the power injection s 0 at the slack bus 0. This can model a distribution system with a substation at bus 0 and n constant-power loads at the other buses. For optimal power flow problems p j and V j on generator buses or s j on load buses can be variables as well. For instance economic dispatch optimizes real power generations p j at generator buses; demand response optimizes demands s j at load buses; and volt/var control optimizes reactive powers q j at capacitor banks, tap changers, or inverters. These remarks also apply to the branch flow model presented next. B. Branch flow model In the branch flow model we adopt a connected directed graph G =(N +,Ẽ) where each node in N + := {0,1,...,n}

3 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH represents a bus and each edge in Ẽ N + N + represents a transmission or distribution line. Fix an arbitrary orientation for G and let m := Ẽ be the number of directed edges in G. Denote an edge by ( j,k) or j! k if it points from node j to node k. For each edge ( j,k) 2 Ẽ let z jk := 1/y jk be the complex impedance on the line; let I jk be the complex current and S jk = P jk + iq jk be the sending-end complex power from buses j to k. For each bus j 2 N + let V j be the complex voltage at bus j. Assume without loss of generality that V 0 = 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 the following set of power flow equations:  S jk k: j!k =  S ij z ij I ij 2 + s j, j 2 N + (2a) i:i! j I jk = y jk (V j V k ), j! k 2 Ẽ (2b) S jk = V j I H jk, j! k 2 Ẽ (2c) where (2b) is the Ohm s law, (2c) defines branch power, and (2a) imposes power balance at each bus. The quantity z ij I ij 2 represents line loss so that S ij z ij I ij 2 is the receiving-end complex 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 C 2m+n+1 x satisfies (2)} For convenience we include V 0 in the vector variable V := (V j, j 2 N + ) with the understanding that V 0 := 1\0 is fixed. C. Equivalence Even though the bus injection model (1) and the branch flow model (2) are defined by different sets of equations in terms of their own variables, both are models of the Kirchhoff s laws and therefore must be related. We now clarify the precise sense in which these two mathematical models are equivalent. We say two sets A and B are equivalent, denoted by A B, if there is a bijection between them [58]. Theorem 1: V(s) X(s) for any power injections s. Remark 2: Two models. Given the bijection between the solution sets V(s) and X(s) any result in one model is in principle derivable in the other. Some results however are much easier to state or derive in one model than the other. For instance BIM allows a much cleaner formulation of the semidefinite program (SDP) relaxation. BFM for radial networks has a convenient recursive structure that allows a more efficient computation of power 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 hard to formulate or prove in BIM. Finally, since BFM directly models branch flows S jk and currents I jk, it is easier to use for some applications. We will therefore freely use either model depending on which is more convenient for the problem at hand. A. Bus injection model III. OPTIMAL POWER FLOW As 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 apple V j 2 apple v j, j 2 N + (3) where v j and v j are given lower and upper bounds on voltage magnitudes. Throughout this paper we assume v j > 0 to avoid triviality. The power injections are also constrained: s j apple s j apple 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 where s j is fixed (e.g. a constant-power load), by setting s j = s j to the specified value. For the slack bus 0, unless otherwise specified, we always assume v 0 = v 0 = 1 and s 0 = i, s 0 = + i. Therefore we sometimes replace j 2 N + in (3) and (4) by j 2 N. We can eliate the variables s j from the OPF formulation by combining (1) and (4) into s j apple  y H jk V j(v j H k:( j,k)2e V H k ) apple s j, j 2 N + (5) Then OPF in the bus injection model can be defined just in terms of the complex voltage vector V. Define V := {V 2 C n+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 over the network. All these costs can be expressed as functions of V. Then the problem of interest is: OPF: V C(V ) subject to V 2 V (7) Since (5) is quadratic, V is generally a nonconvex set. OPF is thus 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 C 2(m+n+1). We can also eliate the variables s j as for the bus injection model by combining (2a) and (4) but it will prove convenient to retain s :=(s j, j 2 N + ) as part of the variables. Define the feasible set in the branch flow model: X := { x 2 C 2(m+n+1) x satisfies (2),(3),(4)} (8) Let the cost function in the branch flow model be C( x). Then the optimal power flow problem in the branch flow model is: OPF: x C( x) subject to x 2 X (9) Since (2) is quadratic, X is generally a nonconvex set. OPF is thus a nonconvex problem and NP-hard to solve in general. Remark 4: OPF equivalence. By Theorem 1 there is a bijection between V and X. Throughout this paper we assume that the cost functions in BIM and BFM are equivalent under this bijection and we abuse notation to denote them by the

4 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH same symbol C( ). Then OPF (7) in BIM and (9) in BFM are equivalent. Remark 5: OPF variants. OPF as defined in (7) and (9) is a simplified version that ignores other important constraints such 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] and a recent discussion in [61] on real-life OPF problems. Some of these can be incorporated without any change to the results in this paper (e.g. see [57], [62] for models that include shunt elements and line limits). Indeed a shunt element y j at bus j can be easily included in BIM by modifying (1) into: s j = Â y H jk V j(v j H k: j k V H k )+yh j V j 2 or included in BFM by modifying (2a) into: Â S jk + y H j V j 2 k: j!k C. OPF as QCQP = Â S ij z ij I ij 2 + s j i:i! j Before we describe convex relaxations of OPF we first show that, when C(V ) := V H CV is quadratic in V for some Hermitian matrix C, OPF is indeed a quadratically constrained quadratic program (QCQP) by converting it into the standard form. We will use the derivation in [62] for OPF (7) in BIM. OPF (9) in BFM can similarly be converted into a standard form QCQP. Define the (n + 1) (n + 1) admittance matrix Y by 8 >< Â y ik, if i = j k:k i Y ij = y ij, if i 6= j and i j >: 0 otherwise Y is symmetric but not necessarily Hermitian. Let I j be the net injection current from bus j to the rest of the network. Then the current vector I and the voltage vector V are related by the Ohm s law I = YV. BIM (1) is equivalent to: s j = V j I H j = (e H j V )(I H e j ) where e j is the (n + 1)-dimensional vector with 1 in the jth entry and 0 elsewhere. Hence, since I = YV, we have s j = tr e H j VV H Y H e j = tr Y H e j e H j VV H = V H Y H j V where Y j := e j e H j Y is an (n+1) (n+1) matrix with its jth row equal to the jth row of the admittance matrix Y and all other rows equal to the zero vector. Y j is in general not Hermitian so that V H Yj H V is in general a complex number. Its real and imaginary parts can be expressed in terms of the Hermitian and skew Hermitian components of Yj H defined as: Then F j := 1 2 Y H j +Y j and Y j := 1 2i Y H j Y j Re s j = V H F j V and Im s j = V H Y j V 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 J j := e j e H j denote the Hermitian matrix with a single 1 in the ( j, j)th entry and 0 everywhere else. Then OPF (7) can be written as a standard form QCQP: V H CV (10a) V 2C n+1 s.t. V H F j V apple p j, V H ( F j )V apple p j (10b) where j 2 N + in (10). V H Y j V apple q j, V H ( Y j )V apple q j (10c) V H J j V apple v j, V H ( J j )V apple v j (10d) IV. FEASIBLE SETS AND RELAXATIONS: BIM In this and the next section we derive semidefinite relaxations of OPF and clarify their relations. The cost function C of OPF is usually assumed to be convex in its variables. The difficulty of OPF formulated here thus arises from the nonconvex feasible sets V for BIM and X for BFM. The basic approach to deriving convex relaxations of OPF is to design convex supersets of (equivalent sets of) V or X and imize the same cost function over these supersets. Different choices of convex supersets lead to different relaxations, but they all provide a lower bound to OPF. If every optimal solution of a convex relaxation happens to lie in V or X then it is also feasible and hence optimal for the original OPF. In this case we say the realxation is exact. In this section we present three characterizations of the feasible set V in BIM. These characterizations naturally suggest convex supersets and semidefinite relaxations of OPF, and we prove equivalence relations among them. In the next section we treat BFM. In Part II of the paper we discuss sufficient conditions that guaranteed exact relaxations. A. Preliaries Since OPF is a nonconvex QCQP there is a standard semidefinite relaxation through the equivalence relation: for any Hermitian matrix M, V H MV = tr MVV H = tr MW for a psd rank-1 matrix W. Applying this transformation to the QCQP formulation (10) leads to an equivalent problem of the form: W2S n+1 tr CW s.t. tr C l W apple b l, W 0, rank W = 1 for appropriate Hermitian matrices C l and real numbers b l. This problem is equivalent to (10) because given a psd rank- 1 solution W, a unique solution V of (10) can be recovered through rank-1 factorization W = VV H. Unlike (10) which is quadratic in V this problem is convex in W except the nonconvex rank-1 constraint. Removing the rank-1 constraint yields the standard SDP relaxation. We now generalize this intuition to characterize the feasible set V in (6) in terms of partial matrices. These characterizations lead naturally to SDP, chordal, and second-order cone program (SOCP) relaxations of OPF in BIM, as shown in [58], [28].

5 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH We start with some basic definitions on partial matrices and their completions; see e.g. [63], [64], [65] for more details. Fix any connected undirected graph F with n vertices and m edges connecting distinct vertices. 1 A partial matrix W F is a set of 2m + n complex numbers defined on F: W F := {[W F ] jj,[w F ] jk,[w F ] kj nodes j and edges ( j,k) of F} W F can be interpreted as a matrix with entries partially specified by these complex numbers. If F is a complete graph (in which there is an edge between every pair of vertices) then W F is a fully specified n n matrix. A completion W of W F is any fully specified n n matrix that agrees with W F on graph F, i.e., [W] jj = [W F ] jj, [W] jk = [W F ] jk for j,( j,k) 2 F Given an n n matrix W we use W F to denote the submatrix of W on F, i.e., the partial matrix consisting of the entries of W defined on graph F. If q is a clique (a fully connected subgraph) of F then let W F (q) denote the fully-specified principal submatrix of W F defined on q. We extend the definitions of Hermitian, psd, and rank-1 for matrices to partial matrices, as follows. A partial matrix W F is Hermitian, denoted by W F = WF H, if [W F ] jk =[W F ] H kj for all ( j,k) 2 F; it is psd, denoted by W F 0, if W F is Hermitian and the principal submatrices W F (q) are psd for all cliques q of F; it is rank-1, denoted by rank W F = 1, if the principal submatrices W F (q) are rank-1 for all cliques q of F. We say W F is 2 2 psd (rank-1), denoted by W F ( j,k) 0 (rank W F ( j,k) =1) if, for all edges ( j,k) 2 F, the 2 2 principal submatrices apple [WF ] [W F ]( j,k) := jj [W F ] jk [W F ] kj [W F ] kk are psd (rank-1). F is a chordal graph if either F has no cycle or all its imal cycles (ones without chords) are of length three. A chordal extension c(f) of F is a chordal graph that contains F, i.e., c(f) has the same vertex set as F but an edge set that is a superset of F s edge set. In that case we call the partial matrix W c(f) a chordal extension of the partial matrix W F. Every graph F has a chordal extension, generally nonunique. In particular a complete supergraph of F is a trivial chordal extension of F. For our purposes chordal graphs are important because of the result [63, Theorem 7] that every psd partial matrix has a psd completion if and only if the underlying graph is chordal. When a positive definite completion exists, there is a unique positive definite completion, in the class of all positive definite completions, whose deterant is maximal. Theorem 2 below extends this to rank-1 partial matrices. B. Feasible sets We can now characterize the feasible set V of OPF defined in (6). Recall the undirected connected graph G =(N +,E) that models a power network. Given a voltage vector V 2 V define a partial matrix W G := W G (V ): for j 2 N + and ( j,k) 2 E, [W G ] jj := V j 2 (11a) [W G ] jk := V j Vk H =: [W G ] H kj (11b) 1 In this subsection we abuse notation and use n,m to denote general integers unrelated to the number of buses or lines in a power network. Then the constraints (5) and (3) imply that the partial matrix W G satisfies 2 s j apple  y H jk [W G ] jj [W G ] jk apple s j, j 2 N + (12a) k:( j,k)2e v j apple [W G ] jj apple v j, j 2 N + (12b) Following Section III-C these constraints can also be written in a (partial) matrix form as: p j apple tr F j W G apple p j q j apple tr Y j W G apple q j v j apple tr J j W G apple v j The converse is not always true: given a partial matrix W G that satisfies (12) it is not always possible to recover a voltage vector V in V. Indeed this is possible if and only if W G has a completion W that is psd rank-1, because in that case W satisfies (12) since y jk = 0 if ( j,k) 62 E and it can be uniquely factored as W = VV H with V 2 V. We hence seek conditions additional to (12) on the partial matrix W G that guarantee that it has a psd rank-1 completion W from which V 2 V can be recovered. Our first key result provides such a characterization. We say that a partial matrix W G satisfies the cycle condition if for every cycle c in G  ( j,k)2c \W jk = 0 mod 2p (13) When \W jk represent voltage phase differences across each line then the cycle condition imposes that they sum to zero (mod 2p) around any cycle. The next theorem, proved in [58, Theorem 3] and [28], implies that W G has a psd rank-1 completion W if and only if W G is 2 2 psd rank-1 on G and satisfies the cycle condition (13), if and only if it has a chordal extension W c(g) that is psd rank-1. 3 Consider the following conditions on (n + 1) (n + 1) matrices W and partial matrices W c(g) and W G : W 0, rank W = 1 (14) W c(g) 0, rank W c(g) = 1 (15) W G ( j,k) 0, rank W G ( j,k)=1, ( j,k) 2 E, (16) Theorem 2: Fix a graph G on n + 1 nodes and any chordal extension c(g) of G. Assug W jj > 0, W c(g) jj > 0 and [W G ] jj > 0, j 2 N +, we have: (1) Given an (n + 1) (n + 1) matrix W that satisfies (14), its submatrix W c(g) satisfies (15). (2) Given a partial matrix W c(g) that satisfies (15), its submatrix W G satisfies (16) and the cycle condition (13). (3) Given a partial matrix W G that satisfies (16) and the cycle condition (13), there is a completion W of W G that satisfies (14). 2 The constraint (12a) can also be written compactly in terms of the admittance matrix Y as in [66]: s apple diag WY H apple s 3 The theorem also holds with psd replaced by negative semidefinite.

6 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH Informally Theorem 2 says that (14) is equivalent to (15) is equivalent to (16)+(13). It characterizes a property of the full matrix W (rank W = 1) in terms of its submatrices W c(g) and W G. This is important because the submatrices are typically much smaller than W for large sparse networks and much easier to compute. The theorem thus allows us to solve simpler problems in terms of partial matrices as we now explain. Define the set of Hermitian matrices: W := {W 2 S n+1 W satisfies (12),(14)} (17) Fix any chordal extension c(g) of G and define the set of Hermitian partial matrices W c(g) : W c(g) := {W c(g) W c(g) satisfies (12),(15)} (18) Finally define the set of Hermitian partial matrices W G : W G := {W G W G satisfies (12),(13),(16)} (19) Note that the definition of psd for partial matrices implies that W c(g) and W G are Hermitian. The assumption v j > 0, j 2 N + implies that all matrices or partial matrices have strictly positive diagonal entries. Theorem 2 implies that given a partial matrix W c(g) 2 W c(g) or a partial matrix W G 2 W G there is a psd rank-1 completion W 2 W from which a solution V 2 V of OPF can be recovered. In fact we know more: given any Hermitian partial matrix W G (not necessarily in W G ), the set of all completions of W G that satisfies the condition in Theorem 2(3) consists of a single psd rank-1 matrix and infinitely many indefinite non-rank-1 matrices; see [58, Theorems 5 and 8] and discussions therein. Hence the psd rank-1 completion W of a W G 2 W G is unique. Corollary 3: Given a partial matrix W c(g) 2 W c(g) or W G 2 W G there is a unique psd rank-1 completion W 2 W. Recall that two sets A and B are equivalent (A B) if there is a bijection between them. Even though W,W c(g),w G are different kinds of spaces Theorem 2 and Corollary 3 imply that they are all equivalent to the feasible set of OPF. Theorem 4: V W W c(g) W G. Theorem 4 suggests three equivalent problems to OPF. We assume the cost function C(V ) in OPF depends on V only through the partial matrix W G defined in (11). For example if the cost is total real line loss in the network then C(V )= j Re s j =  j  k:( j,k)2e Re [W G ] jj [W G ] jk y H jk. If the cost is a weighted sum of real generation power then C(V )= j c j Re s j + p d j where p d j are the given real power demands at buses j; again C(V ) is a function of the partial matrix W G. Then Theorem 4 implies that OPF (7) is equivalent to W C(W G) s.t. W 2 Ŵ (20) where Ŵ is any one of the sets W,W c(g),w G. Specifically, given an optimal solution W opt in W, it can be uniquely decomposed into W opt = V opt (V opt ) H. Then V opt is in V and an optimal solution of OPF (7). Alternatively given an optimal solution W opt F in W c(g) or W G, Corollary 3 guarantees that W opt F has a unique psd rank-1 completion W opt in W from which an optimal V opt 2 V can be recovered. In fact given a partial matrix W G 2 W G (or W c(g) 2 W c(g) ) there is a more direct construction of a feasible solution V 2 V of OPF than through its completion; see Section IV-D. C. Semidefinite relaxations Hence solving OPF (7) is equivalent to solving (20) over any of W,W c(g),w G for an appropriate matrix variable. The difficulty with solving (20) is that the feasible sets W, W c(g), and W G are still nonconvex due to the rank-1 constraints and the cycle condition (13). Their removal leads to SDP, chordal, and SOCP relaxations of OPF respectively. Relax W, W c(g) and W G to the following convex supersets: W + := {W 2 S n+1 W G satisfies (12),W 0} W + c(g) := {W c(g) W G satisfies (12),W c(g) 0} W + G := {W G W G satisfies (12),W G ( j,k) 0, ( j,k) 2 E} Define the problems: OPF-sdp: OPF-ch: OPF-socp: W C(W G) s.t. W 2 W + (21) C(W G ) s.t. W c(g) 2 W + W c(g) (22) c(g) W G C(W G ) s.t. W G 2 W + G (23) The condition W G ( j,k) 0 in the definition of W + G is equivalent to [W G ] jk =[W G ] H kj and (recall the assumption v j > 0, j 2 N + ) [W G ] jj> 0, [W G ] kk > 0, [W G ] jj [W G ] kk [W G ] jk 2 This is a second-order cone and hence OPF-socp is indeed an SOCP in the rotated form. Remark 6: Literature. SOCP relaxation for OPF seems to be first proposed in [22] for the bus injection model (1), and in [51], [57] for the branch flow model (2) as explained in the next section. By defining a new set of variables v j := V j 2, R jk := V j V k cos(q j q k ), and I jk := V j V k sin(q j q k ) where q j := \V j, [22] rewrites the bus injection model (1) in the complex domain as a set of linear equations in these new variables in the real domain and the following quadratic equations: v j v k = R 2 jk + I2 jk Relaxing these equalities to v j v k R 2 jk + I2 jk enlarges the solution set to a second-order cone that is equivalent to W + G in this paper. SDP relaxation is first proposed in [23] for the bus injection model and analyzed in [24]. Chordal relaxation for OPF is first proposed in [25], [26] and analyzed in [27], [28]. D. Solution recovery When the convex relaxations OPF-sdp, OPF-ch, OPF-socp are exact, i.e., if their optimal solutions W sdp, Wch ch, W socp G happen to lie in W, W c(g), W G respectively, then an optimal solution V opt of the original OPF can be recovered from these

7 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH solutions. Indeed the recovery method works not just for an optimal solution, but any feasible solution that lies in W, W c(g) or W G. Moreover, given a W 2 W or a W c(g) 2 W c(g), the construction of V depends on W or W c(g) only through their submatrix W G. We hence describe the method for recovering the unique V from a W G, which may be a partial matrix in W G or the submatrix of a (partial) matrix in W or W c(g). Let T be an arbitrary spanning tree of G rooted at bus 0. Let P j denote the unique path from node 0 to node j in T. Recall that V 0 = 1\0 without loss of generality. For i = 1,...,n, let q V i := [W G ] ii \V i :=  \[W G ] jk ( j,k)2p i Then it can be checked that V is in (6) and feasible for OPF. E. Tightness of relaxations Since W W +, W c(g) W + c(g), W G 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 compare in terms of computational efficiency and tightness? OPF-socp is the simplest computationally. OPF-ch usually requires more computation than OPF-socp but much less than OPF-sdp for large sparse networks (even though OPF-ch can be as complex as OPF-sdp in the worse case [64], [65]). The relative tightness of the relaxations depends on the network topology. For a general mesh network OPF-sdp is as tight a relaxation as OPF-ch and they are strictly tighter than OPFsocp. For a tree (radial) network the hierarchy collapses and all three are equally tight. We now make this precise. Consider their feasible sets W +, W + c(g) and W+ G. We say that a 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 B that 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 implies A ' 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 the network is radial. This is a slightly different formulation of the same results in [58], [28]. Theorem 5: V v W + ' W + c(g) v W+ G. If G is a tree then V v W + ' W + c(g) ' W+ G. Let C opt,c sdp,c ch,c socp be the optimal values of OPF (7), OPF-sdp (21), OPF-ch (22), OPF-socp (23) respectively. Theorem 4 and Theorem 5 directly imply Corollary 6: C opt C sdp = C ch C socp. If G is a tree then C opt C sdp = C ch = C socp. Remark 7: Tightness. Theorem 5 and Corollary 6 imply that for radial networks one should always solve OPF-socp since it is the tightest and the simplest relaxation of the three. For mesh networks there is a tradeoff between OPF-socp and OPF-ch/OPF-sdp: the latter is tighter but requires heavier computation. Between OPF-ch and OPF-sdp, OPF-ch is usually preferable as they are equally tight but OPF-ch is usually much faster to solve for large sparse networks. See [25], [26], [28], [27], [67] for numerical studies that compare these relaxations. F. Chordal relaxation Theorem 2 through Corollary 6 apply to any chordal extension c(g) of G. The choice of c(g) does not affect the optimal value of the chordal relaxation but deteres its complexity. Unfortunately the optimal choice that imizes the complexity of OPF-ch is NP-hard to compute. This difficulty is due to two conflicting factors in choosing a c(g). Recall that the constraint W c(g) 0 in the definition of W + consists of multiple constraints that the principal c(g) submatrices W c(g) (q) 0, one for each (maximal) clique q of c(g). When two cliques q and q 0 share a node their submatrices W c(g) (q) and W c(g) (q 0 ) share entries that must be decoupled by introducing auxiliary variables and equality constraints on these variables. The choice of c(g) deteres the number and sizes of these submatrices W c(g) (q) as well as the numbers of auxiliary variables and additional decoupling constraints. On the one hand if c(g) contains few cliques q then the submatrices W c(g) (q) tend to be large and expensive to compute (e.g. if c(g) is the complete graph then there is a single clique, but W c(g) = W and OPF-ch is identical to OPF-sdp). On the other hand if c(g) contains many small cliques q then there tends to be more overlap and chordal relaxation tends to require more decoupling constraints. Hence choosing a good chordal extension c(g) of G is important but nontrivial. See [64], [65] and references therein for methods to compute efficient chordal relaxations of general QCQP. For OPF [27] proposes effective techniques to reduce the number of cliques in its chordal relaxation. To further reduce the problem size [67] proposes to carefully drop some of the decoupling constraints, though the resulting relaxation can be weaker. V. FEASIBLE SETS AND RELAXATIONS: BFM We now present an SOCP relaxation of OPF in BFM proposed in [51], [57] in two steps. We first relax the phase angles of V and I in (2) and then we relax a set of quadratic equalities to inequalities. This derivation pinpoints the difference between radial and mesh topologies. It motivates a recursive version of BFM for radial networks (Section VI) and the use of phase shifters for convexification of mesh networks (Part II [30]). A. Feasible sets Consider the following set of equations in the variables x := (S,`,v,s) in R 3(m+n+1) : 4  S jk =  (S ij z ij`ij )+s j, j 2 N + (24a) k: j!k i:i! j v j v k = 2Re z H jk S jk z jk 2` jk, j! k 2 Ẽ (24b) v j` jk = S jk 2, j! k 2 Ẽ (24c) 4 The use of complex variables is only a shorthand and should be interpreted as operations in real variables. For instance (24a) is a shorthand for  P jk =  k: j!k i:i! j P ij r ij`ij + p j  Q jk =  Q ij x ij`ij + q j k: j!k i:i! j

8 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH and define the solution set as: X nc := {x 2 R 3(m+n+1) x satisfies (3),(4),(24)} Note that the vector v includes v 0 and s includes s 0. The model (24) is first proposed in [45], [46]. 5 It can be derived as a relaxation of BFM (2) as follows. Taking the squared magnitude of (2c) and replacing V j 2 and I jk 2 by v j and ` jk respectively yield (24c). To obtain (24b), use (2b) (2c) to write V k = V j z jk S jk Vj 1 and take the squared magnitude on both sides to eliate the phase angles of V and I. These operations define a mapping h : C 2(m+n+1)! R 3(m+n+1) by: for any x =(S,I,V,s), h( x) :=(S,`,v,s) with ` jk = I jk 2 and v j = V j 2. Throughout this paper we assume the cost function C( x) in OPF (9) depends on x only through x := h( x). For example for total real line loss C( x)=â Rez ( j,k)2ẽ jk` jk. If the cost is a weighted sum of real generation power then C( x)=â j (c j p j + p d j ) where p j are the real parts of s j and p d j 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 subset of X nc, X v X nc, since h( X) X nc. We now characterize the subset of X nc that is equivalent to X. Given an x :=(S,`,v,s) 2 R 3(m+n+1) define b(x) 2 R m by b jk (x) := \ v j z H jk S jk, j! k 2 Ẽ (25) Even though x does not include phase angles of V, x implies a phase difference across each line j! k 2 Ẽ given by b jk (x). The subset of X nc that is equivalent to X are those x for which there exists q such that q j q k = b jk (x). To state this precisely let B be the m n (transposed) reduced incidence matrix of G: 8 >< 1 if edge l 2 Ẽ leaves node j B lj = 1 if edge l 2 Ẽ enters node j >: 0 otherwise where j 2 N. Consider the set of x 2 X nc 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 R 3(m+n+1) x satisfies (3),(4),(24),(26)} The following result characterizes the feasible set X of OPF in BFM and follows from [57, Theorems 2, 4]. Theorem 7: X X X nc. The bijection between X and X is given by h defined above restricted to X. Its inverse h 1 (S,`,v,s)=(S,I,V,s) is defined on X in terms of q(x) by: V j := p v j e iq j(x), j 2 N (27a) I jk := p` jk e i(q j(x) \S jk ), j! k 2 Ẽ (27b) The condition (26) is equivalent to the cycle condition (13) in the bus injection model. To see this fix any spanning tree 5 The original model, called the DistFlow equations, in [45], [46] is for radial (distribution) networks, but its extension here to mesh networks is trivial. T =(N,E T ) of the (directed) graph G. We can assume without loss of generality (possibly after re-labeling the links) that E T consists of links l = 1,...,n. Then B can be partitioned into B = apple BT where the n n submatrix B T 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) = apple bt (x) b? (x) The next result, proved in [57, Theorems 2 and 4], provides a more explicit characterization of (26) in terms of b(x). When it holds this characterization has the same interpretation of the cycle condition in (13): the voltage angle differences implied by x sum to zero (mod 2p) around any cycle. Formally let b be the extension of b from directed to undirected links: for each j! k 2 Ẽ let b jk (x) := b jk (x) and b kj (x) := b jk (x). We say c :=(j 1,..., j K ) is an undirected cycle if, for each k = 1,...,K, either j k! j k+1 2 Ẽ or j k+1! j k 2 Ẽ with the interpretation that j K+1 := j 1 ; ( j k, j k+1 ) 2 c denotes one of these links. Theorem 8: An x 2 X nc satisfies (26) if and only if around each undirected cycle c we have B? Â b jk (x) = 0 mod 2p (28) ( j,k)2c In that case q(x) =P B 1 T b T (x) is the unique solution of (26) in ( p,p] n, where P(f) projects f to ( p,p] n. Theorem 8 deteres when the voltage magnitudes v of a given x can be assigned phase angles q(x) so that the resulting x := h 1 (x) is a power flow solution in X. B. SOCP relaxation The set X nc that contains the (equivalent) feasible set X of OPF is still nonconvex because of the quadratic equalities in (24c). Relax them to inequalities: and define the set: v j ` jk S jk 2, ( j,k) 2 Ẽ (29) X + :={x 2 R 3(m+n+1) x satisfies (3),(4),(24a),(24b),(29)} Clearly X h X X nc X + ; see Figure 1. Moreover X + is a second-order cone in the rotated form. The three sets X, X nc, X + define the following problems: OPF: OPF-nc: OPF-socp: x C(x) subject to x 2 X (30) x C(x) subject to x 2 X nc (31) x C(x) subject to x 2 X + (32) The next theorem follows from the results in [57] and implies

9 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH C 2(m+n+1) X h h 1 R 3(m+n+1) Fig. 1: Feasible sets X of OPF (9) in BFM, its equivalent set X (defined by h) and its relaxations X nc and X +. If G is a tree then X = X nc. that OPF (9) is equivalent to imization over X and OPFsocp is its SOCP relaxation. Moreover for radial networks voltage and current angles can be ignored and OPF (9) is equivalent to OPF-nc. Theorem 9: X X X nc X +. If G is a tree then X X = X nc X +. Let C opt be the optimal cost of OPF (9) in the branch flow model. Let C opf, C nc, C socp be the optimal costs of OPF (30), OPF-nc (31), OPF-socp (32) respectively defined above. Theorem 9 implies Corollary 10: C opt = C opf C nc C socp. If G is a tree then C opt = C opf = C nc C socp. Remark 8: SOCP relaxation. Suppose one solves OPF-socp and obtains an optimal solution x socp :=(S,`,v,s) 2 X +. For radial networks if x socp attains equality in (29) then x socp 2 X nc and Theorem 9 implies that an optimal solution x opt := (S,I,V,s) 2 X of OPF (9) can be recovered from x socp. Indeed x opt = h 1 (x socp ) where h 1 is defined in (27). Alternatively one can use the angle recovery algorithms in [57, Part I] to recover x opt. For mesh networks x socp needs to both attain equality in (29) and satisfy the cycle condition (26) in order for an optimal solution x opt to be recoverable. Our experience with various practical test networks suggests that x socp usually attains equality in (29) but, for mesh networks, rarely satisfes (26) [51], [57], [56], [28]. Hence OPF-socp is effective for radial networks but not for mesh networks (in both BIM and BFM). C. Equivalence Theorem 9 establishes a bijection between X and the feasible set X of OPF (9) in BFM. Theorem 4 establishes a bijection between W G and the feasible set V of OPF (7) in BIM. Theorem 1 hence implies that X X V W G. Moreover their SOCP relaxations are equivalent in these two models [58], [28]. Define the set of partial matrices defined on G that are 2 2 psd rank-1 but do not satisfy the cycle condition (13): X X nc W nc := {W nc W G satisfies (12),W G ( j,k) 0, X + rank W G ( j,k)=1 for all ( j,k) 2 E} Clearly W G W nc W + G in general and W G = W nc W + G for radial networks. Theorem 11: X W G, X nc W nc and X + W + G. The bijection between X + and W + G is defined as follows. Let W G C 2m+n+1 denote the set of Hermitian partial matrices (including [W G ] 00 = v 0 which is given). Let x :=(S,`,v,s) denote vectors in R 3(m+n+1). Define the linear mapping g : W G! X + by x = g(w G ) where S jk := y H jk [W G] jj [W G ] jk, j! k ` jk := y jk 2 [W G ] jj +[W G ] kk [W G ] jk [W G ] kj, j! k v j := [W G ] jj, j 2 N + s j := Â y H jk [W G] jj [W G ] jk, j 2 N + k: j k Its inverse g 1 : X +! W G is W G = g 1 (x) where [W G ] jj := v j for j 2 N + and [W G ] jk := v j z H jk S jk =: [W G ] H kj for j! k. The mapping g and its inverse g 1 restricted to W G (W nc ) and X (X nc ) define the bijection between them. VI. BFM FOR RADIAL NETWORKS Theorem 9 implies that for radial networks the model (24) is exact. This is because the reduced incident matrix B in (26) is n n and invertible, so the cycle condition is always satisfied [57, Theorem 4]. Hence a solution in X nc can be mapped to a branch flow solution in X by the mapping h 1 defined in (27). For radial networks this model has two advantages: (i) it has a recursive structure that simplifies computation, and (ii) it has a linear approximation that provides simple bounds on branch powers S jk and voltage magnitudes v j, as we now show. A. Recursive equations and graph orientation The model (24) holds for any graph orientation of G. It has a recursive structure when G is a tree. In that case different orientations have different boundary conditions that initialize the recursion and may be convenient for different applications. Without loss of generality we take bus 0 as the root of the tree. We discuss two different orientations: one where every link points away from bus 0 and the other where every link points towards bus 0. 6 Case I: Links point away from bus 0. Model (24) reduces to: Â S jk = S ij z ij`ij + s j, j 2 N + (33a) k: j!k v j v k = 2Re z H jk S jk z jk 2` jk, j! k 2 Ẽ (33b) v j` jk = S jk 2, j! k 2 Ẽ (33c) where bus i in (33a) denotes the unique parent of node j (on the unique path from node 0 to node j), with the understanding that if j = 0 then S i0 := 0 and `i0 := 0. Similarly when j is a leaf node 7 all S jk = 0 in (33a). The model (33) is called the DistFlow equations and first proposed in [45], [46]. Its recursive structure is exploited in [47] to analyze the power flow solutions given an (s j, j 2 N), as we now explain using the special case of a linear network with n + 1 buses that represents a main feeder. To simplify notation denote 6 An alternative model is to use an undirected graph and, for each link ( j,k), the variables (S jk,`jk ) and (S kj,`kj ) are defined for both directions, with the additional equations S jk + S kj = z jk` jk and `kj = ` jk. 7 A node j is a leaf node if there exists no i such that i! j 2 Ẽ.

10 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH 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 (v 0 is given): S j+1 = S j z j` j + s j+1, j = 0,...,n 1 (34a) v j+1 = v j 2Re(z H j S j ) z j 2` j, j = 0,...,n 1 (34b) v j` j = S j 2, j = 0,...,n 1 (34c) S 0 = s 0, S n = 0 (34d) Let x j :=(S j,`j,v j ), j 2 N +. If s 0 were known then one can start with (v 0,s 0 ) and use the recursion (34a) (34c) to compute x j in terms of s 0 = S 0, i.e., (34) can be collapsed into functions of the scalar variable s 0 (recall that (s j, j 2 N) are given): x j = f j (s 0 ), j 2 N + (35) Use the boundary condition (34d) S n = f n (s 0 )=0 to solve for the scalar variable s 0. The other variables x j can then be computed from (35). This method can be extended to a 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 the forward/backward sweep for radial networks. Case II: Links point towards bus 0. Model (24) reduces to: Ŝ ji =  k:k! j Ŝ kj z kjˆ`kj + s j, j 2 N + (36a) ˆv k ˆv j = 2Re z H kjŝkj z kj 2 ˆ`kj, k! j 2 Ẽ (36b) ˆv k ˆ`kj = Ŝ kj 2, k! j 2 Ẽ (36c) where i in (36a) denotes the node on the unique path between node 0 and node j. The boundary condition is defined by S ji = 0 in (36a) when j = 0 and S kj = 0,`kj = 0 in (36a) when j is a leaf node. An advantage of this orientation is illustrated in the next subsection in proving a simple bound on ˆv j. B. Linear approximation and bounds By setting ` jk = 0 in (33) we obtain a linear approximation of the the branch flow model, with the graph orientation where all links point away from bus 0:  S lin jk = Sij lin + s j, j 2 N + (37a) k: j!k v lin j v lin k = 2Re z H jk Slin jk, j! k 2 E (37b) where bus i in (37a) denotes the unique parent of bus j. The boundary condition is: Si0 lin := 0 in (37a) when j = 0, and Slin jk = 0 in (37a) when j is a leaf node. This is called the simplified DistFlow equations in [46], [70]. It is a good approximation of (33) because the loss z jk` jk is typically much smaller than the branch power flow S jk. The next result provides simple bounds on (S,v) in terms of their linear approximations (S lin,v lin ). Denote by T j the subtree rooted at bus j, including j. We write k 2 T j to mean node k of T j and (k,l) 2 T j to mean edge (k,l) of T j. Denote by P k the set of links on the unique path from bus 0 to bus k. Lemma 12: Fix any v 0 and s 2 R 2(n+1). Let (S,`,v) and (S lin,v lin ) be solutions of (33) and (37) respectively with the given v 0 and s. Then (1) For i! j 2 E S lin ij =  k2t j s k 0 S ij =  s k ij`ij +  z kl`kl A k2t j (k,l)2t j (2) For i! j 2 E, S ij Sij lin with equality if only if `ij and all `kl in T j are zero. (3) For j 2 N + v lin j = v 0  2Re z H ik Slin ik (i,k)2p j v j = v 0  (i,k)2p j 2Re z H ik S ik z ik 2`ik (4) For j 2 N +, v j apple v lin j. Lemma 12 says that the power flow S ij on line (i, j) equals the total load  k2t j s k in the subtree rooted at node j plus the total line loss in supplying these loads. The linear approximation Sij lin neglects the line loss and underestimates the required power to supply these loads. Lemma 12(1) (3) can be easily proved by recursing on (33a) (33b) and (37). Since S ij Sij lin but z ik 2`ik 0, a direct proof of Lemma 12(4) is not obvious. Instead, one can make use of Lemma 13 below and define a bijection between the solutions (S,`,v) of (33) and the solutions (Ŝ, ˆ`, ˆv) of (36) in which v = ˆv. It can be checked that the solutions of (37) and those of (38) are related by S lin = Lemma 13(4) implies Lemma 12(4). A linear approximation of (36) is (setting ˆ`kj = 0): ˆv lin k Ŝ lin ji =  k:k! j ˆv lin j = 2Re 1 Ŝ lin and v lin = ˆv lin. Then Ŝ lin kj + s j, j 2 N + (38a) z H kjŝlin, k! j 2 Ẽ (38b) Lemma 13: Fix any v 0 and s 2 R 2(n+1). Let (Ŝ, ˆ`, ˆv) and (Ŝ lin, ˆv lin ) be solutions of (36) and (38) respectively with the given v 0 and s. Then (1) For all j! i 2 E Ŝ lin ji =  k2t j s k Ŝ ji =  k2t j s k  (k,l)2t j z kl ˆ`kl (2) For all j! i 2 E, Ŝ ji apple Ŝ lin ji with equality if and only if all `kl in T j are zero. (3) For j 2 N + ˆv lin j = v 0 +  2Re z H ikŝlin (i,k)2p j ˆv j = v 0 +  (i,k)2p j 2Re z H ikŝik z ik 2 ˆ`ik (4) For j 2 N +,ˆv j apple ˆv lin j. Lemma 13 says that the branch power Ŝ ji (towards bus 0) equals the total injection  k2t j s k in the subtree rooted at bus

11 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15 27, MARCH j us the line loss in that subtree. The linear approximation Ŝ lin ji neglects the loss and hence overestimates the branch power flow. It can be proved by recursing on (36a) (36b) and (38). Remark 9: Bounds for SOCP relaxation. Lemmas 12 and 13 do not depend on the quadratic equalities (33c) and (36c) as long as ` jk 0. In particular the lemmas hold if the equalities have been relaxed to inequalities v j` jk S jk 2. These bounds are used in [56] to prove a sufficient condition for exact SOCP relaxation for radial networks. Remark 10: Linear approximations. For radial networks the linear approximations (37) and (38) of BFM have two advantages over the (linear) DC approximation of BIM. First they have a simple recursive structure that leads to simple bounds on power flow quantities. Second DC approximation assumes 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 much higher than in transmission systems, voltages can fluctuate significantly and reactive powers are used to regulate them. On the other hand (37) and (38) are applicable only for radial networks whereas DC approximation applies to mesh networks as well. See also [21] for a more accurate linearization of BIM that addresses the shortcogs of DC OPF. VII. CONCLUSION We have presented a bus injection model and a branch flow model, formulated several relaxations of OPF, and proved their relations. These results suggest a new approach to solving OPF summarized in Figure 2. For radial networks we recommend OPFBsocp) W G socp 2x2)rankB1) Y,)mesh) Y) radial) cycle) condi2on) Y) OPFBch) ch W c(g) V opt rankb1) Y) Recover) or OPF)solu2on) OPFBsdp) x opt W sdp Y) radial) Y) OPFBsocp) x socp equality) Y,)mesh) cycle) condi2on) Fig. 2: Solving OPF through semidefinite relaxations. solving OPF-socp in either BIM or BFM though there is preliary evidence that BFM can be more stable numerically. For mesh networks we recommend solving OPF-ch for small networks and OPF-socp followed by a heuristic search for a feasible point for large networks. The key for this solution strategy is that the relaxations are exact so that an optimal solution of the original OPF can be recovered. In Part II of this paper [30] we summarize sufficient conditions that guarantee exact relaxation. Acknowledgment. We thank the support of NSF through NetSE CNS , ARPA-E through GENI DE-AR , the National Science Council of Taiwan through NSC P , Southern California Edison, the Los Alamos National Lab and Caltech s Resnick Institute. We thank the anonymous reviewers for their helpful suggestions. REFERENCES [1] S. H. Low. Convex relaxation of optimal power flow: a tutorial. In IREP Symposium Bulk Power System Dynamics and Control (IREP), Rethymnon, Greece, August [2] J. Carpentier. Contribution to the economic dispatch problem. Bulletin de la Societe Francoise des Electriciens, 3(8): , In French. [3] H.W. Dommel and W.F. Tinney. Optimal power flow solutions. Power Apparatus and Systems, IEEE Transactions on, PAS-87(10): , Oct [4] J. A. Momoh. Electric Power System Applications of Optimization. Power Engineering. Markel Dekker Inc.: New York, USA, [5] M. Huneault and F. D. Galiana. A survey of the optimal power flow literature. IEEE Trans. on Power Systems, 6(2): , [6] J. A. Momoh, M. E. 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Quadratically constrained quadratic programs on acyclic graphs with application to power flow. arxiv: v1, March [63] R. Grone, C. R. Johnson, E. M. Sa, and H. Wolkowicz. Positive definite completions of partial Hermitian matrices. Linear Algebra and its Applications, 58: , [64] Mituhiro Fukuda, Masakazu Kojima, Kazuo Murota, and Kazuhide Nakata. Exploiting sparsity in semidefinite programg via matrix completion I: General framework. SIAM Journal on Optimization, 11: , [65] K. Nakata, K. Fujisawa, M. Fukuda, M. Kojima, and K. Murota. Exploiting sparsity in semidefinite programg via matrix completion II: Implementation and numerical results. Mathematical Programg, 95(2): , [66] Baosen Zhang and David Tse. Geometry of the injection region of power networks. IEEE Trans. Power Systems, 28(2): , [67] Martin S. Andersen, Anders Hansson, and Lieven Vandenberghe. Reduced-complexity semidefinite relaxations of optimal power flow problems. arxiv: v1, August [68] R.D. Zimmerman and Hsiao-Dong Chiang. Fast decoupled power flow for unbalanced radial distribution systems. Power Systems, IEEE Transactions on, 10(4): , [69] M.S. Srinivas. Distribution load flows: a brief review. In Power Engineering Society Winter Meeting, IEEE, volume 2, pages vol.2, [70] M.E. Baran and F.F. Wu. Network reconfiguration in distribution systems for loss reduction and load balancing. IEEE Trans. on Power Delivery, 4(2): , Apr Steven H. Low (F 08) is a Professor of the Computing+Mathematical Sciences and Electrical Engineering at Caltech, and a Changjiang Chair Professor of Zhejiang University. Before that, he was with AT&T Bell Labs, Murray Hill, NJ, and the University of Melbourne, Australia. He was a co-recipient of IEEE best paper awards, the R&D 100 Award, and an Okawa Foundation Research Grant. He is on the Technical Advisory Board of Southern California Edison and was a member of the Networking and Information Technology Technical Advisory Group for the US President s Council of Advisors on Science and Technology (PCAST). He is a Senior Editor of the IEEE Journal on Selected Areas in Communications, the IEEE Trans. Control of Network Systems, and the IEEE Trans. Network Science & Engineering, and is on the editorial board of NOW Foundations and Trends in Networking, and in Electric Energy Systems. He is an IEEE Fellow and received his BS from Cornell and PhD from Berkeley both in EE. His research interests are in power systems and communication networks.

13 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE Convex Relaxation of Optimal Power Flow Part II: Exactness Steven H. Low EAS, Caltech slow@caltech.edu Abstract This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact. I. INTRODUCTION The optimal power flow (OPF) problem is fundamental in power systems as it underlies many applications such as economic dispatch, unit commitment, state estimation, stability and reliability assessment, volt/var control, demand response, etc. OPF seeks to optimize a certain objective function, such as power loss, generation cost and/or user utilities, subject to Kirchhoff s laws as well as capacity, stability and security constraints on the voltages and power flows. There has been a great deal of research on OPF since Carpentier s first formulation in 1962 [1]. Recent surveys can be 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 large number of optimization algorithms and relaxations have been proposed. To the best of our knowledge solving OPF through semidefinite relaxation is first proposed in [14] as a secondorder cone program (SOCP) for radial (tree) networks and in [15] as a semidefinite program (SDP) for general networks in a bus injection model. It is first proposed in [16], [17] as an SOCP for radial networks in the branch flow model of [18], [19]. While these convex relaxations have been illustrated numerically in [14] and [15], whether or when they will turn out to be exact is first studied in [20]. Exploiting graph sparsity to simplify the SDP relaxation of OPF is first proposed in [21], [22] and analyzed in [23], [24]. Solving OPF through convex relaxation offers several advantages, as discussed in Part I of this tutorial [25, Section I]. In particular it provides the ability to check if a solution is globally optimal. If it is not, the solution provides a lower bound on the imum cost and hence a bound on Acknowledgment: We thank the support of NSF through NetSE CNS , ARPA-E through GENI DE-AR , Southern California Edison, the National Science Council of Taiwan through NSC P , the Los Alamos National Lab (DoE), and Caltech s Resnick Institute. A preliary and abridged version has appeared in Proceedings of the IREP Symposium - Bulk Power System Dynamics and Control - IX, Rethymnon, Greece, August 25-30, how far any feasible solution is from optimality. Unlike approximations, if a relaxed problem is infeasible, it is a certificate that the original OPF is infeasible. This tutorial presents main results on convex relaxations of OPF developed in the last few years. In Part I [25], we present the bus injection model (BIM) and the branch flow model (BFM), formulate OPF within each model, and prove their equivalence. The complexity of OPF formulated here lies in the quadratic nature of power flows, i.e., the nonconvex quadratic constraints on the feasible set of OPF. We characterize these feasible sets and design convex supersets that lead to three different convex relaxations based on semidefinite programg (SDP), chordal extension, and second-order cone programg (SOCP). When a convex relaxation is exact, an optimal solution of the original nonconvex OPF can be recovered from every optimal solution of the relaxation. In Part II we summarize main sufficient conditions that guarantee the exactness of these relaxations. Network topology turns out to play a critical role in detering whether a relaxation is exact. In Section II we review the definitions of OPF and their convex relaxations developed in [25]. We also define the notion of exactness adopted in this paper. In Section III we present three types of sufficient conditions for these relaxations to be exact for radial networks. These conditions are generally not necessary and they have implications on allowable power injections, voltage magnitudes, or voltage angles: A Power injections: These conditions require that not both constraints on real and reactive power injections be binding at both ends of a line. B Voltages magnitudes: These conditions require that the upper bounds on voltage magnitudes not be binding. They can be enforced through affine constraints on power injections. C Voltage angles: These conditions require that the voltage angles across each line be sufficiently close. This is needed also for stability reasons. These conditions and their references are summarized in Tables I and II. Some of these sufficient conditions are proved using BIM and others using BFM. Since these two models are equivalent (in the sense that there is a linear bijection between their solution sets [24], [25]), these sufficient conditions apply to both models. The proofs of these conditions typically do not require that the cost function be convex (they focus

14 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE type condition model reference remark A power injections BIM, BFM [26], [27], [28], [29], [30] [31], [16], [17] B voltage magnitudes BFM [32], [33], [34], [35] allows general injection region C voltage angles BIM [36], [37] makes use of branch power flows TABLE I: Sufficient conditions for radial (tree) networks. network condition reference remark with 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 voltages type B [40], [41] assumes nonnegative voltages TABLE II: Sufficient conditions for mesh networks on the feasible sets and usually only need the cost function to be monotonic). Convexity is required however for efficient computation. Moreover it is proved in [35] using BFM that when the cost function is convex then exactness of the SOCP relaxation implies uniqueness of the optimal solution for radial networks. Hence the equivalence of BIM and BFM implies that any of the three types of sufficient conditions guarantees that, for a radial network with a convex cost function, there is a unique optimal solution and it can be computed by solving an SOCP. Since the SDP and chordal relaxations are equivalent to the SOCP relaxation for radial networks [24], [25], these results apply to all three types of relaxations. Empirical evidences suggest some of these conditions are likely satisfied in practice. This is important as most power distribution systems are radial. These conditions are insufficient for general mesh networks because they cannot guarantee that an optimal solution of a relaxation satisfies the cycle condition discussed in [25]. In Section IV we show that these conditions are however sufficient for mesh networks that have tunable phase shifters at strategic locations. The phase shifters effectively make a mesh network behave like a radial network as far as convex relaxation is concerned. The result can help detere if a network with a given set of phase shifters can be convexified and, if not, where additional phase shifters are needed for convexification. These conditions are also sufficient for direct current (dc) mesh networks where all variables are in the real rather than complex domain. Counterexamples are known where SDP relaxation is not exact, especially for AC mesh networks without tunable phase shifters [42], [43]. We discuss three recent approaches for global optimization of OPF when the semidefinite relaxations discussed in this tutorial fail. We conclude in Section V. All proofs are omitted and can be found in the original papers or the arxiv version of this tutorial. II. OPF AND ITS RELAXATIONS We use the notations and definitions from Part I of this paper. In this section we summarize the OPF problems and their relaxations developed there; see [25] for details. We adopt in this paper a strong sense of exactness where we require the optimal solution set of the OPF problem and that of its relaxation be equivalent. This implies that an optimal solution of the nonconvex OPF problem can be recovered from every optimal solution of its relaxation. This is important because it ensures any algorithm that solves an exact relaxation always produces a globally optimal solution to the OPF problem. Indeed interior point methods for solving SDPs tend to produce a solution matrix with a maximum rank [44], so can miss a rank-1 solution if the relaxation has non-rank-1 solutions as well. It can be difficult to recover an optimal solution of OPF from such a nonrank-1 solution, and our definition of exactness avoids this complication. See Section II-C for detailed justifications. A. Bus injection model The BIM adopts an undirected graph G 1 and can be formulated in terms of just the complex voltage vector V 2 C n+1. The feasible set is described by the following constraints: s j apple  y H jk V j(v j H k:( j,k)2e V H k ) apple s j, j 2 N + (1a) v j apple V j 2 apple v j, j 2 N + (1b) where s j,s j,v j,v j, possibly ± ± i, are given bounds on power injections and voltage magnitudes. Note that the vector V includes V 0 which is assumed given (v 0 = v 0 and \V 0 = 0 ) unless otherwise specified. The problem of interest is: OPF: C(V ) subject to V satisfies (1) (2) V 2Cn+1 1 We will use bus and node interchangeably and line and link interchangeably.

15 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE For relaxations consider the partial matrix W G defined on the network graph G that satisfies s j apple  y H jk [W G ] jj [W G ] jk apple s j, j 2 N + (3a) k:( j,k)2e v j apple [W G ] jj apple v j, j 2 N + (3b) We say that W G satisfies the cycle condition if for every cycle c in G  ( j,k)2c \[W G ] jk = 0 mod 2p (4) We assume the cost function C depends on V only through VV H and use the same symbol C to denote the cost in terms of a full or partial matrix. Moreover we assume C depends on the matrix only through the submatrix W G defined on the network graph G. See [25, Section IV] for more details including the definitions of W c(g) 0 and W G ( j,k) 0. Define the convex relaxations: OPF-sdp: C(W G) subject to W G satisfies (3),W 0 (5) W2S n+1 OPF-ch: C(W G ) subject to W G satisfies (3),W c(g) 0 (6) W c(g) OPF-socp: W G C(W G ) subject to W G satisfies (3), W G ( j,k) 0, ( j,k) 2 E (7) For BIM, we say that OPF-sdp (5) is exact if every optimal solution W sdp of OPF-sdp is psd rank-1; OPF-ch (6) is exact if every optimal solution Wc(G) ch of OPF-ch is psd rank-1 (i.e., the principal submatrices Wc(G) ch (q) of W ch c(g) are psd rank-1 for all maximal cliques q of the chordal extension c(g) of graph G); OPF-socp (7) is exact if every optimal solution W socp G of OPF-socp is 2 2 psd rank-1 and satisfies the cycle condition (4). To recover an optimal solution V opt of OPF (2) from W sdp or Wc(G) ch socp or WG, see [25, Section IV-D]. B. Branch flow model The BFM adopts a directed graph G and is defined by the following set of equations:  S jk k: j!k =  S ij z ij I ij 2 + s j, j 2 N + (8a) i:i! j I jk = y jk (V j V k ), j! k 2 Ẽ (8b) S jk = V j I H jk, j! k 2 Ẽ (8c) Denote the variables in BFM (8) by x := (S, I,V, s) 2 C 2(m+n+1). Note that the vectors V and s include V 0 (given) and s 0 respectively. Recall from [25] the variables x := (S,`,v,s) 2 R 3(m+n+1) that is related to x by the mapping x = h( x) with ` jk := I jk 2 and v j := V j 2. The operational constraints are: v j apple v j apple v j, j 2 N + (9a) s j apple s j apple s j, j 2 N + (9b) We assume the cost function depends on x only through x = h( x). Then the problem in BFM is: OPF: x C(x) subject to x satisfies (8),(9) (10) For SOCP relaxation consider:  S jk =  (S ij z ij`ij )+s j, j 2 N + (11a) k: j!k i:i! j v j v k = 2Re z H jk S jk z jk 2` jk, j! k 2 Ẽ (11b) v j` jk S jk 2, j! k 2 Ẽ (11c) We say that x satisfies the cycle condition if 9q 2 R n such that Bq = b(x) mod 2p (12) where B is the m n reduced incidence matrix and, given x :=(S,`,v,s), b jk (x) := \(v j z H jk S jk) can be interpreted as the voltage angle difference across line j! k implied by x (See [25, Section V]). The SOCP relaxation in BFM is OPF-socp: x C(x) subject to x satisfies (11),(9) (13) For BFM, OPF-socp (13) in BFM is exact if every optimal solution x socp attains equality in (11c) and satisfies the cycle condition (12). See [25, Section V-A] for how to recover an optimal solution x opt of OPF (10) from any optimal solution x socp of its SOCP relaxation. C. Exactness The definition of exactness adopted in this paper is more stringent than needed. Consider SOCP relaxation in BIM as an illustration (the same applies to the other relaxations in BIM and BFM). For any sets A and B, we say that A is equivalent to B, denoted by A B, if there is a bijection between these two sets. Let M(A) denote the set of imizers when a certain function is imized over A. Let V and W + G denote the feasible sets of OPF (2) and OPF-socp (7) respectively: V := {V 2 C n+1 V satisfies (1)} W + G := {W G W G satisfies (3),W G ( j,k) 0,( j,k) 2 E} Consider the following subset of W + G : W G := {W G W G satisfies (3),(4),W G ( j,k) 0, rank W G ( j,k)=1,( j,k) 2 E} Our definition of exact SOCP relaxation is that M(W + G ) W G. In particular, all optimal solutions of OPF-socp must be 2 2 psd rank-1 and satisfy the cycle condition (4). Since

16 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE W G V (see [25]), exactness requires that the set of optimal solutions of OPF-socp (7) be equivalent to that of OPF (2), i.e., M(W + G )=M(W G) M(V). If M(W + G ) )M(W G) M(V) then OPF-socp (7) is not exact according to our definition. Even in this case, however, every sufficient condition in this paper guarantees that an optimal solution of OPF can be easily recovered from an optimal solution of the relaxation that is outside W G. The difference between M(W + G )=M(W G) and M(W + G ) )M(W G) is often or, depending on the objective function; see Remarks 1 and 2 and comments after Theorems 5 and 8 in Section III. Hence we adopt the more stringent definition of exactness for simplicity. III. RADIAL NETWORKS In this section we summarize the three types of sufficient conditions listed in Table I for semidefinite relaxations of OPF to be exact for radial (tree) networks. These results are important as most distribution systems are radial. For radial networks, if SOCP relaxation is exact then SDP and chordal relaxations are also exact (see [25, Theorems 5, 9]). We hence focus in this section on the exactness of OPF-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 and OPF-socp (13) is exact if all of its optimal solutions attain equalities in (11c). We will freely use either BIM or BFM in discussing these results. To avoid triviality we make the following assumption throughout the paper: The voltage lower bounds satisfy v j > 0, j 2 N +. The original problems OPF (2) and (10) are feasible. A. Linear separability We will first present a general result on the exactness of the SOCP relaxation of general QCQP and then apply it to OPF. This result is first formulated and proved using a duality argument in [27], generalizing the result of [26]. It is proved using a simpler argument in [31]. Fix an undirected graph G =(N +,E) where N + := {0,1,...,n} and E N + N +. Fix Hermitian matrices C l 2 S n+1, l = 0,...,L, defined on G, i.e., [C l ] jk = 0 if ( j,k) 62 E. Consider QCQP: x H C 0 x (14a) x2c n+1 subject to x H C l x apple b l, l = 1,...,L (14b) where C 0,C l 2 C (n+1) (n+1), b l 2 R, l = 1,...,L, and its SOCP relaxation where the optimization variable ranges over Hermitian partial matrices W G : W G tr C 0 W G (15a) subject to tr C l W G apple b l, l = 1,...,L (15b) W G ( j,k) 0, ( j,k) 2 E (15c) The following result is proved in [27], [31]. It can be regarded as an extension of [45] on the SOCP relaxation of QCQP from the real domain to the complex domain. Consider: 2 A1: The cost matrix C 0 is positive definite. A2: For each link ( j,k) 2 E there exists an a jk such that \[C l ] jk 2 [a ij,a ij + p] for all l = 0,...,L. Let C opt and C socp denote the optimal values of QCQP (14) and SOCP (15) respectively. Theorem 1: Suppose G is a tree and A2 holds. Then C opt = C socp and an optimal solution of QCQP (14) can be recovered from every optimal solution of SOCP (15). Remark 1: The proof of Theorem 1 prescribes a simple procedure to recover an optimal solution of QCQP (14) from any optimal solution of its SOCP relaxation (15). The construction does not need the optimal solution of SOCP (15) to be 2 2 rank-1. Hence the SOCP relaxation may not be exact according to our definition of exactness, i.e., some optimal solutions of (15) may be 2 2 psd but not 2 2 rank-1. If the objective function is strictly convex however then the optimal solution sets of QCQP (14) and SOCP (15) are indeed equivalent. Corollary 2: Suppose G is a tree and A1 A2 hold. Then SOCP (15) is exact. We now apply Theorem 1 to our OPF problem. Recall that OPF (2) in BIM can be written as a standard form QCQP [27]: x2c n V H C 0 V s.t. V H F j V apple p j, V H ( F j )V apple p j (16a) V H Y j V apple q j, V H ( Y j )V apple q j (16b) V H J j V apple v j, V H ( J j )V apple v j for some Hermitian matrices C 0,F j,y j,j j where j 2 N +. A2 depends only on the off-diagonal entries of C 0, F j, Y j (J j are diagonal matrices). It implies a simple pattern on the power injection constraints (16a) (16b). Let y jk = g jk ib jk with g jk > 0,b jk > 0. Then we have (from [27]): 8 1 >< 2 Y ij = 1 2 (g ij ib ij ) if k = i [F k ] ij = 1 2 >: Y ij H = 1 2 (g ij+ ib ij ) if k = j 0 if k 62 {i, j} [Y k ] ij = 8 1 >< 2i Y ij = 1 2 (b ij+ ig ij ) if k = i 1 2i >: Y ij H = 1 2 (b ij ig ij ) if k = j 0 if k 62 {i, j} Hence for each line ( j,k) 2 E the relevant angles for A2 are ( p,p]. 2 All angles should be interpreted as mod 2p, i.e., projected onto

17 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE those of [C 0 ] jk and [F j ] jk = [F k ] jk = [Y j ] jk = 1 2 (g jk ib jk ) 1 2 (g jk + ib jk ) 1 2 (b jk + ig jk ) [Y k ] jk = 1 2 (b jk ig jk ) as well as the angles of [F j ] jk, [F k ] jk and [Y j ] jk, [Y k ] jk. These quantities are shown in Figure 1 with their magnitudes normalized to a common value and explained in the caption of the figure. "# Φ j $ %jk [ Φ k ] jk Im #$ Φ j % &jk [ Φ k ] jk #$ % &jk [ Ψ k ] Ψ j jk [C 0 ] jk α jk "# Ψ j $ %jk [ Ψ k ] jk upper)bounds) on))p j, q j, p k, q k Re lower)bounds) on))p j, q j, p k, q k Fig. 1: Condition A2 on a line ( j,k) 2 E. The quantities ([F j ] jk,[f k ] jk,[y j ] jk,[y k ] jk ) on the left-half plane correspond to finite upper bounds on (p j, p k,q j,q k ) in (16a) (16b); ( [F j ] jk, [F k ] jk, [Y j ] jk, [Y k ] jk ) on the right-half plane correspond to finite lower bounds on (p j, p k,q j,q k ). A2 is satisfied if there is a line through the origin, specified by the angle a jk, so that the quantities corresponding to finite upper or lower bounds on (p j, p k,q j,q k ) lie on one side of the line, possibly on the line itself. The load oversatisfaction condition in [26], [30] corresponds to the Imaxis that excludes all quantities on the right-half plane. The sufficient condition in [29, Theorem 2] corresponds to the red line in the figure that allows a finite lower bound on the real power at one end of the line, i.e., p j or p k but not both, and no finite lower bounds on reactive powers q j and q k. 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 complex plane through the origin such that [C 0 ] jk as well as those ±[F i ] jk and ±[Y i ] jk corresponding to finite lower or upper bounds on (p i,q i ), for i = j,k, are all on one side of the line, possibly on the line itself. Let C opt and C socp denote the optimal values of OPF (2) and OPF-socp (7) respectively. Corollary 3: Suppose G is a tree and A2 holds. 1) C opt =C socp. Moreover an optimal solution V opt of OPF (2) can be recovered from every optimal solution W socp G 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 satisfied if there is a line where both the real and reactive power injections at both ends are both lower and upper bounded (8 combinations as shown in the figure). A2 requires that some of them be unconstrained even though in practice they are always bounded. It should be interpreted as requiring that the optimal solutions obtained by ignoring these bounds turn out to satisfy these bounds. This is generally different from solving the optimization with these constraints but requiring that they be inactive (strictly within these bounds) at optimality, unless the cost function is strictly convex. The result proved in [27] also includes constraints on real branch power flows and line losses. Corollary 3 includes several sufficient conditions in the literature for exact relaxation as special cases; see the caption of Figure 1. Corollary 3 also implies a result first proved in [16], using a different technique, that SOCP relaxation is exact in BFM for radial networks when there are no lower bounds on power injections s j. The argument in [16] is generalized in [17, Part I] to allow convex objective functions, shunt elements, and line limits in terms of upper bounds on ` jk. Assume A3: The cost function C(x) is convex, strictly increasing in `, nondecreasing in s =(p, q), and independent of branch flows S =(P,Q). A4: For j 2 N +, s j = i. Popular cost functions in the literature include active power loss over the network or active power generations, both of which satisfy A3. The next result is proved in [16], [17]. Theorem 4: Suppose G is a tree and A3 A4 hold. Then OPF-socp (13) is exact. Remark 2: If the cost function C(x) in A3 is only nondecreasing, rather than strictly increasing, in `, then A3 A4 still 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 maintains strict inequalities in (11c) and hence is infeasible for OPF. Even though OPF-socp is not exact in this case, the proof of 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 subsection) require that some power injection constraints not be binding, type B conditions require non-binding voltage upper bounds. They are proved in [32], [33], [34], [35] using BFM.

18 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE For radial networks the model originally proposed in [18], [19], which is (11) with the inequalities in (11c) replaced by equalities, is exact. This is because the cycle condition (12) is always satisfied as the reduced incidence matrix B is n n and invertible for radial networks. Following [35] we adopt the graph orientation where every link points towards node 0. Then (11) for a radial network reduces to: S jk = Â (S ij z ij`ij )+s j, j 2 N + (17a) i:i! j v j v k = 2Re z H jk S jk z jk 2` jk, j! k 2 Ẽ (17b) v j` jk S jk 2, j! k 2 Ẽ (17c) where v 0 is given and in (17a), k denotes the node on the unique path from node j to node 0. The boundary condition is: S jk := 0 when j = 0 in (17a) and S ij = 0, `ij = 0 when j is a leaf node. 3 As before the voltage magnitudes must satisfy: v j apple v j apple 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 bounded above: s j 2 S j {s j 2 C s j apple s j }, j 2 N (18b) for some given s j, j 2 N. 4 Then the SOCP relaxation is OPF-socp: x C(x) subject to (17),(18) (19) As defined in Section II-C, OPF-socp (19) is exact if every optimal solution x socp attains equality in (17c). In that case an optimal solution of BFM (10) can be uniquely recovered from x socp. We make two comments on the constraint sets S j in (18b). First S j need not be convex nor even connected for convex relaxations to be exact. They (only) need to be convex to be efficiently computable. Second such a general constraint on s is useful in many applications. It includes the case where s j are subject to simple box constraints, but also allows constraints of the form s j 2 apple a, \s j applef j that is useful for volt/var control [46], or q j 2{0,a} for capacitor configurations. Geometric insight. To motivate our sufficient condition, we first explain a simple geometric intuition using a two-bus network on why relaxing voltage upper bounds guarantees exact SOCP relaxation. Consider bus 0 and bus 1 connected by a line with impedance z := r + ix. Suppose without loss of generality that v 0 = 1 pu. Eliating S 01 = s 0 from (17), the model reduces to (dropping the subscript on `01 ): p 0 r` = p 1, q 0 x` = q 1, p q 2 0 = ` (20) 3 A node j 2 N is a leaf node if there is no i such that i! j 2 Ẽ. 4 We assume here that s 0 is unconstrained, and since V 0 := 1\0 pu, the constraints (18) involve only j in N, not N +. and v 1 v 0 = 2(rp 0 + xq 0 ) z 2` (21) Suppose s 1 is given (e.g., a constant power load). Then the variables are (`,v 1, p 0,q 0 ) and the feasible set consists of solutions of (20) and (21) subject to additional constraints on (`,v 1, p 0,q 0 ). The case without any constraint is instructive p 0 r = p 1 q 0 x = q 1 c high v 1 = p q 0 2 low v 1 Fig. 2: Feasible set of OPF for a two-bus network without any constraint. It consists of the (two) points of intersection of the line with the convex surface (without the interior), and hence is nonconvex. SOCP relaxation includes the interior of the convex surface and enlarges the feasible set to the line segment joining these two points. If the cost function C is increasing in ` or (p 0,q 0 ) then the optimal point over the SOCP feasible set (line segment) is the lower feasible point c, and hence the relaxation is exact. No constraint on ` or (p 0,q 0 ) will destroy exactness as long as the resulting feasible set contains c. and shown in Figure 2. The point c in the figure corresponds to a power flow solution with a large v 1 (normal operation) whereas the other intersection corresponds to a solution with a small v 1 (fault condition). As explained in the caption, SOCP relaxation is exact if there is no voltage constraint and as long as constraints on (`, p 0,q 0 ) does not remove the high-voltage (normal) power flow solution c. Only when the system is stressed to a point where the high-voltage solution becomes infeasible will relaxation lose exactness. This agrees with conventional wisdom that power systems under normal operations are well behaved. Consider now the voltage constraint v 1 apple v 1 apple v 1. Substituting (20) into (21) we obtain v 1 = (1 + rp 1 + xq 1 ) z 2` translating the constraint on v 1 into a box constraint on `: 1 z 2 (rp 1 + xq v 1 ) apple ` apple 1 z 2 (rp 1 + xq v 1 ) Figure 2 shows that the lower bound v 1 (corresponding to an upper bound on `) does not affect the exactness of SOCP relaxation. The effect of upper bound v 1 (corresponding to a lower bound on `) is illustrated in Figure 3. As explained in the caption of the figure SOCP relaxation is exact if the q 0 p 0

19 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE upper bound v 1 does not exclude the high-voltage power flow solution c and is not exact otherwise. c (a) Voltage constraint not binding q 0 p 0 op2mal)solu2on)of)socp)) (infeasible)for)opf)) c (b) Voltage constraint binding Fig. 3: Impact of voltage upper bound v 1 on exactness. (a) When v 1 (corresponding to a lower bound on `) is not binding, the power flow solution c is in the feasible set of SOCP and hence the relaxation is exact. (b) When v 1 excludes c from the feasible set of SOCP, the optimal solution is infeasible for OPF and the relaxation is not exact. To state the sufficient condition for a general radial network, recall from [25, Section VI] the linear approximation of BFM for radial networks obtained by setting ` jk = 0 in (17): for each s S lin jk (s) =  i2t j s i v lin j (s) = v  (i,k)2p j Re z H ik Slin ik (s) q 0 p 0 (22a) (22b) where T j denotes the subtree at node j, including j, and P 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 and Remark 9]: S jk Define the 2 2 matrix function apple S lin jk (s) and v j apple v lin j (s) (23) A jk (S jk,v j ) := I 2 v j z jk S jk T (24) where z jk :=[r jk x jk ] T is the line impedance and S jk := [P jk Q jk ] T is the branch power flows, both taken as 2- T dimensional real vectors so that z jk S jk is a 2 2 matrix with rank less or equal to 1. The matrices A jk (S jk,v j ) describe how changes in the real and reactive power flows propagate towards the root node 0; see comments below. Evaluate the Jacobian matrix A jk (S jk,v j ) at the boundary values: h i +, A jk := A jk S lin jk (s) v j := I 2 h i + T z jk S lin jk v (s) (25) j Here [a] + T is the row vector [[a1 ] + [a 2 ] + ] 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 refer to a link interchangeably by ( j,k) or j and use A j, A j, z j etc. in place of A jk, A jk, z jk etc. respectively. Assume B1: The cost function is C(x) :=  n j=0 C j (Res j ) with C 0 strictly increasing. There is no constraint on s 0. B2: The set S j of injections satisfies v lin j (s) apple v j, j 2 N, where v lin j (s) is given by (22). B3: For each leaf node j 2 N let the unique path from j to 0 have k links and be denoted by P j := ((i k,i k 1 ),...,(i 1,i 0 )) with i k = j and i 0 = 0. Then A it A z it 0 i t 0 +1 > 0 for all 1 apple t apple 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 requires that the cost functions C j depend only on the injections s j. For instance, if C j (Res j )=p j, then the cost is total active power loss over the network. It also requires that C 0 be strictly increasing but makes no assumption on C j, j > 0. Common cost functions such as line loss or generation cost usually satisfy B1. If C 0 is only nondecreasing, rather than strictly increasing, in p 0 then B1 B3 still guarantee that all optimal solutions of OPF (10) are (effectively) optimal for OPF-socp (19), but OPF-socp may not be exact, i.e., it may have an optimal solution that maintains strict inequalities in (17c). In this case the proof of Theorem 5 can be used to recursively construct from it another optimal solution that attains equalities in (17c). B2 is affine in the injections s := (p,q). It enforces the upper bounds on voltage magnitudes because of (23). B3 is a technical assumption and has a simple interpretation: the branch power flow S jk on all branches should move in the same direction. Specifically, given a marginal change in the complex power on line j! k, the 2 2 matrix A jk is (a lower bound on) the Jacobian and describes the effect of this marginal change on the complex power on the line immediately upstream from line j! k. The product of A i in B3 propagates this effect upstream towards the root. B3 requires that a small change, positive or negative, in the power flow on a line affects all upstream branch powers in the same direction. This seems to hold with a significant margin 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 have the following simple and practical interpretation: OPF-socp is 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

20 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE 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 convexity, i.e., the cost C(x) =Â n j=0 C j(res j ) need not be a convex function and the injection regions S j need not be convex sets. Convexity allows polynomial-time computation. Moreover when it is convex the exactness of SOCP relaxation also implies the uniqueness of the optimal solution, as the following result from [35] shows. Theorem 6: Suppose the costs C j, j = 0,...,n, are convex functions and the injection regions S j, j = 1,...,n, are convex sets. If the relaxation OPF-socp (19) is exact then its optimal solution is unique. Consider the model of [18] for radial networks, which is (17) with the inequalities in (17c) replaced by equalities. Let X denote an equivalent feasible set of OPF, 5 i.e., those x 2 R 3(m+n+1) that satisfy (17), (18) and attain equalities in (17c). The proof of Theorem 6 reveals that, for radial networks, the feasible set X has a hollow interior. Corollary 7: Suppose G is a tree. If ˆx and x are distinct solutions in X then no convex combination of ˆx and x can be in X. In particular X is nonconvex. This property is illustrated vividly in several numerical examples for mesh networks in [47], [48], [49], [50]. C. Angle differences The sufficient conditions in [29], [36], [37] require that the voltage angle difference across each line be small. We explain the intuition using a result in [36] for an OPF problem where V j are fixed for all j 2 N + and reactive powers are ignored. Under these assumptions, as long as the voltage angle difference is small, the power flow solutions form a locally convex surface that is the Pareto front of its relaxation. This implies that the relaxation is exact. This geometric picture is apparent in earlier work on the geometry of power flow solutions, see e.g. [47], and underlies the intuition that the dynamics of a power system is usually benign until it is pushed towards the boundary of its stability region. The geometric insight in Figures 2 and 3 for BFM and later in this subsection for BIM says that, when it is far away from the boundary, the local convexity structure also facilitates exact relaxation. Reactive power is considered in [37, Theorem 1] with fixed V j where, with an additional constraint on the lower bounds of reactive power injections that ensure these lower bounds are not tight, it is proved that if the original OPF problem is feasible then its SDP relaxation is exact. The case of variable V j without reactive power is considered in [36, Theorem 7] but the simple geometric structure is lost. 5 There 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 V j = V j e iq j and suppose V j are given. Consider: p,p,q C(p) (26a) s.t. p j apple p j apple p j, j 2 N + (26b) q jk apple q jk apple q jk, ( j,k) 2 E (26c) p j = Â P jk, j 2 N + (26d) k:k j P jk = V j 2 g jk V j V k g jk cosq jk + V j V k b jk sinq jk ( j,k) 2 E (26e) where q jk := q j q k are the voltage angle differences across lines ( j,k). We comment on the constraints on angles q jk in (26). When the voltage magnitudes V i are fixed, constraints on real power flows, branch currents, line losses, as well as stability constraints can all be represented in terms of q jk. Indeed a line flow constraint of the form P jk applep jk becomes a constraint on q jk using the expression for P jk in (26e). A current constraint of the form I jk applei jk is also a constraint on q jk since I jk 2 = y jk ( V j 2 + V k 2 2 V j V k cosq jk ). The line loss over ( j,k) 2 E is equal to P jk + P kj which is again a function of q jk. Stability typically requires q jk to stay within a small threshold. Therefore given constraints on branch power or current flows, losses, and stability, appropriate bounds q jk,q jk can be detered in terms of these constraints, assug V j are fixed. We can eliate the branch flows P jk and angles q jk from (26). Since V j, j 2 N +, are fixed we assume without loss of generality that V j = 1 pu. Define the injection region ( P q := p 2 R n p j = Â g jk g jk cosq jk + b jk sinq jk, k:k j ) q jk apple q jk apple q jk, j 2 N +,( j,k) 2 E Let P p := {p 2 R n p j apple p j apple p j, j 2 N}. Then (26) is: OPF: (27) p C(p) subject to p 2 P q \ P p (28) This problem is hard because the set P q is nonconvex. To avoid triviality we assume OPF (28) is feasible. For a set A let conva denote the convex hull of A. Consider the following problem that relaxes the nonconvex feasible set P q \ P p of (28) to a convex superset: OPF-socp: p C(p) subject to p 2 conv(p q ) \ P p (29) We will show below that (29) is indeed an SOCP. It is said to be exact if every optimal solution of (29) lies in P q \ P p and is therefore also optimal for (28). We say that a point x 2 A R n is a Pareto optimal point

21 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE in A if there does not exist another x 0 2 A such that x 0 apple x with at least one strictly smaller component x 0 j < x j. The Pareto front of A, denoted by O(A), is the set of all Pareto optimal points in A. The significance of O(A) is that, for any increasing function, its imizer, if exists, is necessarily in O(A) whether A is convex or not. If A is convex then x opt is a Pareto optimal point in O(A) if and only if there is a nonzero vector c :=(c 1,...,c n ) 0 such that x opt is a imizer of c T x over A [51, pp ]. Assume C1: C(p) is strictly increasing in each p j. C2: For all ( j,k) 2 E, tan 1 b jk g < q jk jk apple q jk < tan 1 b jk g. jk The following result, proved in [36], [37], says that (29) is exact provided q jk are suitably bounded. Theorem 8: Suppose G is a tree and C1 C2 hold. 1) P q \ P p = O(conv(P q ) \ P p ). 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 not strictly increasing in all p j, then P q \P p O(conv(P q ) \ P p ) and OPF-socp may not be exact according to our definition. Even in that case it is possible to recover an optimal solution of OPF from any optimal solution of OPF-socp. Theorem 8 is illustrated in Figures 4 and 5. As explained ( p j, p k ) p k 2b jk Fig. 4: Feasible set of OPF (28) for a two-bus network without any constraint when V j are fixed and reactive powers are ignored. It is an ellipse without the interior, hence nonconvex. OPF-socp (29) includes the interior of the ellipse and is hence convex. If the cost function C is strictly increasing in (p j, p k ) then the Pareto front of the SOCP feasible set will lie on the lower part of the ellipse, O(P q )=P q, and hence OPF-socp is exact. in the caption of Figure 4, if there are no constraints then SOCP relaxation (29) is exact under condition C1. It is clear from the figure that upper bounds on power injections do not affect exactness whereas lower bounds do. The purpose of condition C2 is to restrict the angle q jk in order to eliate the upper half of the ellipse from P q. As explained in the caption of Figure 5, under C2, P q \ P p = O(conv(P q ) \ P p ) and hence the relaxation is exact. Otherwise it may not. 2g jk p j Pareto)front) ( p j, p k ) (a) Exact relaxation with constraint Pareto)front) ( p j, p k ) (b) Inexact relaxation with constraint Fig. 5: With lower bounds p on power injections, the feasible set of OPF-socp (29) is the shaded region. (a) When the feasible set of OPF (28) is restricted to the lower half of the ellipse (small q jk ), the Pareto front remains on the ellipse itself, P q \P p = O(conv(P q ) \ P p ), and hence the relaxation is exact. (b) When the feasible set of OPF includes upper half of the ellipse (large q jk ), the Pareto front may not lie on the ellipse if p is large, making the relaxation not exact. When the network is not radial or V j are not constants, then the feasible set can be much more complicated than ellipsoids [48], [49], [50]. Even in such settings the Pareto fronts might still coincide, though the simple geometric picture is lost. See [47] for a numerical example on an Australian system or [24] on a three-bus mesh network. D. Equivalence Since BIM and BFM are equivalent, the results on exact SOCP relaxation and uniqueness of optimal solution apply in both models. Recall the linear bijection g from BIM to BFM defined in [25, end of Section V] by x = g(w G ) where S jk := y H jk [W G] jj [W G ] jk, j! k ` jk := y jk 2 [W G ] jj +[W G ] kk [W G ] jk [W G ] kj, j! k v j := [W G ] jj, j 2 N + s j := Â y H jk [W G] jj [W G ] jk, j 2 N + k: j k The mapping g allows us to directly apply Theorem 6 to BIM. We summarize all the results for type A and type B conditions for radial networks. 6 Theorem 9: Suppose G and G are trees. Suppose conditions A1 A2, or A3 A4, or B1 B3 hold. Then 1) BIM: SOCP relaxation (7) is exact. Moreover if C(W G ) is convex in ([W G ] jj,[w G ] jk ) then the optimal solution is unique. 2) BFM: SOCP relaxation (13) is exact. Moreover if C(x) := Â j C j (p j ) is convex in p then the optimal solution is unique. 6 To apply type C conditions to BFM, one needs to translate the angles q jk to the BFM variables x := (S,`,v,s) through b jk (x), though this will introduce additional nonconvex constraints into OPF of the form q jk apple b jk (x) apple q jk.

22 IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, JUNE Since both the SDP and the chordal relaxations are equivalent to the SOCP relaxation for radial networks, these results apply to SDP and chordal relaxations as well. IV. MESH NETWORKS In this section we summarize a result of [17, Part II] on mesh 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 an optimal solution W socp G /xsocp of SOCP relaxation, W socp G /xsocp must satisfy both a local condition and a global cycle condition ((4) for BIM and (12) for BFM); see the definition of exactness in Section II. The conditions of Section III guarantee that every SOCP optimal solution will satisfy the local condition (i.e., W socp G is 2 2 psd rank-1 and x socp attains equalities in (11c)), whether the network is radial or mesh, but do not guarantee that it satisfies the cycle condition. For radial networks, the cycle condition is vacuous and therefore the conditions of Section III are sufficient for SOCP relaxation to be exact. The result of [17, Part II] implies that these conditions are sufficient also for a mesh network that has tunable phase shifters at strategic locations. Similar conditions also extend to dc networks where all variables are real and the voltages are assumed nonnegative. A. AC networks with phase shifters For 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 it does then it can be uniquely mapped to an optimal solution of OPF (10), according to [17, Theorem 2]. If it does not then the solution is not physically implementable because it does 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 of the SOCP relaxation that attains equalities in (11c) always satisfies the cycle condition [17, Theorem 4]. This is not the case for a mesh network where B is m n with m > n. It is proved in [17, Part II] however that if the network has tunable phase shifters then any SOCP solution that attains equalities in (11c) becomes implementable even if the solution does not satisfy the cycle condition. This extends the sufficient 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 to introducing a free variable f c in (4) for each basis cycle c so that the cycle condition can always be satisfied for any W G. The results presented here however start with a simple power flow model (30) for networks with phase shifters. This model makes transparent the effect of the spatial distribution of phase shifters and how they impact the exactness of SOCP relaxation and can be useful in other contexts, such as the design of a network of FACTS (Flexible AC Transmission Systems) devices. BFM with phase shifters. We consider an idealized phase shifter that only shifts the phase angles of the sending-end voltage and current across a line, and has no impedance nor limits on the shifted angles. Specifically consider an idealized phase shifter parametrized by f jk across line j! k as shown in Figure 6. As before let V j denote the sending-end voltage at j φ jk i Fig. 6: Model of a phase shifter in line j! k. node j. Define I jk to be the sending-end current leaving node j towards node k. Let i be the point between the phase shifter f jk and line impedance z jk. Let V i and I i be the voltage at i and the current from i to k respectively. Then the effect of an idealized phase shifter, parametrized by f jk, is summarized by the following modeling assumptions: z jk V i = V j e if jk and I i = I jk e if jk The power transferred from nodes j to k is still (defined to be) S jk := V j I H jk, which is equal to the power V iii H from nodes i to k since the phase shifter is assumed to be lossless. Applying Ohm s law across z jk, we define the branch flow model with phase shifters as the following set of equations: Â S jk = Â S ij z ij I ij 2 + s j, j 2 N + (30a) k: j!k i:i! j I jk = y jk V j V k e if jk k, j! k 2 Ẽ (30b) S jk = V j I H jk, j! k 2 Ẽ (30c) Without phase shifters (f jk = 0), (30) reduces to BFM (8). Let x := (S,I,V,s) 2 C 2(m+n+1) denote the variables in (30). Let x :=(S,`,v,s) 2 R 3(m+n+1) denote the variables in SOCP relaxation (13). These variables are related through the mapping x = h( x) where ` jk = I jk 2 and v j = V j 2. In particular, 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 the cycle condition changes from (12) to: given any x that satisfies (11) with equalities in (11c), 9(q,f) 2 R n+m such that Bq = b(x) f mod 2p (31) It is proved in [17, Part II] that, given any x that attains equalities in (11c), there always exists a q in ( p,p] n and a f in ( p,p] m that solve (31). Moreover phase shifters are needed only on lines not in a spanning tree. Exact SOCP relaxation. Recall the OPF problem (10) where the feasible set X without phase shifters is: X := { x x satisfies (30) with f = 0 and (9)}