Convexification of Generalized Network Flow Problem with Application to Power Systems

Transcription

1 1 Convexification of Generalized Network Flow Problem with Alication to Power Systems Somayeh Sojoudi and Javad Lavaei + Deartment of Comuting and Mathematical Sciences, California Institute of Technology + Deartment of Electrical Engineering, Columbia University Abstract This aer is concerned with the minimum-cost flow roblem over an arbitrary flow network. In this roblem, each node is associated with some ossibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This roblem, named generalized network flow (GNF, is highly non-convex due to its nonlinear equality constraints. Under the ractical assumtion of monotonicity and convexity of the flow and cost functions, a convex relaxation is roosed, which always finds the otimal injections. This relaxation may fail to find otimal flows because the maing from injections to flows might lead to an exonential number of solutions. However, once otimal injections are found in olynomial time, other techniques can be used to find a feasible set of flows corresonding to the injections. A rimary alication of this work is in otimization over ower networks. Recent work on the otimal ower flow (OPF roblem has shown that this non-convex roblem can be solved efficiently using semidefinite rogramming (SDP after two aroximations: relaxing angle constraints (by adding virtual hase shifters and relaxing ower balance equations to inequality constraints. The results of this work rove two facts for the OPF roblem: (i the second relaxation (on balance equations is not needed in ractice under a very mild angle assumtion, (ii if the SDP relaxation fails to find a rank-one solution, the otimal injections (and not flows may still be recovered from an undesirable high-rank solution. I. INTRODUCTION The area of network flows lays a central role in oerations research, comuter science and engineering [1], [2]. This area is motivated by many real-word alications in assignment, transortation, communication networks, electrical ower distribution, roduction scheduling, financial budgeting, and aircraft routing, to name only a few. Started by the classical book [3] in 1962, network flow roblems have been studied extensively [4], [5], [6], [7]. The minimum-cost flow roblem aims to otimize the flows over a flow network that is used to carry some commodity from suliers to consumers. In a flow network, there is an injection of some commodity at every node, which leads to two flows over each line (arc at its endoints. The injection deending on being ositive or negative, corresonds to suly or demand at the node. The minimum-cost flow roblem has been studied thoroughly for a lossless network, where the amount of flow entering a line equals the amount of flow leaving the line. However, since many real-world flow networks are lossy, the minimum-cost flow roblem has also attracted much attention for generalized networks, also known as networks with gain [2], [8]. In this tye of network, each line is associated with a constant gain relating the two flows of the line through a s: sojoudi@caltech.edu and lavaei@ee.columbia.edu linear function. From the otimization ersective, network flow roblems are convex and can be solved efficiently unless there are discrete variables involved [9]. There are several real-world network flows that are lossy, where the loss is a nonlinear function of the flows. An imortant examle is ower distribution networks for which the loss over a transmission line (with fixed voltage magnitudes at both ends is given by a arabolic function due to Kirchhoff s circuit laws [10]. The loss function could be much more comlicated deending on the ower electronic devices installed on the transmission line. To the best of our knowledge, there is no theoretical result in the literature on the olynomial-time solvability of network flow roblems with nonlinear flow functions, excet in very secial cases. This aer is concerned with this general roblem, named Generalized Network Flow (GNF. Note that the term GNF has already been used in the literature for networks with linear losses, but it corresonds to arbitrary lossy networks in this work. GNF aims to otimize the nodal injections subject to flow constraints for each line and box constraints for both injections and flows. A flow constraint is a nonlinear equality relating the flows at both ends of a line. To solve GNF, this aer makes the ractical assumtion that the cost and flow functions are all monotonic and convex. The GNF roblem is still highly non-convex due to its equality constraints. Relaxing the nonlinear equalities to convex inequalities gives rise to a convex relaxation of GNF. It can be easily observed that solving the relaxed roblem may likely lead to a solution for which the new inequality flow constraints are not binding. One may seculate that this observation imlies that the convex relaxation is not tight. However, the objective of this work is to show that as long as GNF is feasible, the convex relaxation is tight. More recisely, the convex relaxation always finds the otimal injections (and hence the otimal objective value, but robably roduces wrong flows leading to non-binding inequalities. However, once the otimal injections are obtained at the nodes, a feasibility roblem can be solved to find a set of feasible flows corresonding to the injections. Note that the reason why the convex relaxation does not necessarily find the correct flows is that the maing from injections to flows is not invertible. For examle, it is known in the context of ower systems that the ower flow equations may not have a unique solution. The main contribution of this work is to show that although GNF is NP-hard (since the flow equations can have an exonential number of solutions, the otimal injections can be found in olynomial time.

2 2 A. Alication of GNF in Power Systems The oeration of a ower network deends heavily on various large-scale otimization roblems such as state estimation, otimal ower flow (OPF, contingency constrained OPF, unit commitment, sizing of caacitor banks and network reconfiguration. These roblems are highly non-convex due to the nonlinearities imosed by laws of hysics [11], [12]. For examle, each of the above roblems has the ower flow equations embedded in it, which are nonlinear equality constraints. The nonlinearity of OPF, as the most fundamental otimization roblem for ower systems, has been studied since 1962 leading to develoing various heuristic and localsearch algorithms [13], [14], [15]. These algorithms suffer from sensitivity and convergence issues, and more imortantly they may convergence to a local otimum that is noticeably far from a global solution. Recently, it has been shown in [16] that the semidefinite rogramming (SDP relaxation is able to find the global solution of the OPF roblem under a sufficient condition, which is satisfied for IEEE benchmark systems with 14, 30, 57, 118 and 300 buses and many randomly generated ower networks. The aers [16] and [12] show that this condition holds widely in ractice due to the assivity of transmission lines and transformers. In articular, [12] shows that in the case when this condition is not satisfied (see [17] for counterexamles, OPF can always be solved globally in olynomial time after two aroximations: (i relaxing angle constraints by adding a sufficient number of actual/virtual hase shifters to the network, (ii relaxing ower balance equalities at the buses to inequality constraints. OPF under Aroximation (ii was also studied in [18] and [19] for distribution networks. The aer [20] studies the otimization of active ower flows over distribution networks under fixed voltage magnitudes and shows that SDP relaxation works without Aroximation (i as long as a very ractical angle condition is satisfied. The idea of convex relaxation develoed in [21] and [16] can be alied to many other ower roblems, such as voltage regulation [22], state estimation [23], calculation of voltage stability margin [24], charging of electric vehicles [25], SCOPF with variable ta-changers and caacitor banks [26], dynamic energy management [10] and electricity market [27]. Energy-related otimizations with embedded ower flow equations can be regarded as nonlinear network flow roblems, which are analogous to GNF. The results derived in this work for a general GNF roblem lead to the following conclusions for OPF (and some of the abovementioned OPF-based otimizations: They generalize the result of [20] to networks with virtual hase shifters. This roves that in order to use SDP relaxation for OPF over an arbitrary ower network, it may not be needed to relax ower balance equalities to inequality constraints under a very mild angle assumtion. If the SDP relaxation does not work exactly for an OPF roblem, it means that the obtained bus voltages are wrong. However, it can be inferred from the results of this work on GNF that the obtained bus injection owers may still be correct, in which case the otimal voltages can be recovered by solving a searate ower flow roblem. B. Notations The following notations will be used throughout this aer: R and R + denote the sets of real numbers and nonnegative numbers, resectively. Given two matrices M and N, the inequality M N means that M is less than or equal to N element-wise. Given a set T, its cardinality is shown as T. Lowercase, bold lowercase and uercase letters are used for scalars, vectors and matrices (say x, x and X. II. PROBLEM STATEMENT AND CONTRIBUTIONS Consider an undirected grah (network G with the vertex set N := {1, 2,..., m} and the edge set E N N. For every i N, let N(i denote the set of the neighboring vertices of node i. Assume that every edge (i, j E is associated with two unknown flows ij and ji belonging to R. The arameters ij and ji can be regarded as the flows entering the edge (i, j from the endoints i and j, resectively. Define i = ij, i N (1 j N(i The arameter i is called nodal injection at vertex i or simly injection, which is equal to the sum of the flows leaving vertex i through the edges connected to this vertex. Given an edge (i, j E, we assume that the flows ij and ji are related to each other via a function f ij ( to be introduced later. To secify which of the flows ij and ji is a function of the other, we give an arbitrary orientation to every edge of the grah G and denote the resulting grah as G. Denote also its directed edge set as E. If an edge (i, j E belongs to E, we then exress ji as a function of ij. Definition 1: Define the vectors n, e and d as follows: n = { i i N } e = { ij (i, j E} d = { ij (i, j E} (2a (2b (2c (the subscrits n, e and d stand for nodes, edges and directed edges. The terms n, e and d are referred to as injection vector, flow vector and semi-flow vector, resectively (note that e contains two flows er each line, while d has only one flow er line. Definition 2: Given two arbitrary oints x,y R n, the box B(x,y is defined as follows: B(x,y = {z R n x z y} (3 (note that B(x,y is non-emty only if x y. Assume that each nodal injection i must be within the given interval [ min i, max i ] for every i N. We use the shorthand notation B for the box B( min n, max n, where min n and max n are the vectors of the lower bounds min i s and the uer bounds max i s, resectively. This aer is concerned with the following roblem.

3 3 Generalized network flow (GNF: f i ( i min P n B,P e R E i N subject to i = j N(i (4a ij, i N (4b ji = f ij ( ij, (i, j E (4c ij [ min ij, max ij ], (i, j E (4d where 1 f i ( is convex and monotonically increasing for every i N. 2 f ij ( is convex and monotonically decreasing for every (i, j E. 3 The limits min ij and max ij are given for every (i, j E. In the case when f ij ( ji is equal to ij for all (i, j E, the GNF roblem reduces to the network flow roblem for which every line is lossless. A few remarks can be made here: Given an edge (i, j E, there is no exlicit limit on ji in the formulation of the GNF roblem because restricting ji is equivalent to limiting ij. Given a node i N, the assumtion of f i ( i being monotonically increasing is motivated by the fact that increasing the injection i normally elevates the cost in ractice. Given an edge (i, j E, ij and ji can be regarded as the inut and outut flows of the line (i, j, which travel in the same direction. The assumtion of f ij ( ij being monotonically decreasing is motivated by the fact that increasing the inut flow normally makes the outut flow higher in ractice (note that ji = f ij ( ij. Definition 3: Define P as the set of all vectors n for which there exists a vector e such that ( n, e satisfies equations (4b, (4c and (4d. The set P and P B are referred to as injection region and box-constrained injection region, resectively. Regarding Definition 3, the box-constrained injection region is indeed the rojection of the feasible set of GNF onto the sace of the injection vector n. Now, one can exress GNF geometrically as follows: Geometric GNF : min f i ( i (5 n P B i N Note that e has been eliminated in Geometric GNF. It is hard to solve this roblem directly because the injection region P is non-convex in general. This non-convexity can be observed in Figure 2(a, which shows P for the two-node grah drawn in Figure 1. To address this non-convexity issue, the GNF roblem will be convexified naturally next. Convexified generalized network flow (CGNF: min P n B,P e R E f i ( i i N subject to i = j N(i (6a ij, i N (6b ji f ij ( ij, (i, j E (6c ij [ min ij, max ij ], (i, j E (6d ( (2 12 (1 21 (2 21 Fig. 1: The grah G studied in Section III-A. where ( min ij, max ij = (f ji ( max ji, f ji ( min ji for every (i, j E such that (j, i E. Note that CGNF has been obtained from GNF by relaxing equality (4c to inequality (6c and adding limits to ij for every (j, i E. One can write: Geometric CGNF : min f i ( i (7 n P c B i N where P c denotes the set of all vectors n for which there exists a vector e such that ( n, e satisfies equations (6b, (6c and (6d. Two main results to be roved in this aer are: Geometry of injection region: Given any two oints n and n in the injection region, the box B( n, n is entirely contained in the injection region. Similar result holds true for the box-constrained injection region. Relationshi between GNF and CGNF: If ( n, e and ( n, e denote two arbitrary solutions of GNF and CGNF, then n = n. Hence, although CGNF may not be able to find a feasible flow vector for GNF, it always finds the correct otimal injection vector for GNF. The alication of these results in ower systems will also be discussed. Note that this work imlicitly assumes that every two nodes of G are connected via at most one edge. However, the results to be derived later are all valid in the resence of multile edges between two nodes. To avoid comlicated notations, the roof will not be rovided for this case. However, Section III-A studies a simle examle with arallel lines. III. MAIN RESULTS In this section, a detailed illustrative examle will first be rovided to clarify the issues and highlight the contribution of this work. In Subsections III-B and III-C, the main results for GNF will be derived, whose alication in ower systems will be later discussed in Subsection III-D. A. Illustrative Examle In this subsection, we study the articular grah G deicted in Figure 1. This grah has two vertices and two arallel edges. Let ( (1 12, (1 21 and ((2 12, (2 21 denote the flows associated with the first and second edges of the grah, resectively. Consider the following GNF roblem: min f 1 ( 1 + f 2 ( 2 (8a ( 2 subject to (i 21 = (i , i = 1, 2 (8b 0.5 ( , 1 (2 12 1, (8c 1 = ( (2 12, 2 = ( (2 21 (8d

4 4 (a (b (c Fig. 2: (a Injection region P for the GNF roblem given in (8; (b The set P c corresonding to the GNF roblem given in (8; (c This figure shows the set P c corresonding to the GNF roblem given in (8 together with a box constraint ( 1, 2 B for four different ositions of B. with the variables 1, 2, (1 12, (1 21, (2 12, (2 21, where f 1( and f 2 ( are both convex and monotonically increasing. The CGNF roblem corresonding to this roblem can be obtained by relacing (8b with (i 21 ((i and adding the limits ( and ( One can write: Geometric GNF: Geometric CGNF: min f 1( 1 + f 2 ( 2 (9a ( 1, 2 P min f 1 ( 1 + f 2 ( 2 (9b ( 1, 2 P c where P and P c are indeed the rojections of the feasible sets of GNF and CGNF over the injection sace ( 1, 2. The green area in Figure 2(a shows the injection region P. As exected, this set is non-convex. In contrast, the set P c is a convex set containing P. This set is shown in Figure 2(b, which includes two arts: (i the green area is the same as P, (ii the blue area is the art of P c that does not exist in P. Thus, the transition from GNF to CGNF extends the injection region P to a convex set by adding the blue area. Notice that P c has three boundaries: (1 straight line on the to, (2 straight line on the right side, and (3 a lower curvy boundary. Since f 1 ( and f 2 ( are both monotonically increasing, the unique solution of Geometric CGNF must lie on the lower curvy boundary of P c. Since this lower boundary is in the green area, it is contained in P. As a result, the unique solution of Geometric CGNF is a feasible oint of P and therefore it is a solution of Geometric GNF. This means that CGNF finds the otimal injection vector for GNF. To make the roblem more interesting, we add the box constraint ( 1, 2 B to GNF (and corresondingly to CGNF, where B is an arbitrary rectangular convex set in R 2. The effect of this box constraint will be investigated in four different scenarios: Assume that B corresonds to Box 1 (including its interior in Figure 2(c. In this case, P B = P c B = φ, meaning that Geometric GNF and Geometric CGNF are both infeasible. Assume that B corresonds to Box 2 (including its interior in Figure 2(c. In this case, the solution of Geometric CGNF lies on the lower boundary of P c and therefore it is also a solution of Geometric GNF. Assume that B corresonds to Box 3 (including its interior in Figure 2(c. In this case, the solutions of Geometric GNF and Geometric CGNF are identical and both corresond to the lower left corner of the box B. Assume that B corresonds to Box 4 (including its interior in Figure 2(c. In this case, P B = φ but P c B φ. Hence, Geometric GNF is infeasible while Geometric CGNF has an otimal solution In summary, it can be argued that indeendent of the osition of the box B in R 2, CGNF finds the otimal injection vector for GNF as long as GNF is feasible. B. Geometry of Injection Region In order to study the relationshi between GNF and CGNF, it is beneficial to exlore the geometry of the feasible set of GNF. Hence, we investigate the geometry of the injection region P and the box-constrained injection region P B in this art. Theorem 1: Consider two arbitrary oints ˆ n and n in the injection region P. The box B(ˆ n, n is contained in P. The roof of this theorem is based on four lemmas, and will be rovided later in this subsection. To understand this theorem, consider the injection region P deicted in Figure 2(a corresonding to the illustrative examle given in Section III-A. If any arbitrary box is drawn in R 2 in such a way that its uer right corner and lower left corner both lie in the green area, then the entire box must lie in the green area comletely. This can be easily roved in this secial case and is true in general due to Theorem 1. The result of Theorem 1 can be generalized to the box-constrained injection region, as stated below. Corollary 1: Consider two arbitrary oints ˆ n and n belonging to the box-constrained injection region P B. The box B(ˆ n, n is contained in P B. Proof: The roof follows immediately from Theorem 1. The rest of this subsection is dedicated to the roof of Theorem 1, which is based on a series of definitions and lemmas. Definition 4: Define B d as the box containing all vectors d introduced in (2c satisfying the condition ij [ min ij, max ij ] for every (i, j E. Definition 5: Given two arbitrary oints d, d B d, define M( d, d as follows: Let M( d, d be a matrix with N rows indexed by the vertices of G and with E columns indexed by the edges in E. For every vertex k N and edge (i, j E, set the (k, (i, j th entry of M( d, d (the one in the intersection

5 Edge (1,2 Edge (2,3 Edge (3,1 ji (a Node 1 Node 2 Node 3 1 * 0 (b 0 1 * Fig. 3: (a A articular grah G; (b The matrix M( (, d, d corresonding to the grah G in Figure (a; (c: The (j, (i, j th entry of M( d, d (shown as * is equal to the sloe of the line connecting the oint ( ij, ji to ( ij, ji. (, ~ of row k and column (i, j as 1 minif k = i max f ij( ij f ij( ij ij ij if k = j and ij ij f min ij ( (10 ij if k = j and ij = ij 0 otherwise where f ij ( ij denotes the right derivative of f ij ( ij if ij < max ij and the left derivative of f ij ( ij if ij = max ij. To illustrate Definition 5, consider the three-node grah G deicted in Figure 3(a. The matrix M( d, d associated with this grah has the structure shown in Figure 3(b, where the * entries deend on the secific values of d and d. Consider an edge (i, j E. The (j, (i, j th entry of M( d, d is equal to f ij ( ij f ij ( ij ij ij, (11 rovided ij ij. As can be seen in Figure 3(c, this is equal to the sloe of the line connecting the oint ( ij, ji to the oint ( ij, ji on the arameterized curve ( ij, ji, where ji = f ij ( ij. Moreover, f ij ( ij is the limit of this sloe as the oint ( ij, ji aroaches ( ij, ji. It is also interesting to note that M( d, d has one ositive entry, one negative entry and m 2 zero entries in each column (note that the sloe of the line connecting ( ij, ji to ( ij, ji is always negative. The next lemma exlains how the matrix M( d, d can be used to relate the semi-flow vector to the injection vector. Lemma 1: Consider two arbitrary injection vectors n and n in P, associated with the semi-flow vectors d and d (defined in (2. The relation holds. Proof: One can write: i i = By using the relations n n = M( d, d ( P d P d (12 j N(i ( ij ij, i N (13 ji = f ij ( ij, ji = f ij ( ij, (i, j E, (14 * 0 1 min ij ( ij, ji (c ~, ~ ij ( ji max ij ij it is straightforward to verify that (12 and (13 are equiva- lent. The next lemma investigates an imortant roerty of the matrix M( d, d. Lemma 2: Given two arbitrary oints d, d B d, assume that there exists a nonzero vector x R m such that x T M( d, d 0. If x has at least one strictly ositive entry, then there exists a nonzero vector y R m + such that y T M( d, d 0. Proof: Consider an index i 0 N such that x i0 > 0. Define V(i 0 as the set of all vertices i N from which there exists a directed ath to vertex i 0 in the grah G. Note that V(i 0 includes vertex i 0 itself. The first goal is to show that x i 0, i V(i 0 (15 To this end, consider an arbitrary set of vertices i 1,..., i k in V(i 0 such that {i 0, i 1..., i k } forms a direct ath in G as i k i k 1 i 1 i 0 (16 To rove (15, it suffices to show that x i1,..., x ik 0. For this urose, one can exand the roduct x T M( d, d and use the fact that each column of M( d, d has m 2 zero entries to conclude that x i1 + f i 1i 0 ( i1i 0 f i1i 0 ( i1i 0 i1i 0 i1i 0 x i0 0 (17 Since x i0 is ositive and f i1i 0 ( is a decreasing function, x i1 turns out to be ositive. Now, reeating the above argument for i 1 instead of i 0 yields that x i2 0. Continuing this reasoning leads to x i1,..., x ik 0. Hence, inequality (15 holds. Now, define y as { xi if i V(i y i = 0, i N (18 0 otherwise In light of (15, y is a nonzero vector in R m +. To comlete the roof, it suffices to show that y T M( d, d 0. Similar to the indexing rocedure used for the columns of the matrix M( d, d, we index the entries of the E dimensional vector y T M( d, d according to the edges of G. Now, given an arbitrary edge (α, β E, the following statements hold true: If α, β V(i 0, then the (α, β th entries of y T M( d, d and x T M( d, d (i.e., the entries corresonding to the edge (α, β are identical.

6 6 If α V(i 0 and β V(i 0, then the (α, β th entry of y T M( d, d is equal to y α. If α V(i 0 and β V(i 0, then the (α, β th entry of y T M( d, d is equal to zero. Note that the case α V(i 0 and β V(i 0 cannot haen, because if β V(i 0 and (α, β E, then α V(i 0 by the definition of V(i 0. It follows from the above results and the inequality x T M( d, d 0 that y T M( d, d 0. The next lemma studies the injection region P in the case when f ij ( s are all iecewise linear. Lemma 3: Assume that the function f ij ( is iecewise linear for every (i, j E. Consider two arbitrary oints ˆ n, n P and a vector n R m satisfying the relations ˆ n n n n (19 There exists a strictly ositive number ɛ max with the roerty n ε n P, ε [0, ɛ max ] (20 Proof: The roof is based on Lemmas 1 and 2. Since the roof is lengthy and comlicated, it has been moved to [28]. The next lemma roves Theorem 1 in the case when f ij ( s are all iecewise linear. Lemma 4: Assume that the function f ij ( is iecewise linear for every (i, j E. Given any two arbitrary oints ˆ n, n P, the box B(ˆ n, n is a subset of the injection region P. Proof: With no loss of generality, assume that ˆ n n (because otherwise B(ˆ n, n is emty. To rove the lemma by contradiction, suose that there exists a oint n B(ˆ n, n such that n P. Consider the set { } γ γ [0, 1], n + γ( n n P (21 and denote its maximum as γ max (the existence of this maximum number is guaranteed by the closedness and comactness of P. Note that n + γ( n n is equal to n at γ = 1. Since n P by assumtion, we have γ max < 1. Denote n +γ max ( n n as n. Hence, n P and ˆ n n n (recall that γ max < 1. Define n as n n. One can write: ˆ n n n n, ˆ n, n P (22 By Lemma 3, there exists a strictly ositive number ɛ max with the roerty or equivalently n ε n P, ε [0, ɛ max ] (23 n +(γ max +ε(1 γ max ( n n P, ε [0, ɛ max ] (24 Notice that γ max + ε(1 γ max > γ max, ε > 0 (25 Due to (24, this violates the assumtion that γ max is the maximum of the set given in (21. Lemma 4 will be deloyed next to rove Theorem 1 in the general case. Proof of Theorem 1: Consider an arbitrary aroximation of f ij ( by a iecewise linear function for every (i, j E. As a counterart of P, let P s denote the injection region in the iecewise-linear case. By Lemma 4, we have B(ˆ n, n P s (26 Since the iecewise linear aroximation can be made in such a way that the sets P and P s become arbitrarily close to each other, the above relation imlies that the interior of B(ˆ n, n is a subset of P. On the other hand, P is a closed set. Hence, the box B(ˆ n, n must entirely belong to P. C. Relationshi Between GNF and CGNF In this subsection, the relationshi between GNF and CGNF will be exlored. Theorem 2: Assume that the GNF roblem is feasible. Let ( n, e and ( n, e denote arbitrary solutions of GNF and CGNF, resectively. The relation n = n holds. Observe that since ( n, e is a feasible oint of CGNF, one can write i min i, i N (27 The roof of Theorem 2 is lengthy and involved in the general case, but it simlifies greatly in the secial case i = min i, i N (28 Hence, the general roof has been moved to [28], but its secial case is rovided below. Proof of Theorem 2 under Condition (28: ( n, e being a feasible oint of GNF imlies that Equations (28 and (29 lead to Define the vector n as i = ij + (i,j E i min i, i N (29 n n (30 (j,i E f ij ( ij, i N (31 Notice that n belongs to P. It can be inferred from the definition of CGNF that n n (32 Since n, n P, it follows from Theorem 1, (30 and (32 that n P. On the other hand, n B. Therefore, n P B, meaning that n is a feasible oint of Geometric GNF. Since the feasible set of Geometric CGNF includes that of Geometric GNF, n must be a solution of Geometric GNF as well. The roof follows from equation (30 and the fact that n is another solution of Geometric GNF. An otimal solution of CGNF comrises two arts: injection vector and flow vector. Theorem 2 states that CGNF always finds the correct otimal injection vector solving GNF. An examle is rovided in [28] to understand the reason why CGNF may not be able to find the correct otimal flow vector solving GNF.

7 7 D. Otimal Power Flow in Electrical Power Networks In this subsection, the results derived earlier for GNF will be alied to ower networks. Consider a grou of generators (sources of energy, which are connected to a grou of electrical loads (consumers via an electrical ower network (grid. This network comrises a set of transmission lines connecting various nodes to each other (e.g., a generator to a load. Figure 4(a exemlifies a four-node ower network with two generators and two loads. Each load requests certain amount of energy, and the question of interest is to find the most economical ower disatch by the generators so that the demand and network constraints are met. To formulate the roblem, let G denote the flow network corresonding to the electrical ower network, where Each injection i, i G, reresents either the active ower roduced by a generator and injected to the network or the active ower absorbed from the network by an electrical load. Each ij, (i, j E, reresents the active ower entering the transmission line (i, j from its i endoint. The roblem of otimizing the flows in a ower network is called otimal ower flow (OPF. In this art, the goal is to otimize only active ower, but most of the results to be develoed later can be generalized to reactive ower as well. Let v i denote the comlex (hasor voltage at node i N of the ower network. Denote the hase of v i as θ i. Given an edge (j, k G, we denote the admittance of the transmission line between nodes j and k as g ib, where the symbol i denotes the imaginary unit. g and b are nonnegative numbers due to the assivity of the line. There are two flows entering the transmission line from its both ends. These flows are given by: = v j 2 g + v j v k b sin(θ v j v k g cos(θ, kj = v k 2 g v j v k b sin(θ v j v k g cos(θ where θ = θ j θ k. As traditionally done in the ower area, assume that v j and v k are fixed at their nominal values, while θ is a variable to be designed. If θ varies from π to π, then the feasible set of (, kj becomes an ellise, as illustrated in Figure 4(b. It can be seen from this figure that kj cannot be written as a function of. This observation is based on the imlicit assumtion that there is no limit on θ. Suose that θ must belong to an interval [ θ max, θmax ] for some angle θ max. If the new feasible set for (, kj resembles the artial ellise drawn in Figure 4(c, then kj can be exressed as f ( for a monotonically decreasing, convex function f (. This haens if θ max tan 1 ( b g (33 It is interesting to note that the right side of the above inequality is equal to 45.0, 63.4 and 78.6 for b g equal to 1, 2 and 5, resectively. Note that b g is normally greater than 5 (due to the secifications of transmission lines and θ max is normally less than 15 and very rarely as high as 30 due to stability and thermal limits (this angle constraint is forced either directly or through min and max in ractice. Hence, Condition (33 is very ractical. By assuming that this condition is satisfied, there exists a monotonically decreasing, convex function f ( such that kj = f (, [ min, max ], (34 where min and max corresond to θ max and θ max, resectively. Given two disarate edges (j, k and (j, k, the hase differences θ and θ j k may not be varied indeendently if the grah G is cyclic (because the sum of the hase differences over a cycle must be zero. This is not an issue if the grah G is acyclic (corresonding to distribution networks or if there is a sufficient number of hase-shifting transformers in the network. If none of these cases is true, then one could add virtual hase shifters to the ower network at the cost of aroximating the OPF roblem. As soon as the flows (or hase differences on various lines can be varied indeendently, equation (34 yields that the roblem of otimizing active flows reduces to GNF. In this case, Theorems 1 and 2 can be used to study the corresonding aroximated OPF roblem. As a result, the otimal injections for the aroximated OPF can be found via the corresonding CGNF roblem. This imlies two facts about the SDP and SOCP relaxations roosed in [16] and [12] for solving the OPF roblem: The relaxations are exact without using the concet of load over-satisfaction (i.e., relaxing the flow constraints. This is the generalization of the result derived in [20]. The relaxations always yields the otimal injections, but the roduced flow vector can be wrong (meaning that the flow inequality constraints are not all binding. It is easy to contrive such examles. In addition to active owers, voltage magnitudes and reactive owers are often variables in ower systems. The following remarks can be made for a general OPF roblem: Reactive flows can be written as linear functions of active flows. This imlies that the above conclusions on OPF are valid even if the reactive ower at each bus is uer bounded by a given number. In the case when the voltage magnitudes are variable, the flow constraint kj = f ( needs to be relaced by kj = f (,x, where x is an exogenous inut containing the voltage magnitudes at all buses. The technique roosed in this aer can be used to show that there is a region for x over which the above conclusions on OPF are valid. Due to sace restrictions, the details are omitted here. IV. CONCLUSIONS Network flow lays a central role in oerations research, comuter science and engineering. Due to the comlexity of this roblem, the main focus has been on lossless flow networks and more recently on networks with a linear loss function. This aer studies the generalized network flow (GNF roblem, which aims to otimize the flows over a lossy flow network. It is assumed that the two flows over a line are related to each other via an arbitrary convex monotonic

8 8 Generator v 1 g 12 -b 12 i v 2 Load kj kj ( 1 (1 g14-b14 i g23-b23 i * (2 Generator (2 v 4 g 34 -b 34 i v 3 (a ~ (, Load Fig. 4: (a: An examle of electrical ower network; (b Feasible set for (, kj ; (c Feasible set for (, kj after imosing lower and uer bounds on θ. function. The GNF roblem is hard due to the resence of nonlinear equality flow constraints. If the flow constraints are relaxed to convex inequalities, these constraints may not be~ binding at otimality (as verified in simulations. This imlies that a natural convex relaxation of GNF may lead to wrong flows. Nonetheless, this aer roves that the nodal injections min obtained by solving the convex relaxation are otimal, as long as GNF is feasible. In other words, this aer rooses a olynomial-time algorithm for finding the otimal injections. Obtaining a set of flows associated with the otimal injections is a searate roblem and has been considered as future work. An immediate alication of this work is in ower systems, where the goal is to otimize the ower flows at buses and over transmission lines. Recent work on the otimal ower flow roblem has shown that this non-convex roblem can min max be solved via a convex relaxation after two aroximations: relaxing angle constraints (by adding virtual hase shifters and relaxing ower balance equations to inequality flow constraints. The results on GNF roves that the second relaxation (on ower balance equations is redundant under a very mild angle assumtion. REFERENCES [1] A. V. Goldberg, E. Tardos, and R. E. Tarjan, Network flow algorithms, Flows, Paths and VLSI (Sringer, Berlin, , [2] W. S. Jewell, Otimal flow through networks with gains, Oerations Research, vol. 10, , [3] L. R. Ford and D. R. Fulkerson, Flows in networks, Princeton University Press, [4] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network flows: theory, algorithms, and alications, Prentice-Hall, [5] J. Edmonds and R. M. Kar, Theoretical imrovements in algorithmic efficiency for network flow roblems, Journal of the ACM, vol. 19, , [6] D. Goldfarb and J. Hao, Polynomial-time rimal simlex algorithms for the minimum cost network flow roblem, Algorithmica, vol. 8, , [7] D. Bienstock, S. Chora, O. Gunluk, and C. Y. Tsai, Minimum cost caacity installation for multicommodity network flows, Mathematical Programming, vol. 81, , [8] J. L. Goffin, J. Gondzio, R. Sarkissian, and J. P. Vial, Solving nonlinear multicommodity flow roblems by the analytic center cutting lane method, Mathematical Programming, vol. 76, , [9] S. Boyd and L. Vandenberghe, Convex Otimization. Cambridge, [10] M. Kraning, E. Chu, J. Lavaei, and S. Boyd, Dynamic network energy management via roximal message assing, To aear in Foundations and Trends in Otimization, htt:// boyd/ aers/df/msg ass dyn.df. * max (b [11] I. A. Hiskens and R. J. Davy, Exloring the ower flow solution sace boundary, IEEE Transactions on Power Systems, vol. 16, no. 3, , [12] S. Sojoudi and J. Lavaei, Physics of ower networks makes hard otimization roblems easy to solve, IEEE Power & Energy Society General Meeting, [13] J. Carentier, Contribution to the economic disatch roblem, Bulletin Society Francaise Electriciens, [14] J. A. Momoh, M. E. El-Hawary, and R. Adaa, A review of selected otimal ower flow literature to Part I: Nonlinear and quadratic rogramming aroaches, IEEE Transactions on Power Systems, [15] J. A. Momoh, M. E. El-Hawary, and R. Adaa, A review of selected otimal ower flow literature to Part II: Newton, linear rogramming and interior oint methods, IEEE Transactions on Power Systems, [16] J. Lavaei and S. H. Low, Zero duality ga in otimal ower flow roblem, IEEE Transactions on Power Systems, vol. 27, no. 1, , [17] B. Lesieutre, D. Molzahn, A. Borden, and C. L. DeMarco, Examining the limits of the alication of semidefinite rogramming to ower flow roblems, 49th Annual Allerton Conference, [18] S. Bose, D. F. Gayme, S. Low, and M. K. Chandy, Otimal ower flow over tree networks, Proceedings of the Forth-Ninth Annual Allerton Conference, [19] B. Zhang and D. Tse, Geometry of injection regions of ower networks, To aear in IEEE Transactions on Power Systems, [20] J. Lavaei, B. Zhang, and D. Tse, Geometry of ower flows in tree networks, IEEE Power & Energy Society General Meeting, [21] J. Lavaei and S. H. Low, Convexification of otimal ower flow roblem, 48th Annual Allerton Conference, [22] A. Y. S. Lam, B. Zhang, A. Dominguez-Garcia, and D. Tse, Otimal distributed voltage regulation in ower distribution networks, Submitted for ublication, [23] Y. Weng, Q. Li, R. Negi, and M. Ilic, Semidefinite rogramming for ower system state estimation, IEEE Power & Energy Society General Meeting, [24] D. K. Molzahn, B. C. Lesieutre, and C. L. DeMarco, A sufcient condition for ower flow insolvability with alications to voltage stability margins, htt://arxiv.org/df/ df, [25] S. Sojoudi and S. H. Low, Otimal charging of lug-in hybrid electric vehicles in smart grids, IEEE Power & Energy Society General Meeting, [26] J. Lavaei, Zero duality ga for classical of roblem convexifies fundamental nonlinear ower roblems, American Control Conference, [27] J. Lavaei and S. Sojoudi, Cometitive equilibria in electricity markets with nonlinearities, American Control Conference, [28] S. Sojoudi and J. Lavaei, Convexification of generalized network flow roblem with alication to ower systems, htt:// edu/ lavaei/generalized Net Flow.df, (c