THE optimal transmission network expansion planning. AC Transmission Network Expansion Planning: A Semidefinite Programming Branch-and-Cut Approach

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1 1 AC Transmission Network Expansion Panning: A Semidefinite Programg Branch-and-Cut Approach Bissan Ghaddar and Rabih A. Jabr, Feow, IEEE arxiv: v1 [math.oc] 9 Nov 2017 Abstract Transmission network expansion panning is a mixed-integer optimization probem, whose soution is used to guide future investment in transmission equipment. An approach is presented to find the goba soution of the transmission panning probem using an AC network mode. The approach buids on the semidefinite reaxation of the AC optima power fow probem (ACOPF); its computationa engine is a new speciaized branch-and-cut agorithm for transmission expansion panning to dea with the underying mixed-integer ACOPF probem. Vaid inequaities that are based on specific knowedge of the expansion probem are empoyed to improve the soution quaity at any node of the search tree, and thus significanty reduce the overa computationa effort of the branch-and-bound agorithm. Additionay, sparsity of the semidefinite reaxation is expoited to further reduce the computation time at each node of the branch-and-cut tree. Despite the vast number of pubications on transmission expansion panning, the proposed approach is the first to provide expansion pans that are gobay optima using a soution approach for the mixed-integer ACOPF probem. The resuts on standard networks serve as important benchmarks to assess the soution quaity from existing techniques and simpified modes. Index Terms Design optimization, mathematica programg, noninear network anaysis, optimization methods, power system panning. I. INTRODUCTION THE optima transmission network expansion panning (TNEP) probem seeks to compute the specifications and ocations of eectrica transmission equipment so that the expanded system can meet the expected future oad and generation patterns [1]. The panning probem can be carried out over a singe stage, or it can be exted over mutipe stages that represent a onger panning period; the optima decision in mutistage panning inks the required investment in transmission equipment to a particuar time period [2], [3]. In its most accurate representation, TNEP is a mixed-integer noninear program whose soution is computationay chaenging. The appication of computing techniques to sove the TNEP probem has been steadiy increasing since the 1970s, and has passed through various accuracy eves of network representation that were adapted to the existing optimization sover technoogy [4]. This paper considers the soution of TNEP with the system being accuratey represented via the AC B. Ghaddar is with the Department of Management Sciences, University of Wateroo, Canada (emai: bissan.ghaddar@uwateroo.ca). R. A. Jabr is with the Department of Eectrica & Computer Engineering, American University of Beirut, P.O. Box , Riad E- Soh / Beirut , Lebanon (emai: rabih.jabr@aub.edu.b). network mode; it empoys the power network semidefinitebased reaxation [5] in a branch-and-cut optimization methodoogy that is specificay taiored to the TNEP probem. The outcome is a TNEP pan empoying the AC network mode, that is provaby the gobay optima soution. It is worth noting that none of the existing AC-TNEP techniques can caim the goba optimay of their soutions. The TNEP iterature reports expansion soutions using different network modes: (i) the transportation mode that enforces Kirchhoff s current aw (KCL) at a nodes [1], [6], [7], (ii) the hybrid mode that adds to the transportation mode the Kirchhoff s votage aw (KVL) equations ony for existing circuits [8], (iii) the DC network mode that adds to the hybrid mode the KVL equations for new circuits [9], and (iv) the AC network mode [10]. Both the DC and AC network modes when empoyed in TNEP resut in terms that have products of integer and continuous variabes, which further contribute to the chaenge in computing the soution. This can be circumvented by the use of the disjunctive reformuation in TNEP probems that are based on DC [11] and AC [12] power fow modes. The soution techniques for TNEP can be cassified into three categories: (i) the ocay optima search methods, (ii) the meta-heuristics, and (iii) the compete search methods based on conventiona optimization theory. The constructive heuristic agorithm [1], [10] is one of the eariest oca search methods appied to TNEP; it can give good quaity soutions but which are unikey to be optima. The discrepancy-bounded oca search [13] is a generaization of the constructive heuristic that gives better soutions with ACOPF modes. The metaheuristic methods are rooted in the processes of natura and physica systems [14]; these incude simuated anneaing, tabu search, and severa variants of genetic agorithms that mimic the mechanisms of natura seection and evoution [2], [15] [20]. Athough meta-heuristics can avoid being trapped in ocay optima TNEP soutions, they do not give a rigorous indicator on the quaity of the soution. The cassica optimization agorithms t to fa into two categories: (i) Bers decomposition type agorithms and (ii) Branchand-Bound/Branch-and-Cut. Due to the non-convexity of the DC/AC-TNEP formuations, it is possibe for the Bers cut to chop off a part of the feasibe region that incudes the goba soution; for DC-TNEP, this can be circumvented either via the hierarchica decomposition method [21], [22] or by reformuating the TNEP probem as a disjunctive mode [23], [24]. State-of-the-art impementations of branch-and-cut

2 2 approaches are currenty provided in commercia packages for mixed-integer inear programg; these have been used in disjunctive modes of DC-TNEP [3], in addition to enhanced inear representations that approximate network osses and reactive power constraints [25] [28]; a recent extension incudes modeing the choice of both HVAC and HVDC transmission equipment [29]. Whie the inear network modes have been extensivey used in TNEP, the strict AC mode is rarey discussed due to the computationa compexity that it introduces; ony [10], [12], [13], [19], [20] from amongst the above surveyed methods use the AC power fow equations. However, TNEP soutions that are based on the DC network mode can yied arbitrariy poor designs that require further reinforcement to satisfy the AC network feasibiity [13], [27]. The main impediment to the use of the AC network mode in TNEP based on cassica optimization is the non-convexity of the AC power fow constraints; this has been circumvented in [30] by expoiting (i) the conic reaxation of the AC optima power fow (OPF) over networks with virtua controabe phase shifters, and (ii) a state-of-the-art branch-and-cut agorithm for mixed-integer conic programg. Athough [30] reports soutions that satisfy the AC network mode constraints, these soutions may not be gobay optima due to the use of inear equations that are needed to aeviate the effect of the virtua phase shifters. To ascertain goba optimaity of the AC-TNEP soutions, this paper proposes the use of the semidefinite programg (SDP) reaxation of the AC optima power fow probem [5]; the SDP reaxation does not entai the disadvantage associated with the virtua phase shifters, but it requires the deveopment of a speciaized branch-and-cut sover to hande SDP constraints. This paper contributes an SDP branch-and-cut method for AC-TNEP, provides detais of vaid inequaities, and proposes to utiize sparsity and probem structure to speed up the soution. The resuts are provaby goba, and serve as benchmarks against which the quaity of approximated methods and heuristic soutions coud be gauged. II. MATHEMATICAL FORMULATION In this section, the same notation is used as in [30]. The topoogy of the power system P = (N, E) is represented as an undirected graph, where each vertex n N is caed a bus and each edge e E is caed a branch inking buses to one another. The parameter N denotes the number of buses and E denotes the number of branches. Let G N be the set of generators and E N N be the set of a branches and N() are buses adjacent to bus. Let S d = P d + jq d be the active and reactive oad (demand) at each bus N and P g + jq g represent the compex power of the generator at bus G. Define V = RV + jiv as the votage at each bus N and S m = P m + jq m as the apparent power fow on the ine (, m) E. The edge set L E contains the branches (, m) such that the apparent power fow imit is ess than a certain given toerance ε. The next section describes the formuation of the transmission network expansion panning probem in detais. A. SDP Formuation To formuate the TNEP mode, the ACOPF probem is first presented as it is the main buiding bock of the TNEP formuation. In this work, the focus is on the rectanguar powervotage formuation of the OPF probem. In the rectanguar formuation, the bus votages are represented by the rea and imaginary votage components. The rea and reactive power fows are quadratic functions of rea and imaginary parts of the votage. This resuts in the generator imits, the fixed oads and the apparent power ine imits being non-convex constraints, and in addition the ower imits of bus votage magnitudes are non-convex. Hence, the rectanguar formuation is aso nonconvex. The foowing parameters are needed to formuate the probem: P and P are the imits on active generation capacity at bus, where P = P = 0 for a N/G. Q and Q are the imits on reactive generation capacity at bus, where Q = Q = 0 for a N/G. P d is the active power demand at bus. Q d is the reactive power demand at bus. V bus. S and V are the imits on the votage at a given m is the imit on the absoute vaue of the apparent power of a branch (, m) L. Additionay, et y m = g m +jb m be the series admittance in the π-mode of ine (, m) and b m be the shunt susceptance in the π-mode of ine (, m). Given a compex votage V at bus, et RV denote the rea part of V and IV denote the imaginary part. In terms of power ine fows, the foowing equations hod: P m = b m (RV IV m RV m IV ) (1) + g m (RV 2 + IV m 2 IV IV m RV RV m ) Q m = b m (RV RV m + IV IV m RV 2 IV 2 ) + g m (RV IV m RV m IV ) b m 2 (RV 2 + IV 2 ) (2) Let RV and IV be the variabes in addition to the variabes P g, P m and Q m. The aim of the optima power fow probem is to satisfy demand at a buses with the imum tota production costs of generators such that the soution obeys the physica aws and other operationa constraints such as transmission ine fow imit constraints. The OPF can be formuated as the foowing quadratic non-convex probem: s.t. G P Q P g (OPF) c 2 (P g )2 + c 1 (P g ) + c0 (3) P g P, G, (4) Q g Q, G, (5) P d = m N() P m, N, (6)

3 3 Q g Qd = m N() Q m, N, (7) (V ) 2 RV 2 + IV 2 (V ) 2, N, (8) Pm 2 + Q 2 m (Sm ) 2, (, m) E, (9) (1) (2) (10) The objective function (3) imizes the cost of power generation. Constraints (4), (5), and (6) set imits on the active power, reactive power and represent the rea and reactive noda baance. Constraints (8) restricts the votage on each bus. Constraints (9) set a imit on the apparent power fow at every branch. Whie constraints (10) define the power generated and the active and reactive power fow. To appy SDP reaxation to the rectanguar formuation (OPF), define a hermitian matrix X = V V that is X = X, where X is the conjugate transpose of X. The Hermitian matrix X is defined as: V 1 2 V 1 V V 1 V n X = V V1 V 2 V Vn V n V1 V n V V n 2 with X m the corresponding eement in the th row and mth coumn. By ignoring the rank constraints, the standard SDP reaxation of the OPF probem is obtained with the foowing set of constraints: P Q P g P g (SDP-OPF) P, G, (11) Q g Q, G, (12) P d = Q g Qd = m N() m N() P m, N (13) Q m, N (14) (V ) 2 X ii (V ) 2, N, (15) Pm 2 + Q 2 m (Sm ) 2, (, m) E, (16) P m = g m X g m RX m b m IX m (17) Q m = (b m + b m 2 )X + b m RX m g m IX m (18) X = X 0 (19) The SDP reaxation in the compex domain is formuated in Bose et a. [31] and is widey used in the iterature now for its notationa simpicity. The SDP reaxation for OPF has been widey used since it was originay proposed by Bai et a. [32] and Lavaei and Low [5]. Because convex conic programs are poynomiay sovabe, the SDP reaxation offers an effective way for obtaining goba optima soutions to OPF probems whenever the reaxation is exact. Motivated by the success of SDP reaxations for OPF probems, an SDP approach is proposed to sove the TNEP probem. TNEP consists of finding the imum cost pan for the eectrica system expansion so that the network adequatey serves the forecasted system oad over a given horizon; the OPF probem appears as the backbone of TNEP. In addition to the variabes defined above, et { αm t 1 if ine t is instaed in transmission corridor (, m) = 0 otherwise. Two additiona compex matrices are defined, Xm t and Xt (m), the vaues of X m and X for the t ine in transmission corridor (, m) respectivey. When ine t is not instaed in the transmission corridor (, m), the corresponding X(m) t = 0 and the rea and imaginary parts of Xm t = 0. Whie when ine t is instaed, the corresponding X(m) t = X = V 2 and the rea and imaginary parts of Xm t are equa to that of X m and are bounded by: 0 R{Xm} t V Vm V Vm I{Xm} t V Vm. Thus, the TNEP optimization mode can be formuated as a mixed integer semidefinite program (MISDP) that is a reaxation of the mixed-integer ACOPF probem: f(α, P g ) = n n c m α t m + cp P g (,m) E t=n =1 s.t. P g P d = P m m N() Q g Qd = Q m m N() P P g P Q Q g Q n m P m = Pm t t=1 n m Q m = Q t m t=1 Pm t = g mx(m) t g mr{xm t } b mi{xm t } (TNEP) (20a) (20b) (20c) (20d) (20e) (20f) (20g) (20h) Q t m = (b m + b m 2 )Xt (m) + b mr{x t m } g mi{x t m } (20i) n m nm t=1 α t m n m (20j) α t m αt 1 m (20k) (V ) 2 α t m Xt (m) (V ) 2 α t m (20) 0 R{Xm t } V Vm α t m (20m) V Vm m I{Xt m } V Vm m (20n) (V ) 2 (1 α t m ) X X(m) t (V ) 2 (1 α t m ) (20o) 0 R{X m } R{Xm t } V Vm (1 α t m ) (20p) V Vm m ) I{X m} I{Xm t } (20q) I{X m } I{Xm t } V Vm m ) (20r) (Pm 1 )2 + (Q 1 m )2 (Sm α 1 m (20s) α t m {0, 1}, X = X 0. (20t) The objective function (20a) imizes the expansion cost with a penaty term on the power osses [30]. Constraints (20b)-(20e) set imits on the active power and reactive power on each bus. Constraints (20f)-(20i) compute the tota rea/reactive power fow aong a transmission corridor which is obtained by sumg the power fow over the distinct ines.

4 4 Constraints (20j) and (20k) enforce a imit on the number of ines added and ensure a sequentia instaation of circuits in each transmission ine respectivey. Constraints (20)-(20r) enforce that when ine t is not instaed in the transmission corridor (, m) then αm t = 0 and the corresponding Xt (m) =0. In addition, the rea and imaginary parts of Xm t = 0 and hence Pm t = Qt m = 0. Whie when ine t is instaed then αm t = 1 and the corresponding Xt (m) = X = V 2. In addition, the rea and imaginary parts of Xm t are equa to that of X m and are bounded by the votage constraints. Finay, constraints (20s) set a imit on the apparent power fow at every ine. The presence of the binary variabes in addition to the SDP constraint in the TNEP formuation, (TNEP), resuts in a mixed integer semidefinite program that is difficut to sove. III. VALID INEQUALITIES To sove the above formuation of the TNEP probem, (TNEP), a speciaized branch-and-bound agorithm is deveoped where the binary condition is reaxed and at every node a SDP reaxation is soved. Since SDP reaxations are typicay expensive to sove, having a strong bound at the root node is critica to the computationa efficiency of the agorithm. Thus, a set of vaid inequaities is introduced to attain a stronger reaxation of the feasibe region of the TNEP probem. In order to obtain the inequaities, resuts from [33] are used to generate fencing constraints. Additionay, conic constraints are added to improve the quaity of the reaxation. The resuting convex program is a strengthened SDP program with improved ower bounds. The first set of constraints that are added are conic constraints: (P t m) 2 + (Q t m) 2 (S m ) 2 α t m t, (, m) E. (21a) The second set of constraints are referred to as the fence constraints. Fencing constraints are a generaization of Kirchhoff Current Law, KCL and form part of a heuristic methodoogy for transmission expansion panning [33]. Three kinds of fences are generated and added to the SDP reaxation: around a singe node, around one node and neighboring node, and around a node and its entire neighborhood. To generate a fencing constraint: Pace an imaginary fence around a portion of the power system. Cacuate power deivered into or out of the fenced area. Compare the transmission capacity with the net oad or net generation within the fenced area to detere if the transmission is adequate. The constraints are written as: n m = {n, m N() n m t=n α t m P d P S m P d P S } (22a) (22b) where. is the ceiing operator and S = (,m) {S m }. Fig. 1. Exampe of a 6 bus network. Exampe 3.1: Consider the exampe with 6 nodes as shown in Figure 1 [10]. A fence is set on nodes 1-5 of the power network. The sum of the oads at buses 1-5 is P d =760 whie the sum of power generated at the same buses is P =530 and n = 5. The power fow imitation is given as S16 = 90, S56 = 98, S6 = 120 ( = 2, 3, 4). The transmission deficit can be represented by the foowing fencing constraint α1,6 1 + α2,6 1 + α3,6 1 + α4,6 1 + α5,6 1 + α1,6 2 + α2,6 2 + α3,6 2 + α4,6 2 + α5,6 2 + α1,6 3 + α5, IV. SPARSITY OF THE SDP RELAXATION Expoiting sparsity has been one of the essentia toos for soving arge-scae optimization probems in genera and optima power fow probems in particuar [34], [35]. Soving existing semidefinite reaxations of arge optima power fow probems requires expoiting power system sparsity. Using a matrix competion decomposition, existing semidefinite reaxations of optima power fow probems are computationay tractabe for probems with hundreds of buses. Since the optima power fow probem is a buiding bock of transmission expansion network panning, to sove the atter a framework for expoiting the sparsity characterized in terms of a chorda graph structure via positive semidefinite matrix competion is used [36]. The computation time of soving the SDP reaxation can be significanty improved using SparseCoO that is used as a preprocessor, which reduces the dimension of matrix variabes in an SDP reaxation before appying the SDP sover. When appied to the semidefinite reaxation, SparseCoO enhances the structured sparsity of the probem, i.e., the correative sparsity. As a resut, the resuting SDP can be soved more effectivey by appying the sparse SDP reaxation. In a branchand-cut framework, expoiting sparsity is important as it reduces the computationa time at each node of the tree resuting in a significant decrease in the tota soving time. V. SDP BRANCH-AND-CUT To deveop an efficient technique, we adopt a branchand-bound approach to sove the TNEP probem using a SDP reaxation, combined with vaid inequaities. The success and the computationa efficiency of the branch-and-bound procedure strongy deps on the quaity of the reaxation

5 5 bounds, the eary generation of binary feasibe soutions to get good bounds, and the branching rues used to obtain the subprobems [37]. Appying the SDP reaxation in addition to the vaid inequaities can hep speed up the branch-and-bound process by improving the bounds at each node, thus reducing the number of nodes of the tree. The foowing sections, describe the branch-and-bound agorithm that is impemented to sove TNEP. The genera agorithm, feasibe soution, and branching rues are described in detais. A. Feasibe Soution During a branch-and-bound procedure, it is of particuar importance to find a feasibe soution to be abe to fathom nodes in the tree reducing the search space. In the described branch-and-bound agorithm, rounding the optima soution αm t to obtain ˆαt m is utiized for a feasibe soution. The obtained rounded soution is checked for feasibiity and the rank of the SDP matrix is checked. The incumbent is updated in case the soution is feasibe and the rank of the SDP matrix is 1. For the branching strategy, depth first is used, that is, the next node to be soved of the branch-and-bound tree is one of the chid nodes of the current node soved. Depth-first node seection goes deep into the branch and bound tree at each iteration, so it reaches the eaf nodes quicky. This is one way of achieving an eary feasibe soution and hence an incumbent. B. Inequaity Generation Scheme Branch-and-cut methods combine the cassica branch-andbound with cutting pane techniques; they define vaid inear inequaities to improve the ower bound of the soution and speed up the search. The branch-and-cut approach can be sped up consideraby by the empoyment of an inequaity generation scheme, either just at the root node of the tree, or at every node of the tree. In the case of the TNEP probem, the inequaities are ony generated at the root node, but since they are vaid to a the chidren nodes, they are added to each node of the tree. C. Branch-and-Cut Agorithm As discussed in the previous sections, the branch-and-cut agorithm for TNEP probem consists of soving SDP reaxations at every node of the tree. Since the variabes are binary, the set of possibe configurations of the variabes is finite and equa to 2 n E thus the branch-and-bound procedure stops after a finite number of nodes. Given (TNEP) and the binary index set J {1,..., n }, a sketch of the branch-and-cut agorithm is given as foows: Vaid inequaities are added at the root node and are reused at the nodes of the branch-andbound tree as they are aso vaid for the chidren nodes. VI. COMPUTATIONAL RESULTS In this section, the performance of the proposed branch-andcut approach is evauated for the TNEP probem on different test cases. The method was programmed in MATLAB and the computationa experiments were conducted on a Lenovo Agorithm 1: Branch-and-Cut Agorithm for TNEP input: (TNEP) output: α t m and z T NEP set: G 0 = G {αm t {0, 1} : t J}, µ G 0 =, z T NEP =, ˆα m t = [ ], Nodes = {N 0} whie Nodes φ do Choose node N i Nodes; set: Nodes = Nodes\N i ; Sove (TNEP i ) to obtain z i and αm t ; if z i = then Fathom N i ; ese if z i z T NEP, then Fathom N i ; ese if αm t {0, 1} t (, m), then if f(αm t ) z T NEP then z T NEP = f(αm t ), ˆαt m = αt m ; Fathom N i ; ese Appy rounding to αm t to obtain ᾱt m ; if feasibe and f(ᾱm t ) z T NEP then z T NEP = f(ᾱm t ), ˆαt m = ᾱt m ; Choose (G 1, G 2 ) as a branching rue for G; set: Nodes = Nodes {N G1, N G2 }; Thinkstation P300 with 32GB of RAM. The MOSEK [38] sover was used to sove the SDP reaxations and SparseCoO [39] was used for expoiting sparsity at each node of the branch-and-bound tree. Four networks are considered, two 6-bus [10], 24-bus [10], and 46-bus systems [30]. The 6- bus Graver system has six buses and 75 potentia ines for the addition of new circuits. From the basic topoogy, two scenarios are taken into account: expansion with existing ines (instance 6-bus), and expansion without existing ines (instance 6gf-bus). The third instance is the IEEE 24-bus and the fourth test system is the 46-bus south Braziian network. Tabe I, presents the computationa resuts for the proposed branch-and-bound approach for the SDP reaxation with and without vaid inequaities as we as with and without expoiting sparsity. The root node ower bound as we as the optima soution (scaed objective vaue (20a)) are reported in the tabe. Additionay, the computationa time (in seconds) and the number of nodes of the branch-and-bound tree are given. From Tabe I, it can be observed that the strengthened SDP reaxation with vaid inequaities has a stronger ower bound at the root node and this transates to ower number of nodes as we as computationa time. At the of the branch-and-bound agorithm, the optima soution is obtained for the origina non-convex quadratic probems as the SDP reaxation produces a rank-1 soution. Notice the significant improvement in the root bound for 6gf instance where the root bound improved from to Furthermore,

6 6 the computationa time of soving TNEP using sparsity is significanty ower than the computationa time of soving the probem without expoiting sparsity. The soution time of the SDP reaxation of the instances with 24 and 46 buses are reduced by a factor of around 10 when expoiting sparsity. Tabe II summarizes the number of transmission corridors (TC) and the tota number of existing (EL), potentia (PL), and optima ines (OL) for a test systems. The expansion pans for the four test systems are given in Tabes III-V, together with their expansion costs. By comparing with the resuts in [30], it becomes evident that the mixed-integer conic sover [30] did give a gobay optima soution for the same TNEP instances athough it coud not rigorousy prove the optimaity of the designs. Aso as in [30], a comparison with the scaed objective vaues in Tabe I shows that the osses are a sma fraction of the tota objective (20a) and are therefore not expected to affect the optima design; in fact, the oss term is incuded in the objective function so as to induce a rank-1 soution. VII. CONCLUSION This paper presented the impementation detais of a mixedinteger SDP method that is speciaized to sove the AC-TNEP probem. The method is demonstrated on standard TNEP probems with 6, 24, and 46 nodes. Vaid inequaities and expoiting the probem structure through sparsity are proposed to speed up the branch-and-bound search; without these cuts, the soution of the argest network is not even possibe. The panning resuts are feasibe with respect to the AC power fow constraints, and they are gobay optima; this is in contrast to many of the resuts that are obtained from simpified inear modes, and which are not even feasibe from an AC network standpoint. Athough previous AC-TNEP methods reported some test resuts that are identica to the ones herein, the importance of this work is that it is the first to rigorousy ascertain goba optimaity. This has vaue in checking the quaity of heuristic and oca search approaches, in addition to soutions that are based on simpified modeing. Future directions incude exting the mode to dea with potentia changes in the demand as we as uncertainties in the renewabe power generation in transmission expansion panning. Furthermore, the same approach can be appied to operationa probems such as operations with transmission switching. ACKNOWLEDGMENT Bissan Ghaddar was supported by NSERC Discovery Grant RGPIN REFERENCES [1] L. L. Garver, Transmission network estimation using inear programg, IEEE Trans. Power Appar. Syst., vo. PAS-89, no. 7, pp , Sep [2] A. H. Escobar, R. A. Gaego, and R. Romero, Mutistage and coordinated panning of the expansion of transmission systems, IEEE Trans. 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