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IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15–27, MARCH 2014 (WITH PROOFS) 1 Convex Relaxation of Optimal Power Flow Part I: Formulations and Equivalence Steven H. Low Electrical Engineering, Computing+Mathematical Sciences Engineering and Applied Science, Caltech [email protected] April 15, 2014 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. Citation: IEEE Transactions on Control of Network Systems, 15(1): 15–27, March 2014. This is an extended version with Appendices VIII and IX that provide some mathematical preliminaries and proofs of the main results. All proofs can be found in their original papers. We provide proofs here because (i) it is convenient to have all proofs in one place and in a uniform notation, and (ii) some of the formulations and presentations here are slightly different from those in the original papers. A preliminary and abridged version has appeared in Proceedings of the IREP Symposium - Bulk Power System Dynamics and Control - IX, Rethymnon, Greece, August 25-30, 2013. arXiv:1405.0766v1 [math.OC] 5 May 2014
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IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15–27, MARCH 2014 (WITH PROOFS) 2 C ONTENTS I Introduction 4 I-A Outline of paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I-B Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 II Power flow models 6 II-A Bus injection model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 II-B Branch flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 II-C Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 III Optimal power flow 8 III-A Bus injection model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 III-B Branch flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 III-C OPF as QCQP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 IV Feasible sets and relaxations: BIM 11 IV-A Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 IV-B Feasible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 IV-C Semidefinite relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 IV-D Solution recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 IV-E Tightness of relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 IV-F Chordal relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 V Feasible sets and relaxations: BFM 17 V-A Feasible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 V-B SOCP relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 V-C Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 VI BFM for radial networks 21 VI-A Recursive equations and graph orientation . . . . . . . . . . . . . . . . . . . . . . . . 21 VI-B Linear approximation and bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 VII Conclusion 25 Appendix: VIII: Mathematical preliminaries 26 A QCQP, SDP, SOCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 B Graph, partial matrix and completion . . . . . . . . . . . . . . . . . . . . . . . . . . 28 C Chordal relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
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IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15–27, MARCH 2014 (WITH PROOFS) 3 Appendix: IX: Proofs of main results 32 A Proof of Theorem 1: equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 B Proof of Theorem 2: rank-1 characterization . . . . . . . . . . . . . . . . . . . . . . . 33 C Proof of Corollary 3: uniqueness of completion . . . . . . . . . . . . . . . . . . . . . 34 D Proof of Theorem 5: BIM feasible sets . . . . . . . . . . . . . . . . . . . . . . . . . 34 E Proof of Theorem 7: BFM feasible sets . . . . . . . . . . . . . . . . . . . . . . . . . 34 F Proof of Theorem 8: BFM cycle condition . . . . . . . . . . . . . . . . . . . . . . . 36 G Proof of Theorem 9: radial networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 H Proof of Theorem 11: equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 I Proof of Lemma 12: voltage bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 References 42 Acknowledgment. We thank the support of NSF through NetSE CNS 0911041, ARPA-E through GENI DE-AR0000226, Southern California Edison, the National Science Council of Taiwan through NSC 103- 3113-P-008-001, the Los Alamos National Lab (DoE), and Caltech’s Resnick Institute.
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IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 1(1):15–27, MARCH 2014 (WITH PROOFS)
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