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arxiv: 2604.00664 · v2 · submitted 2026-04-01 · 🧮 math.OC

Theoretical Perspectives on Jabr-Type Convex Relaxations for AC Optimal Power Flow

Gabor Riccardi , Ambrogio Maria Bernardelli , Stefano Gualandi This is my paper

Pith reviewed 2026-05-13 22:25 UTC · model grok-4.3

classification 🧮 math.OC
keywords AC optimal power flowJabr relaxationsecond-order cone programmingmultilinear equalitiescycle constraintsMcCormick envelopesconvex hullnetwork optimization
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The pith

Jabr relaxations for AC optimal power flow can be unified through multilinear equalities that reinterpret cycle constraints and distinguish graph from set convexification.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This review paper examines techniques that tighten the second-order cone relaxation introduced by Jabr for the alternating current optimal power flow problem. It develops a single framework by recasting cycle-based strengthening methods as multilinear consistency conditions whose convex hulls follow from classical theory. The analysis compares primal McCormick envelopes with dual extended formulations and locates the precise structural conditions under which the resulting relaxations become identical. A reader cares because the synthesis separates the geometric task of convexifying network interactions from the task of convexifying the nonlinear power-flow equations, offering clearer guidance for constructing tighter yet tractable approximations on large networks.

Core claim

The paper establishes that strengthening techniques for Jabr-type relaxations share common geometric and graph-theoretic foundations that admit unification via multilinear equalities. Cycle constraints are reinterpreted as multilinear consistency conditions, their convexification is analyzed through classical convex hull theory, and the relationship between primal McCormick relaxations and dual extended formulations is clarified. Structural conditions are identified under which these relaxations coincide, and a distinction is drawn between convexifying the interaction graph and convexifying the feasible set of the ACOPF.

What carries the argument

The unifying perspective based on multilinear equalities, which recasts cycle constraints as consistency conditions on multilinear terms and connects primal and dual convexification procedures.

If this is right

  • Under the identified structural conditions, primal McCormick and dual extended formulations generate equivalent tightenings of the Jabr relaxation.
  • Cycle constraints can be systematically replaced by their multilinear convex envelopes without loss of tightness when the network satisfies the stated conditions.
  • Design of stronger relaxations can proceed by separately addressing graph topology and power-flow nonlinearity rather than treating them as a single object.
  • The framework supplies a common language for comparing existing cycle-based, envelope-based, and reformulation-based strengthenings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same multilinear-unification lens could be applied to other polynomial network problems such as gas flow or water network optimization to obtain analogous tightening results.
  • On very large grids the distinction between graph convexification and set convexification may guide hybrid solvers that apply different strengthening levels to different subgraphs.
  • Empirical checks of the coincidence conditions on benchmark instances could quantify the computational savings achieved by switching between primal and dual representations.

Load-bearing premise

The strengthening techniques for Jabr relaxations share common geometric and graph-theoretic foundations that are amenable to unification via multilinear equalities.

What would settle it

A concrete power network on which the reinterpretation of cycle constraints as multilinear consistency conditions produces a relaxation whose feasible set differs from the set obtained by direct convex-hull computation on the original cycle polynomials.

Figures

Figures reproduced from arXiv: 2604.00664 by Ambrogio Maria Bernardelli, Gabor Riccardi, Stefano Gualandi.

Figure 1
Figure 1. Figure 1: Decomposition of a cycle into 3- and 4- cycles. [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
read the original abstract

The alternating current optimal power flow problem is a fundamental yet highly nonconvex optimization problem whose structure reflects both nonlinear power flow physics and the topology of the underlying network. Among convex relaxations, the second-order cone relaxation introduced by Jabr has proven particularly influential, serving as a computationally efficient alternative to semidefinite relaxations and a foundation for numerous strengthening techniques. In recent years, a variety of approaches have been proposed to tighten Jabr-type relaxations, including cycle-based constraints, convex envelopes of multilinear terms, and dual reformulations. However, these developments are often presented independently, concealing their common geometric and graph-theoretic foundations. This paper provides a structured review of strengthening techniques for the Jabr relaxation and develops a unifying perspective based on multilinear equalities. We reinterpret cycle constraints as multilinear consistency conditions, analyze their convexification through classical convex hull theory, and investigate the relationship between primal McCormick relaxations and dual extended formulations. In particular, we identify structural conditions under which these relaxations coincide and clarify the distinction between convexifying the interaction graph and convexifying the feasible set of the ACOPF. The resulting framework connects graph structure, multilinear convexification, and conic relaxations in a unified manner, offering both a conceptual synthesis of existing results and new insights for the design of stronger relaxations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper reviews strengthening techniques for Jabr-type convex relaxations of the AC Optimal Power Flow problem and develops a unifying perspective based on multilinear equalities. It reinterprets cycle-based constraints as multilinear consistency conditions, applies classical convex-hull results to obtain convex envelopes, and investigates the relationship between primal McCormick relaxations and dual extended formulations, identifying structural conditions under which these coincide while distinguishing graph convexification from feasible-set convexification.

Significance. If the structural conditions and reinterpretations hold, the work offers a useful conceptual synthesis connecting graph structure, multilinear convexification, and conic relaxations for ACOPF. It credits the common geometric foundations across cycle constraints, multilinear envelopes, and extended formulations, providing a lens that may aid design of stronger relaxations without introducing new computational methods.

minor comments (3)
  1. [§2] The abstract and introduction use 'interaction graph' without an early formal definition; add a dedicated paragraph or figure in §2 clarifying its relation to the power network graph and the multilinear terms.
  2. [§3] Notation for the McCormick envelopes and extended dual variables is introduced piecemeal; consolidate the variable definitions and their domains into a single table or subsection for readability.
  3. [§4] Several citations to prior cycle-based strengthenings appear in the review section but lack explicit cross-references to the corresponding equations in the unifying framework; add forward pointers when the reinterpretation is applied.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its unifying perspective on Jabr-type relaxations, and the recommendation for minor revision. We appreciate the acknowledgment that the work connects graph structure, multilinear convexification, and conic relaxations without introducing new computational methods. Since no specific major comments were raised, we interpret the minor revision as addressing any editorial or presentational suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper develops a unifying theoretical perspective on Jabr-type relaxations by reinterpreting cycle constraints as multilinear consistency conditions and applying classical convex-hull results to derive convex envelopes. Structural conditions for coincidence of primal McCormick and dual extended formulations are stated explicitly in terms of the interaction graph, following directly from the multilinear reformulation without reduction to fitted parameters, self-definitions, or load-bearing self-citations. The distinction between graph convexification and feasible-set convexification is obtained via standard convex analysis rather than ansatzes or renamed known results from the authors' prior work. The derivation chain remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper draws on standard results from convex optimization and graph theory without introducing new free parameters or entities; the unification relies on reinterpreting prior work.

axioms (1)
  • standard math Convex hull theory for multilinear functions
    Applied to convexify cycle constraints and multilinear terms as described in the abstract.

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    Al-Khayyal and James E

    Ffaiz A. Al-Khayyal and James E. Falk. Jointly Constrained Biconvex Programming.Mathematics of Operations Research, 8(2):273–286, 1983

    work page 1983

  2. [2]

    Sogol Babaeinejadsarookolaee, Adam Birchfield, Richard Christie, Car- leton Coffrin, C.L. Demarco, Ruisheng Diao, Michael Ferris, Stephane Flis- counakis, Scott Greene, Renke Huang, C´ edric Josz, Roman Korab, Bernard Lesieutre, Jean Maeght, Daniel Molzahn, Thomas Overbye, Patrick Pan- ciatici, Byungkwon Park, Jonathan Snodgrass, and Ray Zimmerman. The Po...

    work page 2021

  3. [3]

    Sahinidis, and Mohit Tawarmalani

    Xiaowei Bao, Nikolaos V. Sahinidis, and Mohit Tawarmalani. Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs.Optimization Methods and Software, 24(4-5):485–504, 2009

    work page 2009

  4. [4]

    Math- ematical programming formulations for the alternating current optimal power flow problem.Annals of Operations Research, 314(1):277–315, 2022

    Daniel Bienstock, Mauro Escobar, Claudio Gentile, and Leo Liberti. Math- ematical programming formulations for the alternating current optimal power flow problem.Annals of Operations Research, 314(1):277–315, 2022. 25

    work page 2022

  5. [5]

    Approximate method for AC trans- mission switching based on a simple relaxation for ACOPF problems

    Daniel Bienstock and Gonzalo Mu˜ noz. Approximate method for AC trans- mission switching based on a simple relaxation for ACOPF problems. In 2015 IEEE Power & Energy Society General Meeting, pages 1–5. IEEE, 2015

    work page 2015

  6. [6]

    LP formulations for polynomial optimization problems.SIAM Journal on Optimization, 28(2):1121–1150, 2018

    Daniel Bienstock and Gonzalo Mu˜ noz. LP formulations for polynomial optimization problems.SIAM Journal on Optimization, 28(2):1121–1150, 2018

    work page 2018

  7. [7]

    On linear relaxations of opf prob- lems.arXiv preprint arXiv:1411.1120, 2014

    Daniel Bienstock and Gonzalo Mu˜ noz. On linear relaxations of opf prob- lems.arXiv preprint arXiv:1411.1120, 2014

    work page arXiv 2014

  8. [8]

    Strong NP-hardness of AC power flows feasibility, 2019

    Daniel Bienstock and Abhinav Verma. Strong NP-hardness of AC power flows feasibility, 2019

    work page 2019

  9. [9]

    Accurate linear cutting-plane re- laxations for acopf.Mathematical Programming Computation, pages 1–55, 2025

    Daniel Bienstock and Mat´ ıas Villagra. Accurate linear cutting-plane re- laxations for acopf.Mathematical Programming Computation, pages 1–55, 2025

    work page 2025

  10. [10]

    Anjos, and S´ ebastien Le Digabel

    Christian Bingane, Miguel F. Anjos, and S´ ebastien Le Digabel. Tight-and- Cheap Conic Relaxation for the AC Optimal Power Flow Problem.IEEE Transactions on Power Systems, 33(6):7181–7188, 2018

    work page 2018

  11. [11]

    Globally solving nonconvex quadratic programming problems with box constraints via in- teger programming methods.Mathematical Programming Computation, 10(3):333–382, 2018

    Pierre Bonami, Oktay G¨ unl¨ uk, and Jeff Linderoth. Globally solving nonconvex quadratic programming problems with box constraints via in- teger programming methods.Mathematical Programming Computation, 10(3):333–382, 2018

    work page 2018

  12. [12]

    Subhonmesh Bose, Steven Low, and K. Chandy. Equivalence of Branch Flow and Bus Injection Models.2012 50th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2012, pages 1893–1899, 10 2012

    work page 2012

  13. [13]

    Bugosen, Robert B

    Sergio I. Bugosen, Robert B. Parker, and Carleton Coffrin. Applications of Lifted Nonlinear Cuts to Convex Relaxations of the AC Power Flow Equations.IEEE Transactions on Power Systems, 2024

    work page 2024

  14. [14]

    Michael Bynum, Anya Castillo, Jean-Paul Watson, and Carl D. Laird. Strengthened SOCP Relaxations for ACOPF with McCormick Envelopes and Bounds Tightening. InComputer Aided Chemical Engineering, vol- ume 44, pages 1555–1560. Elsevier, 2018

    work page 2018

  15. [15]

    On convex relaxations of quadri- linear terms.J

    Sonia Cafieri, Jon Lee, and Leo Liberti. On convex relaxations of quadri- linear terms.J. Global Optimization, 47:661–685, 08 2010

    work page 2010

  16. [16]

    Cain, Richard P

    Mary B. Cain, Richard P. O’Neill, and Anya Castillo. History of optimal power flow and formulations.Federal Energy Regulatory Commission, 1:1– 36, 01 2012. 26

    work page 2012

  17. [17]

    Carpentier

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

    work page 1962

  18. [18]

    SOCP relaxations of optimal power flow problem considering current margins in radial networks.Energies, 11(11):3164, 2018

    Yuwei Chen, Ji Xiang, and Yanjun Li. SOCP relaxations of optimal power flow problem considering current margins in radial networks.Energies, 11(11):3164, 2018

    work page 2018

  19. [19]

    Konstantina Christakou, Dan-Cristian Tomozei, Jean-Yves Le Boudec, and Mario Paolone. AC OPF in radial distribution networks–Part I: On the lim- its of the branch flow convexification and the alternating direction method of multipliers.Electric Power Systems Research, 143:438–450, 2017

    work page 2017

  20. [20]

    A linear-programming approximation of AC power flows.INFORMS Journal on Computing, 26(4):718–734, 2014

    Carleton Coffrin and Pascal Van Hentenryck. A linear-programming approximation of AC power flows.INFORMS Journal on Computing, 26(4):718–734, 2014

    work page 2014

  21. [21]

    Relaxations of multilinear convex envelopes: dual is better than primal

    Alberto Costa and Leo Liberti. Relaxations of multilinear convex envelopes: dual is better than primal. InInternational Symposium on Experimental Algorithms, pages 87–98. Springer, 2012

    work page 2012

  22. [22]

    The multilinear polytope for acyclic hypergraphs.SIAM Journal on Optimization, 28(2):1049–1076, 2018

    Alberto Del Pia and Aida Khajavirad. The multilinear polytope for acyclic hypergraphs.SIAM Journal on Optimization, 28(2):1049–1076, 2018

    work page 2018

  23. [23]

    The running intersection relax- ation of the multilinear polytope.Mathematics of Operations Research, 46(3):1008–1037, 2021

    Alberto Del Pia and Aida Khajavirad. The running intersection relax- ation of the multilinear polytope.Mathematics of Operations Research, 46(3):1008–1037, 2021

    work page 2021

  24. [24]

    A polynomial-size extended for- mulation for the multilinear polytope of beta-acyclic hypergraphs.arXiv preprint arXiv:2212.11239, 2023

    Alberto Del Pia and Aida Khajavirad. A polynomial-size extended for- mulation for the multilinear polytope of beta-acyclic hypergraphs.arXiv preprint arXiv:2212.11239, 2023. arXiv:2212.11239 [math.CO]

    work page arXiv 2023

  25. [26]

    arXiv:2501.04805 [math.OC]

    work page arXiv

  26. [27]

    Masoud Farivar and Steven H. Low. Branch flow model: Relaxations and convexification–Part I.IEEE Transactions on Power Systems, 28(3):2554– 2564, 2013

    work page 2013

  27. [28]

    Masoud Farivar and Steven H. Low. Branch Flow Model: Relaxations and Convexification–Part II.IEEE Transactions on Power Systems, 28(3):2565–2572, 2013

    work page 2013

  28. [29]

    Philipp Fortenbacher and Turhan Demiray. Linear/quadratic programming-based optimal power flow using linear power flow and absolute loss approximations.International Journal of Electrical Power & Energy Systems, 107:680–689, 2019

    work page 2019

  29. [30]

    Lingwen Gan, Na Li, Ufuk Topcu, and Steven H. Low. Exact convex relaxation of optimal power flow in radial networks.IEEE Transactions on Automatic Control, 60(1):72–87, 2014. 27

    work page 2014

  30. [31]

    Accuracy enhancement of second-order cone relaxation for AC optimal power flow via linear mapping.Electric Power Systems Research, 212:108646, 2022

    Khalil Gholami, Ali Azizivahed, Li Li, and Jiangfeng Zhang. Accuracy enhancement of second-order cone relaxation for AC optimal power flow via linear mapping.Electric Power Systems Research, 212:108646, 2022

    work page 2022

  31. [32]

    Tightening quadratic convex relaxations for the alternating current optimal transmission switch- ing problem.INFORMS Journal on Computing, 2025

    Cheng Guo, Harsha Nagarajan, and Merve Bodur. Tightening quadratic convex relaxations for the alternating current optimal transmission switch- ing problem.INFORMS Journal on Computing, 2025

    work page 2025

  32. [33]

    Convex quadratic relaxations for mixed-integer nonlinear programs in power sys- tems.Mathematical Programming Computation, 9(3):321–367, 2017

    Hassan Hijazi, Carleton Coffrin, and Pascal Van Hentenryck. Convex quadratic relaxations for mixed-integer nonlinear programs in power sys- tems.Mathematical Programming Computation, 9(3):321–367, 2017

    work page 2017

  33. [34]

    Polynomial SDP cuts for optimal power flow

    Hassan Hijazi, Carleton Coffrin, and Pascal Van Hentenryck. Polynomial SDP cuts for optimal power flow. In2016 Power Systems Computation Conference (PSCC), pages 1–7. IEEE, 2016

    work page 2016

  34. [35]

    A sufficient condition on convex relaxation of AC optimal power flow in distribution networks.IEEE Transactions on Power Systems, 32(2):1359–1368, 2016

    Shaojun Huang, Qiuwei Wu, Jianhui Wang, and Haoran Zhao. A sufficient condition on convex relaxation of AC optimal power flow in distribution networks.IEEE Transactions on Power Systems, 32(2):1359–1368, 2016

    work page 2016

  35. [36]

    Rabih A. Jabr. Radial distribution load flow using conic programming. IEEE Transactions on Power Systems, 21(3):1458–1459, 2006

    work page 2006

  36. [37]

    Rabih A. Jabr. A conic quadratic format for the load flow equations of meshed networks.IEEE Transactions on Power Systems, 22(4):2285–2286, 2007

    work page 2007

  37. [38]

    Rabih A. Jabr. Optimal power flow using an extended conic quadratic formulation.IEEE Transactions on Power Systems, 23(3):1000–1008, 2008

    work page 2008

  38. [39]

    Dey, and Xu Andy Sun

    Burak Kocuk, Santanu S. Dey, and Xu Andy Sun. Strong SOCP Re- laxations for the Optimal Power Flow Problem.Operations Research, 64(6):1177–1196, December 2016

    work page 2016

  39. [40]

    Dey, and Xu Andy Sun

    Burak Kocuk, Santanu S. Dey, and Xu Andy Sun. New formulation and strong MISOCP relaxations for AC optimal transmission switching prob- lem.IEEE Transactions on Power Systems, 32(6):4161–4170, 2017

    work page 2017

  40. [41]

    Dey, and Xu Andy Sun

    Burak Kocuk, Santanu S. Dey, and Xu Andy Sun. Matrix minor reformu- lation and SOCP-based spatial branch-and-cut method for the AC optimal power flow problem.Mathematical Programming Computation, 10(4):557– 596, 2018

    work page 2018

  41. [42]

    AC- feasibility on tree networks is NP-hard.IEEE Transactions on Power Sys- tems, 31(1):798–801, 2015

    Karsten Lehmann, Alban Grastien, and Pascal Van Hentenryck. AC- feasibility on tree networks is NP-hard.IEEE Transactions on Power Sys- tems, 31(1):798–801, 2015

    work page 2015

  42. [43]

    Steven H. Low. Convex relaxation of optimal power flow–Part I: Formula- tions and equivalence.IEEE Transactions on Control of Network Systems, 1(1):15–27, 2014. 28

    work page 2014

  43. [44]

    Steven H. Low. Convex relaxation of optimal power flow–Part II: Exactness. IEEE Transactions on Control of Network Systems, 1(2):177–189, 2014

    work page 2014

  44. [45]

    Some Results on the Strength of Relaxations of Multilinear Functions.Mathematical Pro- gramming, 136, 04 2012

    James Luedtke, Mahdi Namazifar, and Jeff Linderoth. Some Results on the Strength of Relaxations of Multilinear Functions.Mathematical Pro- gramming, 136, 04 2012

    work page 2012

  45. [46]

    Maranas and Christodoulos A

    Costas D. Maranas and Christodoulos A. Floudas. Finding all solutions of nonlinearly constrained systems of equations.Journal of Global Optimiza- tion, 7(2):143–182, September 1995

    work page 1995

  46. [47]

    McCormick

    Garth P. McCormick. Computability of global solutions to factorable non- convex programs: Part I–Convex underestimating problems.Mathematical Programming, 10(1):147–175, December 1976

    work page 1976

  47. [48]

    Meyer and Christodoulos A

    Clifford A. Meyer and Christodoulos A. Floudas. Trilinear Monomials with Positive or Negative Domains: Facets of the Convex and Concave En- velopes. InFrontiers in Global Optimization, pages 327–352. Springer US, Boston, MA, 2004

    work page 2004

  48. [49]

    Meyer and Christodoulos A

    Clifford A. Meyer and Christodoulos A. Floudas. Convex envelopes for edge-concave functions.Mathematical Programming, 103(2):207–224, June 2005

    work page 2005

  49. [50]

    Least squares estimation based SDP cuts for SOCP relaxation of AC OPF

    Zhixin Miao, Lingling Fan, Hossein Ghassempour Aghamolki, and Bo Zeng. Least squares estimation based SDP cuts for SOCP relaxation of AC OPF. IEEE Transactions on Automatic Control, 63(1):241–248, 2017

    work page 2017

  50. [51]

    Molzahn and Ian A

    Daniel K. Molzahn and Ian A. Hiskens. A survey of relaxations and approx- imations of the power flow equations.Foundations and Trends in Electric Energy Systems, 4(1-2):1–221, 2019

    work page 2019

  51. [52]

    An exact convex formulation of the optimal power flow in radial distribution networks including transverse components.IEEE Transactions on Automatic Control, 63(3):682–697, 2017

    Mostafa Nick, Rachid Cherkaoui, Jean-Yves Le Boudec, and Mario Paolone. An exact convex formulation of the optimal power flow in radial distribution networks including transverse components.IEEE Transactions on Automatic Control, 63(3):682–697, 2017

    work page 2017

  52. [53]

    AC Optimal Power Flow: a strengthened SDP relaxation and an iterative MILP scheme for global optimization, February 2022

    Antoine Oustry. AC Optimal Power Flow: a strengthened SDP relaxation and an iterative MILP scheme for global optimization, February 2022

    work page 2022

  53. [54]

    The boolean quadric polytope: Some characteristics, facets and relatives.Mathematical Programming, 45(1):139–172, 1989

    Manfred Padberg. The boolean quadric polytope: Some characteristics, facets and relatives.Mathematical Programming, 45(1):139–172, 1989

    work page 1989

  54. [55]

    Anatoliy D. Rikun. A Convex Envelope Formula for Multilinear Functions. Journal of Global Optimization, 10(4):425–437, June 1997

    work page 1997

  55. [56]

    Tuning successive linear programming to solve AC optimal power flow problem for large networks.International Journal of Electrical Power & Energy Systems, 137:107807, 2022

    Sayed Abdullah Sadat and Mostafa Sahraei-Ardakani. Tuning successive linear programming to solve AC optimal power flow problem for large networks.International Journal of Electrical Power & Energy Systems, 137:107807, 2022. 29

    work page 2022

  56. [57]

    Springer, 2003

    Alexander Schrijver.Combinatorial Optimization: Polyhedra and Effi- ciency, volume 24 ofAlgorithms and Combinatorics. Springer, 2003

    work page 2003

  57. [58]

    Laird, Anya Castillo, and Joseph K

    Yuanxun Shao, Dillard Robertson, Michael Bynum, Carl D. Laird, Anya Castillo, and Joseph K. Scott. Efficient bounds tightening based on SOCP relaxations for AC optimal power flow.Optimization and Engineering, 26(1):83–119, 2025

    work page 2025

  58. [59]

    Convexification of optimal power flow problem by means of phase shifters

    Somayeh Sojoudi and Javad Lavaei. Convexification of optimal power flow problem by means of phase shifters. In2013 IEEE International Conference on Smart Grid Communications (SmartGridComm), pages 756–761. IEEE, 2013

    work page 2013

  59. [60]

    Zimmerman, Carlos E

    Ray D. Zimmerman, Carlos E. Murillo-S´ anchez, and Robert J. Thomas. MATPOWER: Steady-State Operations, Planning, and Analysis Tools for Power Systems Research and Education.IEEE Transactions on Power Systems, 26(1):12–19, 2011

    work page 2011

  60. [61]

    A survey on conic relaxations of optimal power flow problem.European Journal of Operational Research, 287(2):391–409, 2020

    Fariba Zohrizadeh, Cedric Josz, Ming Jin, Ramtin Madani, Javad Lavaei, and Somayeh Sojoudi. A survey on conic relaxations of optimal power flow problem.European Journal of Operational Research, 287(2):391–409, 2020. 30

    work page 2020