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References

Consolidated bibliography for the Bounds, Branches, and Feasibility note and its Trusting the solver capstone. Inline citations throughout the note link directly to the source; this page is the formal list.

Power flow solvability, uniqueness, and the high-voltage solution

  • S. Bolognani and S. Zampieri, "On the existence and linear approximation of the power flow solution in power distribution networks," IEEE Transactions on Power Systems, vol. 31, no. 1, pp. 163–172, 2016. DOI
  • J. W. Simpson-Porco, "A theory of solvability for lossless power flow equations — Part I: Fixed-point power flow," and "Part II: Conditions for radial networks," IEEE Transactions on Control of Network Systems, vol. 5, no. 3, 2018. arXiv (Part I)
  • K. Dvijotham, E. Mallada, and J. W. Simpson-Porco, "High-voltage solution in radial power networks: existence, properties, and equivalent algorithms," IEEE Control Systems Letters, vol. 1, no. 2, pp. 322–327, 2017. arXiv
  • C. Wang, A. Bernstein, J.-Y. Le Boudec, and M. Paolone, "Explicit conditions on existence and uniqueness of load-flow solutions in distribution networks," IEEE Transactions on Smart Grid, vol. 9, no. 2, pp. 953–962, 2018. DOI
  • A. Bernstein, C. Wang, E. Dall'Anese, J.-Y. Le Boudec, and C. Zhao, "Load flow in multiphase distribution networks: existence, uniqueness, non-singularity and linear models," IEEE Transactions on Power Systems, 2018. DOI

Convex relaxations and exactness

  • S. H. Low, "Convex relaxation of optimal power flow — Part I: Formulations and equivalence; Part II: Exactness," IEEE Transactions on Control of Network Systems, vol. 1, no. 1–2, 2014. DOI (Part II)
  • L. Gan, N. Li, U. Topcu, and S. H. Low, "Exact convex relaxation of optimal power flow in radial networks," IEEE Transactions on Automatic Control, vol. 60, no. 1, 2015. arXiv
  • S. Sojoudi and J. Lavaei, "Physics of power networks makes hard optimization problems easy to solve," IEEE PES General Meeting, 2012. DOI
  • B. Kocuk, S. S. Dey, and X. A. Sun, "Inexactness of SDP relaxation and valid inequalities for optimal power flow" (two-bus characterization of the three approximation outcomes), IEEE Transactions on Power Systems, vol. 31, no. 1, pp. 642–651, 2016. DOI
  • Z. Yuan and M. Paolone, "Properties of convex optimal power flow model based on power loss relaxation" (objective-monotonicity ⇒ exactness), Electric Power Systems Research, 2020. arXiv
  • J.-L. Lupien and A. Lesage-Landry, "Ex post conditions for the exactness of optimal power flow conic relaxations," 2023. arXiv
  • D. K. Molzahn and I. A. Hiskens, "A survey of relaxations and approximations of the power flow equations," Foundations and Trends in Electric Energy Systems, vol. 4, no. 1–2, pp. 1–221, 2019. DOI
  • L. Bobo, A. Venzke, and S. Chatzivasileiadis, "Second-order cone relaxations of the optimal power flow for active distribution grids," 2020. arXiv

Voltage collapse and loadability

  • I. Dobson and L. Lu, "New methods for computing a closest saddle-node bifurcation and worst-case load power margin for voltage collapse," IEEE Transactions on Power Systems, vol. 8, no. 3, pp. 905–913, 1993. DOI
  • J. W. Simpson-Porco, F. Dörfler, and F. Bullo, "Voltage collapse in complex power grids," Nature Communications, vol. 7, 10790, 2016. DOI
  • T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems, Springer, 1998 (canonical treatment of ZIP loads, the nose point, and continuation power flow). DOI

Computational complexity of AC OPF

  • K. Lehmann, A. Grastien, and P. Van Hentenryck, "AC-feasibility on tree networks is NP-hard," IEEE Transactions on Power Systems, vol. 31, no. 1, pp. 798–801, 2016. DOI · arXiv
  • D. Bienstock and A. Verma, "Strong NP-hardness of AC power flows feasibility," Operations Research Letters, vol. 47, no. 6, pp. 494–501, 2019. DOI · arXiv

Global optimization and optimality certificates

  • D. K. Molzahn and I. A. Hiskens, "Moment-based relaxation of the optimal power flow problem" (Lasserre / moment SDP hierarchy), Power Systems Computation Conference (PSCC), 2014. arXiv
  • C. Coffrin, H. L. Hijazi, and P. Van Hentenryck, "The QC relaxation: a theoretical and computational study on optimizing optimal power flow," IEEE Transactions on Power Systems, vol. 31, no. 4, pp. 3008–3018, 2016. DOI · arXiv
  • H. Nagarajan, M. Lu, S. Wang, R. Bent, and K. Sundar, "An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs" (Alpine.jl spatial branch-and-bound), Journal of Global Optimization, vol. 74, pp. 639–675, 2019. DOI · arXiv
  • S. Gopinath, H. L. Hijazi, T. Weisser, H. Nagarajan, M. Yetkin, K. Sundar, and R. W. Bent, "Proving global optimality of ACOPF solutions" (SDP bound tightening with valid cuts; closes gaps on PGLib), Electric Power Systems Research, vol. 189, 106688,
    1. DOI ·
    arXiv

Local optima and OPF landscape

  • W. A. Bukhsh, A. Grothey, K. I. M. McKinnon, and P. A. Trodden, "Local solutions of the optimal power flow problem" (documented multiple local AC-OPF solutions and initialization dependence), IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 4780–4788, 2013. DOI · manuscript
  • Z.-Y. Wang and H.-D. Chiang, "A necessary and sufficient condition for computed OPF solutions to be locally optimal" (distinguishing KKT points and local minima), IEEE Transactions on Power Systems, vol. 38, no. 6, pp. 5491–5500, 2023. DOI
  • F. Zhou and S. H. Low, "Conditions for exact convex relaxation and no spurious local optima" (a benign-landscape theorem under explicit path, monotonicity, and OPF-specialization assumptions), IEEE Transactions on Control of Network Systems, vol. 9, no. 3, 2022. DOI · arXiv

Solver behaviour, constraint qualifications, and insolvability

  • A. Wächter and L. T. Biegler, "On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming" (Ipopt; feasibility restoration phase), Mathematical Programming, vol. 106, no. 1, pp. 25–57, 2006. DOI
  • O. Hinder and Y. Ye, "A one-phase interior point method for nonconvex optimization" (first-order certificates of local infeasibility, local optimality, or unboundedness),
    1. arXiv
  • A. Hauswirth, S. Bolognani, G. Hug, and F. Dörfler, "Generic existence of unique Lagrange multipliers in AC optimal power flow" (LICQ holds generically, via differential topology), IEEE Control Systems Letters, vol. 2, no. 4, pp. 791–796, 2018. DOI · arXiv
  • G. Haeser, O. Hinder, and Y. Ye, "On the behavior of Lagrange multipliers in convex and non-convex infeasible interior point methods" (LICQ/MFCQ and multiplier boundedness), Mathematical Programming, vol. 186, pp. 257–288, 2021. DOI · arXiv
  • D. K. Molzahn, B. C. Lesieutre, and C. L. DeMarco, "A sufficient condition for power flow insolvability with applications to voltage stability margins" (Jacobian singularity and zero-voltage degeneracy), IEEE Transactions on Power Systems, 2013. PDF
  • A. U. Raghunathan and L. T. Biegler, "An ℓ1 exact penalty-barrier phase for degenerate nonlinear programming problems in Ipopt" (redundant / dependent constraints and LICQ recovery), IFAC World Congress, 2020. Link
  • D. Ralph and S. J. Wright, "Superlinear convergence of an interior-point method despite dependent constraints," Mathematics of Operations Research, vol. 25, no. 2, pp. 179–194, 2000. DOI
  • J. Hörsch, H. Ronellenfitsch, D. Witthaut, and T. Brown, "Linear optimal power flow using cycle flows" (linear dependence of nodal KCL and the slack-bus equation), Electric Power Systems Research, 2018. arXiv

Software and the PowerModels ecosystem

  • C. Coffrin, R. Bent, K. Sundar, Y. Ng, and M. Lubin, "PowerModels.jl: an open-source framework for exploring power flow formulations," PSCC, 2018. DOI
  • D. M. Fobes, S. Claeys, F. Geth, and C. Coffrin, "PowerModelsDistribution.jl: an open-source framework for exploring distribution power flow formulations," Electric Power Systems Research, 2020. arXiv

Optimization model debugging

  • JuMP, "Debugging" tutorial and "Solutions" manual. Debugging · Solutions
  • E. Kalvelagen, "The best way to debug infeasible models," Yet Another Math Programming Consultant, 2018. Link
  • YALMIP, "Infeasible or unbounded" and "Debugging unbounded models." 1 · 2
  • GAMS, "Execution errors and performance." Link
  • Pyomo, "Model debugging." Link
  • AIMMS, "Debug infeasible or unbounded results." Link