Proving global optimality of ACOPF solutions

Gopinath, Smitha; Hijazi, Hassan; Weisser, Tillmann; Nagarajan, Harsha; Yetkin, Mertcan; Sundar, Kaarthik; Bent, Russell

doi:10.1016/j.epsr.2020.106688

Cited by 27 publications

(23 citation statements)

References 16 publications

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“…For transmission systems and unbalanced distribution systems, networks are usually highly meshed. It has been found that for most of meshed networks, both convex relaxation and local search algorithms can also yield the global optimum for most of testcases [39], [40]. Thus Theorem 3 suggests that there may also exist similar Lyapunov-like function and paths for meshed networks.…”

Section: A Optimal Power Flowmentioning

confidence: 93%

Conditions for Exact Convex Relaxation and No Spurious Local Optima

Zhou

Low

2022

IEEE Trans. Control Netw. Syst.

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Non-convex optimization problems can be approximately solved via relaxation or local algorithms. For many practical problems such as optimal power flow (OPF) problems, both approaches tend to succeed in the sense that relaxation is usually exact and local algorithms usually converge to a global optimum. In this paper, we study conditions which are sufficient or necessary for such non-convex problems to simultaneously have exact relaxation and no spurious local optima. Those conditions help us explain the widespread empirical experience that local algorithms for OPF problems often work extremely well.

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Section: A Optimal Power Flowmentioning

confidence: 93%

Conditions for Exact Convex Relaxation and No Spurious Local Optima

Zhou

Low

2022

IEEE Trans. Control Netw. Syst.

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“…The algorithm iterates between problem (i) and a set of problems of type (ii) obtained by considering different variables and whether it minimizes or maximizes the selected variable, the authors remark this second stage of the iterations can be parallelized through the consideration of the different sub-problems of type (ii). The approach proposed in Gopinath et al (2020) is supported by numerical experiments on cases from benchmark libraries Babaeinejadsarookolaee (2019, v19.05) and Coffrin et al (2014, v0.3). These experiments compare the gap obtained at the end of the algorithm with the gap of the two first levels of Lasserre's hierarchy.…”

Section: Improving the Optimality Gap Of Sdp Relaxationsmentioning

confidence: 99%

“…Some interesting improvements to optimality gap given by SDP relaxations and the corresponding exact formulations are proposed in Gopinath et al (2020). The SDP relaxation is approximated by a sequence of formulations, called the "determinant hierarchy", introduced in Hijazi et al (2016).…”

Section: Improving the Optimality Gap Of Sdp Relaxationsmentioning

confidence: 99%

“…where T (G) is a tree-decomposition (Diestel 2017) of the network G, C is one of the nodes of this tree-decomposition (which corresponds to a clique of G, thus, C ⊆ B), and S is a sub-clique of C. The matrix X S is the submatrix of X resulting from restricting X to the columns and rows of X corresponding to the buses in S. Since real-life networks are usually sparse, it is possible to describe them with a tree-decomposition where the cliques have small size and, therefore, the number of sub-cliques is polynomial on the size of B. SDP relaxations and Reformulation-Linearization Technique (RLT) (Sherali and Alameddine 1992) cuts were shown to be complementary in Anstreicher (2009). This idea is applied to current and power expressions in Gopinath et al (2020). Specifically, if Eqs.…”

Section: Improving the Optimality Gap Of Sdp Relaxationsmentioning

confidence: 99%

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Mathematical programming formulations for the alternating current optimal power flow problem

et al. 2020

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Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

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“…Liu et al propose a method to recover a feasible solution when the relaxation is inexact [14]. Convex relaxations also play an important role AND β (REACTIVE POWER), BY DEVICE TYPE [10] in approaches for global optimization of the non-linear OPF problem [15]. Unbalanced (O)PF generalizes the power flow equations to include the physics of phase unbalance.…”

Section: Introductionmentioning

confidence: 99%

Voltage-Dependent Load Models in Unbalanced Optimal Power Flow Using Power Cones

Claeys

Geth

2021

IEEE Trans. Smart Grid

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Mathematical models representing the behavior of electrical loads are an important part of any optimal power flow problem. The current state-of-the-art in unbalanced optimal power flow mostly considers wye-connected, constant power loads. However, for applications such as conservation voltage reduction, it is crucial to model how the consumption of the loads depends on the voltage of the network. This article develops a unified framework to handle a wide variety of load types: delta-or wye-connected, constant power, constant current and exponential load models. Furthermore, it proposes a novel convex relaxation for the exponential model, using power cones, that is intersected next with a well-known semi-definite relaxation of unbalanced OPF. Finally, numerical results on the LVTestCase feeder are included for both the exact, non-linear equations and the convex relaxation, which show how considering the voltage sensitivity and connection type can lead to different objective values and voltage profiles. Additional results for 128 low-voltage networks, examine the quality of the solutions obtained with the proposed relaxation.

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Proving global optimality of ACOPF solutions

Cited by 27 publications

References 16 publications

Conditions for Exact Convex Relaxation and No Spurious Local Optima

Conditions for Exact Convex Relaxation and No Spurious Local Optima

Mathematical programming formulations for the alternating current optimal power flow problem

Voltage-Dependent Load Models in Unbalanced Optimal Power Flow Using Power Cones

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