Exploiting Sparsity in SDP Relaxations of the OPF Problem

Jabr, Rabih A.

doi:10.1109/tpwrs.2011.2170772

Cited by 148 publications

(140 citation statements)

References 8 publications

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“…A matrix completion decomposition exploits sparsity by converting the positive semidefinite constraint on the large W matrix (9j) to positive semidefinite constraints on many smaller submatrices of W. These submatrices are defined using the cliques (i.e., completely connected subgraphs) of a chordal extension of the power system network graph. See [10], [22], [23] for a full description of a formulation that enables solution of (9) for systems with thousands of buses.…”

Section: Semidefinite Relaxation Of the Opf Problemmentioning

confidence: 99%

“…For instance, the SDP relaxation gaps for the large-scale Polish [24] and PEGASE [25] systems, which represent portions of European power systems, are all less than 0.3%. 23 Further, the SDP relaxation is exact (i.e., zero relaxation gap) for the IEEE 14-, 30-, 39-, 57-bus systems, the 118-bus system modified to enforce a small minimum line resistance [2], and several of the large-scale Polish test cases [10]. 4 (See [24], [26] for case descriptions.)…”

Section: A Generation Cost Constraintmentioning

confidence: 99%

“…8 The submatrices are determined by the maximal cliques of a chordal supergraph of the network; see [10], [19], [23] for further details.…”

Section: An Algorithm For Determining the Laplacian Weightsmentioning

confidence: 99%

See 2 more Smart Citations

A Laplacian-Based Approach for Finding Near Globally Optimal Solutions to OPF Problems

Molzahn

Josz²,

Hiskens

et al. 2017

IEEE Trans. Power Syst.

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Abstract-A semidefinite programming (SDP) relaxation globally solves many optimal power flow (OPF) problems. For other OPF problems where the SDP relaxation only provides a lower bound on the objective value rather than the globally optimal decision variables, recent literature has proposed a penalization approach to find feasible points that are often nearly globally optimal. A disadvantage of this penalization approach is the need to specify penalty parameters. This paper presents an alternative approach that algorithmically determines a penalization appropriate for many OPF problems. The proposed approach constrains the generation cost to be close to the lower bound from the SDP relaxation. The objective function is specified using iteratively determined weights for a Laplacian matrix. This approach yields feasible points to the OPF problem that are guaranteed to have objective values near the global optimum due to the constraint on generation cost. The proposed approach is demonstrated on both small OPF problems and a variety of large test cases representing portions of European power systems.

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Section: Semidefinite Relaxation Of the Opf Problemmentioning

confidence: 99%

Section: A Generation Cost Constraintmentioning

confidence: 99%

See 1 more Smart Citation

A Laplacian-Based Approach for Finding Near Globally Optimal Solutions to OPF Problems

Molzahn

Josz²,

Hiskens

et al. 2017

IEEE Trans. Power Syst.

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“…While these convex relaxations have been illustrated numerically in [17] and [18], whether or when they will turn out to be exact is first studied in [22]. Exploiting graph sparsity to simplify the SDP relaxation of OPF is first proposed in [23], [24] and analyzed in [25], [26]. See a comprehensive tutorial in [27], [28] (also [8]).…”

Section: A Convexificationmentioning

confidence: 99%

“…This approach decomposes the positive semidefinite constraint on the moment matrix into constraints on many smaller matrices. With direct application of this approach, the first-order moment-based relaxation is feasible for problems with thousands of buses [41], [25], [37]. Direct application of the matrix completion decomposition to the second-order relaxation results in computational intractability for problems with more than approximately forty buses [37], [40].…”

Section: A Convexificationmentioning

confidence: 99%

Advanced optimization methods for power systems

Panciatici¹,

Campi

Garatti

et al. 2014

2014 Power Systems Computation Conference

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Abstract-Power system planning and operation offers multitudinous opportunities for optimization methods. In practice, these problems are generally large-scale, non-linear, subject to uncertainties, and combine both continuous and discrete variables. In the recent years, a number of complementary theoretical advances in addressing such problems have been obtained in the field of applied mathematics. The paper introduces a selection of these advances in the fields of non-convex optimization, in mixedinteger programming, and in optimization under uncertainty. The practical relevance of these developments for power systems planning and operation are discussed, and the opportunities for combining them, together with high-performance computing and big data infrastructures, as well as novel machine learning and randomized algorithms, are highlighted.

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