AC-Feasibility on Tree Networks is NP-Hard

Lehmann, Karsten; Grastien, Alban; Hentenryck, Pascal Van

doi:10.1109/tpwrs.2015.2407363

Cited by 199 publications

(135 citation statements)

References 10 publications

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“…Because there are additional relaxations we have introduced in (3a)-(4b), the final solutions of the proposed SOC-ACOPF model are generally not tight for the constraint (1g) in ACOPF. Using interior point method to solve the proposed SOC-ACOPF model in polynomial time does not violate the NP-hardness proof [34], [35] of ACOPF because the proposed SOC-ACOPF is still a relaxed model of ACOPF. …”

Section: Theoremmentioning

confidence: 99%

Distribution Locational Marginal Pricing by Convexified ACOPF and Hierarchical Dispatch

Yuan

Hesamzadeh

Biggar

2018

IEEE Trans. Smart Grid

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Abstract-This paper proposes a hierarchical economic dispatch (HED) mechanism for computing distribution locational marginal prices (DLMPs). The HED mechanism involves three levels: The top level is the national (regional) transmission network, the middle level is the distribution network, while the lowest level reflects local embedded networks or microgrids. Each network operator communicates its generalized bid functions (GBFs) to the next higher level of the hierarchy. The GBFs approximate the true cost function of a network by a series of affine functions. The concept of Benders cuts are employed in simulating the GBFs. The AC optimal power flow (ACOPF) is convexified and then used for dispatching generators and calculating GBFs and DLMPs. The proposed convexification is based on the second order cone reformulation. A sequential optimization algorithm is developed to tighten the proposed second order cone relaxation of ACOPF. The properties of the sequential tightness algorithm are discussed and proved. The HED is implemented in the GAMS grid computing platform. The GBFs and DLMPs are calculated for the modified IEEE 342 node low voltage test system. The numerical results show the utility of the proposed HED and GBF in implementing DLMP.

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Section: Theoremmentioning

confidence: 99%

Distribution Locational Marginal Pricing by Convexified ACOPF and Hierarchical Dispatch

Yuan

Hesamzadeh

Biggar

2018

IEEE Trans. Smart Grid

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“…Lehmann et al showed that the DC power analogue of s ‐ t ‐FlowFeasibility is strongly NP‐hard for planar graphs of maximum degree 3. They also show that a generalization of s ‐ t ‐MaxFlow with at least two sources and sinks cannot be approximated in polynomial time better than

2^{{(\log n)}^{1 - ϵ}}

for an ϵ > 0 unless the problems in NP can be solved in quasi‐polynomial deterministic time.…”

Section: Basic Modelmentioning

confidence: 99%

Algorithmic results for potential‐based flows: Easy and hard cases

et al. 2018

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Potential‐based flows are an extension of classical network flows in which the flow on an arc is determined by the difference of the potentials of its incident nodes. Such flows are unique and arise, for example, in energy networks. Two important algorithmic problems are to determine whether there exists a feasible flow and to maximize the flow between two designated nodes. We show that these problems can be solved for the single source and sink case by reducing the network to a single arc. However, if we additionally consider switches that allow to force the flow to 0 and decouple the potentials, these problems are NP‐hard. Nevertheless, for particular series‐parallel networks, one can use algorithms for the subset sum problem. Moreover, applying network presolving based on generalized series‐parallel structures allows to significantly reduce the size of realistic energy networks.

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“…The SDP relaxation was exact (or proved infeasibility) 2 To obtain satisfactory convergence of the SDP solver, these systems are pre-processed to remove low-impedance lines (i.e., lines whose impedance values have magnitudes less than 1 × 10 −3 per unit) as in [15]. 3 These relaxation gaps are calculated using the objective values from the SDP relaxation (9) and solutions obtained either from the second-order moment relaxation [14] (where possible) or from MATPOWER [24]. 4 Even the minor modifications performed when pre-processing lowimpedance lines and enforcing minimum line resistances are not needed for some test cases.…”

Section: A Generation Cost Constraintmentioning

confidence: 99%

“…The OPF problem is non-convex due to the non-linear power flow equations, may have local optima [1], and is generally NP-Hard [2], even for relatively simple cases such as treetopologies [3]. There is a large literature on solving OPF problems using local optimization techniques (e.g., successive quadratic programs, Lagrangian relaxation, heuristic optimization, and interior point methods [4], [5]).…”

Section: Introductionmentioning

confidence: 99%

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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AC-Feasibility on Tree Networks is NP-Hard

Cited by 199 publications

References 10 publications

Distribution Locational Marginal Pricing by Convexified ACOPF and Hierarchical Dispatch

Distribution Locational Marginal Pricing by Convexified ACOPF and Hierarchical Dispatch

Algorithmic results for potential‐based flows: Easy and hard cases

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

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