A sparse convex AC OPF solver and convex iteration implementation based on 3-node cycles

Ma, Minyue; Fan, Lingling; Miao, Zhixin; Zeng, Bo; Ghassempour, Hossein

doi:10.1016/j.epsr.2019.106169

“…Compared with MATPOWER, the decision variables in (14) are no longer voltage phasors. Instead, R and I can be initialized using the solution of the PSD matrix X from an SDP relaxation AC OPF solver developed in Ma et al…”

Section: Rank‐1 Psd Matrix‐based Nonlinear Programming Formulationmentioning

confidence: 99%

“…To obtain the initial point for the nonlinear solver, we solve OPF problems first through a CVX‐based sparse SOCP/SDP relaxation solver . The objective value of the SOCP/SDP solver and the maximal rank of the corresponding PSD submatrices will be given for each test case.…”

Section: Case Studiesmentioning

confidence: 99%

Rank‐1 positive semidefinite matrix‐based nonlinear programming formulation for AC OPF

Ma

¹

,

Fan

²

2019

Int Trans Electr Energ Syst

Self Cite

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Summary Semidefinite programming (SDP) relaxation offers a tight relaxation to nonconvex alternating current optimal power flow (AC OPF) problems. When the solution obtained from SDP relaxation of AC OPF is a rank‐1 positive semidefinite (PSD) matrix, this solution is exact to the original problem. Research efforts have been devoted to find a rank‐1 PSD matrix. In this paper, a nonlinear programming formulation with the PSD matrix as the decision variable is proposed. The rank‐1 PSD matrix constraint is equivalent to all 2×2 minors of the PSD matrix being zero. The main challenge of the proposed formulation is the large number of the quadratic equality constraints. For a system of N buses, there are CN2CN2 minor related constraints (For a 10‐node system, this number is 2025). Graph decomposition–based approach is then implemented in this research to decompose a power grid into radial lines and three‐node cycles. Enforcing the related submatrices PSD and rank‐1 guarantees a full PSD rank‐1 matrix. Case study results demonstrate that the proposed formulation can provide similar quality results with the original AC OPF formulation.

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“…This requires the study of newer methodologies involving active distribution network modeling [16], adequate mathematical formulation of optimization problems aiming to increase efficiency and solution quality [17] and the proper implementation of optimization techniques to solve the problem [18]. Additionally, recent multidisciplinary research has been directed to increase computational efficiency in the solution of the OPF problem [18], showing results particularly in newer metaheuristic techniques (e.g., the arithmetic optimization algorithm (AOA) [19], the hybrid Harris hawks optimizer-AOA (hHHO-AOA) [20], and the mutation improved grey wolf optimizer (MIGWO) [21]), and in the convexification of the problem with relaxations based on second-order cones [22,23] or positive semi-definite expressions [24]. This is relevant since AC-OPF formulations are more complex than the original power balance problem; they might include nonlinear cost functions (in addition to the quadratic expression for the losses found in the original AC-OPF fomulation),mixed-integer expressions (as in, a reconfiguration [25] or resource allocation problem [26]), and/or uncertainty, which-individually or all together-are not necessarily convex when included to the problem (in addition to the non-convex quadratic constraints in the original AC-Power Flow problem), thus limiting the quality of the solutions and the scope of results.…”

Section: Introductionmentioning

confidence: 99%

Convex Stochastic Approaches for the Optimal Allocation of Distributed Energy Resources in AC Distribution Networks with Measurements Fitted to a Continuous Probability Distribution Function

Osorio

¹

,

Rosero

²

2023

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This paper addresses the optimal stochastic allocation of distributed energy resources in distribution networks. Typically, uncertain problems are analyzed in multistage formulations, including case generation routines, resulting in computationally exhaustive programs. In this article, two probabilistic approaches are proposed–range probability optimization (RPO) and value probability optimization (VPO)–resulting in a single-stage, convex, stochastic optimal power flow problem. RPO maximizes probabilities within a range of uncertainty, whilst VPO optimizes the values of random variables and maximizes their probabilities. Random variables were modeled with hourly measurements fitted to the logistic distribution. These formulations were tested on two systems and compared against the deterministic case built from expected values. The results indicate that assuming deterministic conditions ends in highly underestimated losses. RPO showed that by including ±10% uncertainty, losses can be increased up to 40% with up to −72% photovoltaic capacity, depending on the system, whereas VPO resulted in up to 85% increases in power losses despite PV installations, with 20% greater probabilities on average. By implementing any of the proposed approaches, it was possible to obtain more probable upper envelopes in the objective, avoiding case generation stages and heuristic methods.

show abstract

“…This requires the study of newer methodologies involving active distribution network modeling [16], adequate mathematical formulation of optimization problems aiming to increase efficiency and solution quality [17] and the proper implementation of optimization techniques to solve the problem [18]. Additionally, recent multidisciplinary research has been directed to increase computational efficiency in the solution of the OPF problem [18], showing results particularly in newer metaheuristic techniques (e.g., the arithmetic optimization algorithm (AOA) [19], the hybrid Harris hawks optimizer-AOA (hHHO-AOA) [20], and the mutation improved grey wolf optimizer (MIGWO) [21]), and in the convexification of the problem with relaxations based on second-order cones [22,23] or positive semi-definite expressions [24]. This is relevant since AC-OPF formulations are more complex than the original power balance problem; they might include nonlinear cost functions (in addition to the quadratic expression for the losses found in the original AC-OPF fomulation),mixed-integer expressions (as in, a reconfiguration [25] or resource allocation problem [26]), and/or uncertainty, which-individually or all together-are not necessarily convex when included to the problem (in addition to the non-convex quadratic constraints in the original AC-Power Flow problem), thus limiting the quality of the solutions and the scope of results.…”

Section: Introductionmentioning

confidence: 99%

Convex Stochastic Approaches for the Optimal Allocation of Distributed Energy Resources in AC Distribution Networks with Measurements Fitted to a Continuous Probability Distribution Function

Mendoza-Osorio,

Rosero-García

2023

Preprint

1

0

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This paper addresses the optimal stochastic allocation of Distributed Energy Resources in distribution Networks. Typically, uncertain problems are analyzed in multi-stage formulations including case generation routines, resulting in computationally exhaustive programs. In this article, two probabilistic approaches are proposed resulting in a single-stage, convex, stochastic optimal power flow problem: The Range-Probability Optimization (RPO) and Value-Probability Optimization (VPO). The RPO maximizes probabilities within a range of uncertainty, whilst the VPO optimizes the values of the random variables and maximizes their probabilities. Random variables are modeled with hourly measurements fitted to the logistic distribution. These formulations were tested on two systems and compared against the deterministic case built from expected values. Results indicate that assuming deterministic conditions ends in highly underestimated losses. The RPO showed that by including a ±10 % uncertainty, the losses can be increased up to 40 % with up to −72 % photovoltaic capacity, depending on the system, whereas the VPO resulted in up to 85 % increases in power losses despite PV installations, with 20 % greater probabilities in average. By implementing any of the proposed approaches, it was possible to obtain more probable upper envelopes in the objective, avoiding case generation stages and heuristic methods.

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A sparse convex AC OPF solver and convex iteration implementation based on 3-node cycles

Cited by 9 publications

References 17 publications

Rank‐1 positive semidefinite matrix‐based nonlinear programming formulation for AC OPF

Rank‐1 positive semidefinite matrix‐based nonlinear programming formulation for AC OPF

Convex Stochastic Approaches for the Optimal Allocation of Distributed Energy Resources in AC Distribution Networks with Measurements Fitted to a Continuous Probability Distribution Function

Convex Stochastic Approaches for the Optimal Allocation of Distributed Energy Resources in AC Distribution Networks with Measurements Fitted to a Continuous Probability Distribution Function

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