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Pricing Economic Dispatch With AC Power Flow via Local Multipliers and conic Relaxation

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS 2024年第3期11卷 1704-1716页

作者： Ndrio, Mariola Winnicki, Anna Bose, Subhonmesh Univ Illinois Dept Elect & Comp Engn Urbana IL 61801 USA

We analyze two locational marginal pricing schemes in electricity markets derived from an economic dispatch (ED) problem with ac power flow equations that define a nonconvex feasible set. The first among these prices called ac locational marginal prices (AC-LMPs) is derived from Lagrange multipliers that satisfy Karush-Kuhn-Tucker conditions for stationarity/local optimality of the nonconvex ED problem. The second, called SDP-LMPs, is derived from optimal dual multipliers of the semidefinite programming (SDP)-based convex relaxation of the ED problem. We establish that AC-LMPs and SDP-LMPs are derived from Lagrange dual-equivalent problems. Hence, they coincide under a zero duality gap but may not be equal when the gap is nonzero or when the AC-LMPs are associated with a stationary/locally optimal, but not globally optimal, dispatch. SDP-LMPs share interesting parallels with convex hull prices (CHPs), being derived from the Lagrangian dual of the nonconvex ED problem. However, there are important differences. For example, the epigraphs of the SDP relaxation of the ED problem and the latter, parameterized by the nodal power demands, may not enjoy the relationship that CHPs satisfy and derive their name from. Also, while CHPs minimize a form of side-payments, SDP-LMPs may not. We prove that AC-LMPs (and SDP-LMPs under zero duality gap) guarantee revenue adequacy under a condition that is sufficient but not necessary. Finally, the SDP-LMPs are shown to be equal to SOCP-DLMPs, which are distribution locational marginal prices derived with second-order cone programming-based relaxations of power flow equations over radial distribution networks. We illustrate our theoretical findings through numerical experiments.

关键词： Pricing Costs Mathematical models Reactive power Biological system modeling Network systems Minimization conic programming electricity markets locational marginal pricing

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Real-time microgrid economic dispatch based on model predictive control strategy

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Journal of Modern Power Systems and Clean Energy 2017年第5期5卷 787-796页

作者： Yan DU Wei PEI Naishi CHEN Xianjun GE Hao XIAO Institute of Electrical Engineering Chinese Academy of SciencesBeijing 100190China University of Chinese Academy of Sciences Beijing 100049China China Electric Power Research Institute Beijing 100192China

To deal with uncertainties of renewable energy,demand and price signals in real-time microgrid operation,this paper proposes a model predictive control strategy for microgrid economic dispatch, where hourly schedule is constantly optimized according to the current system state and latest forecast information. Moreover, implicit network topology of the microgrid and corresponding power flow constraints are considered, which leads to a mixed integer nonlinear optimal power flow problem. Given the non-convexity feature of the original problem, the technique of conic programming is applied to efficiently crack the nut. Simulation results from a reconstructed IEEE-33 bus system and comparisons with the routine day-ahead microgrid schedule sufficiently substantiate the effectiveness of the proposed MPC strategy and the conic programming method.

关键词： conic programming Economic dispatch(ED) Microgrid Mixed-integer nonlinear programming(MINLP) Model predictive control(MPC) Optimal power flow(OPF)

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Investment models based on clustered scenario trees

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2013年第2期227卷 314-324页

作者： Wong, Man Hong Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China

Stochastic programming is widely applied in financial decision problems. In particular, when we need to carry out the actual calculations for portfolio selection problems, we have to assign a value for each expected return and the associated conditional probability in advance. These estimated random parameters often rely on a scenario tree representing the distribution of the underlying asset returns. One of the drawbacks is that the estimated parameters may be deviated from the actual ones. Therefore, robustness is considered so as to cope with the issue of parameter inaccuracy. In view of this, we propose a clustered scenario-tree approach, which accommodates the parameter inaccuracy problem in the context of a scenario tree. (C) 2012 Elsevier B.V. All rights reserved.

关键词： conic programming Stochastic programming Interior point methods Robust optimization Scenario tree Portfolio selection

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A Study of Convex Convex-Composite Functions via Infimal Convolution with Applications

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MATHEMATICS OF OPERATIONS RESEARCH 2021年第4期46卷 1324-1348页

作者： Burke, James, V Tim, Hoheisel Nguyen, Quang, V Univ Washington Dept Math Seattle WA 98115 USA McGill Univ Dept Math & Stat Montreal PQ H3A 0B9 Canada

In this paper, we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone convexity, is straightforward. The results are established under a verifiable Slater-type condition, with relaxed monotonicity and without lower semi continuity assumptions on the functions in play. The versatility of our findings is illustrated by a series of applications in optimization and matrix analysis, including conic programming, matrix-fractional, variational Gram, and spectral functions.

关键词： convex-composite function cone-induced ordering K-convexity Fenchel conjugate infimal convolution subdifferential conic programming matrix-fractional function variational Gram function spectral function

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Convex mixed-integer nonlinear programs derived from generalized disjunctive programming using cones

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2024年第1期88卷 251-312页

作者： Neira, David E. Bernal Grossmann, Ignacio E. Purdue Univ Davidson Sch Chem Engn 480 Stadium Mall Dr W Lafayette IN 47907 USA Univ Space Res Assoc Res Inst Adv Comp Sci 425 3rd St SWSuite 950 Washington DC 20024 USA NASA Ames Res Ctr Quantum Artificial Intelligence Lab 5000 Forbes Ave Moffett Field CA 94035 USA Carnegie Mellon Univ Chem Engn Dept 5000 Forbes Ave Pittsburgh PA 15213 USA

We propose the formulation of convex Generalized Disjunctive programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show the reformulation of conic GDPs into Mixed-Integer conic programming (MICP) problems through both the big-M and hull reformulations. These reformulations have the advantage that they are representable using the same cones as the original conic GDP. In the case of the hull reformulation, they require no approximation of the perspective function. Moreover, the MICP problems derived can be solved by specialized conic solvers and offer a natural extended formulation amenable to both conic and gradient-based solvers. We present the closed form of several convex functions and their respective perspectives in conic sets, allowing users to formulate their conic GDP problems easily. We finally implement a large set of conic GDP examples and solve them via the scalar nonlinear and conic mixed-integer reformulations. These examples include applications from Process Systems Engineering, Machine learning, and randomly generated instances. Our results show that the conic structure can be exploited to solve these challenging MICP problems more efficiently. Our main contribution is providing the reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems.

关键词： Mixed-integer nonlinear programming conic programming Generalized disjunctive programming Perspective function

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Approximate Global Minimizers to Pairwise Interaction Problems via Convex Relaxation

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2018年第1期17卷 417-456页

作者： Bandegi, Mandi Shirokoff, David NJIT Dept Math Sci Newark NJ 07102 USA

We present a new approach for computing approximate global minimizers to a large class of nonlocal pairwise interaction problems defined over probability distributions. The approach predicts candidate global minimizers, with a recovery guarantee, that are sometimes exact, and often within a few percent of the optimum energy (under appropriate normalization of the energy). The procedure relies on a convex relaxation of the pairwise energy that exploits translational symmetry, followed by a recovery procedure that minimizes a relative entropy. Numerical discretizations of the convex relaxation yield a linear programming problem over convex cones that can be solved using well-known methods. One advantage of the approach is that it provides sufficient conditions for global minimizers to a nonconvex quadratic variational problem, in the form of a linear, convex, optimization problem for the autocorrelation of the probability density. We demonstrate the approach in a periodic domain for examples arising from models in materials, social phenomena, and flocking. The approach also exactly recovers the global minimizer when a lattice of Dirac masses solves the convex relaxation. An important by-product of the relaxation is a decomposition of the pairwise energy functional into the sum of a convex functional and nonconvex functional. We observe that in some cases, the nonconvex component of the decomposition can be used to characterize the support of the recovered minimizers.

关键词： global minimizers nonconvex energy pairwise interactions convex relaxations conic programming semidefinite programming flocking self-assembly

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A stochastic mixed-integer convex programming model for long-term distribution system expansion planning considering greenhouse gas emission mitigation

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INTERNATIONAL JOURNAL OF ELECTRICAL POWER & ENERGY SYSTEMS 2019年第Jun.期108卷 86-95页

作者： Home-Ortiz, Juan M. Melgar-Dominguez, Ozy D. Pourakbari-Kasmaei, Mandi Sanches Mantovani, Jose Roberto Sao Paulo State Univ UNESP Elect Engn Dept BR-15385000 Ilha Solteira SP Brazil Aalto Univ Dept Elect Engn & Automat Maarintie 8 Espoo 02150 Finland

This paper proposes a multistage convex distribution system planning model to find the best reinforcement plan over a specified horizon. This strategy determines planning actions such as reinforcement of existing substations, conductor replacement of overloaded feeders, and siting and sizing of renewable and dispatchable distributed generation units. Besides, the proposed approach aims at mitigating the greenhouse gas emissions of electric power distribution systems via a monetary form. Inherently, this problem is a non-convex optimization model that can be an obstacle to finding the optimal global solution. To remedy this issue, convex envelopes are used to recast the original problem into a mixed integer conic programming (MICP) model. The MICP model guarantees convergence to optimal global solution by using existing commercial solvers. Moreover, to address the prediction errors in wind output power and electricity demands, a two-stage stochastic MICP model is developed. To validate the proposed model, detail analysis is carried out over various case studies of a 34-node distribution system under different conditions, while to show its potential and effectiveness a 135-node system with two substations is used. Numerical results confirm the effectiveness of the proposed planning scheme in obtaining an economic investment plan at the presence of several planning alternatives and to promote an environmentally committed electric power distribution network.

关键词： conic programming Distributed energy Multistage distribution system expansion planning Renewable energy sources Stochastic programming

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The generalized close enough traveling salesman problem

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2023年第3期310卷 974-991页

作者： Di Placido, Andrea Archetti, Claudia Cerrone, Carmine Golden, Bruce Univ Molise Dept Biosci & Terr Campobasso Italy ESSEC Business Sch Dept Informat Syst Decis Sci & Stat Cergy France Univ Genoa Dept Econ & Business Studies Genoa Italy Univ Maryland Robert H Smith Sch Business College Pk MD USA

This paper studies a generalization of the close enough traveling salesman problem referred to as the generalized close enough traveling salesman problem (GCETSP). The canonical problem contains a set of customers, each associated with an area (neighborhood) that is generally circular. In the GCETSP, each customer is associated with a set of disks with different radii. Having multiple disks around the customer allows us to model several real-world applications, in which a higher benefit is gained by more closely approaching each target. A prize is assigned to each disk and is collected if the disk is traversed. The goal is to determine the route that visits each customer and the depot and maximizes the difference between the total collected prize and the route length. The total collected prize is given by the sum of the customer prices' associated with the innermost disk traversed by the route. We propose a heuristic algorithm and an evolutionary approach, specifically, a genetic algorithm (GA), to solve this problem. We evaluate the GA's performance on instances generated from benchmark CETSP and TSP instances. We then compare GA solutions with CETSP solutions and solutions obtained through an alternative approach based on pre-selecting intersection points with customers' disks. The results show that the GA can identify high-quality solutions with a short computing time.& COPY;2023 Elsevier B.V. All rights reserved.

关键词： Metaheuristics Close enough traveling salesman problem Genetic algorithms conic programming

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Kinematic yield design computational homogenization of micro-structures using the stabilized iRBF mesh-free method

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APPLIED MATHEMATICAL MODELLING 2021年 91卷 322-334页

作者： Ho, Phuc L. H. Le, Canh V. Nguyen, Phuong H. Int Univ VNU HCMC Dept Civil Engn Hanoi Vietnam HCMC Univ Technol & Educ Fac Civil Engn Hanoi Vietnam

This paper presents a novel numerical formulation based on kinematic theorem and homogenization theory for yield design of periodic heterogeneous materials. Macroscopic strength domains and failure mechanisms of such materials can be directly determined by solving an optimization problem formulated on a representative volume element (RVE). The integrated radial basis function (iRBF) mesh-free method is employed to approximate the fluctuating part of microscopic displacement field. The combination between nodal integration and conic programming leads to the fact that the resulting optimization problem can be solved efficiently, and that accurate solutions can be obtained with minimal computational cost. Various examples are investigated, demonstrating that the proposed method is able to approximate macroscopic limit surfaces of heterogeneous or composite materials, and is capable of capturing failure modes at micro-scale. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Computational homogenization Integrated radial basis function Yield design conic programming

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Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2010年第5期73卷 1143-1159页

作者： Fang, D. H. Li, C. Ng, K. F. Jishou Univ Coll Math & Comp Sci Jishou 416000 Peoples R China Zhejiang Univ Dept Math Hangzhou 310027 Zhejiang Peoples R China King Saud Univ Coll Sci Dept Math Riyadh 11451 Saudi Arabia Chinese Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China Chinese Univ Hong Kong IMS Hong Kong Hong Kong Peoples R China

For an inequality system defined by an infinite family of proper convex functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications. Under the new constraint qualifications, we provide necessary and/or sufficient conditions for the KKT rules to hold. Similarly, we provide characterizations for constrained minimization problems to have total Lagrangian dualities. Several known results in the conic programming problem are extended and improved. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Convex inequality system Optimality condition Total Lagrangian duality conic programming

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