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Mixed-integer nonseparable piecewise linear models for the hydropower production function in the Unit Commitment problem

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ELECTRIC POWER SYSTEMS RESEARCH 2020年第0期182卷 106234-000页

作者： Brito, B. H. Finardi, E. C. Takigawa, F. Y. K. Fed Inst Tocantins BR-77020450 Palmas Brazil Univ Fed Santa Catarina INESC P&D Brazil BR-88040900 Florianopolis SC Brazil Fed Inst Santa Catarina BR-88020300 Florianopolis SC Brazil

The hydro unit commitment (HUC) problem seeks to determine, for a short-term horizon with (semi)hourly discretization, the status (on/off) and generation level of each generating unit (GU) to meet plant and GUs constraints. In the HUC problem, the nonlinearities and non-convexities of the hydro production function (HPF), and the presence of binary variables that identify which GUs must be dispatched at each time step make the search for a solution challenging. Due to the recent developments in commercial mixed-integer linear programming (MILP) solvers, it is possible to approximate the nonlinear and nonconvex HPF through piecewise-linear (PWL) models with reasonable accuracy. Throughout this paper we emphasize the potential advantages of seven MILP formulations that can be categorized as parametric and non-parametric methods. Given the complexities of state-of-the-art solvers, it is hard to predict which formulation performs better. Although some guidelines can be found in literature, the formulation that performs best can be strongly dependent on the specific problem structure or data. In this context, we develop and compare seven multidimensional nonseparable PWL models for representing the HPF in the HUC problem. To assess the performance of each PWL model, we present results using a 6-GU hydro plant of two different types.

关键词： Hydro unit commitment piecewise-linear models Hydro production function Mixed-integer linear programming

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Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems

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PHYSICA D-NONlinear PHENOMENA 2014年 269卷 103-119页

作者： Acary, Vincent de Jong, Hidde Brogliato, Bernard INRIA Grenoble Rhone Alpes F-38330 Montbonnot St Martin France

Gene regulatory networks control the response of living cells to changes in their environment. A class of piecewise-linear (PWL) models, which capture the switch-like interactions between genes by means of step functions, has been found useful for describing the dynamics of gene regulatory networks. The step functions lead to discontinuities in the right-hand side of the differential equations. This has motivated extensions of the PWL models based on differential inclusions and Filippov solutions, whose analysis requires sophisticated numerical tools. We present a method for the numerical analysis of one proposed extension, called Aizerman-Pyatnitskii (AP)-extension, by reformulating the PWL models as a mixed complementarity system (MCS). This allows the application of powerful methods developed for this class of nonsmooth dynamical systems, in particular those implemented in the SICONOS platform. We also show that under a set of reasonable biological assumptions, putting constraints on the right-hand side of the PWL models, AP-extensions and classical Filippov (F)-extensions are equivalent. This means that the proposed numerical method is valid for a range of different solution concepts. We illustrate the practical interest of our approach through the numerical analysis of three well-known networks developed in the field of synthetic biology. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Nonsmooth dynamical systems Gene regulatory networks piecewise-linear models Filippov solutions Differential inclusions Complementarity systems

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Binary Voronoi linear trees for the approximate solution of functional optimization problems

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COMPUTATIONAL & APPLIED MATHEMATICS 2025年第2期44卷 1-23页

作者： Cervellera, Cristiano Maccio, Danilo Italian Natl Res Council Inst Marine Engn Via Marini 16 I-16149 Genoa Italy

The need to find solutions in function form that optimize a given nonlinear cost functional arises routinely in many important areas of operations research and applied mathematics. In most practical cases, problems of this kind require a numerical solution based on some suitable class of approximating architectures. This paper introduces the use of binary Voronoi linear trees (BVLTs) for the approximate solution of a general class of functional optimization problems. The main features of the considered trees are (i) a splitting scheme based on a Voronoi bisection criterion and (ii) linear outputs in the leaves, which make the resulting models more flexible compared to classic trees with cuts parallel to the axes and constant outputs. At the same time, due to the binary recursive structure, BVLTs retain the well-known efficiency of decision tree architectures. Consistently with the typical tree construction framework, we provide a greedy algorithm for the approximate solution of the addressed functional optimization problem. Universal approximation capabilities of the proposed class of models are derived in the theoretical analysis, and the consistency of the solution is discussed as well. In order to improve accuracy and robustness, we also consider the use of BVLTs in ensemble fashion, through an aggregation scheme well suited to optimization purposes. Simulation tests involving various optimization problems are presented, showing how the proposed algorithm can cope well in complex multivariate contexts, especially in ensemble form.

关键词： Functional optimization Decision trees piecewise-linear models Ensemble methods Nonlinear programming

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Multi-plant hydro power production function modeling: Application to the stochastic mid-term hydrothermal planning problem

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ELECTRIC POWER SYSTEMS RESEARCH 2024年 235卷

作者： Lira, Pedro T. M. Diniz, Andre Luiz Borges, Carmen L. T. Elect Energy Res Ctr CEPEL Rio De Janeiro Brazil Fed Univ Rio De Janeiro UFRJ Rio De Janeiro Brazil

This work proposes the modeling and application of the so-called "multi-hydro plants"for the stochastic midterm hydrothermal planning problem. The approach consists of aggregating sub-cascades of hydroelectric plants based on the fact that the operation of the cascades is dictated only by decisions on the release of the reservoirs with regularization capacity, and the operation of run-of-the-river plants is a consequence of such decisions. As a result, we obtain a formulation of the problem with a smaller number of variables and constraints, without any loss of representation of the physical and operation constraints of the hydro plants. This yields a reduction in the computational time to solve typical stochastic hydrothermal coordination problems of more than 20%, and results are presented for a real case of the Brazilian system. Furthermore, there is even a gain in accuracy of about 6% in the approximation of the non-concave hydro production function, and an analytical investigation regarding this aspect is conducted.

关键词： Hydro production function Mid-term planning Stochastic programming piecewise-linear models

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piecewise linear approximations for hydropower production function applied on the hydrothermal unit commitment problem

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INTERNATIONAL JOURNAL OF ELECTRICAL POWER & ENERGY SYSTEMS 2022年 135卷 107464-107464页

作者： Santos, K. V. Finardi, E. C. Univ Fed Amazonas BR-69080900 Manaus AM Brazil Univ Fed Santa Catarina LabPlan BR-88040900 Florianopolis SC Brazil INESC P&D Brasil BR-11055300 Santos SP Brazil

In a centralized-based dispatch market, the network-constrained hydrothermal unit commitment (NCHTUC) determines generation decisions that minimize the expected operating cost, usually in a day-ahead planning horizon. In this context, hydropower offers unique flexibility features, and an adequate representation is crucial for overcoming operation challenges. However, due to computational burden, simplifications on hydropower production function (HPF) modeling are necessary since NCHTUC is a large-scale nonlinear discrete optimization problem. The plant-based HPF piecewise linear approach is the most common simplification presented for real life cases. In this case, the generating units (GUs) of a plant are aggregated and represented by a single equivalent generator. Although this approach reduces the size and the complexity of the NCHTUC significantly, several operating issues are not considered adequately, especially the forbidden zones and the nonlinearities of the GUs. In this paper, we present a new approach that considers the nonlinearities and forbidden zones of the HPF via aggregation of the GUs and piecewise mixed-integer linear approximation in an innovative way. Experiments are performed in a modified version of the IEEE 118-bus system to verify the approach's effectiveness, analyzed in terms of the relation between computation burden and the HPF approximation quality.

关键词： Hydro Production Function piecewise-linear models Hydrothermal Unit Commitment Mixed-Integer linear Programming

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A novel approach to cutting decision trees

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CENTRAL EUROPEAN JOURNAL OF OPERATIONS RESEARCH 2014年第3期22卷 553-565页

作者： Uney-Yuksektepe, Fadime Istanbul Kultur Univ Dept Ind Engn TR-34156 Istanbul Turkey

In data mining, binary classification has a wide range of applications. Cutting Decision Tree (CDT) induction is an efficient mathematical programming based method that tries to discretize the data set on hand by using multiple separating hyperplanes. A new improvement to CDT model is proposed in this study by incorporating the second goal of maximizing the distance of the correctly classified instances to the misclassification region. Computational results show that developed model achieves better classification accuracy for Wisconsin Breast Cancer database and Japanese Banks data set when compared to existing piecewise-linear models in literature. Furthermore, remarkable results are obtained for the well-known benchmarking data sets (Buba Liver Disorders, Blood Tranfusion and Pima Indian Diabetes) when compared to the original CDT model.

关键词： Discriminant analysis Mathematical programming Data mining Decision trees piecewise-linear models

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Discontinuity correction in piecewise-linear models of oscillators for phase noise characterization

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INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS 2007年第1期35卷 93-104页

作者： Carbone, A. Palma, F. Univ Roma La Sapienza Dipartimento Ingn Elettron I-00144 Rome Italy

Decomposition of noise perturbation along Floquet eigenvectors has been extensively used in order to achieve a complete analysis of phase noise in oscillator. piecewise-linear approximation of nonlinear devices is usually adopted in numerical calculation based on multi-step integration method for the determination of unperturbed oscillator solution. In this case, exact determination of the monodromy matrix can be hampered by the presence of discontinuities between models introduced by the approximation. In this paper we demonstrate that, without the proper corrections, relevant errors occur in the determination of eigenvalues and eigenvectors, if adjacent linear models presents discontinuities. We obtain this result by the analysis of a simple 2-D oscillator with piecewise-linear parameter. We also demonstrate that a correct calculation can be achieved introducing properly calculated state vector boundary conditions by the use of interface matrices. This correction takes into account the effects of discontinuities between the linear models, leading to exact calculation of eigenvalues and eigenvectors, and, consequently, of the phase noise spectrum. Copyright (c) 2006 John Wiley & Sons, Ltd.

关键词： oscillator noise monodromy matrix interface matrices piecewise-linear models spectral distribution

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Classification by vertical and cutting multi-hyperplane decision tree induction

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DECISION SUPPORT SYSTEMS 2010年第3期48卷 430-436页

作者： Better, Marco Glover, Fred Samorani, Michele OptTek Syst Inc Boulder CO 80302 USA Univ Colorado Leeds Sch Business Boulder CO 80309 USA

Two-group classification is a key task in decision making and data mining applications. We introduce two new mixed integer programming formulations that make use of multiple separating hyperplanes. They represent a generalization of previous piecewise-linear models that embed rules having the form of hyperplanes, which are used to successively separate the two groups. In fact, the classifiers obtained are particular types of decision trees which are allowed to grow in depth and not in width. Computational results show that our models achieve better classification accuracy in less time than previous approaches. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Data mining Discrimination analysis piecewise-linear models Mathematical programming

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On the use of multivariable piecewise-linear models for predicting human response to anesthesia

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING 2004年第11期51卷 1876-1887页

作者： Lin, HH Beck, CL Bloom, MJ Univ Illinois Dept Gen Engn Urbana IL 61801 USA Univ Illinois Dept Mech & Ind Engn Urbana IL 61801 USA NYU Sch Med Dept Anesthesiol New York NY 10016 USA

The standard modeling paradigm used to describe the relationship between input anesthetic agents and output patient endpoint variables are single-input single-output pharmacokinetic-pharmacodynamic (PK-PD) compartment models. In this paper, we propose the use of multivariable piecewise-linear models to describe the relations between inputs that include anesthesia, surgical stimuli and disturbances to a variety of patient output variables. Subspace identification methods are applied to clinical data to construct the models. A comparison of predicted and measured responses is completed, which includes predictions from PK-PD models, and piecewise-linear time-invariant models.

关键词： anesthesia compartment models piecewise-linear models subspace identification

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Analytical solutions for peeling using beam-on-foundation model and cohesive zone

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JOURNAL OF ADHESION 2004年第4期80卷 313-331页

作者： Plaut, RH Ritchie, JL Virginia Polytech Inst & State Univ Dept Civil & Environm Engn Ctr Adhes & Sealant Sci Blacksburg VA 24061 USA

When fibrillation occurs during peeling, the normal stress in the adhesive may gradually reduce to zero at the peel front. The shear stress also reduces to zero. Classical beam-spring (or beam-on-elastic-foundation) models do not yield solutions that have these properties. With the use of a beam-on-foundation model combined with a cohesive zone in the neighborhood of the peel front, these properties can be satisfied. In order to obtain analytical solutions, peel tests are considered in which the backing has a small slope and is linearly elastic in the adhered region, and the traction law is assumed to be piecewise linear. Cases are considered with only normal stresses in the adhesive (mode I), only shear stresses (mode II), and both stresses coupled (mixed-mode behavior). Analytical solutions are obtained for displacements of the backing, forces in the backing, and stresses between the adhesive and the backing.

关键词： cohesive zone models elastic peel arm elastic-plastic foundation piecewise-linear models peel test

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