Solving Stochastic Hydrothermal Unit Commitment with a New Primal Recovery Technique Based on Lagrangian Solutions

The high penetration of intermittent renewable generation has prompted the development of Stochastic Hydrothermal Unit Commitment (SHUC) models, which are more difficult to be solved than their thermal-based counterparts due to hydro generation constraints and inflow uncertainties. This work presents a SHUC model applied in centralized cost-based dispatch. The SHUC is represented by a two-stage stochastic model, formulated as a large-scale mixed-binary linear programming problem. The solution strategy is divided into two steps. The first step is the Lagrangian Relaxation (LR) approach, which is applied to solve the dual problem and generate a lower bound for SHUC. The second step is given by a Primal Recovery where we use the solution of the LR dual problem with heuristics based on Benders’ Decomposition to obtain the primal-feasible solution. Both steps benefit from each other, exchanging information over the iterative process. We assess our approach in terms of the quality of the solutions and running times on space and scenario LR decompositions. The computational instances use various power systems, considering the different configuration of plants (capacity and number of units). The results show the advantage of our primal recovery technique compared to solving the problem via MILP solver. This is true already for the deterministic case, and the advantage grows as the problem’s size (number of plants and/or scenarios) does. The space decomposition provides better solutions, although scenario one provides better lower bounds, but the main idea is to encourage researchers to explore LR decompositions and heuristics in other relevant problems

Solving Stochastic Hydrothermal Unit Commitment with a New Primal RecoverycTechnique Based on Lagrangian Solutions

The high penetration of intermittent renewable generation has prompted the development of Stochastic Hydrothermal Unit Commitmentc(SHUC) models, which are more difficult to be solved than their thermal-basedccounterparts due to hydro generation constraints and inflow uncertainties.cThis work presents a SHUC model applied in centralized cost-based dispatch, where the uncertainty is related to the water availability in reservoirs and demand. The SHUC is represented by a two-stage stochastic model, formulated as a large-scale mixed-binary linear programming problem. The solution strategy is divided into two steps, performed sequentially, with intercalated iterations to find the optimal generation schedule. The first step is the Lagrangian Relaxation (LR) approach. The second step is given by a Primal Recovery based on LR solutions and a heuristic based on Benders' Decomposition. Both steps benefit from each other, exchanging information over the iterative process. We assess our approach in terms of the quality of the solutions and running times on space and scenario LR decompositions. The results show the advantage of our primal recovery technique compared to solving the problem via MILP solver. This is true already for the deterministic case, and the advantage grows as the problem’s size (number of plants and/or scenarios) does

Different Decomposition Strategies to Solve Stochastic Hydrothermal Unit Commitment Problems

Solving very-large-scale optimization problems frequently require to decompose them in smaller subproblems, that are iteratively solved to produce useful information. One such approach is the Lagrangian Relaxation (LR), a broad range technique that leads to many different decomposition schemes. The LR supplies a lower bound of the objective function and useful information for heuristics aimed at constructing feasible primal solutions. In this paper, we compare the main LR strategies used so far for Stochastic Hydrothermal Unit Commitment problems, where uncertainty mainly concerns water availability in reservoirs and demand (weather conditions). This problem is customarily modeled as a two-stage mixed-integer optimization problem. We compare different decomposition strategies (unit and scenario schemes) in terms of quality of produced lower bound and running time. The schemes are assessed with various hydrothermal systems, considering different configuration of power plants, in terms of capacity and number of units

Comparing Spatial and Scenario Decomposition for Stochastic Hydrothermal Unit Commitment Problems

Solving very-large-scale optimization problems frequently require to decompose them in smaller subproblems, that are iteratively solved to produce useful information. One such approach is the Lagrangian Relaxation (LR), a general technique that leads to many different decomposition schemes. The LR produces a lower bound of the objective function and useful information for heuristics aimed at constructing feasible primal solutions. In this paper, we compare the main LR strategies used so far for Stochastic Hydrothermal Unit Commitment problems, where uncertainty mainly concerns water availability in reservoirs and demand (weather conditions). The problem is customarily modeled as a two-stage mixed-integer optimization problem. We compare different decomposition strategies (unit and scenario schemes) in terms of quality of produced lower bound and running time. The schemes are assessed with various hydrothermal systems, considering different configuration of power plants, in terms of capacity and number of units

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject

Mathematical Programming bounds for Large-Scale Unit Commitment Problems in Medium-Term Energy System Simulations

We consider a large-scale unit commitment problem arising in medium-term simulation of energy networks, stemming from a joint project between the University of Milan and a major energy research centre in Italy. Optimal plans must be computed for a set of thermal and hydroelectric power plants, located in one or more countries, over a time horizon spanning from a few months to one year, with a hour-by-hour resolution. We propose a mixed-integer linear programming model for the problem. Since the complexity of this unit commitment problem and the size of real-world instances make it impractical to directly optimise this model using general purpose solvers, we devise ad-hoc heuristics and relaxations to obtain approximated solutions and quality estimations. We exploit an incremental approach: at first, a linear relaxation of an aggregated model is solved. Then, the model is disaggregated and the full linear relaxation is computed. Finally, a tighter linear relaxation of an extended formulation is obtained using column generation. At each stage, metaheuristics are run to obtain good integer solutions. Experimental tests on real-world data reveal that accurate results can be obtained by our framework in affordable time, making it suitable for efficient scenario simulations

ALGORITHMS FOR THE LARGE-SCALE UNIT COMMITMENT PROBLEM IN THE SIMULATION OF POWER SYSTEMS

Lo Unit Commitment Problem (UCP) \ue8 un problema di programmazione matematica dove un insieme di impianti termoelettrici deve essere programmato per soddisfare la domanda di energia e altri vincoli di sistema. Il modello \ue8 impiegato da decenni per supportare la pianificazione operazionale di breve termine dei sistemi elettrici. In questo lavoro affrontiamo il problema di risolvere UCP lineari di larga-scala per realizzare simulazioni accurate di sistemi elettrici, con i requisiti aggiuntivi di impiegare capacit\ue0 di calcolo convenzionali, ad esempio i personal computers, ed un tempo di soluzione di poche ore. Il problema, sotto le medesime condizioni, \ue8 affrontato abitualmente dal nostro partner industriale RSE S.p.A. (Ricerche Sistema Energetico), uno dei principali centri ricerche industriali su sistemi energetici in Italia. L\u2019ottimizzazione diretta di queste formulazioni con solutori generici \ue8 impraticabile. Nonostante sia possibile calcolare buone soluzioni euristiche, ovvero con un gap di ottimalit\ue0 sotto il 10%, in tempi ragionevoli per UCP di larga scala, si richiedono soluzioni pi\uf9 accurate, per esempio con gap sotto l\u20191%, per migliorare l\u2019affidabilit\ue0 delle simulazioni ed aiutare gli esperti di dominio, che potrebbero non essere familiari con i dettagli dei metodi di programmazione matematica, a supportare meglio le loro analisi. Tra le idee che abbiamo esplorato i seguenti metodi risultano i pi\uf9 promettenti: una mateuristica per calcolare efficientemente buone soluzioni e due metodi esatti di bounding: column generation e Benders decomposition. Questi metodi decompongono il problema disaccoppiando il commitment degli impianti termoelettrici, rappresentati da variabili discrete, e il loro livello di produzione, rappresentato da variabili continue. I nostri esperimenti dimostrano che il modello possiede propriet\ue0 intrinseche come degenerazione e forma della funzione obbiettivo piatta che ostacolano o impediscono la convergenza in risolutori allo stato dell\u2019arte. Tuttavia, i metodi che abbiamo sviluppato, sfruttando efficacemente le propriet\ue0 strutturali del modello, permettono di raggiungere soluzioni quasi ottime in poche iterazioni per la maggior parte delle istanze.The Unit Commitment Problem (UCP) is a mathematical programming problem where a set of power plants needs to be scheduled to satisfy energy demand and other system-wide constraints. It has been employed for decades to support short-term operational planning of power plants. In this work we tackle the problem of solving large-scale linear UCPs to perform accurate medium-term power systems simulations, with the additional requirements of employing conventional computing power, such as personal computers, and a solution time of a few hours. The problem, under such conditions, is routinely faced by our industry partner, the Energy Systems Development department at RSE S.p.A. (Ricerche Sistema Energetico), a major industrial research centre on power systems in Italy. The direct optimization of these formulations via general-purpose solvers is impractical. While good heuristic solutions, that is with an optimality gap below 10%, can be found for large-scale UCPs in affordable time, more accurate solutions, for example with a gap below 1%, are sought to improve the reliability of the simulations and help domain experts, who may not be familiar with the details of mathematical programming methods, to better support their analysis. Among the ideas we explored, the following methods are the most promising: a matheuristic to efficiently compute good solutions and two exact bounding methods: column generation and Benders decomposition. These methods decompose the problem by decoupling the commitment of thermal plants, represented by discrete variables, and their level of production, represented by continuous variables. Our experiments proved that the model posses inherent properties as degeneracy and objective flatness which hinder or prevent convergence in state-of-the-art solvers. On the other hand, the methods we devised, by effectively exploiting structural properties of the model, allow to reach quasi-optimal solutions within a few iterations on most instances

Proposta de um modelo para alocação ótima de unidades hidrelétricas para usinas em cascata

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia Elétrica, Florianópolis, 2011Este trabalho propõe um modelo para o problema de alocação de unidades hidrelétricas, cujo objetivo consiste em determinar quais unidades devem operar, e os respectivos níveis de geração de usinas hidrelétricas em cascata, a cada hora, em um horizonte de um dia. Como uma contribuição apresenta-se uma nova modelagem para a função de produção das unidades geradoras, com destaque para as perdas mecânicas e elétricas presentes nos conjuntos turbina gerador. Para levar em consideração as complexidades inerentes deste problema de maneira condizente com as necessidades do caso brasileiro, o modelo da alocação é representado matematicamente como um problema de programação não linear binário-misto. Com o objetivo de resolver este problema eficientemente este trabalho faz uso de uma estratégia de decomposição baseada nos métodos da Relaxação Lagrangeana e do Lagrangeano Aumentado. Diferentes análises em torno da modelagem e da estratégia de solução propostas neste trabalho são realizadas mediante o uso de um sistema composto por quatro usinas hidrelétricas em cascata, cuja capacidade de potência instalada é de 4.170 MW

Técnicas de dualidade e programação não-linear inteira-mista aplicadas ao programa diário da operação eletroenergética

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-Graduação em Engenharia Elétrica.O problema da programação diária da operação eletroenergética é tratado, neste trabalho, como um problema de otimização não-linear inteiro-misto. Nesse sentido, é considerada uma modelagem detalhada das componentes do sistema visando representá-lo de maneira realista. Para resolver o problema, são utilizadas técnicas de dualidade. Essas técnicas são baseadas na Relaxação Lagrangeana com duplicação de variáveis, permitindo decompor o problema em uma série de subproblemas mais simples de serem resolvidos. Como resultado da utilização da Relaxação Lagrangeana em problemas não convexos, obtém-se uma solução primal inviável, sendo necessário realizar uma recuperação da solução primal do problema. Assim, heurísticas podem ser empregadas para encontrar uma solução primal viável, as quais são baseadas em um conjunto de regras que devem ser definidas com antecedência. No entanto, padronizar esses conjuntos de regras não é uma tarefa fácil. Nesse contexto, são testadas duas metodologias que precisam de heurísticas para contornar isso. A primeira baseada no Lagrangeano Aumentado Inexato e a segunda baseada no Primal Proximal. Além disso, como resultado do acoplamento entre essas duas metodologias, surgem dois modelos híbridos que também são testados. Adicionalmente, existem na atualidade pacotes comerciais de Programação Não-Linear Inteira-Mista capazes de lidar com este tipo de problema. Neste trabalho, o problema da programação diária da operação eletroenergética também é solucionado utilizando um desses pacotes. Os resultados obtidos com esse pacote são utilizados para avaliar as técnicas de dualidade em torno de uma estimativa do gap de dualidade (qualidade da solução) e do esforço computacional. O sistema base utilizado para os testes é composto por 11 usinas, 18 barras e 25 linhas.The Electroenergetic Daily Operation Programming problem is treated, in this work, as a mixed integer nonlinear programming problem. In that way, a detailed modeling of the system components, in order to represent it realistically, is considered. To solve the problem, duality techniques are used. These techniques are based in Lagrangean Relaxation with variables duplication, allowing decompose the original problem in a set of subproblems more simples to be solved. As a result of using Lagrangean Relaxation in nonconvex problems, an unfeasible solution is founded, been necessary to realize a recovery of a primal feasible solution. Thus, heuristic can be used to find feasible primal solutions, which are based in a set of rules that must be defined previously. However, standardize those sets of rules is not an easy task. In this context, two methodologies that use heuristics are tested to overcome this difficult. The first one is based on Inexact Augmented Lagrangean and the second one is based on Primal Proximal. Furthermore, as a result of coupling those two methodologies arise two hybrid models that also are tested. Additionally, exist commercial solvers of Mixed Integer Nonlinear Programming that are able to deal with this kind of problem and thereby it is also used in this work for solve the Electroenergetic Daily Operation Programming problem. The solver results are used to evaluate the dual techniques around a duality gap estimative (quality of solution) and computational performance. The base system used in tests is composted of 11 plants, 18 buses and 25 transmission lines