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A Mixed-Integer Linear Programming Approach to the Ac Optimal Power Flow in Distribution Systems

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A MIXED-INTEGER LINEAR PROGRAMMING APPROACH TO THE AC OPTIMAL POWER FLOW IN DISTRIBUTION SYSTEMS Rafael de Sá Ferreira Dissertação de Mestrado apresentada ao Programa de Pós-graduação em Engenharia Elétrica, COPPE, da Universidade Federal do Rio de Janeiro, como parte dos requisitos necessários à obtenção do título de Mestre em Engenharia Elétrica. Orientador: Carmen Lucia Tancredo Borges Rio de Janeiro Março de 2013 A MIXED-INTEGER LINEAR PROGRAMMING APPROACH TO THE AC OPTIMAL POWER FLOW IN DISTRIBUTION SYSTEMS Rafael de Sa Ferreira DISSERTA<;Ao SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ COIMBRA DE POS-GRADUA<;Ao E PESQUISA DE ENGENHARIA (COPPE) DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS NECESSARIOS PARA A OBTEN<;Ao DO GRAU DE MESTRE EM CIENCIAS EM ENGENHARIA ELETRICA. Examinada por: Prof. Carmen Lucia Tancredo Borges, D.Sc. Prof. Djal a osqueira Falcao, Ph.D. Eng. Mario Veiga Ferraz Pereira, D.Sc. RIO DE JANEIRO, RJ - BRASIL MAR<;O DE 2013 Ferreira, Rafael de Sá A mixed-integer linear programming approach to the AC optimal power flow in distribution systems/ Rafael de Sá Ferreira. Rio de Janeiro: UFRJ/COPPE, 2013. XVIII, 245 p.: il.; 29,7 cm. Orientador: Carmen Lucia Tancredo Borges Dissertação (mestrado) – UFRJ/ COPPE/ Programa de Engenharia Elétrica, 2013. Referencias Bibliográficas: p. 171-179. 1. Sistemas de distribuição. 2. Fluxo de potência ótimo. 3. Programação inteira mista. I. Borges, Carmen Lucia Tancredo. II. Universidade Federal do Rio de Janeiro, COPPE, Programa de Engenharia Elétrica. III. Título. iii AGRADECIMENTOS Gostaria de expressar minha gratidão a todos que me apoioaram, direta ou indiretamente, na elaboração desta dissertação: Aureo Ferreira, Tiago Ferreira, Maria Izabel Sá, Beatriz Amorim, Carmen Borges, Mario Pereira, Luiz Augusto Barroso, Martha Carvalho, Luiz Mauricio Thomé, André Dias, Sergio Granville, Rafael Kelman, Lujan Latorre, Raphael Chabar, Gerson Oliveira, Luiz Carlos da Costa, Fernanda Thomé, Priscila Lino, Silvio Binato, Djalma Falcão, Wadaed da Costa, Alessandro Moreira, Maria Helena Vale, Antônio Braga, Daniele Oliveira, Ronald Voelzke, Andreas Ettlinger, Ralph Hendriks, Guntram Schultz, Estevão Cruz, Antônio José Lima, Vitor Haase, Raul Duarte, Gabriel Cunha e todos os familiares, amigos, colegas de trabalho e de estudo que merecem sinceros agradecimentos. iv Resumo da Dissertação apresentada à COPPE/UFRJ como parte dos requisitos necessários para a obtenção do grau de Mestre em Ciências (M.Sc.) UMA ABORDAGEM DE PROGRAMAÇÃO INTEIRA MISTA PARA O FLUXO DE POTÊNCIA ÓTIMO CA EM REDES DE DISTRIBUIÇÃO Rafael de Sá Ferreira Março/2013 Orientador: Carmen Lucia Tancredo Borges Programa: Engenharia Elétrica O problema de fluxo de potência ótimo em redes de corrente alternada (FPO-CA) está dentre as ferramentas computacionais necessárias para o suporte à tomada de decisão no contexto do planejamento da operação e expansão de sistemas de distribuição. Nesta dissertação, emprega-se técnicas de linearização e convexificação para obter uma reformulação da versão não-linear do FPO-CA como um problema de programação inteira linear mista (PLIM). A formulação proposta: (i) captura o comportamento não-linear do sistema de distribuição através de aproximação cuja acurácia pode ser arbitrada pelo usuário; (ii) dá suporte a decisões discretas e contínuas; (iii) é construída com base em variáveis convencionalmente utilizadas para a descrição do comportamento da rede elétrica, o que resulta em flexibilidade na definição de funções objetivo e estende a aplicabilidade da formulação proposta a um conjunto elevado de problemas; e (iv) pode ser tratada por meio de pacotes comerciais para a solução de problemas de programação inteira mista, podendo-se obter soluções ótimas globais. Características físicas específicas de sistemas de distribuição são extensamente exploradas para obter-se uma formulação PLIM que concilie acurácia e desempenho computacional. A aplicabilidade e as características principais da formulação proposta são demonstradas com o auxílio de estudos de caso. v Abstract of Dissertation presented to COPPE/UFRJ as a partial fulfillment of the requirements for the degree of Master of Science (M.Sc.). A MIXED-INTEGER LINEAR PROGRAMMING APPROACH TO THE AC OPTIMAL POWER FLOW IN DISTRIBUTION SYSTEMS Rafael de Sá Ferreira March/2013 Advisor: Carmen Lucia Tancredo Borges Department: Electrical Engineering The alternating current (AC) optimal power flow (ACOPF) is among the computational tools required to support decision making in distribution system operations and expansion planning. In this dissertation, linearization and convexification techniques are employed in order to reformulate the non-linear version of the ACOPF for distribution systems, and a mixed-integer linear programming reformulation of this problem is proposed. The proposed formulation: (i) captures the non-linear behavior of the distribution system with an arbitrarily accurate approximation, with attention to the AC nature of the distribution system; (ii) supports both continuous and discrete decisions; (iii) is constructed with basis on conventional physical variables that describe network behavior, yielding significant flexibility in the definition of objective functions and extending its applicability to a number of different problems; and (iv) can be solved to global optimality with the use of widely employed and commercially available mixed-integer linear optimization solvers. Specific physical characteristics of distribution systems are extensively explored for achieving a MILP formulation that conciliates the desired attributes of accuracy and computational performance. The applicability and the main characteristics of the proposed formulation are showcased with help of several case studies. vi TABLE OF CONTENTS NOMENCLATURE ...................................................................................................... XII Indices and sets ....................................................................................................... xii Parameters .............................................................................................................. xiii Continuous decision variables................................................................................ xvi Binary decision variables ..................................................................................... xviii 1 INTRODUCTION .................................................................................................... 1 1.1 Background and motivation ....................................................................... 1 1.2 Bibliographic review .................................................................................. 3 1.3 Objective and contributions of this dissertation ....................................... 10 1.4 Organization of the dissertation ............................................................... 12 2 THE (NON-LINEAR) ACOPF IN DISTRIBUTION SYSTEM OPERATIONS AND EXPANSION PLANNING ........................................................................... 14 2.1 Relevant characteristics of distribution systems ...................................... 14 2.1.1 Shunt susceptance of overhead distribution lines ..................................... 15 2.1.2 Resistance-to-reactance ratio .................................................................... 15 2.1.3 Radiality constraints and reconfiguration ................................................. 16 2.1.4 Unbalance between phases ....................................................................... 17 2.2 The ACOPF for distribution systems ....................................................... 18 2.2.1 Constraints: modeling electrical behavior and enforcing operating limits......................................................................................................... 18 2.2.1.1 Kirchhoff’s Laws............................................................................... 19 2.2.1.2 Generators ......................................................................................... 20 2.2.1.3 Loads ................................................................................................. 22 2.2.1.4 Operating limits ................................................................................. 24 2.2.1.5 Voltage reference buses .................................................................... 26 2.2.1.6 Slack buses and buses without generators and/or loads .................... 27 2.2.1.7 Radiality constraints .......................................................................... 28 2.2.2 Objective functions for selected distribution system operations and expansion planning applications .............................................................. 29 2.2.2.1 Minimization of costs of load shedding ............................................ 30 2.2.2.2 Minimization of costs of curtailment of non-controllable generation ......................................................................................... 31 vii 2.2.2.3 Minimization of generation costs ...................................................... 33 2.2.2.4 Minimization of costs of power imports ........................................... 33 2.2.2.5 Minimization of costs of ohmic losses .............................................. 35 2.2.2.6 Minimization of costs of reinforcements to the distribution system . 36 2.2.2.7 Minimization of costs of capacitor placement .................................. 37 2.2.2.8 Minimization
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