UNIVERSIDADE ESTADUAL DE CAMPINASFaculdade de Engenharia Elétrica e de Computação

Juan Sebastian Giraldo Chavarriaga

Mathematical Programming Models for the Optimal EnergyManagement of Modern Electric Distribution Systems

Considering Uncertainty

Modelos Matemáticos para o Gerenciamento Ótimo deEnergia dos Sistemas Modernos de Distribuição Considerando

Incerteza

Campinas, SP2019

UNIVERSIDADE ESTADUAL DE CAMPINASFaculdade de Engenharia Elétrica e de Computação

Juan Sebastian Giraldo Chavarriaga

Mathematical Programming Models for the OptimalEnergy Management of Modern Electric Distribution

Systems Considering Uncertainty

Modelos Matemáticos para o Gerenciamento Ótimo deEnergia dos Sistemas Modernos de Distribuição

Considerando Incerteza

Thesis presented to the School of Electricaland Computer Engineering of the Universityof Campinas in partial fulfillment of therequirements for the degree of Doctor inElectrical Engineering, in the area of Elec-tric Energy.Tese apresentada à Faculdade de EngenhariaElétrica e de Computação da UniversidadeEstadual de Campinas como parte dosrequisitos exigidos para a obtenção do títulode Doutor em Engenharia Elétrica, na Áreade Energia Elétrica.

Supervisor: Prof. Dr. Carlos Alberto de Castro Junior

Este exemplar corresponde à versãofinal da tese defendida pelo alunoJuan Sebastian Giraldo Chavar-riaga, e orientada pelo Prof. Dr.Carlos Alberto de Castro Junior

Campinas, SP2019

Ficha catalográficaUniversidade Estadual de Campinas

Biblioteca da Área de Engenharia e ArquiteturaLuciana Pietrosanto Milla - CRB 8/8129

Giraldo Chavarriaga, Juan Sebastian, 1989- G441m GirMathematical programming models for the optimal energy management of

modern electric distribution systems considering uncertainty / Juan SebastianGiraldo Chavarriaga. – Campinas, SP : [s.n.], 2019.

GirOrientador: Carlos Alberto de Castro. GirTese (doutorado) – Universidade Estadual de Campinas, Faculdade de

Engenharia Elétrica e de Computação.

Gir1. Otimização robusta. 2. Geração distribuída de energia elétrica. 3.

Programação estocástica. I. Castro, Carlos Alberto de. II. UniversidadeEstadual de Campinas. Faculdade de Engenharia Elétrica e de Computação.III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Modelos de programação matemática para o gerenciamento ótimoda energia em sistemas modernos de distribuição considerando incertezasPalavras-chave em inglês:Robust optimizationDistributed generation of electrical energyStochastic programmingÁrea de concentração: Energia ElétricaTitulação: Doutor em Engenharia ElétricaBanca examinadora:Carlos Alberto de Castro [Orientador]Edimar José de OliveiraEduardo Nobuhiro AsadaFernanda Caseño Trindade ArioliLuiz Carlos Pereira da SilvaData de defesa: 12-09-2019Programa de Pós-Graduação: Engenharia Elétrica

Identificação e informações acadêmicas do(a) aluno(a)- ORCID do autor: https://orcid.org/0000-0003-2154-1618- Currículo Lattes do autor: http://lattes.cnpq.br/8326560668079851

Powered by TCPDF (www.tcpdf.org)

COMISSÃO JULGADORA – TESE DE DOUTORADO

Candidato: Juan Sebastian Giraldo Chavarriaga RA: 143684

Data da Defesa: 12 de setembro de 2019

Título da Tese: Mathematical Programming Models for the Optimal EnergyManagement of Modern Electric Distribution Systems Considering Uncertainty

Prof. Dr. Carlos Alberto de Castro JuniorProf. Dr. Edimar José de OliveiraProf. Dr. Eduardo Nobuhiro AsadaProfa. Dra. Fernanda Caseño Trindade ArioliProf. Dr. Luiz Carlos Pereira da Silva

A ata de defesa, com as respectivas assinaturas dos membros da Comissão Jul-gadora, encontra-se no SIGA (Sistema de Fluxo de Dissertação/Tese) e na Secretaria dePós-Graduação da Faculdade de Engenharia Elétrica e de Computação.

Dedicado a mi madre Martha Cecilia Chavarriaga.

Acknowledgements

I would like to express my special appreciation and thanks to my advisor ProfessorDr. Carlos Alberto Castro who has been a tremendous mentor for me. Thank you forencouraging my research and for allowing me to grow as a research scientist. I would alsolike to extend my deepest gratitude to Professor Dr. Marcos Julio Rider for his valuableinsights and advises.

My most sincere thanks to my colleges, friends, and professors of the Department ofSystems and Energy of the School of Electrical and Computing Engineering, UNICAMP,specially to Jhon Alexander Castrillon and Juan Camilo López for their invaluable con-tribution and support.

I also had great pleasure of working with Professor Dr. Federico Milano and all hisresearch team at the School of Electrical and Electronic Engineering, University CollegeDublin. Thank you very much.

Words can not express how grateful I am to my family. To my mother in particular,for all of the sacrifices that you have made on my behalf.

Last but not least important, I would like to acknowledge the financial support Ihave received from different Brazilian public agencies, without which, this research andmore importantly, this personal achievement would not have been possible. I will alwaysbe grateful.

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoalde Nível Superior - Brasil (CAPES) - Finance Code 001

“... if I tried to give you a clue at the cost of your own experience, I should be the worstof teachers... ”

Zen in the Art of Archery, Eugen Herrigel, 1953

AbstractThe modernization of traditional distribution power systems is an unavoidable transitioninto a more efficient, reliable, and environmentally friendly electric power system. Currenttrends in research indicate that the insertion of distributed energy resources and energystorage systems into modern distribution systems could be facilitated by creating flexiblemicrogrids, in which, the energy management system is a fundamental piece. Availableapproaches are mainly focused on balanced equivalents, which is a debatable assumptionfor medium-voltage and low-voltage networks. Moreover, the insertion of renewable energysources and the natural behavior of loads involves the inclusion of intrinsically stochas-tic exogenous parameters, creating the need for methodologies suitable for handling un-certainty. Considering the aforementioned challenges, this thesis proposes four differentmathematical programming models for the optimal energy management of modern dis-tribution systems. The proposed models introduce balanced and unbalanced approaches;grid-connected and islanded microgrid operation modes; and two formulations consideringuncertainty. Each model has been validated using benchmark test systems depending ontheir specific characteristics.

Keywords: Optimal energy management; distributed energy resources; microgrids; ro-bust optimization; chance-constraints.

ResumoA modernização dos sistemas tradicionais de distribuição de energia é uma transição ine-vitável para um sistema de energia elétrica mais eficiente, confiável e ambientalmenteamigável. As tendências atuais de pesquisa indicam que a inserção de recursos energé-ticos distribuídos e sistemas de armazenamento de energia em sistemas de distribuiçãomodernos poderia ser facilitada pela criação de microrredes flexíveis, nas quais, o sis-tema de gerenciamento de energia é uma peça fundamental. As abordagens disponíveistêm sido principalmente focadas em equivalentes balanceados, o que é uma suposiçãodiscutível para redes de média e baixa tensão. Além disso, a inserção de fontes de ener-gia renováveis e o comportamento natural das cargas envolve a inclusão de parâmetrosexógenos intrinsecamente estocásticos, criando a necessidade de metodologias adequadaspara lidar com a incerteza. Considerando o exposto, esta tese propõe quatro modelos deprogramação matemática diferentes para o gerenciamento ótimo de energia de sistemasmodernos de distribuição. Os modelos propostos introduzem abordagens equilibradas edesequilibradas; microrredes operando conectadas à rede ou ilhadas; e duas formulaçõesconsiderando incerteza. Cada modelo foi validado usando sistemas de teste dependendodas suas características específicas.

Palavras-chave: Gerenciamento ótimo de energia; recursos energéticos distribuidos; mi-crorredes; otimização robusta; restrições probabilísticas.

List of Figures

Figure 1.1 – Traditional power system representation. Source: Author . . . . . . . . 20Figure 1.2 – Microgrid representation and commands from the microgrid operator.

Source: Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 1.3 – Number of publications and citations using “Microgrid” in the title +

“EMS” as topic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 1.4 – “Microgrid” in title + “EMS” + “Unbalanced” as topic. . . . . . . . . 26Figure 1.5 – “Microgrid” in title + “EMS” + “Uncertainty” as topic. . . . . . . . . 27Figure 1.6 – Number of references used in the thesis per year and percentage of

papers by publisher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 1.7 – Structure of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.1 – 2-bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 2.2 – Number of iterations to converge and voltage magnitude at operating

point using the NR algorithm with different initial values. . . . . . . . 42Figure 2.3 – Surface and contour of the constraints for the 2-bus system. . . . . . . 45Figure 2.4 – Quadratic cone and rotated quadratic cone. . . . . . . . . . . . . . . . 48Figure 2.5 – Reduced decision tree - Feasible path (continuous), infeasible path

(dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 2.6 – Cumulative density function of random variable 𝑋. . . . . . . . . . . . 52Figure 2.7 – Monte Carlo simulation flow chart. . . . . . . . . . . . . . . . . . . . . 53Figure 2.8 – Representation of the Point Estimate Method - Adapted from [225] . . 54Figure 2.9 – Evolution of the average value of the voltage angle. . . . . . . . . . . . 57Figure 3.1 – Three-bus example distribution network. . . . . . . . . . . . . . . . . . 67Figure 3.2 – Modified 34-bus test system. Adapted from [258] . . . . . . . . . . . . 71Figure 3.3 – Solar irradiance and wind speed . . . . . . . . . . . . . . . . . . . . . . 72Figure 3.4 – Active and reactive power injected by dispatchable distributed genera-

tors - Unlimited fuel 5 periods. . . . . . . . . . . . . . . . . . . . . . . 72Figure 3.5 – Load shedding percentage - Unlimited fuel 24 periods. . . . . . . . . . 73Figure 3.6 – Injected power from Energy Storage Systems and State of Charge -

Unlimited fuel 24 periods. . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 3.7 – Load shedding percentage - Limited fuel 5 periods. . . . . . . . . . . . 74Figure 3.8 – Injected power from energy storage systems and State of Charge - Lim-

ited fuel 5 periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 3.9 – Injected power from distributed generation units and fuel reserve - Lim-

ited fuel 24 periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 3.10–Nominal and total served energy for all tested scenarios. . . . . . . . . 75Figure 4.1 – Current balance at node 𝑖. . . . . . . . . . . . . . . . . . . . . . . . . . 82

Figure 4.2 – Dispatchable distributed generator, Model 1. . . . . . . . . . . . . . . . 83Figure 4.3 – Dispatchable distributed generator Model 2. . . . . . . . . . . . . . . . 84Figure 4.4 – Dispatchable distributed generator, Model 3. . . . . . . . . . . . . . . . 85Figure 4.5 – Energy storage system model. . . . . . . . . . . . . . . . . . . . . . . . 85Figure 4.6 – Operational limits for dispatchable distributed generation units and

energy storage systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 4.7 – Case study microgrid based on the modified IEEE 123 node test feeder. 88Figure 4.8 – Voltage magnitude at bus 114 and current magnitude at line 150-149

considering different synchronous machine models for distributed gen-eration units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Figure 4.9 – Three-phase active and reactive power injected by the distributed gen-eration units with different machine models: Model 1 (left bar), Model 2(center bar), and Model 3 (right bar). . . . . . . . . . . . . . . . . . . . 89

Figure 4.10–Active and reactive powers per phase injected by the distributed gen-eration unit at bus 450 using: a) Model 1, b) Model 2, and c) Model3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Figure 4.11–State of charge and three-phase active power injected by the energystorage system at node 47 considering different synchronous machinemodels for distributed generation units. . . . . . . . . . . . . . . . . . . 91

Figure 4.12–Total energy injected by sources and consumed by loads - Model 1. . . 91Figure 5.1 – Meaning of the robustness adjustment parameter 𝜁 for loads (a) and

renewable energy sources (b). . . . . . . . . . . . . . . . . . . . . . . . 102Figure 5.2 – Illustrative example for choosing 𝜁, based on ϒ = 0.6 and different

types of cumulative density functions. . . . . . . . . . . . . . . . . . . 104Figure 5.3 – Representation of the models used for energy storage systems and dis-

tributed generation units. . . . . . . . . . . . . . . . . . . . . . . . . . 106Figure 5.4 – Limited power variation from a specified set-point for the Monte Carlo

simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Figure 5.5 – Modified microgrid based on the 136-node system. . . . . . . . . . . . . 110Figure 5.6 – Energy balance for grid-connected mode, with 𝜁 = 0.10 for an arbitrary

scenario of the Monte Carlo simulations. . . . . . . . . . . . . . . . . . 111Figure 5.7 – Robustness assessment using Monte Carlo simulations for 𝜁 ∈ [−0.15, 0.30].

Grid-connected mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.8 – Grid-connected mode: Global robustness, objective function, and en-

ergy losses for 𝜁 ∈ [−0.15, 0.30] and the sample average approximationmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 5.9 – Energy balance for isolated mode, with 𝜁 = 0.10 for an arbitrary sce-nario of the Monte Carlo simulations. . . . . . . . . . . . . . . . . . . . 113

Figure 5.10–Percentage of curtailed energy for different values of 𝜁. . . . . . . . . . 113

Figure 5.11–Isolated mode: Objective function and load shedding for different valuesof 𝜁 and the sample average approximation model. . . . . . . . . . . . 114

Figure 5.12–Histograms of the objective function and robustness for different valuesof 𝜁. Islanded operation. . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 5.13–Load shedding at each node for 𝜁 = 0.10. Isolated mode. . . . . . . . . 115Figure 6.1 – Uncertainty sets for a normal probability distribution function. . . . . . 126Figure 6.2 – Convolution of different PDFs - Central limit theorem. . . . . . . . . . 127Figure 6.3 – Representation of the prediction intervals for current and voltage mag-

nitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Figure 6.4 – Flowchart of the proposed accuracy and robustness assessment. . . . . 129Figure 6.5 – IEEE 13-bus test feeder diagram. . . . . . . . . . . . . . . . . . . . . . 131Figure 6.6 – Maximum relative errors in: a) Voltage magnitude expected values.

b) Voltage magnitude standard deviations. c) Current magnitude ex-pected values. d) Current magnitude standard deviations. . . . . . . . . 131

Figure 6.7 – Percentage of rated current and voltage magnitudes in each phase ofthe system with 𝛼 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 6.8 – Cumulative density functions for voltage magnitude at node 675: . . . . 135Figure 6.9 – Average value of the objective function and percentage of feasible sce-

narios for different values of . . . . . . . . . . . . . . . . . . . . . . . . 136Figure 6.10–IEEE-123 bus test feeder diagram. . . . . . . . . . . . . . . . . . . . . 137Figure 6.11–Cumulative density function of the current magnitude at line 150–149,

and for voltage magnitude at node 114 considering different number ofrandom variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Figure 6.12–Mean squared error of higher order moments and percentage of outputvariables of interest with 𝑃 > 0.05. a) Current magnitudes. b) Voltagemagnitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 6.13–Sample cumulative density functions of the current magnitude at line150-149 and the voltage magnitude at node 114, considering differentforecast errors of the standard deviations. . . . . . . . . . . . . . . . . 140

Figure 6.14–Sample cumulative density functions of the current magnitude at line150-149 and the voltage magnitude at node 114, considering differentforecast errors of the expected values. . . . . . . . . . . . . . . . . . . . 141

List of Tables

Table 2.1 – Comparison of power flow solution methods. . . . . . . . . . . . . . . . 40Table 2.2 – Parameters data - example unit commitment. . . . . . . . . . . . . . . . 49Table 2.3 – Possible commitment - example unit commitment. . . . . . . . . . . . . 49Table 2.4 – Optimal solution - example unit commitment. . . . . . . . . . . . . . . . 50Table 3.1 – Objective function cost for tested scenarios . . . . . . . . . . . . . . . . 76Table 4.1 – Parameters of energy storage systems and distributed generation units . 88Table 4.2 – Performance of the tested simulations . . . . . . . . . . . . . . . . . . . 92Table 5.1 – Values and Units of Parameters . . . . . . . . . . . . . . . . . . . . . . 111Table 5.2 – Execution Times of Each Test Case in Seconds. . . . . . . . . . . . . . . 115Table 6.1 – Maximum Relative Errors for the Expected Values and Standard Devi-

ations of Voltage and Current Magnitudes in the IEEE 13-bus System. . 132Table 6.2 – Three-phase Apparent Power Generation (and Power Factor) from the

Dispatchable Distributed Generation Units, Values of the Objective Func-tion, and Execution Times. . . . . . . . . . . . . . . . . . . . . . . . . . 132

Table 6.3 – Expected Values and Standard Deviations for the Three-Phase Voltageand Current Magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Table 6.4 – Percentages of scenarios where voltage limits, current limits, and bothwere guaranteed – IEEE 13-bus system. . . . . . . . . . . . . . . . . . . 135

Table 6.5 – Percentages of scenarios where voltage limits, current limits, and bothwere guaranteed – IEEE 123-bus system . . . . . . . . . . . . . . . . . . 137

Table 6.6 – Groups considering different number of random variables . . . . . . . . 138Table 6.7 – Overview of the System’s Normality and Performance Under Different

Number of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 139Table 6.8 – Percentage of Scenarios in Which Robust Constraints were Guaranteed

Considering Forecast Errors . . . . . . . . . . . . . . . . . . . . . . . . . 141

Acronyms

CDF cumulative distribution function.

CLT central limit theorem.

CSC current-source converter.

DER distributed energy resource.

DG distributed generation.

DSM demand side management.

EMS energy management system.

ESS energy storage systems.

LP linear programming.

MCS Monte Carlo simulation.

MILP mixed-integer linear programming.

MINLP mixed-integer nonlinear programming.

MISOCP mixed-integer second-order cone programming.

MSE mean squared error.

NLP nonlinear programming.

OEM optimal energy management.

PCC point of common coupling.

PDF probability density function.

PEM point estimate method.

POPF probabilistic optimal power flow.

PV photovoltaic.

RES renewable energy sources.

RV random variable.

SAA sample average approximation.

SOC state of charge.

WT wind turbine.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.1 State of Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.1 Power Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1.1 Node Current Balance . . . . . . . . . . . . . . . . . . . . . . . . . 372.1.2 Branch Current Method . . . . . . . . . . . . . . . . . . . . . . . . 392.1.3 Node Power Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 392.1.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2 Basic Concepts on Optimization . . . . . . . . . . . . . . . . . . . . . . . . 422.2.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2.2 Local and Global Optimality . . . . . . . . . . . . . . . . . . . . . . 43

2.2.2.1 Example - Power Flow . . . . . . . . . . . . . . . . . . . . 452.2.3 Mixed-Integer Programming - MIP . . . . . . . . . . . . . . . . . . 46

2.2.3.1 Mixed-Integer Second-Order Cone Programming (MISOCP) 472.2.3.2 Example - Unit Commitment . . . . . . . . . . . . . . . . 48

2.3 Basic Techniques for Considering Uncertainty . . . . . . . . . . . . . . . . 502.3.1 Monte Carlo Simulation - MCS . . . . . . . . . . . . . . . . . . . . 512.3.2 Point Estimate Method - PEM . . . . . . . . . . . . . . . . . . . . 532.3.3 Example - Probabilistic DC Power Flow . . . . . . . . . . . . . . . 56

2.4 Uncertainty in Optimization Problems . . . . . . . . . . . . . . . . . . . . 572.4.1 Stochastic Optimization . . . . . . . . . . . . . . . . . . . . . . . . 582.4.2 Robust Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 592.4.3 Chance-Constrained Optimization . . . . . . . . . . . . . . . . . . . 60

Deterministic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Optimal Energy Management of Isolated Microgrids . . . . . . . . . . . . . 63

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.1.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3 Test System and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3.1 Unlimited Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3.2 Limited Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.4 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Optimal Energy Management of Unbalanced Grid-Connected Microgrids . 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Proposed Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2.1 Loads and Renewable Energy Sources . . . . . . . . . . . . . . . . . 834.2.2 Dispatchable Distributed Generators . . . . . . . . . . . . . . . . . 834.2.3 Energy Storage Systems . . . . . . . . . . . . . . . . . . . . . . . . 854.2.4 Operational Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Models Considering Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 935 Microgrids Energy Management Using a Robust Approach . . . . . . . . . 94

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 Characterization of Robust Parameters . . . . . . . . . . . . . . . . . . . . 1005.2.1 Robust Convex Optimization . . . . . . . . . . . . . . . . . . . . . 1015.2.2 Robust Equivalents for RES . . . . . . . . . . . . . . . . . . . . . . 1025.2.3 Robust Equivalents for Demands . . . . . . . . . . . . . . . . . . . 1035.2.4 Robustness Adjustment Parameter 𝜁 . . . . . . . . . . . . . . . . . 103

5.3 Proposed Robust Micogrids EMS . . . . . . . . . . . . . . . . . . . . . . . 1045.3.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3.2 Microgrid’s Steady-state Operation . . . . . . . . . . . . . . . . . . 1055.3.3 ESS Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3.4 Proposed MISOCP Model . . . . . . . . . . . . . . . . . . . . . . . 107

5.4 Monte Carlo Simulation and Robustness Assessment . . . . . . . . . . . . 1075.5 Tests and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.5.1 Grid-connected Mode . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.5.2 Isolated Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.6 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156 Probabilistic Optimal Power Flow for Unbalanced Distribution Systems . . 117

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2 Probabilistic Optimal Power Flow Model for Unbalanced Three-phase Dis-

tribution Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.2.1 Probabilistic Optimal Power Flow as a Two-stage Problem . . . . . 1236.2.2 Moments of the State RVs and Robust Constraints . . . . . . . . . 124

6.3 Selection of the Robustness Parameter . . . . . . . . . . . . . . . . . . . . 1266.4 Accuracy and Robustness Assessment . . . . . . . . . . . . . . . . . . . . . 128

6.4.1 Sample Average Approximation - SAA . . . . . . . . . . . . . . . . 1296.5 Tests and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.5.1 IEEE 13-bus system – Without Dispatchable Distributed Genera-tion Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.5.2 IEEE 13-bus System – Control Test . . . . . . . . . . . . . . . . . . 1326.5.3 IEEE 13-bus System – Different Probability Limits . . . . . . . . . 1336.5.4 IEEE 123-bus System – Different Robustness Parameters . . . . . . 1366.5.5 IEEE 123-bus System – Different Number of Random Variables . . 1376.5.6 IEEE 123-bus System – Out-of-sample Monte Carlo Simulations . . 140

6.6 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417 Conclusions and Future work . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.1 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

19

1 Introduction

Electric power systems have played a fundamental roll in the development of themodern society and its technological advances. Power systems have been subjected tosubstantial technological changes from their first traceable beginnings in the 1880s whereDC generators fed small groups of local loads, up to becoming the highly meshed, complex,and reliable networks working nowadays [1]. Power systems are once more experiencingbig changes, this time, towards the modernization of the grid and the diversification ofthe energy matrix. These changes have been motivated mostly by economical reasons, butalso strengthened by an arising environmental awareness over the last few decades [2].

The technological transition is already evident in many networks around the worldand it is expected to spread even more in the future, which is considered by many as the“modernization” of the electric power grid [3, 4]. For example, Latin America is consid-ered as a promising region for the penetration of distributed energy resources (DERs),motivated mostly by the uncertainty in hydroelectric generation, which is increased bymeteorological phenomena like El Niño, economic factors, and the creation of regula-tory incentives [5]. Although not entirely accepted by the power system community, theproduct of this modernization has been popularized in recent years with the term smartgrids [6], creating a global expectation. A smart grid is a wide concept however, goingfrom improvements in the infrastructure of the system, such as the use of smart switchesand smart meters for the automation, control, and reliability of the grids, to communi-cations and data analytics. The integration of DERs, energy storage systems (ESS), andplug-in electric vehicles into existing networks are some of the key motivators for this tran-sition, creating new challenges for system operators concerning its operation, planning,and control [7].

Electrical distribution systems conform the final tie between bulk, interconnected,high-voltage (HV) systems, usually operating at voltages higher than 110 kV, and endusers at low-voltage (LV) levels, commonly lower than 1.0 kV. Distribution systems aretraditionally conformed by medium-voltage (MV) feeders working at voltage levels lowerthan 69 kV, supplied by a single energy source identified as the substation, feeding a setof electric loads connected mostly through a radial network [8]. This definition, althoughaccurate for most existing distribution systems, relies on the assumption that the networkis intrinsically passive, hence, the power flow direction is always unidirectional as shownin Figure 1.1.

The modernization of current power systems highly involves the distribution level,leading to the emergence of new concepts, technologies, and paradigms. For example,

Chapter 1. Introduction 20

Figure 1.1 – Traditional power system representation. Source: Author

sociopolitical and economic motivations have driven the increasing insertion of DERsinto distribution systems in recent years, i.e., dispatchable distributed generation (DG)units and non-dispatchable sources, such as solar photovoltaic (PV) and wind turbines(WTs) [2]. The insertion of energy storage systems (ESS) into power systems brings ad-ditional challenges too, from their modeling and control to their bidding and optimaloperation. The use of ESS in power systems has motivated the growth of a variety of stor-age technologies, such as electric batteries, flywheels, super-capacitors, pumped storage,etc., since ESS can improve power system’s operation [9].

The increasing participation of DER at the distribution level plays a key role inthe future of power systems, considering that system operators still face challenges fortheir integration into existing networks. The insertion of additional energy sources intoexisting distribution systems leads to a series of technical difficulties that must be con-sidered and analyzed, such as modifications on the power flows of feeders and its possibleinversion, possible overvoltages, modification of short-circuit levels, stability problems,among others [10–12]. One way for integrating these additional sources has been pro-posed by introducing the concept of microgrids, which are part of the expected futuresmart grids [6]. A microgrid is defined as a cluster of DERs, ESS, and loads, that can beor not connected to a main grid through a point of common coupling (PCC), providingreliable and secure electrical supply to a local community [13].

There are several ideas for implementing microgrids. They can be either AC [14],DC [15], hybrid AC/DC [16], or even using high frequency (500 Hz) AC transmission [17].Moreover, they are able to work interconnected to the power grid or isolated. When themicrogrid has a connection to the main power grid, all power deficit or surplus can be

Chapter 1. Introduction 21

absorbed or delivered to the power grid. On the other hand, the balance must be satisfiedlocally by the microgrid itself if it is autonomous or if it is operating in island mode [18].

Microgrids have gained reputation in recent years due to its positive impacts, notonly limited to the integration of renewable energy sources (RES) into existing networks,but also to the improvement of the overall performance of the whole electrical grid [19,20].Researchers all over the world are studying the effects of microgrids and important in-vestments have been made to construct test-beds and demonstration sites. According tothe latest report on grid-connected and isolated microgrids in [21], there are 2,258 micro-grid projects representing 19.5 GW of planned and installed power capacity, with NorthAmerica being its primary market, followed by Asia Pacific and the Middle East & Africa.Over the last decade, some research entities have developed test-beds around the world.For example, the Consortium for Electric Reliability Technology Solutions (CERTS) inthe USA, the New Energy and Industrial Technology Development Organization (NEDO)in Japan, The Institute for Systems and Computer Engineering, Technology and Science(INESC TEC) in Europe, the Korean Energy Research Institute (KERI) in South Korea,among others [22–24].

The microgrid scenario looks promising in Brazil for the near future as well. Micro-grids operating in island mode have been explored in the northern part of the country byEletrobras, CELPA, and Siemens in communities that are not part of the national inter-connected system [25,26]. These projects are part of the PRODEEM program (Portuguesefor National Program for Energy Development of States and Municipalities), which is in-tended to provide electrical energy to communities that are not supplied by the inter-connected system; with a total PV capacity of 3,94 MWp and an investment of US$3.94million up to 2003 [27]. In fact, according to [21], isolated microgrids represented nearly40% (7.6 GW) of the total global installed capacity, followed by commercial/industrialmicrogrids with 5.5 GW, and utility distribution with 2.3 GW. Investments in this kindof microgrids are estimated to grow at a rate of 17% p.a. by 2020 in Brazil, increasingfrom US$149 million in 2012 to US$518 million in 2020 [28]. Some small grid-connectedmicrogrids can be also found in Paraná [29] and Ceará [30], as well as some test bedscurrently working, such as the one from the Laboratório de Microrredes Inteligentes atthe Federal University of Santa Catarina (UFSC) with 20 kWp of PV generation, 11 kWof eolic generation, and 10 kWh of energy storage [31] and the Usina Distrital in Flori-anópolis, SC, with 8 kWp of renewable energy, a diesel generator of 5.5 kW, and 10 kWhof energy storage [32].

One major advantage of microgrids is their flexibility to operate in grid-connectedor island mode by controlling the connection status of the PCC. A microgrid can inten-tionally island itself from the grid utility without affecting the local power supply duringa period in which the power quality may be deteriorated, e.g., voltage fluctuations, har-

Chapter 1. Introduction 22

Figure 1.2 – Microgrid representation and commands from the microgrid operator.Source: Author

monic distortions, frequency deviations and voltage flickers, or after network contingenciesproducing a disruption of the main energy supply [33]. In this context, a microgrid needsto have a centralized control defining, among other matters, the status of the PCC andsatisfying the power balance. This centralized control is usually composed by an energymanagement system (EMS) either if the microgrid has a connection to a main grid or ifit is isolated.

An EMS is comprised by a set of hardware/software components used to efficientlyoperate the energy resources within the microgrid to achieve selected objectives. These ob-jectives are accomplished by typically, but not exclusively, using optimization techniquesfor scheduling the commitment of dispatchable DG units, such as cogenerators; control-ling the charge/discharge patterns of available ESS, and managing controllable loads [34].Overall, the EMS optimizes and manages the interchange of energy between the microgridand the main grid [35], in the case they are connected, or the local balance in isolatedmicrogrids [25]. Without loss of generality, this problem can be interchangeably referredto as optimal energy management (OEM). A schematic representation of a microgrid isshown in Figure 1.2, where the control commands from the microgrid operator are alsodisplayed.

Uncertainties on system’s operational conditions increase the complexity of theEMS problem. Uncertainties are due to the volatility of some exogenous parameters, e.g.,solar irradiation for PV power generation, wind velocities for WTs, and power consump-

Chapter 1. Introduction 23

tion in conventional loads. From an operational point of view, a secure operation of thesystem requires that voltage magnitude limits and thermal ratings of conductors be re-spected. However, guaranteeing those limitations may not be possible by using determin-istic, average-based approaches. Hence, considering uncertainties is a major concern whenoptimizing the operation of microgrids and modern distribution systems in a given plan-ning horizon [36]. There are several approaches that can be used for involving uncertaintiesin EMS problems, namely, stochastic programming, chance-constrained optimization, ro-bust optimization, and the classical probabilistic optimal power flow (POPF):

∙ Stochastic programming models are a useful and flexible way of addressing thisissue, mainly because of its solid mathematical foundations and theoretical richnesson probability and stochastic processes [37]. In a general approach, it consists onsolving a single mathematical model comprising a set of plausible scenarios andfinding the optimal decision that minimizes the average objective function whilebeing feasible for all scenarios [38]. Still, the main drawback of currently proposedstochastic-based methods is the number of required scenarios to obtain satisfactoryresults, which is highly dependent on the number of random variables (RVs) evenif scenario reduction techniques are applied [39]. Furthermore, reducing the numberof scenarios can lead to inaccurate solutions, thus, the final results are a function ofthe quantity and the quality of the selected scenarios.

∙ Robust optimization and chance-constrained optimization are also suitable tools forconsidering the volatility on random variables within optimization processes. Com-pared to stochastic programming, these approaches are less challenging to be solvedbecause they do not require to operate over multiple probable scenarios. In this mat-ter, robust approaches create a deterministic equivalent of an essentially stochasticoptimization problem, rather than a discrete representation of the probability den-sity functions (PDFs) of the random phenomena [40]. Robust problems tend to betoo conservative nonetheless, and the robustness of the final solution is usually diffi-cult to be assessed beforehand. Similarly, chance-constrained formulations are clas-sically based on deterministic approximations representing the stochastic behaviorof the output variables, however, chance-constrained models are more flexible thanrobust optimization approaches, since the hardness of some constraints can be con-trolled. As in stochastic programming approaches, the stochastic moments of theoutput (or state) RVs are not directly assessed in the model or available for lateranalysis using robust or chance-constrained optimization.

∙ The POPF is a classical problem where the optimal operation point of a networkis obtained while considering the uncertainties and the statistical behavior of thestate variables [41]. Most formulations for the POPF are based on solving a set of

Chapter 1. Introduction 24

deterministic optimal power flows to obtain the stochastic behavior of the outputRVs at the end of the algorithm. However, this kind of algorithm does not guaran-tee a single optimal decision, since it is intended for solving a set of independentdeterministic optimal power flows. This means that output variables are taken asstochastic outputs rather than as single decisions, as in [42]; therefore, the feasibilityunder uncertainty cannot be guaranteed.

This thesis proposes different approaches for the EMS of modern distributionsystems, including microgrids, regarding the integration of dispatchable DG units, non-dispatchable RES, and ESS into balanced and unbalanced electric networks. From a math-ematical outlook, the proposed models cover deterministic approaches, robust optimiza-tion, and a novel framework using stochastic programming and probabilistically-robustconstraints. Convex and non-convex formulations are proposed, such as mixed-integer lin-ear programming (MILP), mixed-integer second-order cone programming (MISOCP), andnonlinear programming (NLP). The addressed models are implemented using a commer-cial mathematical programming language and solved with commercial numerical solvers.

1.1 State of ArtThe concept of microgrid in power systems has been highly explored during the

last 15 years in the specialized literature. This can be seen in Figure 1.3, where the numberof publications and the number of citations per year involving the word “Microgrid” in thetitle plus the “EMS” topic have been condensed. The data was collected from the Web ofScience principal collection and filtered for electrical and electronic engineering [43], witha total of 1,510 results and 19,512 citations to the date of consultation. It can be seenthat the number of publications has increased exponentially since its first appearing in2,005, with a small reduction in 2,018, while the number of citations has been increasingmonotonically every year.

Several approaches have been proposed for solving the EMS problem of moderndistribution systems, including microgrids. In [44], a centralized EMS is proposed based onforecasted values of loads and non-dispatchable generation units. In [45], a multi-objectivesingle-step formulation dispatch of DG and ESS is proposed using a niching evolutionaryalgorithm. A load management model is proposed in [46] to improve microgrid resiliencefollowing islanding. Authors in [47] propose an algorithm to minimize the microgrid totaloperation cost by decomposing the problem into a grid-connected operation master prob-lem and an islanded operation subproblem. In [48], the dispatch for emergency electricservice restoration in the aftermath of a natural disaster in a microgrid is proposed using amixed-integer nonlinear programming (MINLP) model. In [49], the authors use an MILPapproach for an online optimal energy/power control method for the operation of energy

Chapter 1. Introduction 25

Year

2005

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

Nu

mb

er

of

Pu

bli

cati

on

s

0

50

100

150

200

250

300

350

Year

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

Nu

mb

er

of

Cit

ati

on

s

0

1000

2000

3000

4000

5000

6000

Figure 1.3 – Number of publications and citations using “Microgrid” in the title + “EMS”as topic.

storage in grid-connected microgrids, while the islanded operation mode is also consid-ered in [50]. Authors in [51] propose a heuristic approach to solve the power dispatch ofmicrosources. An MILP model for the modeling and experimental verification of an EMSwas proposed in [52], and authors in [53] introduce an EMS for isolated microgrids con-sidering equivalent CO2 emissions. An efficient planning algorithm is proposed for remoteislanded microgrids in [54], minimizing the capital and operational costs while ensuringtechnical feasibility. Analyses of multi-microgrids are also available in the literature, suchas in [55] where a control strategy for coordinated operation of networked microgridsin distribution systems is proposed, and in [56] where a multiple-objective constrainedoptimization is presented for solving the microgrids/nanogrids EMS.

The aforementioned models are focused entirely on single-phase equivalents, andmost of them consider active power balance only.

Unlike HV networks, distribution systems and microgrids, which are characterizedby MV and LV levels, cannot be usually assumed to be balanced. Unbalances are as-sociated to line configurations, i.e., untransposed lines, with two-phase and single-phaselaterals, and to the characteristics of the loads, where single-phase and two-phase connec-tions prevail. These features require the use of three-phase models for the network and itsdevices, thus, increasing the size and complexity of the problem [57]. Furthermore, froman operational point of view, a secure operation of the distribution network requires thatvoltage magnitudes and current ratings of conductors remain within their limits; thus, anAC approach is also necessary.

Unbalanced microgrids have also been the subject of study in the last years, asshown in Figure 1.4. Again taking into account the number of publications and citations

Chapter 1. Introduction 26

Year

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

Nu

mb

er

of

Cit

ati

on

s

0

100

200

300

400

500

600

700

Year

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

Nu

mb

er

of

Pu

bli

cati

on

s

0

5

10

15

20

25

30

35

40

Figure 1.4 – “Microgrid” in title + “EMS” + “Unbalanced” as topic.

per year obtained from [43], but including the topic “EMS Unbalanced” to the search,resulting in a total of 198 results (13% of the first search) with 2,563 citations. It canbe seen that since 2,015, the number of publications has been more or less constantwith more than 30 publications per year; and the citations have been increasing yearafter year. In [58], an EMS for phase balancing in distribution systems is proposed basedon the automated mapping/facilities management/geographic information system. AnEMS for isolated, three-phase microgrids considering unbalanced synchronous machinesis proposed in [59]. However, this approach does not consider reactive power balance toobtain the optimal solution, while in [60], an approximation is performed in order to obtaina linear model and to neglect the network topology. In [61], authors propose an algorithmfor the capacity constrained management of DERs in unbalanced distribution networks.Authors in [62] propose a centralized dispatch for a set of non-synchronous microgridspursuing loss reduction and unbalance compensation. Finally, reference [63] proposes aMILP model for unbalanced microgrids considering unexpected main grid outages.

The EMS can be sometimes merged or modeled using the optimal power flowproblem including DERs. A methodology for unbalanced three-phase optimal power flowfor distribution systems is proposed in [64] based on a quasi-Newton method, while in [65]authors used a genetic algorithm to solve the three-phase distribution optimal power flowin smart grids. An economic dispatch problem for unbalanced, three-phase power distri-bution networks with distributed generation units is proposed in [66] using a semidefiniterelaxation technique. Authors in [67] present an application of the glowworm swarm op-timization method to solve the optimal power flow problem in three-phase islanded mi-crogrids. Reference [68] proposes a multi-period, three-phase, unbalanced optimal powerflow method for distribution systems. Authors in [69] model a three-phase optimal power

Chapter 1. Introduction 27

Year

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

Nu

mb

er

of

Pu

bli

cati

on

s

0

10

20

30

40

50

Year

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

Nu

mb

er

of

Cit

ati

on

s

0

200

400

600

800

1000

1200

1400

Figure 1.5 – “Microgrid” in title + “EMS” + “Uncertainty” as topic.

flow applied to distribution networks using a nonlinear approach to minimize power losses.Finally, authors in [70] formulate a non-linear programming model for the optimal schedul-ing of distributed resources in unbalanced microgrids applying a multi-objective approachconsidering cost minimization, power quality improvements, and energy savings.

Note that the works mentioned above consider deterministic models; hence, theintrinsic uncertainty of exogenous parameters was not considered.

Several approaches have been proposed for considering uncertainty within the op-eration planning of power systems. Similarly to what has been previously done in thischapter, a research was conducted using the Web of Science principal collection [43], nowincluding the topic “Uncertainty”. The number of publications and citations accumulatedin each year is shown in Figure 1.5. A total of 198 results were found with 3,579 citations.Popular approaches include the use of artificial neural networks and fuzzy logic as in [71],the use of evolutionary methods like the imperialist competitive algorithm in [72], the ge-netic algorithm for the day-ahead scheduling of ESS in microgrids [73], and mathematicalprogramming models [74]. Some of the techniques applying mathematical programmingconsidering uncertainty can be listed as stochastic programming, robust optimization,chance-constrained optimization, and the classical POPF.

Stochastic programming consists of solving a single mathematical model compris-ing a set of plausible scenarios and finding the optimal decision that minimizes the averageobjective function while being feasible for all scenarios [38]. It is usually implemented inpower system applications by means of two-stage models, as in [75] considering risk levelsin distribution systems, in [76,77] for unit commitment problems, in [78] for the Volt/Varcontrol in distribution networks, in [79] for stochastic optimal power flow, and for optimalscheduling considering wind farms in [80]. Stochastic programming has also been applied

Chapter 1. Introduction 28

for solving the EMS problem, as in [81], for example, where authors present a stochasticEMS for isolated microgrids. In [82], a stochastic framework is proposed to investigatethe effect of uncertainty on the optimal operation management of microgrids by usinga scenario reduction process, and in [83] the EMS is implemented along a demand re-sponse program. Authors in [74] propose a two-stage stochastic model for the EMS ingrid-connected operation mode. Similarly, authors in [84] use a two-stage stochastic pro-gramming model for the nonconvex economic dispatch problem using a sample averageapproximation (SAA) formulation and introduce two outer approximation algorithms toobtain global solutions. Reference [85] proposes a multi-objective stochastic programmingmodel considering RES and demand response, based on a simplified active power flow.In [86], an approximated dynamic programming algorithm is used for the stochastic eco-nomic dispatch of microgrids. An energy management and reserve scheduling scheme isproposed in [87] considering hydrogen storage, and in [88], the authors propose a stochas-tic optimization framework to address the microgrid energy dispatching problem withrandom RES and vehicle activity pattern using a Markov decision process. However,stochastic programming problems are computationally demanding, since the number ofscenarios that must to be considered is usually high, even if scenario reduction techniquesare applied [39]. Furthermore, reducing the number of scenarios can lead to inaccuratesolutions. Thus, the final results are highly dependent on the quantity and the qualityof the selected scenarios. This has been recognized as a crucial drawback of stochasticoptimization.

Robust optimization is another suitable tool for considering volatility on randomvariables within the optimization process. Compared to stochastic programming, robustoptimization problems are less challenging to be solved since they do not require to operateover multiple probable scenarios. In this matter, robust approaches create deterministicequivalents of an essentially stochastic optimization problem, called robust counterpart,rather than a discrete representation of the PDFs of the random phenomena [40]. Ro-bust optimization has been used for the EMS of modern distribution systems as in [89],for example, where authors propose a linear robust EMS with high penetration of RESconsidering the worst-case transaction cost. In [90], authors propose a framework for amicrogrid EMS based on agent-based modelling, by introducing robust optimization andneural networks. A cost minimization problem is formulated in [91] using chance constraintapproximations and robust optimization algorithms to schedule the energy generationsfor microgrids equipped with uncertain renewable sources and combined heat and powergenerators. A distributed robust EMS for smart grids is introduced in [92] using a non-linear approach. Authors in [93] formulate a robust EMS using a fuzzy predictive controlmodel for wind production. A robust two-stage model was proposed in [94] using a MILPformulation. A scenario-robust, MILP using ensemble weather forecasts for hybrid micro-grids is presented in [95]. Authors in [96] introduce an integrated scheduling approach for

Chapter 1. Introduction 29

microgrids based on robust multi-objective optimization considering demand response.Finally, A robust model for islanding-aware microgrids EMS with RES and co-generationis formulated in [97].

Chance-constrained optimization deals with problems were some selected outputRVs, which are functions of the input RVs of the problem with known PDF, maximize theobjective function while being subject to constraints on these variables. These constraintsmust be maintained at prescribed levels of probability [98,99]. In other words, this meansthat chance-constrained problems “admit random data variations and permits constraintviolations up to specified probability limits” [100]. Several variants have been proposed,like mathematical properties of chance-constrained programming problems concerned onjoint probabilities [101], or reference [102], where a general class of convex approxima-tions to chance-constrained problems are proposed. The first use of chance-constrainedoptimization in power systems can be traced down to [103] where a generation invest-ment planning model has been proposed to include transmission reliability constraintsas chance-constraints. Later, in [104], chance-constraints were used for solving the volt-ampere reactive compensation problem under uncertain operating conditions. Authorsin [105] formulate the unit commitment problem as a chance-constrained optimizationproblem requiring power balance to be met with a specified probability over the entiretime horizon. Reference [106] proposes a model for the optimal transmission system ex-pansion planning based on chance-constrained programming, considering the locationsand capacities of new power plants, and demand growth as uncertain. Authors in [107]present a unit commitment problem with uncertain wind power output, formulated as achance-constrained two-stage stochastic program using the SAA method. The use of arobust-convex, chance-constrained formulation for the optimal power flow problem wasproposed in [108]. Reference [109] formulates a robust chance constrained optimal powerflow that accounts for uncertainty in the parameters of the probability distributions ofthe RVs, by allowing them to be within an uncertainty set. A data-driven approach isutilized in [110] to develop a distributionally robust conservative convex approximation ofthe chance-constraints to enforce voltage regulation with predetermined probability viaChebyshev-based bounds.

Authors in [111] introduce a chance-constrained programming model for the EMSof grid-connected microgrids, while in [112] authors proposed a chance-constrained EMSfor islanded microgrids. Authors in [113] proposed a chance-constrained AC optimal powerflow for distribution systems considering RES, while authors in [114] proposed a non-linear, non-convex, robust framework for active and reactive power management in distri-bution networks using electric vehicles. Authors in [115] formulate a chance-constrainedAC optimal power flow, where probabilistic constraints are utilized to enforce voltageregulation with prescribed probability on distribution systems featuring RES and ESS.Reference [116] proposes a joint chance-constraint relaxation method to solve the AC

Chapter 1. Introduction 30

optimal power flow problem while preventing over voltages in distribution grids underhigh penetrations of PV systems. Reference [117] proposes an MILP model for microgridoperation considering the probability that a microgrid maintains enough spinning reserveto meet local power balance after instantaneously islanding from the main grid. Authorsin [118] present a chance-constrained information gap decision model for multi-periodmicrogrid expansion planning for the optimal sizing, type selection, and installation timeof DERs in microgrids. A chance-constrained two-stage stochastic programming modelis proposed in [119] to evaluate the impacts of variability in renewable resources in themicrogrid operation. In [120], a semidefinite relaxation of a chance-constrained AC op-timal power flow is proposed, which is claimed to provide global optimality. Authorsin [121] propose a data driven distributionally robust chance constrained optimal powerflow model, which ensures that the worst-case probability of violating both the upperand lower limit of a line/bus capacity under a wide family of distributions is small. Fi-nally, authors in [122] formulate a constructive approach to chance-constrained optimalpower flow problems that does not assume a specific distribution, e.g. Gaussian, for theuncertainties.

Last but not less important, uncertainties can be considered using the POPF. ThePOPF is a classical problem where the optimal operation point of a network is obtainedwhile considering the uncertainties and the statistical behavior of the state variables [41].The POPF is an extension of the deterministic optimal power flow problem, which wasoriginally proposed in the 50’s [123–127] for minimizing power losses and obtaining theoptimal dispatch of generators. The optimal power flow problem has been analyzed inthree main subjects in [128], namely deterministic, risk-based, and considering uncer-tainty. Several formulations for the POPF are based on solving a set of deterministicoptimal power flows to obtain the stochastic behavior of the output RVs at the end ofthe algorithm, like in [129], where the quasi-Monte Carlo method is used to solve thePOPF problem. A number of approximate methods exist in the literature for estimatingthe statistical moments of a random function. The point estimate method (PEM), forexample, was first proposed in [130] for functions of only one variable, and later in [131]for multivariable functions. It has been used in several works, as in [132], where a POPFwas proposed using different schemes. Reference [133] presents a POPF that takes intoaccount load variation, wind’s stochastic behavior and variable line’s thermal rating usingthe PEM. Correlations between RVs have been also taken into account, as in [134], whereit is proposed an algorithm to consider the correlation amongst input RVs with the PEM,and in [135], where a POPF is proposed based on the PEM, considering the correlationsof wind speeds following arbitrary PDFs by transforming them into independent Gaussiandistributions.

A review of optimal power flow approaches mainly related to smart distributiongrids is proposed in [136]. In [137], authors use a linear optimal power flow considering

Chapter 1. Introduction 31

the uncertainty of load and RES. Reference [138] uses a genetic algorithm and a modi-fied bacteria foraging algorithm to determine the optimal power flow in a power systemwith considerable wind energy penetration. Authors in [139] introduce an affine arith-metic method to solve the optimal power flow problem with uncertain generation sources.Authors in [140] propose a POPF model with chance constraints that considers the uncer-tainties of wind power generation and load. In [141], it is presented an efficient approach forsolving stochastic, multi-period optimal power flow problems using a family of stochasticnetwork and device constraints based on convex approximations of chance constraints. Anstochastic framework to investigate the impact of correlated wind generators on the EMSof microgrids is proposed in [142]. Reference [143] proposes a dynamic optimal power flowin active distribution networks using affine arithmetic and interval Taylor expansion usingsuccessive linear approximation as solving method. Reference [144] proposes an intervaloptimal power flow method employing affine arithmetic and interval Taylor expansion indistribution systems. In [145], it is proposed a multi-period AC optimal power flow fordistribution systems, considering wind and PV uncertainty. However, this kind of algo-rithms does not guarantee a single optimal decision for dispatchable DGs, since they areintended for solving a set of independent deterministic optimal power flows. Therefore,the feasibility under uncertainty cannot be guaranteed when using this approach. It istraditional that POPF approaches are only focused on balanced problems, leaving an im-portant gap for the analysis of three-phase unbalanced distribution systems. Three-phaseoptimal power flows considering DG units have been proposed in the specialized litera-ture, as in [146], as well as probabilistic power flows for unbalanced distribution systems,as in [147]. However, a mathematical programming model combining both methods hasyet to be proposed.

The power injection from DG units in unbalanced systems has been commonlymodeled using balanced power sources, as in [146, 148, 149]; as single-phase machinesin [150], or considering each phase’s power as controllable variables [63, 151, 152], just tomention some. These models do not consider the physical coupling of power flows betweenphases for calculating the power injected to the system by synchronous machines. Anenergy management system for isolated, three-phase microgrids considering unbalancedsynchronous machines is proposed in [59]. However, this approach does not consider re-active power balance to obtain the optimal solution, while in [60], an approximation isperformed in order to obtain a linear model and to neglect the network topology.

1.2 MotivationThe modernization of traditional distribution power systems is an unavoidable

transition into a more efficient, reliable, and environmentally friendly electric power sys-tem. Some gaps regarding mathematical models involving the short-term planning and

Chapter 1. Introduction 32

53%

17%

7%

23%

IEEE

Elsevier

Springer

Others

1930 1940 1950 1960 1970 1980 1990 2000 2010 20200

5

10

15

20

25

30

Figure 1.6 – Number of references used in the thesis per year and percentage of papersby publisher

operation of modern distribution systems were identified after a comprehensive literatureresearch, comprising a total of 294 references from different publishers. This has been sum-marized in Figure 1.6, where the number of references used in this thesis has been sorteddepending on its year of publication, along with the percentage of references according tothe publisher.

Current trends indicate that the insertion of DERs and ESS into modern distri-bution systems could be achieved by creating flexible microgrids, in which the EMS ismandatory. Available approaches for the EMS of modern distribution systems and micro-grids are mainly focused on balanced equivalents, which is an arguable approximation formedium-voltage and low-voltage networks. Moreover, the insertion of RES and the naturalbehavior of loads involves the inclusion of intrinsically stochastic exogenous parameters,creating the need for methodologies able of handling uncertainty.

1.3 Objectives

∙ To develop deterministic mathematical programming models for the short-term op-eration planning of balanced and unbalanced three-phase microgrids operating ingrid-connected and isolated modes.

∙ To develop mathematical programming models able to consider the uncertaintyof some exogenous parameters for the short-term operation planning of moderndistribution systems, including microgrids operating in grid-connected and isolatedmode.

Chapter 1. Introduction 33

1.4 Contributions

∙ A deterministic mixed-integer second-order cone programming model for the en-ergy management of balanced isolated microgrids. The novelty of this formulationrelies on the consideration of fuel limits and demand side management control in amicrogrid operating in island mode using an AC convex formulation.

∙ A deterministic nonlinear mathematical programming model for the energy manage-ment of unbalanced, three-phase, grid-connected microgrids. This model introducesa formulation for DG based on operating in unbalanced networks. It shows a newstudy aimed at analyzing the effects on the ESS patterns and operational costs af-ter using different common representations for synchronous machines in unbalancedoperation.

∙ An energy management system for balanced microgrids operating in grid-connectedand isolated mode, using robust convex mixed-integer programming. The novelty ofthis model is the introduction of a new robust convex formulation for the microgridEMS operating in grid-connected and islanded modes.

∙ A probabilistic optimal power flow for unbalanced three-phase modern electricaldistribution systems considering probabilistically-robust constraints. The noveltyof this model is the formulation of the POPF as a two-stage optimization modelindependent of the technique used for the scenario generation. The addition ofprobabilistically-robust constraints to the problem is also a novelty, since this im-plies the calculation of useful statistical moments embedded in the model to controlthe robustness of the final solution.

1.5 OutlineThis thesis is divided into two main parts and composed by six chapters, as shown

in Figure 1.7. After the introduction and the definition of the problem in Chapter 1, Chap-ter 2 introduces some basic definitions and concepts used throughout the thesis regardingthe operation of electrical power systems, used simulation techniques, and optimizationprinciples.

The first part of the thesis includes two deterministic approaches:

∙ Chapter 3 presents an optimal energy management system for balanced microgridsoperating in isolated mode using a mixed-integer second-order cone programmingmodel. The model considers dispatchable DG and ESS, fuel availability and theoperational constraints of the system. This chapter is mainly based on the followingconference paper:

Chapter 1. Introduction 34

Figure 1.7 – Structure of thesis

Juan S. Giraldo, Jhon A. Castrillon and Carlos A. Castro “Energy managementof isolated microgrids using mixed-integer second-order cone programming,” in 2017IEEE Power & Energy Society General Meeting. IEEE, 2017, Chicago, IL.DOI: https://doi.org/10.1109/PESGM.2017.8274353

∙ Chapter 4 introduces a nonlinear optimization problem based on nodal currentinjections to solve the optimal energy management of unbalanced, three-phase, grid-connected microgrids, focused particularly on the modelling of small synchronousmachines under unbalanced operation and its effect on the management of ESS.This chapter is mainly based on the following conference paper:

Juan S. Giraldo, Jhon A. Castrillon, Federico Milano, and Carlos A. Castro “Opti-mal Energy Management of Unbalanced Three-Phase Grid-Connected Microgrids,”in 2019 IEEE Power & Energy Society Powertech. IEEE, 2019, Milan, Italy.Preprint https://doi.org/10.1109/PTC.2019.8810498

The second part of the thesis embraces two approaches considering uncertainty:

∙ Chapter 5 presents an energy management system for single-phase or balanced three-phase microgrids in grid-connected and isolated operation modes using a new ro-bust convex optimization approach. The proposed model integrates dispatchableDG units, ESS, load shedding, and stochasticity over conventional demands andnon-dispatchable RES. This chapter is mainly based on the following journal paper:

Juan S. Giraldo, Jhon A. Castrillon, Juan Camilo López, Marcos J. Rider andCarlos A. Castro “Microgrids Energy Management Using Robust Convex Program-

https://doi.org/10.1109/PESGM.2017.8274353https://doi.org/10.1109/PTC.2019.8810498

Chapter 1. Introduction 35

ming,” IEEE Transactions on Smart Grid, vol. 10, no. 4, pp. 4520-4530, July 2019.DOI: https://doi.org/10.1109/TSG.2018.2863049

∙ Chapter 6 presents a new mathematical model for the POPF of unbalanced three-phase modern distribution systems considering probabilistically-robust constraints.The model optimizes the operation of dispatchable DG units, assuming conventionaldemands and RES as input RVs with dissimilar probability distribution functions.This chapter is mainly based on the following journal paper:

Juan S. Giraldo, Juan Camilo López, Jhon A. Castrillon , Marcos J. Rider andCarlos A. Castro “Probabilistic OPF Model for Unbalanced Three-phase Electri-cal Distribution Systems Considering Robust Constraints,” IEEE Transactions onPower Systems, vol. 34, no. 5, pp. 3443-3454, Sept. 2019.DOI: https://doi.org/10.1109/TPWRS.2019.2909404

https://doi.org/10.1109/TSG.2018.2863049https://doi.org/10.1109/TPWRS.2019.2909404

36

2 General Definitions

This chapter introduces some basic definitions and concepts used throughout this the-sis. It embraces the definition for the power flow formulation and its numericalcharacteristics as well as some mathematical programming concepts. In addition, it isimportant to emphasize that this chapter is merely a brief introduction to complex andwide subjects. It is specifically intended to contextualize useful topics for a better under-standing of the forthcoming chapters. Further information can be found in the referencescited in each section.

2.1 Power Flow AnalysisThe power flow analysis, also known as load flow analysis, consists of finding the

steady-state operation of an electrical power network by determining the voltage at everybus, the distribution of power flows, power losses, and in general, any other modelledvariable of interest [153]. Power flows are a fundamental part of almost all tasks in powersystems, such as design, expansion and operation planning, dynamic analysis, and short-circuit analysis.

Nowadays it is common to think about the power flow analysis as a computationaltask. However, analyses over interconnected power networks were not always performedby using simulation tools. For example, network analyzers were very popular in the 30’s,such as the one installed in the Massachusetts Institute of Technology [154], consisting ona scaled replica of a power system, in which, all analyses were performed. A key calculatingmachine was used for the first time for solving the power flow problem in the late 40’s [155],but it was not until the mid 50’s when digital computers were first used [156], opening anew path for hundreds of power flow formulations and applications [157].

The idea behind a load flow is conceptually the same as solving a steady-state ACcircuit, i.e., obtaining the voltages at all nodes and currents at all branches. However,there is a key difference in how some input data is managed:

∙ In circuit analysis the values of all impedances are given (including loads), as wellas all active sources in the circuit, thus, all nodal voltages and branch currents canbe calculated linearly by means of Ohm’s laws.

∙ In load flow analysis loads are commonly expressed in terms of their consumedactive and reactive powers (PQ load) due to the action of under-load tap chang-ers in distribution substations. Generators are defined in terms of constant voltage

Chapter 2. General Definitions 37

magnitude and active power injection, due to the automatic voltage regulator andthe automatic generation control. Finally, transmission lines and transformers areconventionally modelled as lumped 𝜋-circuits with constant parameters [158]. Therelationship between voltage, power and impedances is non-linear, hence, appropri-ate methods for solving systems of non-linear equations are usually required [159].

Detailed models for electrical power systems are based on circuit theory followingMaxwell’s equations, thus, involving integro-differential equations. A common represen-tation relates using ordinary differential equations, as

ℎ (𝑥) = �̇� (2.1)

𝑔 (𝑦) = 0. (2.2)

where 𝑥 and 𝑦 represent the dynamic and algebraic variables of the system, respectively.However, some assumptions can be made depending on the time-scale of interest [160].The main assumption for power flow analysis, and in general the one used in this thesis,demands the steadiness of the system, i.e., �̇� = 0, which can be translated as ignoring alldynamic phenomena. Hence, the power flow analysis is based on algebraic equations only,represented by (2.2). The nature of the functions defining 𝑔 can be expressed in differentforms, some of which will be explained as follows.

2.1.1 Node Current Balance

From circuit theory, the net current at node 𝑖, represented by 𝐼𝑁𝑖 , can be expressedas I𝑁𝑖 = 𝐼𝐺𝑖 − 𝐼𝐿𝑖 , where 𝐼𝐺𝑖 stands for the current injected by active sources and 𝐼𝐿𝑖 thecurrent consumed by loads at that node. The voltages at all nodes of the system, 𝑗, arerepresented by V𝑗, and the relationship between the net currents at bus 𝑖 and voltagesare related by the elements of the admittance matrix, as

I𝑁𝑖 −∑︁

𝑗∈ΩBY𝑖,𝑗V𝑗 = 0 ∀𝑖 ∈ ΩB (2.3)

where ΩB stands for the set of nodes of the system. Notice that nodal voltages can bedetermined by solving the linear system in (2.3) if all nodal currents are constant, orlinearly dependent on the voltage. However, as explained in Section 2.1, loads and activecomponents are commonly modeled as power injections, leading to nonlinear relationships.This formulation has been used in this thesis in Chapter 4 as part of the optimal energymanagement of unbalanced microgrids.

Assume the 2-bus system shown in Figure 2.1, composed by a constant voltagegenerator, a 𝜋-modelled transmission line, and a PQ load. Voltage magnitude and an-gle at bus 𝑖 are unknown variables, as well as the nodal current, adding the nonlin-ear relationship to the problem since it is a function of the inverse of the voltage, i.e.,

Chapter 2. General Definitions 38

Figure 2.1 – 2-bus system.

−I𝑁𝑖 = 𝐼𝐿𝑖 =(︁𝑆𝐿𝑖 /V𝑖

)︁*. Several numerical methods, usually inspired by the fixed-point

iteration method [161], can be performed in order to solve the nonlinear system of equa-tions. One of the most common techniques when using the current balance approach isthe Gauss-Seidel (GS) iterative method. The basic idea of the GS method, applied to solv-ing nonlinear electric circuits, is to assume values for voltages and approximate the loadand generation to ideal current sources by converting powers into current injections. Themapping is then carried out using injected complex power until nodal voltages convergeup to a predefined tolerance [158].

On the bright side, the operations performed over 𝑔 in (2.2) are simple in the GSmethod, e.g., its Jacobian must not be computed in contrast to other methods, facilitatingits implementation and speed per iteration. On the down side, even though the GS methodcould be employed for large networks, its convergence is usually slow (linear) and it canbe numerically unstable. As a disclaimer, note that (2.3) can be solved with any othernonlinear root-finder algorithm, not exclusively with the GS method, which has been usedin this section only as example.

Performing approximations is another approach for solving the power flow problemusing the net current representation, as in [162], where the current injections from PQloads are approximated using the first order Taylor expansion around 1∠0 pu, resultingin

I𝑁𝑖 ≈(︁𝑆𝐿𝑖 (2− V𝑖)

)︁*for the above mentioned example. It should be stated that an expression for constantvoltage buses is not available in [162].

Equation (2.3) can be rewritten using the above approximation as follows:(︁𝑆𝐿𝑖 (2− V𝑖)

)︁*−∑︁

𝑗∈ΩBY𝑖,𝑗V𝑗 = 0 ∀𝑖 ∈ ΩB (2.4)

which is linear, guaranteeing the uniqueness of the solution, if it exists. This formulationhas been used in [163], for example, for the optimal reactive compensation and voltagecontrol of distribution power systems.

Chapter 2. General Definitions 39

2.1.2 Branch Current Method

The branch current method uses both Kirchhoff’s laws to find the current in eachbranch of a circuit by generating a system of equations as a function of nodal voltages.Take the two-bus system of Figure 2.1, for which, the impedance connecting buses 𝑖 and𝑗 can be written as Z𝑖,𝑗 = Z𝑗,𝑖 = 𝑟𝑖𝑗 + 𝑗𝑥𝑖𝑗. Then, by applying Kirchhoff’s laws one gets

V𝑗 − V𝑖 − Z𝑗,𝑖I𝑗𝑖 = 0 ∀𝑗𝑖 ∈ Ωl (2.5)∑︁𝑗𝑖∈Ωl

I𝑗𝑖 − I𝑁𝑖 = 0 ∀𝑖 ∈ ΩB (2.6)

where ΩB and Ωl stand for the set of buses and branches of the system, respectively. Noticethat I𝑗𝑖 is a function of the net current at node 𝑖, including the connection with otherbuses both upstream and downstream. If currents are constant or linear with voltage, i.e.,no constant power component is considered, then, nodal voltages can be determined bysolving the linear system in (2.6). Otherwise, the system of equations is nonlinear andmust be solved iteratively, similarly to the case explained in Section 2.1.1. A three-phaseversion of the branch current method has been used in this thesis in Chapter 6 for moderndistribution systems considering uncertainty.

2.1.3 Node Power Balance

Another common approach for defining function 𝑔 in (2.2) is by using the powerbalance at each bus as:

S𝑁𝑖* − V*𝑖

∑︁𝑗∈ΩB

Y𝑖,𝑗V𝑗 = 0 ∀𝑖 ∈ ΩB (2.7)

where S𝑁𝑖 = 𝑆𝐺𝑖 − 𝑆𝐿𝑖 , is the total complex power injection at bus 𝑖. Equation (2.7) isobtained after multiplying (2.3) by V𝑖, which is usually expressed in polar or rectangularcoordinates for transmission systems [164], but it can also be rearranged in terms of powerflows, as in [165], [166], or [167] for distribution networks. A similar formulation as theone proposed in [167] has been used in Chapter 3 and Chapter 5 in this thesis.

One of the most commonly used iterative methods for solving the set of nonlinearequations in (2.7) is the Newton-Raphson (NR) method, whose origins for solving thepower flow problem date back to the late 60’s [168]. Fast decoupled power flow (FDPF)approximations have been used since the 70’s, first proposed as the XB version [169], andlatter as the BX version [170], depending on the considerations used. Both approachesprovide outstanding performance in terms of convergence and computational burden com-pared to the complete NR method. However, it was not until the 90’s that the goodperformance of the approximations were mathematically demonstrated and unified [171].Nonetheless, NR-based methods require the evaluation of the Jacobian matrix and its

Chapter 2. General Definitions 40

Table 2.1 – Comparison of power flow solution methods.

Number of Buses NR GS FDPFIter. Time [s] Iter. Time [s] Iter. Time [s]

14 3 0.008 141 0.053 6 0.006118 4 0.013 1,379 4.017 7 0.010

1,354 4 0.065 - - 7 0.03113,659 10 1.357 - - 11 0.229

inverse, or an approximation of it in the FDPF, at each iteration. They are characterizedby a good rate of convergence (at least quadratic), if the algorithm converges.

For the sake of comparison, the power flow problem was solved for different testsystems using the classic NR method, the GS method, and the XB version of the FDPFmethod. The number of iterations and the total execution time to converge were obtainedusing the Matpower package [172], assuming a tolerance of 1 · 10−6. Although the timeper iteration was faster in the GS method than using the other two methods, the totalcomputational burden is notably better in the FDPF method, followed by the NR methodeven for small-size systems, as can be concluded from Table 2.1. Moreover, the GS methoddiverged when large systems were tested.

2.1.4 Remarks

Power flow problems can be classified into four main categories [173]. Well con-ditioned, when the power flow solution exists and it is reachable using a flat start andstandard solution methods such as the NR. Ill-conditioned, when the power flow solutionexists, but, due to instabilities of the numerical method, standard solution methods failto get this solution. Bifurcation point, when the solution of the power flow exists but itis either a saddle-node bifurcation or a limit-induced bifurcation, leading to instabilities.Finally, power flow problems can be unsolvable, this is, the power flow solution does notexist due to the parameters used for modelling the system, such as the loading condition.

Nevertheless, it must be reminded that the power flow problem is modelled byintrinsically nonlinear, nonconvex equations. Hence, the uniqueness of the solution (as-suming it exists) cannot be guaranteed. In fact, there are several approaches for findingsolutions of ill-conditioned problems, such as [174] using a step-size optimization methodfor the classical NR method, or recently in [175], using implicit integration solvers. More-over, approaches for finding all solutions for the power flow problem have been an activeresearch field for decades [176–180].

For example, assume the test system in Figure 2.1. Let 𝑗 = 1 and 𝑖 = 2 forsimplicity, where 𝑟12 = 0.01 pu, 𝑥12 = 0.1 pu, 𝑆2 = 𝑃2 + 𝑗𝑄2 = 0.9 + 𝑗0.6 pu, and𝑏𝑠ℎ12 = 0 pu. Assume also that the voltage at the generation node is fixed at 𝑉1 = 1+𝑗0 pu.

Chapter 2. General Definitions 41

Using (2.7), the system of equations describing the state of the system can be written as:

𝑉1𝑉*

2 = |𝑉2|2 + (𝑟12 + 𝑗𝑥12) (𝑆2)* (2.8)

or equivalently:

|𝑉2|4+|𝑉2|2[︁2 (𝑟12𝑃2+𝑥12𝑄2)−|𝑉1|2

]︁+[︁(𝑟12𝑄2−𝑟12𝑃2)2+(𝑟12𝑃2−𝑥12𝑄2)2

]︁= 0 (2.9)

𝜃2 = sin−1 ((𝑟12𝑄2 − 𝑥12𝑃2) /|𝑉1||𝑉2|) (2.10)

Equation (2.9) is biquadratic, hence, it has four solutions. However, two of thesesolutions can be disregarded since they involve negative values for the voltage magnitude,which have no physical meaning. After eliminating the negative solutions, the followingis obtained

|𝑉2| = {0.9209, 0.1180}.

From (2.10), after replacing each solution of |𝑉2| one gets

𝜃2 = {−5.2335∘, −45.3858∘}.

In this case, the solution of interest is easily identified as (𝑉2 = 0.9209∠−5.2335∘).However, obtaining analytic solutions and identifying meaningful results is not always thisstraightforward. As explained before, numerical methods are used for solving the systemof equations.

Take as example the analytic solution for the 2-bus system in (2.8), for which it isknown there are only two positive roots. The solution found and the convergence of thenumerical method depends on the initial value assumed for all variables when solving theproblem. This is a common characteristic of fixed point iteration methods like NR. Theeffect of the initial point on the convergence of the NR method (number of iterations),and the operating point found can be seen in Figure 2.2.

The results obtained were found after solving the power flow problem using 4,000different combinations of initial points with the NR method. As stated before, it is easyto identify the meaningful solution from a physical point of view for this particularsystem. Moreover, note that the attraction region in which the solution converges to|𝑉2| = 0.9209 pu is well defined in yellow in Figure 2.2, while the solution converging to|𝑉2| = 0.1180 pu appears in blue, and divergent initial points in purple. It is importantto note that the region of attraction depends on the parameters of the system, such asthe loading condition, and also on the solution method. Thus, different algorithms arecharacterized by different regions of attraction [160,181]. In practice, however, it is virtu-ally impossible to define the region of attraction for a real-world system, considering thenumber of combinations of initial points for all variables.

Chapter 2. General Definitions 42

(a) Number of iterations to converge. (b) Voltage magnitude at operating point.

Figure 2.2 – Number of iterations to converge and voltage magnitude at operating pointusing the NR algorithm with