Faculdade de Engenharia Elétrica e de Computação
Juan Sebastian Giraldo Chavarriaga
Mathematical Programming Models for the Optimal Energy
Management of Modern Electric Distribution Systems
Considering Uncertainty
Modelos Matemáticos para o Gerenciamento Ótimo de
Energia dos Sistemas Modernos de Distribuição Considerando
Incerteza
Campinas, SP
2019
Faculdade de Engenharia Elétrica e de Computação
Juan Sebastian Giraldo Chavarriaga
Mathematical Programming Models for the Optimal
Energy Management of Modern Electric Distribution
Systems Considering Uncertainty
Modelos Matemáticos para o Gerenciamento Ótimo de
Energia dos Sistemas Modernos de Distribuição
Considerando Incerteza
Thesis presented to the School of Electrical and Computer Engineering of the University of Campinas in partial fulfillment of the requirements for the degree of Doctor in Electrical Engineering, in the area of Elec-tric Energy.
Tese apresentada à Faculdade de Engenharia Elétrica e de Computação da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Doutor em Engenharia Elétrica, na Área de Energia Elétrica.
Supervisor: Prof. Dr. Carlos Alberto de Castro Junior
Este exemplar corresponde à versão final da tese defendida pelo aluno Juan Sebastian Giraldo Chavar-riaga, e orientada pelo Prof. Dr. Carlos Alberto de Castro Junior
Campinas, SP
2019
Biblioteca da Área de Engenharia e Arquitetura Luciana Pietrosanto Milla - CRB 8/8129
Giraldo Chavarriaga, Juan Sebastian,
G441m GirMathematical programming models for the optimal energy management of modern electric distribution systems considering uncertainty / Juan Sebastian Giraldo 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. Universidade Estadual 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 ótimo
da energia em sistemas modernos de distribuição considerando incertezas
Palavras-chave em inglês:
Robust optimization
Distributed generation of electrical energy Stochastic programming
Área de concentração: Energia Elétrica Titulação: Doutor em Engenharia Elétrica Banca examinadora:
Carlos Alberto de Castro [Orientador] Edimar José de Oliveira
Eduardo Nobuhiro Asada
Fernanda Caseño Trindade Arioli Luiz Carlos Pereira da Silva
Data de defesa: 12-09-2019
Programa 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
Candidato: Juan Sebastian Giraldo Chavarriaga RA: 143684 Data da Defesa: 12 de setembro de 2019
Título da Tese: Mathematical Programming Models for the Optimal Energy Management of Modern Electric Distribution Systems Considering Uncertainty
Prof. Dr. Carlos Alberto de Castro Junior Prof. Dr. Edimar José de Oliveira
Prof. Dr. Eduardo Nobuhiro Asada
Profa. Dra. Fernanda Caseño Trindade Arioli Prof. 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 de Pós-Graduação da Faculdade de Engenharia Elétrica e de Computação.
I would like to express my special appreciation and thanks to my advisor Professor Dr. Carlos Alberto Castro who has been a tremendous mentor for me. Thank you for encouraging my research and for allowing me to grow as a research scientist. I would also like to extend my deepest gratitude to Professor Dr. Marcos Julio Rider for his valuable insights and advises.
My most sincere thanks to my colleges, friends, and professors of the Department of Systems 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 his research team at the School of Electrical and Electronic Engineering, University College Dublin. 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 I have received from different Brazilian public agencies, without which, this research and more importantly, this personal achievement would not have been possible. I will always be grateful.
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001
of teachers... ”
The modernization of traditional distribution power systems is an unavoidable transition into a more efficient, reliable, and environmentally friendly electric power system. Current trends in research indicate that the insertion of distributed energy resources and energy storage systems into modern distribution systems could be facilitated by creating flexible microgrids, in which, the energy management system is a fundamental piece. Available approaches are mainly focused on balanced equivalents, which is a debatable assumption for medium-voltage and low-voltage networks. Moreover, the insertion of renewable energy sources 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 different mathematical 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 considering uncertainty. Each model has been validated using benchmark test systems depending on their specific characteristics.
Keywords: Optimal energy management; distributed energy resources; microgrids; ro-bust optimization; chance-constraints.
A modernização dos sistemas tradicionais de distribuição de energia é uma transição ine-vitável para um sistema de energia elétrica mais eficiente, confiável e ambientalmente amigável. As tendências atuais de pesquisa indicam que a inserção de recursos energé-ticos distribuídos e sistemas de armazenamento de energia em sistemas de distribuição modernos poderia ser facilitada pela criação de microrredes flexíveis, nas quais, o sis-tema de gerenciamento de energia é uma peça fundamental. As abordagens disponíveis têm sido principalmente focadas em equivalentes balanceados, o que é uma suposição discutível 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 parâmetros exógenos intrinsecamente estocásticos, criando a necessidade de metodologias adequadas para lidar com a incerteza. Considerando o exposto, esta tese propõe quatro modelos de programação matemática diferentes para o gerenciamento ótimo de energia de sistemas modernos de distribuição. Os modelos propostos introduzem abordagens equilibradas e desequilibradas; microrredes operando conectadas à rede ou ilhadas; e duas formulações considerando incerteza. Cada modelo foi validado usando sistemas de teste dependendo das suas características específicas.
Palavras-chave: Gerenciamento ótimo de energia; recursos energéticos distribuidos; mi-crorredes; otimização robusta; restrições probabilísticas.
Figure 1.1 – Traditional power system representation. Source: Author . . . 20
Figure 1.2 – Microgrid representation and commands from the microgrid operator. Source: Author . . . 22
Figure 1.3 – Number of publications and citations using “Microgrid” in the title + “EMS” as topic. . . 25
Figure 1.4 – “Microgrid” in title + “EMS” + “Unbalanced” as topic. . . 26
Figure 1.5 – “Microgrid” in title + “EMS” + “Uncertainty” as topic. . . 27
Figure 1.6 – Number of references used in the thesis per year and percentage of papers by publisher . . . 32
Figure 1.7 – Structure of thesis . . . 34
Figure 2.1 – 2-bus system. . . 38
Figure 2.2 – Number of iterations to converge and voltage magnitude at operating point using the NR algorithm with different initial values. . . 42
Figure 2.3 – Surface and contour of the constraints for the 2-bus system. . . 45
Figure 2.4 – Quadratic cone and rotated quadratic cone. . . 48
Figure 2.5 – Reduced decision tree - Feasible path (continuous), infeasible path (dashed). . . 50
Figure 2.6 – Cumulative density function of random variable 𝑋. . . . 52
Figure 2.7 – Monte Carlo simulation flow chart. . . 53
Figure 2.8 – Representation of the Point Estimate Method - Adapted from [225] . . 54
Figure 2.9 – Evolution of the average value of the voltage angle. . . 57
Figure 3.1 – Three-bus example distribution network. . . 67
Figure 3.2 – Modified 34-bus test system. Adapted from [258] . . . 71
Figure 3.3 – Solar irradiance and wind speed . . . 72
Figure 3.4 – Active and reactive power injected by dispatchable distributed genera-tors - Unlimited fuel 5 periods. . . 72
Figure 3.5 – Load shedding percentage - Unlimited fuel 24 periods. . . 73
Figure 3.6 – Injected power from Energy Storage Systems and State of Charge -Unlimited fuel 24 periods. . . 74
Figure 3.7 – Load shedding percentage - Limited fuel 5 periods. . . 74
Figure 3.8 – Injected power from energy storage systems and State of Charge - Lim-ited fuel 5 periods. . . 74
Figure 3.9 – Injected power from distributed generation units and fuel reserve - Lim-ited fuel 24 periods. . . 75
Figure 3.10–Nominal and total served energy for all tested scenarios. . . 75
Figure 4.3 – Dispatchable distributed generator Model 2. . . 84 Figure 4.4 – Dispatchable distributed generator, Model 3. . . 85 Figure 4.5 – Energy storage system model. . . 85 Figure 4.6 – Operational limits for dispatchable distributed generation units and
energy storage systems. . . 87 Figure 4.7 – Case study microgrid based on the modified IEEE 123 node test feeder. 88 Figure 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) Model 3. . . 90 Figure 4.11–State of charge and three-phase active power injected by the energy
storage system at node 47 considering different synchronous machine models for distributed generation units. . . 91 Figure 4.12–Total energy injected by sources and consumed by loads - Model 1. . . 91 Figure 5.1 – Meaning of the robustness adjustment parameter 𝜁 for loads (a) and
renewable energy sources (b). . . 102 Figure 5.2 – Illustrative example for choosing 𝜁, based on ϒ = 0.6 and different
types of cumulative density functions. . . 104 Figure 5.3 – Representation of the models used for energy storage systems and
dis-tributed generation units. . . 106 Figure 5.4 – Limited power variation from a specified set-point for the Monte Carlo
simulations. . . 108 Figure 5.5 – Modified microgrid based on the 136-node system. . . 110 Figure 5.6 – Energy balance for grid-connected mode, with 𝜁 = 0.10 for an arbitrary
scenario of the Monte Carlo simulations. . . 111 Figure 5.7 – Robustness assessment using Monte Carlo simulations for 𝜁 ∈ [−0.15, 0.30].
Grid-connected mode. . . 112 Figure 5.8 – Grid-connected mode: Global robustness, objective function, and
en-ergy losses for 𝜁 ∈ [−0.15, 0.30] and the sample average approximation model. . . 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
of 𝜁 and the sample average approximation model. . . . 114
Figure 5.12–Histograms of the objective function and robustness for different values of 𝜁. Islanded operation. . . 115
Figure 5.13–Load shedding at each node for 𝜁 = 0.10. Isolated mode. . . 115
Figure 6.1 – Uncertainty sets for a normal probability distribution function. . . 126
Figure 6.2 – Convolution of different PDFs - Central limit theorem. . . 127
Figure 6.3 – Representation of the prediction intervals for current and voltage mag-nitudes. . . 127
Figure 6.4 – Flowchart of the proposed accuracy and robustness assessment. . . 129
Figure 6.5 – IEEE 13-bus test feeder diagram. . . 131
Figure 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 of the system with 𝛼 = 0. . . 133
Figure 6.8 – Cumulative density functions for voltage magnitude at node 675: . . . . 135
Figure 6.9 – Average value of the objective function and percentage of feasible sce-narios for different values of . . . 136
Figure 6.10–IEEE-123 bus test feeder diagram. . . 137
Figure 6.11–Cumulative density function of the current magnitude at line 150–149, and for voltage magnitude at node 114 considering different number of random variables. . . 138
Figure 6.12–Mean squared error of higher order moments and percentage of output variables of interest with 𝑃 > 0.05. a) Current magnitudes. b) Voltage magnitudes. . . 139
Figure 6.13–Sample cumulative density functions of the current magnitude at line 150-149 and the voltage magnitude at node 114, considering different forecast errors of the standard deviations. . . 140
Figure 6.14–Sample cumulative density functions of the current magnitude at line 150-149 and the voltage magnitude at node 114, considering different forecast errors of the expected values. . . 141
Table 2.1 – Comparison of power flow solution methods. . . 40
Table 2.2 – Parameters data - example unit commitment. . . 49
Table 2.3 – Possible commitment - example unit commitment. . . 49
Table 2.4 – Optimal solution - example unit commitment. . . 50
Table 3.1 – Objective function cost for tested scenarios . . . 76
Table 4.1 – Parameters of energy storage systems and distributed generation units . 88 Table 4.2 – Performance of the tested simulations . . . 92
Table 5.1 – Values and Units of Parameters . . . 111
Table 5.2 – Execution Times of Each Test Case in Seconds. . . 115
Table 6.1 – Maximum Relative Errors for the Expected Values and Standard Devi-ations of Voltage and Current Magnitudes in the IEEE 13-bus System. . 132
Table 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 Voltage and Current Magnitudes . . . 134
Table 6.4 – Percentages of scenarios where voltage limits, current limits, and both were guaranteed – IEEE 13-bus system. . . 135
Table 6.5 – Percentages of scenarios where voltage limits, current limits, and both were guaranteed – IEEE 123-bus system . . . 137
Table 6.6 – Groups considering different number of random variables . . . 138
Table 6.7 – Overview of the System’s Normality and Performance Under Different Number of Random Variables . . . 139
Table 6.8 – Percentage of Scenarios in Which Robust Constraints were Guaranteed Considering Forecast Errors . . . 141
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.
SAA sample average approximation. SOC state of charge.
1 Introduction . . . 19 1.1 State of Art . . . 24 1.2 Motivation . . . 31 1.3 Objectives . . . 32 1.4 Contributions . . . 33 1.5 Outline . . . 33 2 General Definitions . . . 36
2.1 Power Flow Analysis . . . 36
2.1.1 Node Current Balance . . . 37
2.1.2 Branch Current Method . . . 39
2.1.3 Node Power Balance . . . 39
2.1.4 Remarks . . . 40
2.2 Basic Concepts on Optimization . . . 42
2.2.1 Feasibility . . . 42
2.2.2 Local and Global Optimality . . . 43
2.2.2.1 Example - Power Flow . . . 45
2.2.3 Mixed-Integer Programming - MIP . . . 46
2.2.3.1 Mixed-Integer Second-Order Cone Programming (MISOCP) 47 2.2.3.2 Example - Unit Commitment . . . 48
2.3 Basic Techniques for Considering Uncertainty . . . 50
2.3.1 Monte Carlo Simulation - MCS . . . 51
2.3.2 Point Estimate Method - PEM . . . 53
2.3.3 Example - Probabilistic DC Power Flow . . . 56
2.4 Uncertainty in Optimization Problems . . . 57
2.4.1 Stochastic Optimization . . . 58
2.4.2 Robust Optimization . . . 59
2.4.3 Chance-Constrained Optimization . . . 60
Deterministic Approaches . . . 62
3 Optimal Energy Management of Isolated Microgrids . . . 63
3.1 Introduction . . . 65
3.1.1 Motivation . . . 65
3.1.2 Literature Review . . . 66
3.1.3 Contribution . . . 66
3.2 Mathematical Model . . . 66
3.3 Test System and Results . . . 71
3.4 Chapter Conclusions . . . 76
4 Optimal Energy Management of Unbalanced Grid-Connected Microgrids . 77 4.1 Introduction . . . 80
4.1.1 Motivation . . . 80
4.1.2 Literature Review . . . 80
4.1.3 Contributions . . . 81
4.2 Proposed Formulation . . . 81
4.2.1 Loads and Renewable Energy Sources . . . 83
4.2.2 Dispatchable Distributed Generators . . . 83
4.2.3 Energy Storage Systems . . . 85
4.2.4 Operational Limits . . . 86
4.3 Case Study . . . 87
4.4 Chapter Conclusions . . . 92
Models Considering Uncertainty . . . 93
5 Microgrids Energy Management Using a Robust Approach . . . 94
5.1 Introduction . . . 98
5.1.1 Motivation . . . 98
5.1.2 Literature Review . . . 99
5.1.3 Contributions . . . 100
5.2 Characterization of Robust Parameters . . . 100
5.2.1 Robust Convex Optimization . . . 101
5.2.2 Robust Equivalents for RES . . . 102
5.2.3 Robust Equivalents for Demands . . . 103
5.2.4 Robustness Adjustment Parameter 𝜁 . . . 103
5.3 Proposed Robust Micogrids EMS . . . 104
5.3.1 Objective Function . . . 104
5.3.2 Microgrid’s Steady-state Operation . . . 105
5.3.3 ESS Operation . . . 106
5.3.4 Proposed MISOCP Model . . . 107
5.4 Monte Carlo Simulation and Robustness Assessment . . . 107
5.5 Tests and Results . . . 109
5.5.1 Grid-connected Mode . . . 110
5.5.2 Isolated Mode . . . 112
5.6 Chapter Conclusions . . . 115
6 Probabilistic Optimal Power Flow for Unbalanced Distribution Systems . . 117
6.1 Introduction . . . 120
6.1.1 Motivation . . . 120
6.2 Probabilistic Optimal Power Flow Model for Unbalanced Three-phase
Dis-tribution Systems . . . 123
6.2.1 Probabilistic Optimal Power Flow as a Two-stage Problem . . . 123
6.2.2 Moments of the State RVs and Robust Constraints . . . 124
6.3 Selection of the Robustness Parameter . . . 126
6.4 Accuracy and Robustness Assessment . . . 128
6.4.1 Sample Average Approximation - SAA . . . 129
6.5 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 . . . 132
6.5.3 IEEE 13-bus System – Different Probability Limits . . . 133
6.5.4 IEEE 123-bus System – Different Robustness Parameters . . . 136
6.5.5 IEEE 123-bus System – Different Number of Random Variables . . 137
6.5.6 IEEE 123-bus System – Out-of-sample Monte Carlo Simulations . . 140
6.6 Chapter Conclusions . . . 141
7 Conclusions and Future work . . . 143
7.1 General Conclusions . . . 143
7.2 Future Work . . . 144
Bibliography . . . 146
1 Introduction
E
lectric power systems have played a fundamental roll in the development of the modern society and its technological advances. Power systems have been subjected to substantial technological changes from their first traceable beginnings in the 1880s where DC 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 experiencing big changes, this time, towards the modernization of the grid and the diversification of the energy matrix. These changes have been motivated mostly by economical reasons, but also strengthened by an arising environmental awareness over the last few decades [2].The technological transition is already evident in many networks around the world and 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 by meteorological phenomena like El Niño, economic factors, and the creation of regula-tory incentives [5]. Although not entirely accepted by the power system community, the product of this modernization has been popularized in recent years with the term smart
grids [6], creating a global expectation. A smart grid is a wide concept however, going
from improvements in the infrastructure of the system, such as the use of smart switches and 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), and plug-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 end users at low-voltage (LV) levels, commonly lower than 1.0 kV. Distribution systems are traditionally conformed by medium-voltage (MV) feeders working at voltage levels lower than 69 kV, supplied by a single energy source identified as the substation, feeding a set of electric loads connected mostly through a radial network [8]. This definition, although accurate for most existing distribution systems, relies on the assumption that the network is intrinsically passive, hence, the power flow direction is always unidirectional as shown in 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,
Figure 1.1 – Traditional power system representation. Source: Author
sociopolitical and economic motivations have driven the increasing insertion of DERs into 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 optimal operation. 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 storstor-age, etc., since ESS can improve power system’s operation [9].
The increasing participation of DER at the distribution level plays a key role in the future of power systems, considering that system operators still face challenges for their integration into existing networks. The insertion of additional energy sources into existing 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 possible inversion, 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 future
smart grids [6]. A microgrid is defined as a cluster of DERs, ESS, and loads, that can be
or not connected to a main grid through a point of common coupling (PCC), providing reliable 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 the microgrid has a connection to the main power grid, all power deficit or surplus can be
absorbed or delivered to the power grid. On the other hand, the balance must be satisfied locally 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, not only 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 to the 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 North America 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) in the 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 by Eletrobras, 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 (Portuguese
for 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$3.94 million up to 2003 [27]. In fact, according to [21], isolated microgrids represented nearly 40% (7.6 GW) of the total global installed capacity, followed by commercial/industrial microgrids with 5.5 GW, and utility distribution with 2.3 GW. Investments in this kind of microgrids are estimated to grow at a rate of 17% p.a. by 2020 in Brazil, increasing from US$149 million in 2012 to US$518 million in 2020 [28]. Some small grid-connected microgrids can be also found in Paraná [29] and Ceará [30], as well as some test beds currently working, such as the one from the Laboratório de Microrredes Inteligentes at the Federal University of Santa Catarina (UFSC) with 20 kWp of PV generation, 11 kW of 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 kWh of energy storage [32].
One major advantage of microgrids is their flexibility to operate in grid-connected or 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 during a period in which the power quality may be deteriorated, e.g., voltage fluctuations,
har-Figure 1.2 – Microgrid representation and commands from the microgrid operator. Source: Author
monic distortions, frequency deviations and voltage flickers, or after network contingencies producing a disruption of the main energy supply [33]. In this context, a microgrid needs to have a centralized control defining, among other matters, the status of the PCC and satisfying the power balance. This centralized control is usually composed by an energy management system (EMS) either if the microgrid has a connection to a main grid or if it is isolated.
An EMS is comprised by a set of hardware/software components used to efficiently operate the energy resources within the microgrid to achieve selected objectives. These ob-jectives are accomplished by typically, but not exclusively, using optimization techniques for 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 microgrid and the main grid [35], in the case they are connected, or the local balance in isolated microgrids [25]. Without loss of generality, this problem can be interchangeably referred to as optimal energy management (OEM). A schematic representation of a microgrid is shown in Figure 1.2, where the control commands from the microgrid operator are also displayed.
Uncertainties on system’s operational conditions increase the complexity of the EMS 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-tion in convenconsump-tional loads. From an operaconsump-tional point of view, a secure operaconsump-tion of the system 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 when optimizing 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 uncertainties in 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 this issue, mainly because of its solid mathematical foundations and theoretical richness on probability and stochastic processes [37]. In a general approach, it consists on solving a single mathematical model comprising a set of plausible scenarios and finding the optimal decision that minimizes the average objective function while being feasible for all scenarios [38]. Still, the main drawback of currently proposed stochastic-based methods is the number of required scenarios to obtain satisfactory results, which is highly dependent on the number of random variables (RVs) even if scenario reduction techniques are applied [39]. Furthermore, reducing the number of scenarios can lead to inaccurate solutions, thus, the final results are a function of the quantity and the quality of the selected scenarios.
∙ Robust optimization and chance-constrained optimization are also suitable tools for considering the volatility on random variables within optimization processes. Com-pared to stochastic programming, these approaches are less challenging to be solved because they do not require to operate over multiple probable scenarios. In this mat-ter, robust approaches create a deterministic equivalent of an essentially stochastic optimization problem, rather than a discrete representation of the probability den-sity functions (PDFs) of the random phenomena [40]. Robust problems tend to be too 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 behavior of the output variables, however, chance-constrained models are more flexible than robust optimization approaches, since the hardness of some constraints can be con-trolled. As in stochastic programming approaches, the stochastic moments of the output (or state) RVs are not directly assessed in the model or available for later analysis using robust or chance-constrained optimization.
∙ The POPF is a classical problem where the optimal operation point of a network is obtained while considering the uncertainties and the statistical behavior of the state variables [41]. Most formulations for the POPF are based on solving a set of
deterministic optimal power flows to obtain the stochastic behavior of the output RVs 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 independent deterministic optimal power flows. This means that output variables are taken as stochastic outputs rather than as single decisions, as in [42]; therefore, the feasibility under uncertainty cannot be guaranteed.
This thesis proposes different approaches for the EMS of modern distribution systems, 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-robust constraints. Convex and non-convex formulations are proposed, such as mixed-integer lin-ear programming (MILP), mixed-integer second-order cone programming (MISOCP), and nonlinear programming (NLP). The addressed models are implemented using a commer-cial mathematical programming language and solved with commercommer-cial numerical solvers.
1.1
State of Art
The 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 number of publications and the number of citations per year involving the word “Microgrid” in the title plus the “EMS” topic have been condensed. The data was collected from the Web of Science principal collection and filtered for electrical and electronic engineering [43], with a total of 1,510 results and 19,512 citations to the date of consultation. It can be seen that the number of publications has increased exponentially since its first appearing in 2,005, with a small reduction in 2,018, while the number of citations has been increasing monotonically every year.
Several approaches have been proposed for solving the EMS problem of modern distribution systems, including microgrids. In [44], a centralized EMS is proposed based on forecasted values of loads and non-dispatchable generation units. In [45], a multi-objective single-step formulation dispatch of DG and ESS is proposed using a niching evolutionary algorithm. A load management model is proposed in [46] to improve microgrid resilience following islanding. Authors in [47] propose an algorithm to minimize the microgrid total operation cost by decomposing the problem into a grid-connected operation master prob-lem and an islanded operation subprobprob-lem. In [48], the dispatch for emergency electric service restoration in the aftermath of a natural disaster in a microgrid is proposed using a mixed-integer nonlinear programming (MINLP) model. In [49], the authors use an MILP approach for an online optimal energy/power control method for the operation of energy
Year 2005 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Number of Publications 0 50 100 150 200 250 300 350 Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Number of Citations 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 of microsources. An MILP model for the modeling and experimental verification of an EMS was 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 remote
islanded microgrids in [54], minimizing the capital and operational costs while ensuring technical feasibility. Analyses of multi-microgrids are also available in the literature, such as in [55] where a control strategy for coordinated operation of networked microgrids in distribution systems is proposed, and in [56] where a multiple-objective constrained optimization is presented for solving the microgrids/nanogrids EMS.
The aforementioned models are focused entirely on single-phase equivalents, and most of them consider active power balance only.
Unlike HV networks, distribution systems and microgrids, which are characterized by 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-phase laterals, 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 its devices, thus, increasing the size and complexity of the problem [57]. Furthermore, from an operational point of view, a secure operation of the distribution network requires that voltage magnitudes and current ratings of conductors remain within their limits; thus, an AC approach is also necessary.
Unbalanced microgrids have also been the subject of study in the last years, as shown in Figure 1.4. Again taking into account the number of publications and citations
Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Number of Citations 0 100 200 300 400 500 600 700 Year 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Number of Publications 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 can be seen that since 2,015, the number of publications has been more or less constant with more than 30 publications per year; and the citations have been increasing year after year. In [58], an EMS for phase balancing in distribution systems is proposed based on the automated mapping/facilities management/geographic information system. An EMS for isolated, three-phase microgrids considering unbalanced synchronous machines is proposed in [59]. However, this approach does not consider reactive power balance to obtain the optimal solution, while in [60], an approximation is performed in order to obtain a linear model and to neglect the network topology. In [61], authors propose an algorithm for the capacity constrained management of DERs in unbalanced distribution networks. Authors in [62] propose a centralized dispatch for a set of non-synchronous microgrids pursuing loss reduction and unbalance compensation. Finally, reference [63] proposes a MILP model for unbalanced microgrids considering unexpected main grid outages.
The EMS can be sometimes merged or modeled using the optimal power flow problem including DERs. A methodology for unbalanced three-phase optimal power flow for 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 flow in smart grids. An economic dispatch problem for unbalanced, three-phase power distri-bution networks with distributed generation units is proposed in [66] using a semidefinite relaxation 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 power flow method for distribution systems. Authors in [69] model a three-phase optimal power
Year 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Number of Publications 0 10 20 30 40 50 Year 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Number of Citations 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 applyschedul-ing a multi-objective approach considering cost minimization, power quality improvements, and energy savings.
Note that the works mentioned above consider deterministic models; hence, the intrinsic 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 this chapter, a research was conducted using the Web of Science principal collection [43], now including the topic “Uncertainty”. The number of publications and citations accumulated in 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 mathematical programming models [74]. Some of the techniques applying mathematical programming considering 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 findcompris-ing the optimal decision that minimizes the average objective function while being feasible for all scenarios [38]. It is usually implemented in power system applications by means of two-stage models, as in [75] considering risk levels in distribution systems, in [76, 77] for unit commitment problems, in [78] for the Volt/Var control in distribution networks, in [79] for stochastic optimal power flow, and for optimal scheduling considering wind farms in [80]. Stochastic programming has also been applied
for solving the EMS problem, as in [81], for example, where authors present a stochastic EMS for isolated microgrids. In [82], a stochastic framework is proposed to investigate the effect of uncertainty on the optimal operation management of microgrids by using a 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 in grid-connected operation mode. Similarly, authors in [84] use a two-stage stochastic pro-gramming model for the nonconvex economic dispatch problem using a sample average approximation (SAA) formulation and introduce two outer approximation algorithms to obtain global solutions. Reference [85] proposes a multi-objective stochastic programming model 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 is proposed in [87] considering hydrogen storage, and in [88], the authors propose a stochas-tic optimization framework to address the microgrid energy dispatching problem with random RES and vehicle activity pattern using a Markov decision process. However, stochastic programming problems are computationally demanding, since the number of scenarios that must to be considered is usually high, even if scenario reduction techniques are applied [39]. Furthermore, reducing the number of scenarios can lead to inaccurate solutions. Thus, the final results are highly dependent on the quantity and the quality of the selected scenarios. This has been recognized as a crucial drawback of stochastic optimization.
Robust optimization is another suitable tool for considering volatility on random variables within the optimization process. Compared to stochastic programming, robust optimization problems are less challenging to be solved since they do not require to operate over multiple probable scenarios. In this matter, robust approaches create deterministic equivalents 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 RES considering the worst-case transaction cost. In [90], authors propose a framework for a microgrid EMS based on agent-based modelling, by introducing robust optimization and neural networks. A cost minimization problem is formulated in [91] using chance constraint approximations and robust optimization algorithms to schedule the energy generations for microgrids equipped with uncertain renewable sources and combined heat and power generators. 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 control model for wind production. A robust two-stage model was proposed in [94] using a MILP formulation. 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
microgrids based on robust multi-objective optimization considering demand response. Finally, A robust model for islanding-aware microgrids EMS with RES and co-generation is formulated in [97].
Chance-constrained optimization deals with problems were some selected output RVs, which are functions of the input RVs of the problem with known PDF, maximize the objective function while being subject to constraints on these variables. These constraints must be maintained at prescribed levels of probability [98, 99]. In other words, this means that chance-constrained problems “admit random data variations and permits constraint
violations up to specified probability limits” [100]. Several variants have been proposed,
like mathematical properties of chance-constrained programming problems concerned on joint probabilities [101], or reference [102], where a general class of convex approxima-tions to chance-constrained problems are proposed. The first use of chance-constrained optimization in power systems can be traced down to [103] where a generation invest-ment planning model has been proposed to include transmission reliability constraints as chance-constraints. Later, in [104], chance-constraints were used for solving the volt-ampere reactive compensation problem under uncertain operating conditions. Authors in [105] formulate the unit commitment problem as a chance-constrained optimization problem requiring power balance to be met with a specified probability over the entire time horizon. Reference [106] proposes a model for the optimal transmission system ex-pansion planning based on chance-constrained programming, considering the locations and 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 a chance-constrained two-stage stochastic program using the SAA method. The use of a robust-convex, chance-constrained formulation for the optimal power flow problem was proposed in [108]. Reference [109] formulates a robust chance constrained optimal power flow that accounts for uncertainty in the parameters of the probability distributions of the RVs, by allowing them to be within an uncertainty set. A data-driven approach is utilized in [110] to develop a distributionally robust conservative convex approximation of the chance-constraints to enforce voltage regulation with predetermined probability via Chebyshev-based bounds.
Authors in [111] introduce a chance-constrained programming model for the EMS of grid-connected microgrids, while in [112] authors proposed a chance-constrained EMS for islanded microgrids. Authors in [113] proposed a chance-constrained AC optimal power flow 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-constrained AC optimal power flow, where probabilistic constraints are utilized to enforce voltage regulation with prescribed probability on distribution systems featuring RES and ESS. Reference [116] proposes a joint chance-constraint relaxation method to solve the AC
optimal power flow problem while preventing over voltages in distribution grids under high penetrations of PV systems. Reference [117] proposes an MILP model for microgrid operation considering the probability that a microgrid maintains enough spinning reserve to meet local power balance after instantaneously islanding from the main grid. Authors in [118] present a chance-constrained information gap decision model for multi-period microgrid expansion planning for the optimal sizing, type selection, and installation time of DERs in microgrids. A chance-constrained two-stage stochastic programming model is proposed in [119] to evaluate the impacts of variability in renewable resources in the microgrid operation. In [120], a semidefinite relaxation of a chance-constrained AC op-timal power flow is proposed, which is claimed to provide global opop-timality. Authors in [121] propose a data driven distributionally robust chance constrained optimal power flow model, which ensures that the worst-case probability of violating both the upper and 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 optimal power flow problems that does not assume a specific distribution, e.g. Gaussian, for the uncertainties.
Last but not less important, uncertainties can be considered using the POPF. The POPF is a classical problem where the optimal operation point of a network is obtained while 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 was originally proposed in the 50’s [123–127] for minimizing power losses and obtaining the optimal dispatch of generators. The optimal power flow problem has been analyzed in three main subjects in [128], namely deterministic, risk-based, and considering uncer-tainty. Several formulations for the POPF are based on solving a set of deterministic optimal power flows to obtain the stochastic behavior of the output RVs at the end of the algorithm, like in [129], where the quasi-Monte Carlo method is used to solve the POPF problem. A number of approximate methods exist in the literature for estimating the statistical moments of a random function. The point estimate method (PEM), for example, 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 POPF was proposed using different schemes. Reference [133] presents a POPF that takes into account load variation, wind’s stochastic behavior and variable line’s thermal rating using the PEM. Correlations between RVs have been also taken into account, as in [134], where it 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 correlations of wind speeds following arbitrary PDFs by transforming them into independent Gaussian distributions.
A review of optimal power flow approaches mainly related to smart distribution grids is proposed in [136]. In [137], authors use a linear optimal power flow considering
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 system with 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 for solving stochastic, multi-period optimal power flow problems using a family of stochastic network and device constraints based on convex approximations of chance constraints. An stochastic framework to investigate the impact of correlated wind generators on the EMS of microgrids is proposed in [142]. Reference [143] proposes a dynamic optimal power flow in active distribution networks using affine arithmetic and interval Taylor expansion using successive linear approximation as solving method. Reference [144] proposes an interval optimal power flow method employing affine arithmetic and interval Taylor expansion in distribution systems. In [145], it is proposed a multi-period AC optimal power flow for distribution systems, considering wind and PV uncertainty. However, this kind of algo-rithms does not guarantee a single optimal decision for dispatchable DGs, since they are intended for solving a set of independent deterministic optimal power flows. Therefore, the feasibility under uncertainty cannot be guaranteed when using this approach. It is traditional that POPF approaches are only focused on balanced problems, leaving an im-portant gap for the analysis of three-phase unbalanced distribution systems. Three-phase optimal 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 has yet to be proposed.
The power injection from DG units in unbalanced systems has been commonly modeled using balanced power sources, as in [146, 148, 149]; as single-phase machines in [150], or considering each phase’s power as controllable variables [63, 151, 152], just to mention some. These models do not consider the physical coupling of power flows between phases for calculating the power injected to the system by synchronous machines. An energy management system for isolated, three-phase microgrids considering unbalanced synchronous 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 is performed in order to obtain a linear model and to neglect the network topology.
1.2
Motivation
The 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
53% 17% 7% 23% IEEE Elsevier Springer Others 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 0 5 10 15 20 25 30
Figure 1.6 – Number of references used in the thesis per year and percentage of papers by publisher
operation of modern distribution systems were identified after a comprehensive literature research, 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 sorted depending on its year of publication, along with the percentage of references according to the 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 is mandatory. Available approaches for the EMS of modern distribution systems and micro-grids are mainly focused on balanced equivalents, which is an arguable approximation for medium-voltage and low-voltage networks. Moreover, the insertion of RES and the natural behavior 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 in grid-connected and isolated modes.
∙ To develop mathematical programming models able to consider the uncertainty of some exogenous parameters for the short-term operation planning of modern distribution systems, including microgrids operating in grid-connected and isolated mode.
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 formulation relies on the consideration of fuel limits and demand side management control in a microgrid 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 introduces a formulation for DG based on operating in unbalanced networks. It shows a new study aimed at analyzing the effects on the ESS patterns and operational costs af-ter using different common representations for synchronous machines in unbalanced operation.
∙ An energy management system for balanced microgrids operating in grid-connected and isolated mode, using robust convex mixed-integer programming. The novelty of this model is the introduction of a new robust convex formulation for the microgrid EMS operating in grid-connected and islanded modes.
∙ A probabilistic optimal power flow for unbalanced three-phase modern electrical distribution systems considering probabilistically-robust constraints. The novelty of this model is the formulation of the POPF as a two-stage optimization model independent of the technique used for the scenario generation. The addition of probabilistically-robust constraints to the problem is also a novelty, since this im-plies the calculation of useful statistical moments embedded in the model to control the robustness of the final solution.
1.5
Outline
This 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 regarding the operation of electrical power systems, used simulation techniques, and optimization principles.
The first part of the thesis includes two deterministic approaches:
∙ Chapter 3 presents an optimal energy management system for balanced microgrids operating in isolated mode using a mixed-integer second-order cone programming model. The model considers dispatchable DG and ESS, fuel availability and the operational constraints of the system. This chapter is mainly based on the following conference paper:
Figure 1.7 – Structure of thesis
Juan S. Giraldo, Jhon A. Castrillon and Carlos A. Castro “Energy management of isolated microgrids using mixed-integer second-order cone programming,” in 2017
IEEE 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 current injections to solve the optimal energy management of unbalanced, three-phase, grid-connected microgrids, focused particularly on the modelling of small synchronous machines 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 dispatchable DG units, ESS, load shedding, and stochasticity over conventional demands and non-dispatchable RES. This chapter is mainly based on the following journal paper: Juan S. Giraldo, Jhon A. Castrillon, Juan Camilo López, Marcos J. Rider and Carlos A. Castro “Microgrids Energy Management Using Robust Convex
Program-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 conventional demands 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 and Carlos A. Castro “Probabilistic OPF Model for Unbalanced Three-phase Electri-cal Distribution Systems Considering Robust Constraints,” IEEE Transactions on
Power Systems, vol. 34, no. 5, pp. 3443-3454, Sept. 2019.
2 General Definitions
T
his chapter introduces some basic definitions and concepts used throughout this the-sis. It embraces the definition for the power flow formulation and its numerical characteristics as well as some mathematical programming concepts. In addition, it is important to emphasize that this chapter is merely a brief introduction to complex and wide 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 references cited in each section.2.1
Power Flow Analysis
The 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 every bus, the distribution of power flows, power losses, and in general, any other modelled variable of interest [153]. Power flows are a fundamental part of almost all tasks in power systems, 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 computational task. However, analyses over interconnected power networks were not always performed by 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 on a scaled replica of a power system, in which, all analyses were performed. A key calculating machine 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 a new 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 AC circuit, 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 well as all active sources in the circuit, thus, all nodal voltages and branch currents can be calculated linearly by means of Ohm’s laws.
∙ In load flow analysis loads are commonly expressed in terms of their consumed active 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
magnitude and active power injection, due to the automatic voltage regulator and the automatic generation control. Finally, transmission lines and transformers are conventionally modelled as lumped 𝜋-circuits with constant parameters [158]. The relationship 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 following Maxwell’s equations, thus, involving integro-differential equations. A common represen-tation relates using ordinary differential equations, as
ℎ (𝑥) = ˙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., ˙𝑥 = 0, which can be translated as ignoring all
dynamic 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 different forms, 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 𝐼𝑖𝑁, can be expressed as I𝑁𝑖 = 𝐼𝑖𝐺− 𝐼𝐿
𝑖 , where 𝐼𝑖𝐺 stands for the current injected by active sources and 𝐼𝑖𝐿 the
current consumed by loads at that node. The voltages at all nodes of the system, 𝑗, are represented by V𝑗, and the relationship between the net currents at bus 𝑖 and voltages
are related by the elements of the admittance matrix, as I𝑁𝑖 −∑︁
𝑗∈ΩB
Y𝑖,𝑗V𝑗 = 0 ∀𝑖 ∈ ΩB (2.3)
where ΩB stands for the set of nodes of the system. Notice that nodal voltages can be
determined by solving the linear system in (2.3) if all nodal currents are constant, or linearly dependent on the voltage. However, as explained in Section 2.1, loads and active components 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 energy management of unbalanced microgrids.
Assume the 2-bus system shown in Figure 2.1, composed by a constant voltage generator, 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.,
Figure 2.1 – 2-bus system. −I𝑁 𝑖 = 𝐼𝑖𝐿 = (︁ 𝑆𝐿 𝑖 /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 is the 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 load and generation to ideal current sources by converting powers into current injections. The mapping is then carried out using injected complex power until nodal voltages converge up to a predefined tolerance [158].
On the bright side, the operations performed over 𝑔 in (2.2) are simple in the GS method, e.g., its Jacobian must not be computed in contrast to other methods, facilitating its implementation and speed per iteration. On the down side, even though the GS method could be employed for large networks, its convergence is usually slow (linear) and it can be numerically unstable. As a disclaimer, note that (2.3) can be solved with any other nonlinear root-finder algorithm, not exclusively with the GS method, which has been used in this section only as example.
Performing approximations is another approach for solving the power flow problem using the net current representation, as in [162], where the current injections from PQ loads are approximated using the first order Taylor expansion around 1∠0 pu, resulting in
I𝑁𝑖 ≈ (︁𝑆𝑖𝐿(2 − V𝑖) )︁*
for the above mentioned example. It should be stated that an expression for constant voltage buses is not available in [162].
Equation (2.3) can be rewritten using the above approximation as follows:
(︁ 𝑆𝑖𝐿(2 − V𝑖) )︁* −∑︁ 𝑗∈ΩB Y𝑖,𝑗V𝑗 = 0 ∀𝑖 ∈ ΩB (2.4)
which is linear, guaranteeing the uniqueness of the solution, if it exists. This formulation has been used in [163], for example, for the optimal reactive compensation and voltage control of distribution power systems.
