Faculdade de Engenharia Elétrica e de Computação
Pedro Pablo Vergara Barrios
Fault Tolerant Optimal Operation of Droop-Based
Microgrids
Operação Ótima Tolerante a Falhas de uma
Microrede Com Controle Droop
Campinas
2019
Pedro Pablo Vergara Barrios
Fault Tolerant Optimal Operation of Droop-Based
Microgrids
Operação Ótima Tolerante a Falhas de uma Microrede Com
Controle Droop
Thesis presented to the Faculty of Electrical and Comput-ing EngineerComput-ing of the University of Campinas in partial fulfillment of the requirements for the degree of Doctor, in the area of Electric 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 tí-tulo de Doutor em Engenharia Elétrica, na Área de Ener-gia Elétrica, no âmbito do Acordo de Cotutela firmado en-tre a Unicamp e a University of Southern Denmark, SDU (Dinamarca).
Supervisors:
Prof. Dr. Luiz Carlos Pereira da Silva Prof. Dr. Bo Nørregaard Jørgensen
Este exemplar corresponde à versão final da tese defendida pelo aluno Pe-dro Pablo Vergara Barrios, e orientada pelo Prof. Dr. Luiz Carlos Pereira da Silva e o Prof. Dr. Bo Nørregaard Jørgensen.
Campinas
2019
ORCID: https://orcid.org/0000-0003-0852-0169
Ficha catalográfica
Universidade Estadual de Campinas Biblioteca da Área de Engenharia e Arquitetura Elizangela Aparecida dos Santos Souza - CRB 8/8098
Vergara Barrios, Pedro Pablo,
V586f VerFault tolerant optimal operation of droop-based microgrids / Pedro Pablo Vergara Barrios. – Campinas, SP : [s.n.], 2019.
VerOrientadores: Luiz Carlos Pereira da Silva e Bo Norregaard Jorgensen. VerCoorientador: Hamid Reza Shaker.
VerTese (doutorado) – Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação.
VerEm cotutela com: University of Southern Denmark.
Ver1. Energia renovável. 2. Sistemas de energia elétrica. 3. Otimização. I. Silva, Luiz Carlos Pereira da, 1972-. II. Jorgensen, Bo Norregaard. III. Shaker, Hamid Reza. IV. Universidade Estadual de Campinas. Faculdade de
Engenharia Elétrica e de Computação. VI. Título.
Informações para Biblioteca Digital
Título em outro idioma: Operação ótima tolerante a falhas de uma microrede com contole
droop
Palavras-chave em inglês:
Renewable energy Electrical energy systems Optimization
Área de concentração: Energia Elétrica Titulação: Doutor em Engenharia Elétrica Banca examinadora:
Luiz Carlos Pereira da Silva [Orientador] Walmir de Freitas Filho
Bo Norregaard Jorgensen Jan Corfixen Sorensen Zita Maria Almeida do Vale
Data de defesa: 12-02-2019
Programa de Pós-Graduação: Engenharia Elétrica
Candidato: Pedro Pablo Vergara Barrios RA: 153774
Data da Defesa: 12 de Fevereiro de 2019
Título da Tese: Fault Tolerant Optimal Operation of Droop-Based Microgrids.
Título da Tese em Português: Operação Ótima Tolerante a Falha de uma Microrede com Con-trole Droop.
Prof. Dr. Luiz Carlos Pereira da Silva (Presidente, FEEC/UNICAMP)
Prof. Dr. Bo Nørregaard Jørgensen (University of Southern Denmark/Denmark) Prof. Dr. Jan Corfixen Sørensen (University of Southern Denmark/Denmark) Prof. Dr. Walmir de Freitas Filho (FEEC/UNICAMP)
Profa. Dra. Zita Maria Almeida do Vale (Polytechnic of Porto/Portugal)
A ata de defesa, com as respetivas assunaturas dos membros da Comissão Jualgadora, encontra-se no processo de vida acadêmica do aluno.
I believe that one page of this document, and all the words that I want to put on it, are not enough to thank God and life for all the moments, lessons learned, adventures and experiences lived through the development of this Ph.D. thesis. It has been more than three years (and more than five since I left my family in Colombia) pursuing this achievement. It is impossible to deny that through this long (sometimes not so) stage of my life, I have grown not only professionally but also personally. Looking behind, I can see that this would not be possible without all the good and hard moments lived during this time, and more importantly, without the support of so many people in Brazil, in Denmark and in Colombia. I believe that everyone has contributed, in different ways, to help me to get to this point. In their recognition, I would like to share with them some words:
To my supervisor in Brazil, Prof. Luiz Carlos, thank you for all the support and friend-ship during all these years. I will always be grateful for all the opportunities offered. To Prof. Marcos J. Rider, for the time spent during all the discussions, always with the aims of improv-ing this work. In Denmark, to my supervisor Prof. Bo Nørregaard and my co-supervisor Prof. Hamid R. Shaker, for all the support, especially for all the time spent in the coordination of all this: the challenge of doing the Ph.D. in Denmark and in Brazil, at the same time.
To my friends Juan C. López in Brazil and Juan M. Rey in Barcelona, for all the discus-sions during the development of this research and during the process of writing all the papers.
To my colleagues at UNICAMP and SDU, especially to Aisha and Konstantine, for all the chats and coffees.
To so many friends: In Brazil, to Silvia, Marcos, and Jerson, you have always been there. I will always appreciate that. In Colombia, to Anderson, who always laugh at my face and reduce my problems to nothing. In Italy to Dianis, for making me feel special every time we talked, and of course, for the trip to Italy. In Germany, to Jose, for the endless discussions about life. In Denmark, to Athila, for the friendship and endless conversations about almost everything. To my friends Trevor, Pau, Pao, Alex, and Richard; guys, you made the time spent in Odense more than fun.
To E., I cannot thank you enough for all the fun I had with you in Denmark. Thank you for showing me your country, your friends, sharing your culture, and more importantly, your deepest thoughts. Since then, we started to walk together and I want for that to last as much as time allows it.
To my family in Colombia, my dad Martin, my mom Clara, my grandma Elena and my brothers Paris and Martin. I hope you all feel proud of me. Thank you for always believing in me, also for all the words, especially in the hardest moments, and the patience for all the time I have not been there at home.
As we move to a more complex electrical distribution system, as a result of the increas-ing deployment of distributed generation and storage units, many new operational challenges have appeared. In the technical literature, one of the most accepted solution to deal with these operational challenges is the standard hierarchical control framework. This framework com-prises multiple control layers to deal simultaneously with the slowest dynamic associated with the dispatch decisions (high-level layer), and with the fastest dynamic associated with the local control architecture (lower-level layer). Based on this, this thesis focuses on the higher-level layer, presenting new and sophisticated mathematical models to deal with the dispatch prob-lem of an islanded microgrid. The microgrid is composed of different distributed generation (DG) units, including renewable-based units, and battery systems (BSs). The main contribution of these models is related to the modeling of the lower-layer control, implemented using droop control. Moreover, as these mathematical formulations are generally difficult to solve, this thesis also introduces new approximation procedures that simplify the original formulations, allowing the use of commercial solvers.
Although the hierarchical control framework has been accepted as the standard solution, this might not be suitable for large-scale systems. In this sense, a new trend in the research community has recently been established, focused on distributed control frameworks. These frameworks offer attractive features such as fault-tolerance, scalability, plug-and-play, robust-ness, among others. Considering this, a control strategy to operate a microgrid in a distributed fashion is also presented in this thesis. This work has overcome two important drawbacks found in the technical literature: Firstly, the active and reactive power balance is continuously cor-rected, even during the execution of the control strategy; and secondly, the strategy is considered to operate with a distributed version of the hierarchical control framework.
On the other hand, as the operation of the microgrid is essentially different in grid-connected (GC) and islanded (IS) mode, a new and generalized mathematical model is also presented. This model allows the microgrid’s operator to plan in advance the operational mode that guarantee a safe operation. Additionally, a new set of convexification procedures are also introduced, in order to solve the proposed formulation using commercial solvers.
Finally, due to the uncertain operational environment of the renewable-based generation and the consumption, deterministic dispatch approaches might fail to ensure a secure operation. To deal with this, a scenario-based stochastic model for the operation problem of an islanded microgrid is presented. The impact of the level of uncertainty in the data is assessed in the optimal dispatch solution.
Keywords: Microgrids; islanded operation; droop control, optimal power flow, optimal dis-patch.
À medida que avançamos para um sistema de distribuição de energia elétrica mais complexo, como resultado da crescente implantação de sistemas de geração e armazenamento distribuído, novos desafios operacionais tem surgido. Na literatura técnica, uma das soluções aceitas para lidar com estes desafios operacionais corresponde à estrutura de controle hierárquico. Esta estrutura compreende várias camadas de controle, lidando simultaneamente com a dinâmica mais lenta associada às decisões de despacho (camada de alto nível); bem como com a dinâmica mais rápida associada à arquitetura de controle local (camada de nível inferior). Esta tese foca na camada de nível superior apresentando novos e sofisticados modelos matemáticos para lidar com o problema de despacho de uma microrrede ilhada. A microrrede é composta por diferentes unidades de geração distribuída (GD), incluindo unidades baseadas em fontes renováveis, e sis-temas de bateria (SB). A principal contribuição destes modelos está relacionada à modelagem do controle da camada inferior, implementada utilizando controle droop. Além disso, como estas formulações matemáticas são geralmente difíceis de resolver, esta tese também introduz novos procedimentos de aproximação que simplificam as formulações originais, permitindo o uso de solvers comerciais.
Embora a estrutura de controle hierárquico tenha sido aceita como a solução padronizada, essa estrutura de controle pode não ser adequada para sistemas com um grande número de unidades. Nesse sentido, uma nova tendência na comunidade de pesquisa foi recentemente esta-belecida, focada em estruturas de controle distribuídas. Estas estruturas oferecem características atraentes, como tolerância a falhas, escalabilidade, plug-and-play, robustez, entre outras. Diante disso, uma estratégia de controle para operar uma microrrede de forma distribuída é apresen-tada também nesta tese. Este trabalho superou duas desvantagens importantes encontradas na literatura técnica: Em primeiro lugar, o balanço de potência ativa e reativa é continuamente corrigido, mesmo durante a execução da estratégia de controle; e segundo, a estratégia é consid-erada para operar com uma versão distribuída da estrutura de controle hierárquico.
Por outro lado, como a operação da microrrede é essencialmente diferente no modo conectado à rede e no modo ilhado, um novo modelo matemático generalizado também é ap-resentado. Este modelo permite ao operador da microrede planejar antecipadamente o modo operacional, a fim de garantir a operação com segurança. Alem disso, um novo conjunto de pro-cedimentos de convexificação também é introduzido visando resolver a formulação proposta usando solvers comerciais.
Finalmente, devido ao ambiente operacional incerto da geração baseada em renováveis e do consumo, abordagens de despacho determinísticas podem falhar em garantir uma operação segura. Para lidar com essa natureza incerta, também é apresentado um modelo estocástico baseado em cenário para o problema de operação de uma microrrede ilhada. O impacto do nível de incerteza nos dados é avaliado na solução ótima de despacho.
Efterhånden som vi overgår til et mere komplekst elektrisk distributionssystem, som følge af den øget udbredelse af distribueret elektricitet produktion og lagring, er der kom-met mange nye operationelle udfordringer til. I den tekniske litteratur er en af de mest ac-cepterede løsninger til at håndter e disse nye operationelle udfordringer den standardiserede hierarkiske kontrolarkitektur. Denne arkitektur omfatter flere kontrollag der skal behandles sam-tidigt, herunder højniveauslaget forbundet med produktions dispatch beslutningerne, og det hur-tigere lavniveauslag forbundet med den lokale kontrolarkitektur. Denne afhandling fokuserer på højniveauslaget, og præsenterer nye og sofistikerede matematiske modeller til at håndtere produktion dispatch problemet for et mikrogrid i ø-drift. Mikrogridet består af forskellige dis-tribuerede elektricitetsgeneratorer (DG), herunder enheder baseret på vedvarende energikilder og batterisystemer (BS). Hovedmodellen af disse modeller er relateret til modelleringen af lavniveau kontrollen, implementeret ved hjælp af Droop kontrol. Da disse matematiske mod-eller generelt er vanskelige at løse, introducerer denne afhandling også nye approksimation-sprocedurer, der forenkler de oprindelige modeller, hvilket muligøre brugen af kommercielle matematiske problemløsere.
Selvom den hierarkiske kontrolarkitektur er blevet accepteret som standardløsningen, er denne ikke velegnet til brug i storskala systemer. Dette er årsagen til at forskningsmiljøet har ændret fokus til distribuerede kontrolarkitekturer. Distribuerede kontrolarkitekturer tilbyder attraktive egenskaber såsom fejltolerance, skalerbarhed, plug-and-play, robusthed, med mere. På baggrund af dette præsenterer denne afhandling en distribueret kontrolstrategi til drift af et mikrogrid. Den præsenteret distribueret kontrolstrategi overvinder to vigtige ulemper, der findes i den tekniske litteratur: For det første korrigeres den aktive og reaktive effektbalance, selv under udførelsen af kontrolstrategien; og for det andet anses strategien til at fungere med en distribueret version af den hierarkiske kontrolarkitektur.
På den anden side, da driften af et mikrogrid er væsentligt forskellig i henholdsvis netfor-bundet (GC) og ødrift (IS) -tilstand, præsentere afhandlingen også en ny og generaliseret matem-atisk model. Denne model gør det muligt for operatøren at planlægge driften af et mikrogrid, således at der sikres en sikker drift. Derudover introduceres også et nyt sæt af konveksifikation-sprocedurer til at løse den generaliserede model ved anvendelse af kommercielle matematiske problemløsere.
Endeligt, da det er vanskeligt at sikre en deterministisk dispatch af produktionsenheder, baseret på vedvarende energikilder, præsentere denne afhandling en scenariebaseret stokastisk mo-del til løsning af produktion dispatch problemet for et mikrogrid i ø-drift. Virkningen af usikkerhedsniveauet i dataene vurderes i den optimale dispatch løsning.
Figure 1 – Hierarchical control structure of a microgrid composed of three control levels. 24
Figure 2 – Droop control of a DG unit coupled to a microgrid . . . 31
Figure 3 – Control modes of the WT units . . . 34
Figure 4 – 25-bus three-phase microgrid test system . . . 40
Figure 5 – Total active power for the DG units, WTs and the BSs for cases I to IV . . . 41
Figure 6 – Active power generation of the DG units for cases I to IV . . . 42
Figure 7 – Frequency during the planning horizon for cases I to IV . . . 42
Figure 8 – Voltage magnitude profile of the microgrid for cases I to IV . . . 43
Figure 9 – Percentage of deviation of the DG generation cost and active power losses as function of the active droop gain . . . 45
Figure 10 – Active and reactive power generation of the DG units for the large-size case 47 Figure 11 – Commitment decisions of all the DG units for the large-size case . . . 48
Figure 12 – Control modes of the DG units: VCM and PCM . . . 53
Figure 13 – Structure of a DG unit seen as an agent. . . 57
Figure 14 – Flowchart of the proposed distributed dispatch strategy composed of Stage I and II. . . 58
Figure 15 – Illustrative example of the dynamics of the primary, secondary and tertiary control level. . . 59
Figure 16 – Microgrid test system and the communication topology . . . 62
Figure 17 – Microgrid used for the simple numerical example . . . 64
Figure 18 – Total generation cost of the DG units . . . 67
Figure 19 – Local estimation of the incremental cost variable at each DG unit . . . 67
Figure 20 – Total active output power of DG and WT units . . . 68
Figure 21 – Active droop gain of units operating in VCM during operation. . . 69
Figure 22 – Total reactive output power of all the DG units . . . 69
Figure 23 – Voltage profile of the microgrid for all the phases before and after the con-vergence process. . . 70
Figure 24 – Frequency and frequency reference for all units during operation . . . 70
Figure 25 – Additional communication topologies used to test the proposed strategy . . 72
Figure 26 – Comparison of the convergence of the total generation cost of the DG units for different communication topologies. . . 73
Figure 27 – Active output power of the DG units for the cases with 3, 7 and 10 DG units. 73 Figure 28 – Planning horizon T in which the microgrid can operate in grid-connected (GC) or islanded (IS) mode. . . 77
Figure 29 – Flowchart of the procedure used to solve the proposed MINLP model. . . . 84 Figure 30 – 25-bus three-phase microgrid with three DG units, two BSs and three WTs . 85
DG units, WTs and BSs, for the grid-connected, islanded and planned is-landed scenarios for the Case III . . . 88 Figure 32 – Frequency for the grid-connected, islanded and planned islanded scenario
for Case III. . . 88 Figure 33 – Total active profile of the expected load demand, load after curtailment, SE,
DG units, WTs and BS, for the case in which a degradation of the micro-grid’s voltage reference occurs . . . 90 Figure 34 – Voltage magnitude at the PCC and the DG units nodes for the case in which
a degradation of the microgrid’s voltage reference occurs . . . 90 Figure 35 – Bivariate Normal PDF of the forecast error of the active and reactive power
consumption . . . 96 Figure 36 – Scenario generation approach to obtain the set S. . . 97 Figure 37 – 25-bus three-phase microgrid with three DG units, two BSs and three WTs . 101 Figure 38 – Set ˆS of scenarios of active and reactive load consumption and active WT
power generation . . . 102 Figure 39 – Histograms of the DG cost for the MCSs using the optimal schedule defined
by the deterministic and the stochastic models . . . 104 Figure 40 – Average active power for the load consumption, DG units, WTs and the BSs
for the MCSs using the stochastic and deterministic model . . . 105 Figure 41 – Load curtailment decisions defined by the deterministic and stochastic model
for each node . . . 105 Figure 42 – Histograms of the DG cost for the MCSs using the optimal schedule defined
by the the stochastic solutions with forecast error of 10% and 5% . . . 106 Figure 43 – Histograms of the DG cost for the MCSs using the optimal schedule defined
by the the stochastic solutions with forecast error of 15% and 5% . . . 107 Figure 44 – Illustrative example of the active power flow balance at one node . . . 121 Figure 45 – Schematic representation ofL(PG0
Table 1 – DG units information . . . 39
Table 2 – BSs information . . . 40
Table 3 – General comparison of Case I, II, III and IV. . . 44
Table 4 – Comparison of different scenarios in Case IV . . . 44
Table 5 – Comparison of the NLP and MILP solutions for Case IV. . . 46
Table 6 – DG units’ information for the large-size case . . . 47
Table 7 – Active power phase for all DG Units at peak period . . . 48
Table 8 – DG units information . . . 62
Table 9 – DG units information for the numerical example . . . 65
Table 10 – First iterations of the numerical example . . . 66
Table 11 – Comparison of the distributed strategy for different cases . . . 71
Table 12 – Constraint (4.39) for all the possible decisions . . . 83
Table 13 – DG units information . . . 85
Table 14 – BSs information . . . 86
Table 15 – Load level, energy cost and WTs generation . . . 86
Table 16 – Comparison of different cases and scenarios . . . 87
Table 17 – Comparison of cases with degradation of the voltage reference at the PCC . . 89
Table 18 – Total number of variables by additional unit . . . 91
Table 19 – Comparison of Case I for different number of DG units . . . 92
Table 20 – DG Units Information . . . 102
Table 21 – BSs Information . . . 102
Table 22 – Comparison Results of the Stochastic and Deterministic Model . . . 104
BS Battery system
DG Distributed generator
EDS Electrical distribution system
GC Grid-connected
IS Islanded
MCS Monte Carlo Simulations
MILP Mixed-integer linear programming
MINLP Mixed-integer nonlinear programming
MISOCP Mixed-integer second-order conic programming
MNR Modified Newton-Raphson
OPF Optimal power flow
PCM Power control mode
PCC Point of common couple
PDF Probability Distribution Function
VCM Voltage control mode
VSS Value of Stochastic Solution
Sets:
B Set of nodes in which a BS is connected, B ⊂ N .
F Set of phases {A, B, C}.
G Set of nodes in which a DG unit is connected, G ⊂ N .
G1 Set of nodes in which the DG unit m operates in PCM, G1 ⊂ N .
G2 Set of nodes in which the DG unit m operates in VCM, G2 ⊂ N .
L Set of lines.
N Set of nodes of the microgrid.
Om Set of operational constraints of the DG unit m.
ˆ
S Set of initial pool of stochastic scenarios.
S Set of stochastic scenarios.
T Set of time-periods.
W Set of nodes in which a WTs is connected, W ⊂ N .
Indexes: φ Phase φ ∈ F. ψ Phases ψ ∈ F. m Node m ∈ N . mn Line mn ∈ L. n Node n ∈ N .
r r-th block used for the piece-wise linearization of(Pmn,φ,t)2 and(Qmn,φ,t)2.
s Scenario s ∈ S, s ∈ ˆS.
t Time interval t ∈ T .
y y-th block used for the piece-wise linearization of PG m,t
2
. Parameters:
αC Load curtailment cost.
αW WT curtailment cost.
αm Constant parameter associated to the DGs operation cost.
βm Linear parameter associated to the DGs operation cost.
γm Quadratic parameter associated to the DGs operation cost.
∆Gm Discretization step for the piece-wise linearization of PG m,t
2
.
∆mn,t Maximum length of the block for the piece-wise linearization of (Pmn,φ,t)2 and
(Qmn,φ,t)2.
∆t Length of the time-step period.
∆ω Maximum angular frequency deviation .
ε Parameter to control the converge of the active droop protocol. ˆ
ε Parameter to control the converge of the frequency reference protocol.
ηB Efficiency of the BSs.
κ Parameter to control the converge of consensus protocol. ˜
πs True probability of the scenario s.
πs Normalized probability of the scenario s.
ρmn,r Slope of the r-th block for the piece-wise linearization of(Pmn,φ,t)2and(Qmn,φ,t)2.
σm,t,y Slope of the y-th block for the piece-wise linearization of Pm,tG
2
.
σf Forecast error.
ω0 Angular frequency reference.
ω Maximum angular frequency.
ω Minimum angular frequency.
EBm Maximum level of energy of the BSs. EB
m Minimum level of energy of the BSs.
f pW
m Power factor of the WTs.
Imn Maximum lines current limit.
kP
m Active droop gain of DG units.
kQm Reactive droop gain of DG units.
kW
m,t Slope of the WT controller in PC mode.
N Total number of DG units, i.e., N = |G|. PD
m,φ,t Active power demand consumption.
PD
m,φ,t,s Active power demand consumption for the scenario s.
˜ PD
m,φ,t Forecasted active power demand consumption for the stochastic model.
˜
Pm,φ,t Estimated active power flow.
PWm,φ,t Expected maximum active power of the WTs. PW
m,t,s Active power generation of the WTs for the stochastic model.
˜ PW
m,t Forecasted active power generation of the WTs for the stochastic model.
PGm Maximum active generation limit of the DG units. PG
m Minimum active generation limit of the DG units.
PBm Maximum charging/discharging limit of the BSs. QD
m,φ,t Reactive power demand consumption.
QD
m,φ,t,s Reactive power demand consumption for the scenario s.
˜ QD
m,φ,t Forecasted reactive power demand consumption for the stochastic model.
˜
Qm,φ,t Estimated reactive power flow.
QGm Maximum reactive generation limit of the DG units. QG
m Minimum reactive generation limit of the DG units.
R Number of discrete blocks for the piece-wise linearization of(Pmn,φ,t)2and(Qmn,φ,t)2.
˜
Smn,φ,t Estimation of the apparent power flow.
TP Time of response of the primary control level.
˜
Vm,ψ,t Estimated value of the voltage magnitude.
V Maximum voltage magnitude.
V Minimum voltage magnitude.
V0 Voltage magnitude reference.
Y Number of discrete blocks for the piece-wise linearization of PG m,t
2
.
Zmn,φ,ψ Line impedance .
Zmn,φ,ψ′ Transformed line impedance, defined asZmn,φ,ψ′ = Zmn,φ,ψ θψ− θφ.
Continuous Variables:
∆G
m,y Value of the y-th block used for the piece-wise linearization of Pm,tG
2
.
∆P
mn,φ,t,r Value of the r-th block used for the piece-wise linearization of(Pmn,φ,t)2.
∆Qmn,φ,t,r Value of the r-th block used for the piece-wise linearization of(Qmn,φ,t)2.
λ Dual variable associated with the active power balance constraint. λm Local estimation of λ by the DG unit m.
ΛS
m,φ,t Continuous variable used to represent the product betweenPm,φ,tS andxt.
Πm,t Continuous variable used to represent the product betweenPm,tG andxt.
ωm(k) Angular frequency reference of the DG unit m operating in VCM at iteration k.
ωt Angular frequency of the system.
ωt,s Angular frequency of the system for scenario s.
DP
m(k) Active droop gain during iteration k.
DQm Reactive droop gain.
EB
m,t Energy of the BSs.
Imn,φ,tsqr Continuous variable used to represent(Imn,φ,t)2.
PG
m,t Total active output power of the DG units.
PG
m,t,s Total active output power of the DG units for the scenario s.
PG0
m Total scheduled active power of the DG units.
Pm,tGsqr Linear variable used to represent(PG m,t)2.
PG
m,ψ,t Active output power of the DG units at phase ψ.
PG
m,ψ,t,s Active output power of the DG units at phase ψ for the scenario s.
PW
m,t Total active power generation of the WTs.
PS
m,t Active power supplied at the point of common couple (PCC).
PB
m,φ,t Active power injections of the BSs.
Pm,tB+ Charging active power of the BSs. Pm,tB− Discharging active power of the BSs. Pmn,φ,t Active power flow in the lines.
Pmn,φ,t,s Active power flow in the lines for scenario s.
Pmn,φ,t− Negative component for the piece-wise linearization of(Pmn,φ,t)2.
QG
m,t Total reactive output power of the DG units.
QGm,t,s Total reactive output power of the DG units for scenario s.
QGm,ψ,t Reactive output power of the DG units at phase ψ. QG
m,ψ,t,s Reactive output power of the DG units at phase ψ for scenario s.
QW
m,t Total reactive power generation of the WTs.
QS
m,t Reactive power supplied at the point of common couple (PCC).
Qmn,φ,t Reactive power flow in the lines.
Qmn,φ,t,s Reactive power flow in the lines for scenario s.
Qsqrmn,φ,t Linear variable used to represent(Qmn,φ,t)2.
Q+mn,φ,t Positive component for the piece-wise linearization of(Qmn,φ,t)2.
Q−mn,φ,t Negative component for the piece-wise linearization of(Qmn,φ,t)2.
Smn,φ,t Apparent power flow in lines.
Smn,φ,t,s Apparent power flow in lines for scenario s.
SL
mn,φ,t Apparent power losses in lines.
SL
mn,φ,t,s Apparent power losses in lines for scenario s.
Vm,φ,t Voltage magnitude of the nodes.
Vm,φ,t,s Voltage magnitude of the nodes for scenario s.
Vm,φ,tsqr Continuous variable used to represent(Vm,φ,t)2.
Wm,t Continuous variable used to represent the multiplication of variables ωtand d
′
m,t.
Binary Variables:
b+m,t Variable associated with the charging state of the BSs. b−m,t Variable associated with the discharging state of the BSs. cm,t Variable associated with load curtailment.
dm,t Variable associated with the MPP control mode of the WTs.
d′m,t Variable associated with the PC control mode of the WTs. em,t Variable associated with WTs curtailment.
um,t Operational state of the DG units.
xt Variable associated with the operational mode of the microgrid.
Matrix:
C = [cmn] Consensus matrix, with elements cmn.
A = [amn] Adjacency matrix, with elements amn.
Functions:
1 Introduction . . . 23
1.1 Background and Research Motivation . . . 23
1.2 Objectives . . . 25
1.3 Contributions of this Thesis . . . 26
1.4 Thesis Outline . . . 27
2 Centralized Optimal Operation of an Islanded Droop-Based Microgrid . . . 29
2.1 Introduction . . . 29
2.2 Operation of Islanded Microgrids . . . 30
2.2.1 The MINLP Problem Formulation . . . 31
2.3 Linearization of the MINLP Formulation . . . 35
2.3.1 Linearization of the Objective Function . . . 35
2.3.2 Linearization of the AC Power Flow Model . . . 35
2.3.3 Linearization of the WTs Model . . . 36
2.3.4 Linearization of the DG unit Model . . . 37
2.3.5 Linearization of the BSs Model . . . 38
2.3.6 An MILP Model . . . 38
2.4 Simulation Results and Discussions . . . 39
2.4.1 Optimal Operation of the Islanded System . . . 40
2.4.2 Impact of the Droop Gains on the Optimal Dispatch . . . 44
2.4.3 Error Assessment . . . 45
2.4.4 Large-Size Case . . . 46
2.5 Summary . . . 48
3 Distributed Optimal Operation of an Islanded Droop-Based Microgrid . . . 50
3.1 Introduction . . . 50
3.2 Control and Operation of Microgrids . . . 52
3.2.1 Primary and Secondary Control Level . . . 52
3.2.2 Tertiary Control Level . . . 53
3.3 Distributed Optimal Strategy . . . 54
3.3.1 First-Order Consensus Algorithm . . . 54
3.3.2 Distributed Optimal Dispatch Strategy . . . 55
3.3.3 Overview of the Distributed Strategy . . . 58
3.3.4 Operation within the Hierarchical Control Framework . . . 59
3.3.5 Design Considerations . . . 60
3.4 Simulation Results and Discussion . . . 61
3.4.1 Simulation of the Measurement Process . . . 62
3.4.5 Case III: Impact of the Communication Topology on the Performance . 72 3.4.6 Case IV: Scalability and Computational Time . . . 72 3.5 Summary . . . 74 4 Optimal Operation of a Microgrid Considering Grid-Connected and Islanded
Droop-Based Mode . . . 75 4.1 Introduction . . . 75 4.2 The MINLP Formulation . . . 77 4.3 Convexification and Approximation of the MINLP Model . . . 80 4.3.1 Convexification of the MINLP Model . . . 80 4.3.2 Solution Methodology . . . 84 4.4 Simulation Results and Discussions . . . 85 4.4.1 Optimal Operation . . . 86 4.4.2 Degradation of the Voltage Reference at the PCC . . . 88 4.4.3 Error Assessment . . . 90 4.4.4 Scalability . . . 91 4.5 Summary . . . 91 5 Uncertainty Assessment on the Optimal Operation of an Islanded Droop-Based
Microgrid . . . 93 5.1 Introduction . . . 93 5.2 Stochastic Operation Problem of an Islanded Microgrid . . . 95 5.2.1 Uncertainty Modelling . . . 95 5.2.2 Scenario Generation Approach . . . 95 5.2.2.1 Load Consumption . . . 95 5.2.2.2 WTs Generation . . . 96 5.2.2.3 Scenario Generation Process . . . 96 5.2.3 MINLP Model . . . 97 5.3 Linearization of the Stochastic MINLP Model . . . 100 5.4 Simulation Results and Discussions . . . 101 5.4.1 Model: Deterministic vs Stochastic . . . 101 5.4.2 Impact of the Forecast Error . . . 104 5.5 Summary . . . 107 Conclusion . . . 108 References . . . 111 Appendix A – The Three-Phase Power Flow Formulation . . . 120 Appendix B – Convergence Analysis of the Distributed Optimal Strategy . . . 122 Appendix C – Convexification Analysis of the Proposed OPF Formulation . . . 126 Appendix D – Other Publications . . . 130
1 Introduction
1.1 Background and Research Motivation
Driven by environmental awareness, and supported by a strong political agenda; the current electrical energy systems have faced an increased deployment of distributed energy resources. A large portion of these resources are based on renewable generation [1]. This tech-nological advancement has created new operational paradigms, in which microgrids can be found [2]. The US’s Department of Energy defines a microgrid as a «group of interconnected loads and distributed energy resources with clearly defined electrical boundaries that act as a single controllable entity with respect to the grid and can connect and disconnect from the grid to enable it to operate in both grid-connected or islanded mode» [3].
These features (of being able to operate in grid-connected and islanded mode) has al-lowed microgrids to become an interesting alternative to bring electricity to small and isolated communities, especially those localized in islands. In this regards, some benchmark projects can be found around the world. To name a few, in north Brazil, in the Lençóis Island [4], approxi-mately 90 households are supplied by a hybrid PV-wind-Diesel islanded system. In Denmark, the Bornhorm island deploys a wind generation rich system which supplies a 55 MW peak and is able to operate islanded from the transmission system [5].
From the technical point of view, the operation of a microgrid in islanded mode presents many more operational challenges, when compared with the grid-connected mode. Some of these challenges are associated with [6–8]:
• The generation-consumption power balance, which might create bi-directional power flows.
• Stability issues related to the continuous operation of different technology-based dis-tributed generation systems, as well as a lack of an unique voltage magnitude and a frequency reference.
• Low-inertia due to the presence of renewable-based generation systems, connected to the system with electronic interfaces.
• Robustness and reliability, due to the uncertain nature of the renewable-based generation systems, such as PV and wind systems.
In order to allow a proper and reliable integration of the islanded microgrids, these technical challenges need to be overcome. In this sense, in the technical literature, the hierar-chical control framework has been accepted as the standardized approach to operate these sys-tems [9, 10]. Usually, this framework comprises three control levels, as represented in Fig. 1.
Primary Control Primary Control Level Level
DG DG
Electrical Distribution System Secondary Control Level
Tertiary Control Level
Figure 1 – Hierarchical control structure of a microgrid composed of three control levels. The lines represent control links, while the dashed lines represent measurement links. Each of these levels has a clearly defined task, input and output signals, and speed of re-sponse [11]:
Primary Control Level: This corresponds to the fastest level, usually responsible for the local control of the generation systems, setting the voltage magnitude and frequency value. The control infrastructure in this level is usually based on the well-known droop control, im-plemented locally and without any communication. Other recently proposed droop-free control approaches can also be found in literature [12].
Secondary Control Level: The speed of response of this level is slower than the primary level. This is usually in charge of the voltage magnitude and frequency restoration. Thus, its operation can be seen as a correction process, in charge of correcting the voltage magnitude and frequency values in steady-state conditions [13]. The current research trend for this control focuses on developing distributed and robust approaches [14–16] .
Tertiary Control Level: This control adds intelligent to the operation of the microgrid, and has the slowest speed of response. The main task of this control level is related to the coordination of the operation of all the generation and storage units, as well as the controllable loads. This is done with the aim of achieving a reliable, safe and economical operation. The operation of this control level is supported by a communication infrastructure, implemented using advanced metering infrastructure. In literature, centralized and distributed approaches can be found to develop and implement this control level [17, 18].
In the tertiary control, to define the operation of all the distributed energy resources, usually, a unit commitment and economic dispatch problem is stated and solved [19]. If the
technical operational constraints of the electric network are considered, including for instance the voltage magnitude and current flow limits; the economic dispatch problem can be formu-lated as an extended version of an optimal power flow (OPF) formulation. The conventional OPF formulation is valid in grid-connected systems, based on the fact that a slack bus is present and the frequency is considered to be fixed. However, this formulation might not be longer valid for islanded systems since any mismatch between the generation and consumption might create a frequency deviation. Moreover, assumptions such as the consideration of a large-capacity unit might also fail, especially in peak hours.
To overcome this, an extension of the conventional OPF formulation has been recently proposed by the authors in [20] for droop-based islanded systems. The novelty in this OPF formulation is the consideration of the frequency of the system as a power flow variable. To accomplish this, the authors proved that the consideration of the droop expressions are sufficient to define the operation of the DG units, regardless of the control loops used to implement such droop control [21]. The contribution of the work in [20] is considered as the cornerstone of this Ph.D. thesis (and many others recently published works, such as [21–24]), as it allowed the development of new mathematical formulations for the unit commitment and economic dispatch problem of islanded systems, without the need of non-real technical assumptions, as discussed above. Moreover, the use of the islanded OPF formulation allowed to assess the impact of the generation and storage units in the frequency of the system in steady-state conditions; as well as to consider the unbalanced operation of the network through a three-phase power flow formulation. In this sense, the development of these models is an important task in the research community, as there is not currently available a commercial power flow tool, such as OpenDSS or GridLabD, that incorporates the islanded formulation.
On the other hand, although this control structure has been widely accepted, this was developed for centralized systems, where a central entity gathers all the information required by each control level. Based on this, this control structure might not be suitable for large-scale and privacy-preserving systems. In fact, this centralized structure makes difficult to implement the plug-and-play characteristics of the smart microgrids. This feature is important as this reduces the microgrid’s reliability and robustness, due to its poor capacity to respond to an unexpected DG unit fault.
1.2 Objectives
Based on the above-presented discussion, the main objectives pursued in this thesis are related to:
• Contribute to the operation problem of an islanded microgrid with novel theoretical and practical insights, by developing a centralized and a fault tolerance distributed state-of-the-art mathematical model, which must consider the technical operational constraints of
the distribution network.
• Assess the impact of the uncertainty associated with the forecast of the renewable gen-eration and the consumption on the optimal opgen-eration of the microgrid by proposing a stochastic scenario-based optimization formulation.
1.3 Contributions of this Thesis
Motivated by the shortcomings of the conventional OPF formulation, the lack of com-mercial power flow tools to study islanded droop-based microgrids, and the drawbacks of the centralized approaches, the main contributions of this Ph.D. thesis can be summarized as fol-lows:
The first contribution is related to the new mixed-integer nonlinear programming (MINLP) model presented to optimally operate an islanded microgrid. A flexible mixed-integer linear programming (MILP) formulation is also presented in order to solve the proposed formulation using commercial solvers. In this model, an additional control mode for the WTs is considered, which is also modeled using droop control.
The second contribution is related to the proposed fault tolerance distributed strategy. This strategy considers in the optimization approach two control modes for the DG units. In this sense, as one of the control modes is based on droop control, the generation-consumption power balance is continuously corrected, even during the execution of the optimization strategy. As the distributed strategy is supposed to operate within the hierarchical control approach, a distributed version of this control framework is also proposed.
The third contribution is related to the generalization of the islanded OPF formulation to consider in the same mathematical model the grid-connected mode. In order to solve this formulation using commercial solvers, a new mixed-integer second-order cone programming (MISOCP) formulation is presented. A technical discussion of both operational modes (grid-connected and islanded) is also presented.
Finally, the fourth and last contribution is related to the assessment of the uncertainty on the optimal operation of the islanded system. This uncertainty comes from the forecast of the renewable-based generation units and the load consumption. This is done through a new scenario-based MINLP stochastic model. A new approach to generate the stochastic scenarios that capture all the uncertainty involved is also presented.
The publications directly related to the development of this thesis includes:
1. P. P. Vergara, J. C. López, L. C. P. da Silva, M. J. Rider,’Optimal Operation of Unbal-anced Three-Phase Droop-Based Microgrids’, IEEE Trans. on Smart Grid, to be pub-lished, 2017.
2. P. P. Vergara, J. M. Rey, H. R. Shaker, J. M. Guerrero, B. N. Jørgensen, L. C. P. da Silva, ’Distributed Strategy for Optimal Operation of Unbalanced Three-Phase Islanded Microgrids’, IEEE Trans. on Smart Grid, to be published, 2018.
3. P. P. Vergara, J. M. Rey, J. C. Camilo, M. J. Rider, L. C. P. da Silva, H. R. Shaker, , B. N. Jørgensen, ’A Generalized Midel for the Optimal Operation of Microgrids in Grid-Connected and Islanded Droop-Based Mode’, IEEE Trans. on Smart Grid, to be published, 2018.
4. P. P. Vergara, J. C. Camilo, M. J. Rider, H. R. Shaker, L. C. P. da Silva, B. N. Jørgensen, ’A Scenario-Based Model for the Optimal Stochastic Operation of Unbalanced Three-Phase Islanded Microgrids’, submitted to the IEEE Trans. on Smart Grid, 2018.
5. P. P. Vergara, H. R. Shaker, B. N. Jørgensen, L. C. P. da Silva, ’Distributed Consensus-Based Economic Dispatch Considering Grid Operation’, 2017 IEEE Power and Energy Society General Meeting, Chicago, USA, 2017.
6. P. P. Vergara, H. R. Shaker, B. N. Jørgensen, L. C. P. da Silva, ’Generalization of the λ-Method for Decentralized Economic Dispatch Considering Active and Reactive Re-sources’, 2017 Innovative Smart Grid Technologies Conference, ISGT Europe, Torino, Italy, 2017.
The publications 1–4 are incorporated in this thesis as the Chapters 2 to 5, while the rest are cited as references whenever relevant. Other publications during the development of this PhD thesis are listed in the Appendix D.
1.4 Thesis Outline
This thesis is organized in such a way that each chapter can be read independently, fully understandable in terms of their scientific content. The rest of the thesis is structured as follows: In Chapter 2, a new MINLP model is presented to optimally operate an islanded un-balanced microgrid. In this model, the DG units are considered to operate with droop control, which is the main contribution of the proposed formulation. The operation of the BSs and WTs are considered as well. A linearized version of the MINLP model that can be solved using con-vex commercial solvers is also introduced. Simulations are presented in order to validate the proposed model.
In Chapter 3 a distributed strategy to optimally operate an islanded unbalanced micro-grid is presented. This strategy is based on constrained distributed optimization and consensus algorithms, conceived to operate within the standard hierarchical control framework. A techni-cal discussion regarding its implementation is also presented. Simulations are presented in order to validate the effectiveness f the proposed strategy.
As the operation of the microgrid in both operational modes: GC and IS is essentially different, in Chapter 4 a new and generalized mathematical model is presented. This model allows the microgrid’s operator to plan in advance the best operational mode that guarantees a safe operation. Additionally, a set of convexification procedures are also introduced in order to solve the proposed formulation using commercial solvers.
In Chapter 5 the uncertain nature of the renewable-based generation, as well as the forecast of the consumption, are assessed on the optimal operation of an islanded microgrid. To do this, a scenario-based stochastic model is presented. A new approach to generate the stochastic scenarios that capture all the uncertainty involved is also presented. The quality of the defined optimal dispatch is evaluated through Monte Carlo Simulations (MCSs). A comparison between the deterministic and stochastic model is also presented.
2 Centralized Optimal Operation of an
Islanded Droop-Based Microgrid
12.1 Introduction
Microgrids can be seem as small-scale manageable electrical distribution networks, ca-pable of operating in grid-connected and islanded mode. To define the operational schedule of all the distributed generators (DGs) and storage units that these systems might comprises, an optimization problem can be stated [25]. In literature, this problem is generally addressed based on an AC optimal power flow (OPF) formulation [26], proposed to achieve different objectives, including minimization of the operational cost, power losses and environmental impact, or the maximization of the DG penetration, improve voltage regulation, among others [27, 28].
These conventional power flow formulations are valid in grid-connected mode, based on the fact that a slack bus is present and the frequency is considered to be fixed. However, in microgrids operating in islanded mode, these formulations may not be longer valid since [29]: (i)any mismatch between the generation and the load consumption might increase the frequency deviation from the nominal value, i.e. the frequency of the islanded system is not a pre-specified value, (ii) there is no slack-bus for voltage reference, and (iii) all DG units have relatively small generation capacity, thus the utilization of a DG unit as a slack bus may fail, specially in peak hours.
More recently, different approaches have been proposed for power flow analysis in is-landed systems [21–24]. For instance, in [21], a generalized three-phase power flow formulation is presented as a set of non-linear equations and solved using a Newton-trust region method. In [22], the metaheuristic particle swarm optimization (PSO) is used to solve the OPF formu-lation considering the droop gains as variables. In [23] and [24], a modified Newton-Rahpson (MNR) method is presented to consider the frequency of the system as a variable. A similar ap-proach is presented in [30] for hybrid AC/DC islanded microgrids. In [31], other well-studied power flow algorithm, the backward/forward sweep (BFS) algorithm, is adapted for islanded systems. A strategy for the initial back and forward iterations is also proposed without consider-ing a slack bus. A similar BFS approach is presented in [32]. In [33], an OPF formulation is used for inverted-based microgrids. However, the active power losses are obtained using the Kron’s loss formula, aiming to derive an iterative algorithm which provides feasible, but sub-optimal solutions. Other works have focused on the reconfiguration problem of islanded microgrids [34], or cost minimization using a two-stage approach, one for the economic dispatch and other for
1This chapter is based on the paper: P. P. Vergara, J. C. Lopez, M. J. Rider, L. C. P. da Silva, ’Optimal Operation
of Unbalanced Three-Phase Islanded Droop-Based Microgrids’, IEEE Transactions on Smart Grid, to be published, 2018. c IEEE 2018.
the power flow assessment [35].
Although the above-discussed works have succeeded to overcome the limitations of the conventional power flow in islanding systems, i.e., eliminating the need for a slack bus, the power flow analysis is generally restricted to one particular time period. In fact, in these ap-proaches, additional strategies have to be considered to account for time-coupling constraints of the DG units, which limits its application to longer planning horizons. Moreover, binary vari-ables have not been considered in these approaches, generally used to model the commitment decisions of the DG units. Other works, such as [7, 36–38], have carried out multi-period dis-patch analysis for islanded systems. However, these approaches fail to consider frequency as a variable and the droop operation of the DG units in islanded mode.
The main advantage of addressing the operation problem using an OPF formulation is the flexibility to represent the control decision variables (continuous and discrete) of the gener-ation and storage units, as well as other elements, such as voltage regulators, capacitor banks, tap changers transformers, switches, etc. In this context, this chapter extends the unbalanced three-phase OPF formulation for the optimal operation of an islanded droop-based microgrid. In the proposed model, the frequency is considered as one of the power flow variables. First, the islanded OPF model is introduced as a new mixed-integer non-linear programming (MINLP) problem, considering different operational and time-coupling constraints for the DG units and battery systems (BSs). Load curtailment and different modes of operation for the wind tur-bines (WTs) are also considered. Then, a set of efficient linearizations are introduced in order to approximate the original problem into a mixed-integer linear programming (MILP) model. To validate the effectiveness of the proposed formulation, a 25-bus and a 124-bus unbalanced three-phase systems have been used for multiple cases of studies.
2.2 Operation of Islanded Microgrids
In grid-connected mode, the DG units are generally dispatched to follow a pre-specified power reference, while the power flow balance and the voltage magnitude reference are pro-vided by the grid [39]. On the other hand, in islanded mode, the DG units are operated in droop mode, aiming to achieve an appropriate level of active power sharing and voltage reg-ulations considering that no slack-bus is available [11]. This control strategy ensures that any active power unbalance and voltage deviation are shared among all the DG units, in inverse proportion to their droop gains (kP
m, kQm), implemented locally and without any communication
link, as shown in Fig. 2. This operational structure is usually developed based on a hierarchical control scheme, in which a higher-level layer is operated by a central coordinator with main function schedule the operation of all the energy resources, defining the commitment decisions (ON/OFF) for the DG units and WTs, load curtailment decision of users, as well as the oper-ational mode (charging/discharging) and the active input/output power reference for the BSs;
DG
m Droop Control ωt, Vm,φ,t PG m,t, QGm,t ω0, V0Figure 2 – Droop control of a DG unit coupled to a microgrid.
while the droop control is deployed at the lower-level layer [10]. Consequently, if the central coordinator considers the droop operation of the DG units in the economic dispatch process, optimized commitment and dispatch decisions can be scheduled and a feasible day-ahead oper-ation of the microgrid in terms of frequency and voltage reference can be obtained.
To define their active and reactive output power operating with droop control, each DG unit uses the expressions given in (2.1) and (2.2), respectively.
ωt= ω0− kmPPm,tG (2.1)
Vm,φ,t = V0− kmQQGm,t (2.2)
For steady-state analysis, these expressions are sufficient to define the operation of the DG units, regardless of the control structure used to implement such droop characteristics [21]. Equations (2.1) and (2.2) are based on the assumption that the output impedance of the DG unit is inductive, which is valid for synchronous-based units and the majority of converters coupled through inductors to the grid [11]. As discussed in [23], these expressions are in accordance with the IEEE standard 1547.7 for distributed generation in islanded systems [40]. Nevertheless, in a case of non-inductive or complex output impedances, two approaches can be followed: first, a strategy that aims to decouple the active and reactive power regulation can be implemented using a virtual output impedance [41]. In a second approach, the conventional droop equations, given by (2.1) and (2.2), can be replaced by the complex droop equations, stated in (2.3) and (2.4). These equations model the influence of active and reactive power generation over the frequency and voltage regulation, and they are known in the specialized literature as the P − V − ω and Q − V − ω droop equations [23].
ωt = ω0− kmP(Pm,tG − QGm,t) (2.3)
Vm,φ,t = V0− kQm(Pm,tG + QGm,t) (2.4)
2.2.1 The MINLP Problem Formulation
The operation of an islanded droop-based microgrid can be modeled using the MINLP model given by (2.5)–(2.30). The objective function in (2.5) aims at minimizing the total
op-erational cost of the system in the planning horizon T . The first term in (2.5) refers to the total curtailment cost, while the second term refers to the DG units’ generation costs. The latter modeled using a quadratic cost function [42].
min ( X t∈T " X m∈N X φ∈F αCPD m,φ,t(1 − cm,t) + X m∈G (γm(Pm,tG ) 2 + βmPm,tG + αm) #) (2.5) subject to: Smn,φ,tL = X ψ∈F Zmn,φ,ψ′ Smn,ψ,t∗ Vm,φ,tVm,ψ,t ! Smn,φ,t ∀mn ∈ L, ∀φ ∈ F, ∀t ∈ T (2.6) X km∈L Pkm,φ,t− X mn∈L Pmn,φ,t+ R{Smn,φ,tL }+ X m∈G PG m,φ,t = P D m,φ,tcm,t+ X m∈B PB m,φ,t− X m∈W PW m,t/3 ∀m ∈ N , ∀φ ∈ F, ∀t ∈ T (2.7) X km∈L Qkm,φ,t− X mn∈L Qmn,φ,t+ I{Smn,φ,tL } + X m∈G QG m,φ,t = Q D m,φ,tcm,t ∀m ∈ N , ∀φ ∈ F, ∀t ∈ T (2.8) Vm,φ,t2 −V 2 n,φ,t = 2 X ψ∈F R{Zmn,φ,ψ′ }Pmn,ψ,t+ I{Zmn,φ,ψ′ }Qmn,ψ,t− 1 V2 m,φ,t X ψ∈F Zmn,φ,ψ′∗ Smn,φ,t 2 ∀mn ∈ L, ∀φ ∈ F, ∀t ∈ T (2.9) (Pmn,φ,t2 + Q 2 mn,φ,t)/V 2 m,φ,t ≤ I 2 mn ∀mn ∈ L, ∀φ ∈ F, ∀t ∈ T (2.10) PG m,t = X ψ∈F PG m,ψ,t ∀m ∈ G, ∀t ∈ T (2.11) QG m,t = X ψ∈F QG m,ψ,t ∀m ∈ G, ∀t ∈ T (2.12) ωt = ω0− kmPPm,tG ∀m ∈ G, ∀t ∈ T |um,t = 1 (2.13) Vm,φ,t = V0− kQmQGm,t ∀m ∈ G, ∀t ∈ T |um,t = 1 (2.14) Vm,φ,t = Vm,ψ,t ∀m ∈ G, ∀φ, ψ ∈ F, ∀t ∈ T |um,t = 1 (2.15)
PGmum,t ≤ Pm,tG ≤ P G mum,t ∀m ∈ G, ∀t ∈ T (2.16) QGmum,t ≤ QGm,t ≤ Q G mum,t ∀m ∈ G, ∀t ∈ T (2.17) EB m,t = E B m,t−1+ ηBPm,tB ∆t ∀m ∈ B, ∀t ∈ T (2.18) Pm,tB = P B− m,t − P B+ m,t ∀m ∈ B, ∀t ∈ T (2.19) PB+ m,t · Pm,tB−= 0 ∀m ∈ B, ∀t ∈ T (2.20) Pm,tB+ = X ψ∈F Pm,ψ,tB+ ∀m ∈ B, ∀t ∈ T (2.21) Pm,tB− = X ψ∈F Pm,ψ,tB− ∀m ∈ B, ∀t ∈ T (2.22) Pm,φ,tB+ = Pm,ψ,tB+ ∀m ∈ B, ∀φ, ψ ∈ F, ∀t ∈ T (2.23) Pm,φ,tB− = Pm,ψ,tB− ∀m ∈ B, ∀φ, ψ ∈ F, ∀t ∈ T (2.24) 0 ≤ PB+ m,t ≤ P B m ∀m ∈ B, ∀t ∈ T (2.25) 0 ≤ Pm,tB− ≤ P B m ∀m ∈ B, ∀t ∈ T (2.26) EB m ≤ Em,tB ≤ E B m ∀m ∈ B, ∀t ∈ T (2.27) V ≤ Vm,φ,t ≤ V ∀m ∈ N , ∀φ ∈ F, ∀t ∈ T (2.28) ω0− ∆ω ≤ ωt ≤ ω0+ ∆ω ∀t ∈ T (2.29) um,t, cm,t ∈ {0, 1} ∀m ∈ G, ∀t ∈ T (2.30)
In the above formulation, the continuous decision variables are the active and reactive generation power of the DG units (PG
m,t, QGm,t), the active power injection of the BSs (Pm,tB ) and
the active power injection of the WTs (PW
m,t), which depends on the mode of operation. The
binary decision variables correspond to the operational state of the DG units (um,t) and the load
curtailment decisions (cm,t). The grid is modeled by a three-phase power flow model given by
(2.6)–(2.10), derived as a function of the active and reactive power flow through the lines, i.e.,
Smn,φ,t = Pmn,φ,t + jQmn,φ,t (see Appendix 5.5 for a detailed formulation). The impedance of
lines is function of the frequency ωt, the resistance (Rmn,φ,ψ) and the reactance at nominal
fre-quency (Xmn,φ,ψ) of lines. Additionally, a transformation of the line impedance is introduced
and defined as Zmn,φ,ψ′ = Zmn,φ,ψ θψ− θφ. Notice that due to this definition Z
′
mn,φ,ψ is not
symmetric asZmn,φ,ψ. Constraint (2.6) define the apparent power losses through the lines.
Con-straints (2.7) and (2.8) model the active and reactive power balances equations, respectively, considering the output power of the DG units, the input/output power of the BSs and the WTs. Equation (2.9) models the voltage magnitude drop in the lines. The maximum current flow limit is enforced by constraint (2.10).
In islanded systems, time periods with extremely high renewable generation can be considered a critical scenario, specially if the load consumption is lower and the BSs are at its maximum state of charge. To maintain the active power balance, the WTs is equipped with a
MPP PC PW m,t PWm,t ω ωt kW m kW2 m kW1 m
Figure 3 – Control modes of the WT units, where kW1
m > kmW2 and ω = ω0+ ∆ω.
local controller with two operational modes, as shown in Fig. 3. In the Maximum Power Point (MPP) mode, the WTs controller supplies the expected maximum active power (PWm,t), while
in the Power Control (PC) mode, the output power depends on the frequency of the microgrid. Thus, the frequency is used as a feedback signal allowing the WTs to respond to the current operational point of the microgrid. The amount of power that the WTs can supply can be defined tunning the parameter kW
m.
The DG units are modeled using (2.11)–(2.17). The binary variable um,tis used to model
their commitment status, i.e., um,t = 1 ⇔ Pm,tG > 0. Constraints (2.11) and (2.12) define the
total active and reactive power output of the DG units, as a function of the output power of each phase, while (2.13) and (2.14) model the active and reactive droop operation, respectively. It is important to highlight that constraints in (2.13) and (2.14) should be considered in the formulation only if the unit is in operation (i.e., only if um,t = 1). Constraint (2.15) models
the electromotive force of synchronous-based DG units, which is represented by the balanced voltages magnitudes at their internal nodes [21, 36]. Finally, the total generation power limits are defined by constraints in (2.16)–(2.17).
The BSs are modeled using (2.18)–(2.27). In this model, BSs are considered to operate in power control mode, i.e., allowing to schedule their active output power injections in ad-vance. Nevertheless, the inverter used to couple the BSs to the grid could also operate in droop mode in order to support frequency regulation [43]. Constraint in (2.18) models the energy assessment, which depends on the energy stored in the previous time period (EB
m,t−1), while
(2.19) define the total input/output power of the BSs in term of their charging and discharging powers. Constraint in (2.20) is used to model the unimodality operation of the BSs, i.e., the requirement to operate in only one state (charging, discharging or stand-by). The total charging and discharging power are defined by (2.21) and (2.22), respectively, while the balanced active power injection/consumption are enforced by (2.23) and (2.24). Finally, constraints (2.25) to (2.27) define the limits for the charging and discharging active power, and the energy stored in the BSs. Other operational constraints related to the operation of the grid such as voltage magni-tudes limits and maximum frequency deviation are enforced by constraints in (2.28) and (2.29), respectively. Finally, (2.30) defines the binary nature of the decision variables um,tand cm,t.
2.3 Linearization of the MINLP Formulation
MINLP formulations, such as the one presented in Sec. 2.2.1, are difficult to solve and commercial solvers are not able to find optimal solutions in a reasonable amount of time [36]. Hence, in this section, precise linearizations and approximations are used to transform the orig-inal MINLP formulation into an accurate MILP problem.
2.3.1 Linearization of the Objective Function
To linearize the second term in (2.5), the variablePm,tGsqris introduced to model the square
of the active power of the DG unit, i.e.,Pm,tGsqr≃ (PG
m,t)2. Thus,P Gsqr
m,t can be approximated using
a linear piecewise representation. Constraints related to the definition of variables and their limits for this linearization are given in (2.31)–(2.34), where∆Gm= P
G m/Y , ∀m ∈ G. Pm,tGsqr≈ X y∈Y σm,t,y∆Gm,y ∀m ∈ G, ∀t (2.31) PG m,t = X y∈Y ∆G m,t,y ∀m ∈ G, ∀t (2.32) 0 ≤ ∆Gm,t,y ≤ ∆ G m ∀m ∈ G, ∀t, y = 1 . . . Y (2.33) σm,t,y = (2y − 1)∆ G m ∀m ∈ G, ∀t, y = 1 . . . Y (2.34)
2.3.2 Linearization of the AC Power Flow Model
The definition of the apparent power losses in (2.6) is non-linear due to multiplication of the continuous variablesSmn,ψ,t∗ andSmn,φ,t, andVm,ψ,t andVn,φ,t. To linearize Smn,φ,tL ,
esti-mated values of voltage magnitudes, i.e.,V˜m,φ,tandV˜n,ψ,tand, the estimated active and reactive
power flows through lines of each phase φ, i.e.,S˜mn,φ,t= ˜Pmn,φ,t+ j ˜Qmn,φ,t, can be used.
Addi-tionally, the impedance is assumed constant at the nominal frequency value. Thus,SL
mn,φ,t can
be approximated using the linear expression given by (2.35). This approximation introduces a relatively small error in the power flow formulation, since the line power losses are smaller than the real power flow value [44].
Smn,φ,tL = X ψ∈F Zmn,φ,ψ′ Smn,ψ,t∗ ˜ Vm,φ,tV˜m,ψ,t ! ˜ Smn,φ,t (2.35)
The quadratic terms in the left-hand side of constraint (2.9) and (2.10) can be linearized by introducing the variableVm,φ,tsqr to represent the square value ofVm,φ,t. As for the right-hand
side of (2.9), the only non-linear term corresponds to, 1 V2 m,φ,t X ψ∈F Zmn,φ,ψ′∗ Smn,φ,t 2 (2.36) which can be approximated as in (2.37), based on the fact that Zmn,φ,φ ≫ Zmn,φ,ψ, ∀φ, ψ ∈
F|φ 6= ψ. Z ′ mn,φ,φ 2 Smn,φ,t Vm,φ,t 2 = Z ′ mn,φ,φ 2 P2 mn,φ,t+ Q 2 mn,φ,t V2 m,φ,t (2.37) Thus, (2.37) can be linearized introducing the variablesPmn,φ,tsqr andQsqrmn,φ,tto represent
the square values ofPmn,φ,tandQmn,φ,t, respectively. Additionally,Vm,φ,tis approximated based
on its estimation, given byV˜m,φ,t. Then, (2.9) can be rewritten as the linear expression in (2.38).
Vm,φ,tsqr − Vn,φ,tsqr ≈ 2X ψ∈F R{Zmn,φ,ψ′ }Pmn,ψ,t+ I{Zmn,φ,ψ′ }Qmn,ψ,t − Z ′ mn,φ,φ 2 Psqr mn,φ,t+ Q sqr mn,φ,t ˜ V2 m,φ,t ∀mn, ∀φ, ∀t (2.38)
Constraint in (2.10) can be rewritten as the linear expression in (2.39), based also on the continuous variablesPmn,φ,tsqr ,Qsqrmn,φ,tandVm,φ,tsqr .
0 ≤ Pmn,φ,tsqr + Qsqrmn,φ,t ≤ Vm,φ,tsqr I2mn ∀mn, ∀φ, ∀t (2.39)
Finally,Pmn,φ,tsqr and Qsqrmn,φ,t are approximated using a piece-wise linear representation,
given by (2.40)–(2.49), where∆mn,t= V Imn/R, ∀mn ∈ L. Pmn,φ,tsqr + Qsqrmn,φ,t ≈ R X r=1 ρmn,r(∆Pmn,φ,t,r+ ∆ Q mn,φ,t,r) ∀mn, ∀φ, ∀t (2.40) Pmn,φ,t+ + Pmn,t− = R X r=1 ∆Pmn,φ,t,r ∀mn, ∀φ, ∀t (2.41) Q+mn,φ,t+ Q−mn,t = R X r=1 ∆Qmn,φ,t,r ∀mn, ∀φ, ∀t (2.42) Pmn,φ,t = Pmn,φ,t+ − P − mn,φ,t ∀mn, ∀φ, ∀t (2.43) Qmn,φ,t = Q+mn,φ,t− Q−mn,φ,t ∀mn, ∀φ, ∀t (2.44) P+ mn,φ,t, P − mn,φ,t ≥ 0 ∀mn, ∀φ, ∀t (2.45) Q+ mn,φ,t, Q − mn,φ,t ≥ 0 ∀mn, ∀φ, ∀t (2.46) 0 ≤ ∆Pmn,φ,t,r ≤ ∆mn,t ∀mn, ∀φ, ∀t, r = 1 . . . R (2.47) 0 ≤ ∆Qmn,t,φ,r ≤ ∆mn,t ∀mn, ∀φ, ∀t, r = 1 . . . R (2.48) ρmn,r = (2r − 1)∆mn,t ∀mn, ∀t (2.49)
2.3.3 Linearization of the WTs Model
The two control modes of the WTs can be modeled based on the binary variables dm,t
and d′
m,t. Thus, if the WTs operate in the MPP mode, then dm,t = 1 and d
′
analysis can be done if the WTs operate in the PC mode. This model can be represented using (2.50) and (2.51), where variableWm,t is introduced to model the multiplication of variables ωt
and d′
m,t. Thus,Wm,tcan be represented as a linear expression using the disjunctive formulation,
given in (2.53) and (2.52). Finally, (2.54) defines the binary nature of variables dm,t and d
′
m,t.
Notice that the disconnection of the WT can also be considered. In this case, dm,t = d
′ m,t = 0 and PW m,t = 0. dm,t+ d ′ m,t ≤ 1 ∀m ∈ W, ∀t (2.50) Pm,tW = P W m,tdm,t + P W m,td ′ m,t− k W mWm,t ∀m ∈ W, ∀t (2.51) − ω(1 − d′m,t) ≤ Wm,t− ωt≤ ω(1 − d ′ m,t) ∀m ∈ W, ∀t (2.52) ωd′m,t ≤ Wm,t ≤ ωd ′ m,t ∀m ∈ W, ∀t (2.53) dm,t, d ′ m,t ∈ {0, 1} ∀m ∈ W, ∀t (2.54)
In the proposed model, it has been considered that the WTs are source of active power. However, WTs can also supply reactive power. In this case, equation (2.55) can be added to the MINLP model. Here, (2.55) models the reactive power limits as a function of the WTs power factor (pfW
m ), which is considered as an input parameter, and the active power, which depends
on the operation mode (MPP or PC). Finally, notice that equation (2.55) is linear and can be added directly to the MILP model without further adaptation.
|QW
m,t| ≤ Pm,tW tan(arccos(pfmW)) ∀m ∈ W, ∀t (2.55)
2.3.4 Linearization of the DG unit Model
The droop operation of a DG unit, given by (2.13) and (2.14) in the MINLP formulation, and constraint in (2.15), should be considered only if the unit is committed, i.e., if um,t = 1.
This can be enforced by the expression given in (2.56) and (2.57), for the active and reactive droop operation, respectively, and in (2.58) for the balanced voltage magnitude constraint. |kP mPm,tG + (ωt− ω0)| ≤ (1 − um,t)∆ω ∀m ∈ G, ∀t (2.56) |kQ mQ G m,t + (Vm,φ,t− V0)| ≤ (1 − um,t)∆V ∀m ∈ G, ∀t (2.57) |Vm,φ,t− Vm,ψ,t| ≤ (1 − um,t)V ∀m ∈ G, ∀φ, ψ |φ 6= ψ , ∀t (2.58)
Lets consider that the DG unit is not committed, or equivalently that um,t = 0, then
PG
m,t = 0 due to (2.16), while (2.56) is reduced to (2.29), which limits the maximum frequency
deviation (∆ω) from the nominal value (ω0). On the other hand, if the unit is committed, or
equivalently, um,t = 1, constraint (2.56) leads to the active droop operation model in (2.1). In
case of complex output impedances, (2.56) should consider the complex active droop expression given by (2.3) instead. A similar analysis can be done for (2.57) and (2.58).
Finally, notice that the proposed formulation is a function of variableVm,φ,tsqr , while the
reactive power droop model in (2.57) is a function of variable Vm,φ,t. Thus, to approximate
Vm,φ,t, (2.59) can be added to the MILP model. The expression in (2.59) is obtained using an
approximation of the square function through a first-order Taylor series expansion around the nominal voltage (V0).
Vm,φ,t ≈ V0+ (Vm,φ,tsqr − V02)/(2V0) ∀m, ∀t (2.59)
2.3.5 Linearization of the BSs Model
The non-linear expression in (2.20) can be rewritten as a linear expression based on the binary variables b+
m,t and b−m,t, as stated in (2.60). Here, b +
m,t is used to represent the operational
state of the BSs in discharging mode, i.e., if b+
m,t = 1, then P B+
m,t ≥ 0. Similarly, b−m,t is used
to represent the operational state in charging mode. Therefore, if b−
m,t = 1, then P B− m,t ≥ 0.
This is enforced by constraints in (2.61) and (2.62). The stand-by condition is considered when b+m,t = b
+
m,t = 0. Finally, (2.63) defines the binary nature of variables b + m,tand b − m,t. b+m,t+ b − m,t ≤ 1 ∀m ∈ B, ∀t (2.60) 0 ≤ PB+ m,t ≤ P B mb + m,t ∀m ∈ B, ∀t (2.61) 0 ≤ Pm,tB− ≤ P B mb−m,t ∀m ∈ B, ∀t (2.62) b+ m,t, b − m,t ∈ {0, 1} ∀m ∈ B, ∀t (2.63)
2.3.6 An MILP Model
The optimal operation problem of an islanded droop-based microgrid can be stated using an MILP formulation as follows:
min (2.5) where Pm,tGsqr ≃ PG m,t 2 . Subject to: (2.7), (2.8), (2.16)–(2.19), (2.21)–(2.24), (2.29), (2.30), (2.31)–(2.34), (2.35), (2.38), (2.39), (2.40)–(2.49),(2.50)–(2.54), (2.56)–(2.58), (2.59), (2.60)– (2.63) and (2.64). V2 ≤ Vm,φ,tsqr ≤ V2 ∀m ∈ N , ∀φ ∈ F, ∀t ∈ T (2.64)
The application of classical optimization techniques guarantees global optimality of the linearized version of the proposed model. The accuracy of the piecewise linear representation can be improved by increasing the number of discretization blocks (Y , Λ). The methodology used to obtain and analyze the results can be summarized as follows: In a first stage, the esti-mated values for the voltages (V˜m,φ,t), and the active and reactive power flowing through lines
