Universidade Federal de Juiz de Fora
Faculdade de Engenharia
Graduação em Engenharia Elétrica - Energia
Marcelo de Castro Fernandes
Modeling and Simulation of Integrated Transmisson and DistributionSystems to Assess the Impacts of Distributed Energy Resources
Juiz de Fora
2017
Marcelo de Castro Fernandes
Modeling and Simulation of Integrated Transmisson and DistributionSystems to Assess the Impacts of Distributed Energy Resources
Trabalho de Conclusão de Curso apresentadaao programa de Graduação em EngenhariaElétrica - Energia da Universidade Federalde Juiz de Fora, como requisito parcial paraobtenção do grau de Engenheiro Eletricista.
Orientadora: Janaína Gonçalves de Oliveira
Coorientador: Luigi Vanfretti
Juiz de Fora
2017
Ficha catalográfica elaborada através do Modelo Latex do CDC da UFJFcom os dados fornecidos pelo(a) autor(a)
de Castro Fernandes, Marcelo.Modeling and Simulation of Integrated Transmisson and Distribution
Systems to Assess the Impacts of Distributed Energy Resources / Marcelode Castro Fernandes. – 2017.
78 f. : il.
Orientadora: Janaína Gonçalves de OliveiraCoorientador: Luigi VanfrettiTrabalho de Conclusão de Curso – Universidade Federal de Juiz de Fora,
Faculdade de Engenharia. Graduação em Engenharia Elétrica - Energia,2017.
1. Modeling. 2. Simulation. 3. Transmission and distribution systems.I. Gonçalves de Oliveira, Janaína, orient. II. Vanfretti, Luigi, coorient. III.Título.
ACKNOWLEDGEMENTS
I would like to express my gratitude to my supervisor, Prof. Janaína Oliveira, forall advices, support, friendship and confidence in my work.
To my co-supervisor, Luigi Vanfretti, for all incentives and for giving me theoportunity of working on OpenIPSL. Also to Maxime, for his e�orts in helping me in manytasks.
To professors and colleagues from all laboratories I have been worked in. Specialthanks to Gabriel Fogli, Pedro Machado, Pedro Gomes, Dalmo Cardoso and BernardoMusse for valuable advices and for motivating me during this journey.
To my colleagues from university classes, for all the help, discussions and distrac-tions.
To my parents, Rose e Paulo, for giving me endless love and unlimited encourage-ment, crucial for completion of this work.
To my sisters, Luiza and Patrícia, for their a�ection and incentive. Thanks forinspiring me.
To Gabriela, my girlfriend, for giving me unconditional support and love. Thankyou for standing by me.
To my friends and family for being my safe heaven.
To all who somehow contributed for this work, muito obrigado!
“If I have seen further it is by standing on the shoulders of giants”Isaac Newton
RESUMO
A busca por sustentabilidade e pela diversificação da matriz energética trouxe grandesmudanças nos sistemas de energia nos últimos anos. Dentre eles, o sistema elétricode potência é o que apresenta a maior penetração de energias renováveis. As formasalternativas de geração de energia elétrica são, muitas vezes, dispostas na rede de formadistribuída ao invés da tradicional geração centralizada. Esta nova característica impõenovos desafios quanto aos estudos de sistemas elétricos modernos. Inserido neste contexto,esse trabalho investiga a modelagem e simulação de sistemas de transmissão e distribuiçãointegrados, sob diferentes níveis de geração distribuída. O estudo almeja investigar arelevância dos fenômenos observados com a simulação de tais sistemas. Além disso, oestudo utiliza as vantagens da linguagem Modelica para a simulação de sistemas dinâmicos.
Uma rede de transmissão baseado no sistema IEEE 14 barras é conectada a um alimentadorde distribuição baseado no sistema IEEE 13 barras. Para a conexão dos dois sistemas,o presente trabalho faz uso de uma ferramenta de interface híbrida, possibilitando asimulação do sistema de transmissão em monofásico equivalente ao mesmo passo em quea distribuição é simulada como um sistema trifásico. Dois casos bases são apresentados:operação do sistema isolado e operação do sistema conectado a um barramento infinito.Três cenários com diferentes casos de penetração de geração distribuída são montados paracada caso base. A mesma contigência é analisada em todos os cenários.
O trabalho apresenta a modelagem de componentes trifásicos, do dispositivo híbrido deinterface e de um sistema de geração fotovoltaica simples. Os modelos híbrido e trifásicossão validados através de três diferentes testes e foram, posteriormente, agregados a umabiblioteca em linguagem Modelica para simulação de sistemas de potência.
Os resultados obtidos avaliam o impacto do estudo integrado dos sistemas de transmissãoe distribuição com geração fotovoltaica. Foi possível observar diferenças nas frequênciasde máquinas síncronas conectadas ao sistema de transmissão quando estas operavam nocaso base e nos cenários com geração distribuída. Além disso, foi mostrado que os níveisde tensão no sistema de transmissão se mantiveram praticamente os mesmos nos diversoscasos. Já no sistema de distribuição foi observado que a penetração de energia renováveleleva os níveis de tensão. Além disso, picos de corrente injetada pelos painéis solarespuderam ser observados devido à queda brusca de tensão ocasionada pela contingência.O estudo mostra então que análises de sistemas integrados através de elementos híbridospodem ser valiosas ferramentas de estudo para os sistemas elétricos modernos.
Palavras-chave: Simulação. Modelagem. Transmissão. Distribuição. Modelica.
ABSTRACT
The search for sustainability and energy matrix diversification have brought many changesto energy systems in the past few years. The power sector, among them, presents thelargest share of renewables. Alternative sources of electric energy generation are, manytimes, connected to the system as distributed energy resources rather than traditionalcentralized power plants. This new characteristic poses challenges to the study of modernpower systems. In this context, the present work investigates modeling and simulationof integrated transmission and distribution systems, under di�erent levels of distributedgeneration. The study aims to assess the relevance of observed phenomena with simulationof such systems. Besides, this study exploits advantages of Modelica language in simulationdynamic systems.
A transmission grid based on IEEE 14-Bus is connected to a distribution feeder based onIEEE 13-node test system. In order to perform the connection between both systems, ahybrid interface element is used. It allows the simulation of the transmission network asits single-phase equivalent at the same time that distribution system is simulated as athree-phase model. Two base case scenarios are described: one for the system operatingisolated and another one with the system connected to an infinite bus. For each base case,three scenarios are assembled with levels of distributed generation. The same contingencyis studied under the same simulation parameters in all scenarios.
This work presents the modeling of three-phase elements, of hybrid single-phase/three-phase element and of simple solar photovoltaic generation. Three-phase and hybrid modelsare validated through three di�erent tests and included in a Modelica library for simulationof power systems.
Presented results assess the impact of distributed generation in integrated transmissionand distribution systems. It is observed that machines connected to the transmissiongrid work at di�erent frequencies under di�erent PV penetration. Besides, it is shownthat voltage levels at the transmission grid are not influenced by the level of distributedgeneration. On the other hand, distribution systems presents higher voltage levels forincreasing PV installed capacity. Moreover, current peaks are observed due to the suddenvoltage sag caused by the analyzed contingency. Therefore, the study shows that analysisof integrated distribution and transmission grids are valuable tools for the study of modernpower systems.
Key-words: Simulation. Modeling. Transmission. Distribution. Modelica.
List of Figures
Figure 1 – A HVDC simulator owned by ASEA (From IEEE Transactions onPower Apparatus and Systems, Volume: PAS-97, Page 2058, No. 6,Nov./Dec. 1978 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Figure 2 – Diagram for the studied system in its base case. . . . . . . . . . . . . . 20Figure 3 – Diagram for the studied system with distributed PV generation. . . . . 22Figure 4 – Room temperature during simulation of all DER deployment cases. . . 23Figure 5 – Radiance during simulation of all DER deployment cases. . . . . . . . . 24Figure 6 – AVR block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 7 – Diagram for a single-phase fi element. . . . . . . . . . . . . . . . . . . . 28Figure 8 – Diagram for a three-phase fi element. . . . . . . . . . . . . . . . . . . . 29Figure 9 – Representation of a fi element on each sequence for symmetrical com-
ponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 10 – Representation of a hybrid approximate model, where negative and zero
sequence admittances are considered infinite. . . . . . . . . . . . . . . . 32Figure 11 – Complete hybrid model with negative and zero sequence admittances
considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 12 – Three-phase transmission line fi-equivalent model. . . . . . . . . . . . . 35Figure 13 – Representation of a transformer modeled as fi element. . . . . . . . . . 36Figure 14 – Diagram for wye-connected load. . . . . . . . . . . . . . . . . . . . . . 38Figure 15 – Diagram for delta-connected load. . . . . . . . . . . . . . . . . . . . . . 40Figure 16 – Diagram for wye-connected load. . . . . . . . . . . . . . . . . . . . . . 42Figure 17 – Diagram for delta-connected capacitor bank. . . . . . . . . . . . . . . . 43Figure 18 – PV generation diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 19 – Block diagram for DC link voltage control. . . . . . . . . . . . . . . . . 44Figure 20 – IEEE 4 node test feeder diagram. . . . . . . . . . . . . . . . . . . . . . 45Figure 21 – IEEE 13 node test feeder diagram. . . . . . . . . . . . . . . . . . . . . 47Figure 22 – Modified IEEE 14-bus test feeder diagram. . . . . . . . . . . . . . . . . 48Figure 23 – Bus 1 voltage magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 24 – Complete results for other single-phase modeled buses. . . . . . . . . . 53Figure 25 – Results for three-phase bus 650. . . . . . . . . . . . . . . . . . . . . . . 54Figure 26 – Results for three-phase bus 632. . . . . . . . . . . . . . . . . . . . . . . 55Figure 27 – Frequency for synchronous condenser C2 for SG-connected cases. . . . 56Figure 28 – Frequency for synchronous condenser C2 for isolated cases. . . . . . . . 56Figure 29 – Comparison between voltage magnitude in bus 11 for both base cases. . 57Figure 30 – Voltage magnitude in bus 11 for SG-connected scenarios. . . . . . . . . 58Figure 31 – Voltage magnitude in bus 6 for isolated scenarios. . . . . . . . . . . . . 58Figure 32 – Voltage magnitude in bus 11 for isolated scenarios. . . . . . . . . . . . 59Figure 33 – Comparison of base cases scenarios for three-phase bus 632. . . . . . . 60
Figure 34 – Comparison of 50% PV penetration in isolated and SG-connected sce-narios for three-phase bus 632. . . . . . . . . . . . . . . . . . . . . . . . 61
Figure 35 – Comparison of isolated scenarios for three-phase bus 671. . . . . . . . . 62Figure 36 – Comparison of voltage fluctuation on isolated scenarios for three-phase
bus 634. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 37 – Comparison of SG-connected scenarios current peak for three-phase bus
680. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 38 – Current magnitude drained from bus 11 by the transformer connected
to bus 650. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
List of Tables
Table 1 – Base cases characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . 19Table 2 – Di�erences between base cases. . . . . . . . . . . . . . . . . . . . . . . . 20Table 3 – Case with 30% Penetration. . . . . . . . . . . . . . . . . . . . . . . . . . 21Table 4 – Contingency data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Table 5 – Machine parameter description. Adapted from [36]. . . . . . . . . . . . . 25Table 6 – Summary of simulation data. . . . . . . . . . . . . . . . . . . . . . . . . 26Table 7 – Self and mutual admittance matrices for conventional three-phase trans-
former models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Table 8 – Type of modeling for each component. . . . . . . . . . . . . . . . . . . . 46Table 9 – Contingency data for validation test. . . . . . . . . . . . . . . . . . . . . 47Table 10 – Summary of simulation data for both softwares. . . . . . . . . . . . . . . 48Table 11 – Results for IEEE 4-node first case. . . . . . . . . . . . . . . . . . . . . . 49Table 12 – Results for IEEE 4-node second case. . . . . . . . . . . . . . . . . . . . 50Table 13 – Results for the 13-node test case. . . . . . . . . . . . . . . . . . . . . . . 51Table 14 – Branches configuration for tests. . . . . . . . . . . . . . . . . . . . . . . 71Table 15 – Step-Down transformer configuration. . . . . . . . . . . . . . . . . . . . 71Table 16 – Unbalanced load data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Table 17 – Branch data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Table 18 – XFM-1 transformer configuration. . . . . . . . . . . . . . . . . . . . . . 72Table 19 – Capacitor data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Table 20 – Delta-connected load data. . . . . . . . . . . . . . . . . . . . . . . . . . 72Table 21 – Wye-connected load data. . . . . . . . . . . . . . . . . . . . . . . . . . . 73Table 22 – Branch data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Table 23 – Bus data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Table 24 – Transformer parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 75Table 25 – Three-phase load data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Table 26 – Data for synchronous generators and condensers. . . . . . . . . . . . . . 77Table 27 – PV parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Table 28 – Data for static exciter of each synchronous machine. . . . . . . . . . . . 78
Contents
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1 POWER SYSTEMS SIMULATION AND HYBRID TOOLS . . . . . . 131.2 MODELICA LANGUAGE . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 APPLICATION CONTEXT . . . . . . . . . . . . . . . . . . . . . . . . 16
2 OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 SYSTEM OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . 193.1 BASE CASE SCENARIOS . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 DISTRIBUTED ENERGY RESOURCES DEPLOYMENT SCENARIOS 213.3 STUDIED CONTINGENCY . . . . . . . . . . . . . . . . . . . . . . . . 223.4 RADIANCE AND TEMPERATURE CONDITIONS . . . . . . . . . . 233.5 MACHINE MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.6 EXCITATION SYSTEM MODEL . . . . . . . . . . . . . . . . . . . . . 253.7 SIMULATION DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 MODELING ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.1 Hybrid Single-Phase◊Three-Phase Element . . . . . . . . . . . . . . . . 274.1.1.1 Single-Phase Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.1.2 Three-Phase Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.1.3 Hybrid Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1.2 Three-Phase Transmission Line . . . . . . . . . . . . . . . . . . . . . . . 344.1.3 Three-Phase Transformers . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.4 Three-Phase Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.4.1 Wye-Connected Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.4.2 Delta-Connected Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.5 Three-Phase Shunt Capacitor Bank . . . . . . . . . . . . . . . . . . . . 414.1.6 Three-Phase Photovoltaic Generation . . . . . . . . . . . . . . . . . . . 434.2 VALIDATION TEST SYSTEMS . . . . . . . . . . . . . . . . . . . . . . 454.2.1 IEEE 4-Node Test System . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 IEEE 13-Node Test System . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.3 Modified IEEE 14-Bus Test System . . . . . . . . . . . . . . . . . . . . 47
5 RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . 495.1 VALIDATION RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.1 IEEE 4-Node Test System . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1.2 IEEE 13-Node Test System . . . . . . . . . . . . . . . . . . . . . . . . . 505.1.3 Modified IEEE-14 Bus Test System . . . . . . . . . . . . . . . . . . . . 515.2 INTEGRATED SYSTEM RESULTS . . . . . . . . . . . . . . . . . . . 555.2.1 Machines Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2.2 Bus Voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.2.1 Transmission System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.2.2 Distribution System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2.3 Currents Injected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
APPENDIX A – Validation Test Systems Data . . . . . . . . 71A.1 Data for IEEE 4 Node Test System . . . . . . . . . . . . . . . . . . . . 71A.2 Data for IEEE 13-Node Test System . . . . . . . . . . . . . . . . . . . . 72A.3 Data for Modified IEEE 14-Bus Test System . . . . . . . . . . . . . . . 74A.4 Data for IEEE 14-Bus Plus IEEE 13-Node Tests Systems . . . . . . . . 76
APPENDIX B – Machine, PV and Regulator Data . . . . . . 77B.1 Data for Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77B.2 Data for Solar PV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77B.3 Data for AVRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
12
1 INTRODUCTION
In recent years, the world have faced an important transition regarding energysystems. Major concerns with the environment combined with the necessity for diversifyingthe energy matrix have pushed society to look for alternative sources of energy. As aconsequence, many governments have changed their policies in order to encourage invest-ments in renewable energy increasing the system sustainability. The German Energiewendeor Energy Transition [1] is an example of a policy which have bolstered the country’srenewable energy deployment.
This energy transition have major importance in the power sector, in whichrenewable sources deployment seems to have their main domain [2]. As a matter of fact,renewable-based electricity generation accounted for 60% of world’s usage of renewablesources, and renewable technologies accounted for a quarter of the globe’s generation ofelectricity in 2016[2].
This share is expected to increase in the next years due to fast improvement ofcost and performance of renewable sources such as Solar Photovoltaics (PV) and Wind.Indeed, electricity generation by solar photovoltaic (PV) and wind combined was 20%higher in 2016 if compared with generation from these sources in 2015 [2]. Moreover, therenewable energy supply has increased by 4% a year since 2000 showing that the share ofrenewables in the Power Sector will not slowdown in the near future.
Although this transition is almost always associated with benefits, there areproblems inherent to it that must be addressed. Renewable energy sources are, most times,connected to power grid on medium or low voltage levels as distributed generation or DERs(Distributed Energy Resources), causing challenges to the main grid’s safe operation. Forinstance, voltage fluctuation in German’s power grid in 2012 were responsible for damagesin many industries due to instabilities regarding the country’s transition to renewableenergy [3]. These issues in the power grid are consequence of the lack of considerationof DERs, their control and protection when performing studies in main grid’s operation.Therefore, it became evident that losses caused by these instabilities along with theirmain causes can no longer be neglected. In this context, many tools for analysis ofdistribution and transmission systems’ integration have been proposed in order to addressthese problems.
In this chapter, a basic introduction to the development of power system softwaressince early times, along with new tools for integrated simulation of distribution andtransmission systems is presented. The relevance of using Modelica language for studyingpower systems is also discussed.
13
1.1 POWER SYSTEMS SIMULATION AND HYBRID TOOLS
The modeling and simulation of power grids and its electrical components datesfrom before 1930, when engineers needed to build equivalent systems making analogieswith the real power network [4]. For instance, [5] describes a study made for assessing theperformance of multi-terminal high voltage direct current (HVDC) transmission systemby using a simulator. As it is depicted in Figure 1, such simulator needed highly trainedprofessionals for running simulations.
Figure 1 – A HVDC simulator owned by ASEA (From IEEE Transactions on Power Apparatusand Systems, Volume: PAS-97, Page 2058, No. 6, Nov./Dec. 1978 ).
This situation changed gradually due to important and historical marks, such as theconsolidation of the digital computer. It allowed the simulation of more complex systemsand inspired the development of important algorithms and tools still used today. Iterativemethods in order to calculate power-flows in large systems using digital computers foraccelerating the process have been presented [6]. In addition, [7] and [8], motivated by thelimited memory of computers at that time, presented techniques to explore sparsity inmatrices involved in the simulation of power networks. Later on, [9] used such techniquesin computer simulations for studying electromagnetic transients in single and multi-phasesystems.
As the computer developed and its memory increased, important tools becameconsolidated, such as Newton-Raphson-based algorithms for power flow calculations [10].Besides, the memory growth allowed the simulation of successive time steps, in whichdynamic phenomena could be seen and the system stability could be determined [4].Solid algorithms and techniques were, then, combined in order to develop power systemssoftwares which were able to simulate large networks, performing dynamic modeling ofthese complex grids.
14
Traditionally, these softwares consider transmission and distribution networks astwo independent systems, and analysis of each grid is made individually. Each system usesa di�erent approach in order to provide the best results matching more closely to reality.For instance, the transmission system is often based on a single-phase equivalent (positivesequence) modeling [10, 11]. This is due to the assumption of a balanced power grid,which is the result of transposed transmission lines and specific connection on transformersthat minimize any unbalances. On the other hand, distribution networks are not balancedpower grids because the lines are not transposed and the loads are often unbalanced. Inaddition, single-phase DERs worsen the inherent unbalance. Hence, on a distributionnetwork, simulation tools are commonly based on three-phase models [12].
The increasing deployment of DERs motivated the study presented in [13], in2003, showing that impacts from distributed sources cannot be ignored. In that study,distributed generation is connected to buses on the transmission system in order to assesthe transient stability over di�erent penetration scenarios. In the subsequent year, [14]corroborates that distributed generation, at some penetration level, must be considered fortransient stability analysis. The study also shows that oscillations in centralized generatorsare directly influenced by how loaded are the lines connected to it. Therefore, DERs havepositive impact, decreasing lines’ loading and improving the transmission system transientstability.
Back in 2008, the authors of [15] presented a hybrid power flow formulation forintegrating both transmission and distribution systems. The proposed model is able toperform the connection between a system modeled in its single-phase equivalent, withanother one modeled as three-phase system. This idea is extensively studied in [16] alongwith modeling of many di�erent three-phase components in order to provide a completetool for integrating distribution and transmission systems.
The study presented in [17], 2014, proposes parallel processing techniques aimingto avoid the huge computational e�ort that integration of detailed transmission anddistribution system may have. It shows that a large number of dynamic phenomena issolved with accuracy and high numerical convergence.
The authors in [18] presented a master-slave-splitting (MSS) global power flowmethod. The main idea is to describe the transmission system power flow as a mainproblem, and to state distribution systems power flow as sub-problems. The study providesrigorous convergence analysis in order to validate the proposition. Later on, the studiesproposed in [19] and [20] use MSS based algorithms to study contingency analysis andoptimal power flow in integrated transmission and distribution systems. These studieshighlight that versatility of MSS based tools in order to perform steady-state studies incomplex systems.
The lack of tools to study dynamic three-phase unbalanced systems, motivated
15
the authors in [21] to propose the implementation of Three Phase Dynamics Analyzer(TPDA) for studying such networks. The study reveals that hybrid models, which are ableto perform unbalanced situations, show important information that cannot be obtainedfrom balanced-based models. In addition, the work presented in [22] supports this claimby performing simulations of hybrid systems. The analysis reveals that studies consideringboth transmission and distribution systems provides insights into the behavior of the gridthat would be impossible to obtain if these systems were analyzed individually.
In 2017, the study presented in [23] extended the hybrid formulation proposedin [15, 16]. In this recent study, the transmission system presents the three-sequencemodeling, while the distribution grid has three-phase models. This new approach allowsthe representation of negative and zero sequences in dynamic simulations of transmissionsystems, essential feature for studies related to unbalanced faults.
The evolution of power systems simulation shows that tools have always beencreated in order to provide reliable studies regarding the network. As networks increasedits complexity, tools were adapted to meet the demand for new analysis. The discussionmade in this section, based in the presented works, implies that the development of newtools is essential to studies related to modern power systems.
1.2 MODELICA LANGUAGE
As it was discussed in the previous section, simulation tools have major importanceregarding the study of power systems, and thus many softwares were developed duringthe evolution of simulation. However, there are some issues regarding the compatibility ofsimulations among di�erent softwares. Dynamic modeling is often not consistent betweenplatforms used for simulation. This problem is, many times, related to assumptions andsimplifications used in the modeling process.
For instance, conventional modeling based on block diagrams which can result intwo di�erent and inconsistent models [24, 25]. The transparent exchange of equationsused for modeling the grid could mitigate the di�cult task of evaluating verisimilitudeof power system models. Aiming to solve this issue, authors in [24] propose the usage ofopen modeling, equation-based and object oriented Modelica language [26].
Equation-based languages allow the model description to be made by the usageof di�erential and algebraic equations. There is no need to employ simplifications andassumptions that would be made if a block-diagram-based modeling was made. Inaddition, describing models in such way improve the tool’s flexibility since there is nodetermined execution direction to be followed. Since there is no such thing as predeterminedinput/output flow, Modelica language allows the creation of acausal classes, which canadapt to many data flow directions [25].
16
Another advantage is that the o�ered Functional Mock-up Interface (FMI) allowsthe possibility of exploiting powerful engineering tools compliant to the FMI standard.Indeed, Matlab/Simulink has a FMI toolbox where Modelica models may be used; whilein Mathematica, these models are linked directly to the computational kernel [27]. Thestudies presented in [28, 29] show encouraging results regarding the possibilities of usingFMI technologies.
The work flow on a software using Modelica occurs with a compilation of the modelinto machine code which will be executed by a numerical solver for di�erential, algebraicand discrete equations. Therefore, the compiler and the solver are decoupled from eachother, allowing models to be exchanged between any software environment capable ofreading, compiling and executing Modelica code [30].
Due to these exciting characteristics, many studies support the claim that Modelicalanguage is an interesting alternative for modeling the complexity of modern power system.In fact, [31] describes a multi-level approach to model power electronics in power systemsusing Modelica, while studies [32, 33, 34] reveal the complete feasibility of setting upModelica-based tools to solve large power system models.
In order to take advantage of such notable features, power systems libraries usingModelica language were made. This study focuses only on the iTesla Power System Library(iPSL) for phasor time-domain simulations [35]. The package provides a wide set of powersystem components that are made to meet the demand for tools to analyze modern powersystems, which are complex cyber-physical networks.
1.3 APPLICATION CONTEXT
The increasing complexity of modern grids reveals that tools for studying thegrowing impact of DERs on the whole grid are necessary. This claim is clearly supportedby the number of papers that have been published about this subject in the past few yearsand referred in the last two sections. Therefore, the present work focuses on the field ofsimulation tools for modern power systems.
This work is also concerned with providing mathematical modeling and validationof important system components. Further than that, the simulation of a complete systembuilt with these models is presented and tested.
In order to do so, the current study takes advantage of Modelica language and ofopen-source power system libraries made for Modelica-compliant softwares.
This report is structured as follows: Chapter 2, Objectives, which states themajor and minor goals of the present study. Chapter 3, System Overview, provides thedescriptions of the studied power system, consisting of a transmission grid connected toa distribution feeder, in which DERs are connected to. Two base cases scenarios along
17
with three penetration for each base case are described. Chapter 4, Methods, describesall equations used for modeling of elements along with description of three tests made inorder to validate such models. Chapter 5, Results and Discussions, discusses developedmodels’ validation along with the most important results obtained from simulation ofthe main studied system. Chapter 6, Conclusions, summarize the discussions and resultsobtained in this work.
18
2 OBJECTIVES
The current work was made in order to study the interaction between transmissionsystem and distribution systems with di�erent penetration of DERs. In order to do so,three-phase and hybrid elements were modeled using Modelica as modeling language. Theset of three-phase and hybrid models here developed is entitled ThreePhase and it ispart of Open Instance Power System Library (OpenIPSL) [35], located in folder ExampleApplications. In this study, power flows and dynamic simulations were performed usingOpenModelica as software.
Therefore, it is possible to summarize the major goal along with minor goals. Themain goal for the present study is:
• Study the interactions between transmission systems and distribution systems withpresence of distributed energy resources.
Minor goals for the current work are:
• Model and validate the hybrid single-phase/three-phase interface element, usingModelica;
• Model and validate three-phase elements;
• Model three-phase photovoltaic distributed generation.
19
3 SYSTEM OVERVIEW
This chapter aims to present and describe the system used for assessing theinteractions between transmission and distribution systems with distributed generation.The studied system here described consists of a transmission grid connected to a distributionnetwork by a hybrid element. The transmission grid is considered to be completely balanced,and therefore it can be modeled in its single-phase equivalent. On the other hand, thedistribution network must have its unbalanced nature represented, and thus must bemodeled as a three-phase grid. Therefore, the hybrid element which connects both systemsmust be one that can interface a single-phase and three-phase systems, such as theformulation proposed in [15]. Used single-phase models comes from OpenIPSL [35], whilethree-phase and hybrid models are developed in Chapter 4, Methods. Results for theintegrated distribution and transmission system are presented in Chapter 5, Section 5.2,Integrated System Results.
3.1 BASE CASE SCENARIOS
Two di�erent base cases are determined for this study, in order to compare therelevance of operation in an isolated system. Therefore, two base case scenarios are drawn:one for a system operating in an isolated mode, and one for a system operating connectedto a strong grid (SG), which is represented by an infinite bus. Figure 2 represents thediagram used in both tests, Table 1 summarizes main features of this system, while Tables2 shows the main di�erence between the two base case scenarios. The transmission grid isrepresented by the IEEE 14-bus test system, while the distribution network is representedby the IEEE 13-node test system. Parameters for both systems are described in AppendixA.
Table 1 – Base cases characteristics.
Feature DescriptionTransmission System IEEE 14-bus test system
Connection Bus Bus number 11Distribution System IEEE 13-node test system
Power Base 100 [MVA]System Frequency 60 [Hz]
20
Table 2 – Di�erences between base cases.
Element Isolated Case SG Connected CaseG1 Synchronous Machine Infinite Bus
HybridInterface
Three-PhaseModelling
1
2
3 4
56
7
8
9
10
11
12
13
14
G1
G2
C1
C2
C3
650
632
646 645 633 634
611 684 692 675
671
680652
Figure 2 – Diagram for the studied system in its base case.
21
Note that there is no load connected to Bus 11 in the transmission system. Thisload was replaced by the whole IEEE 13-node test system, which has a similar total load tothe original one. Connection between both systems is made by a transformer using �≠Y g
configuration. In addition, no line voltage regulator was included in the distribution grid.All loads were considered to be constant active and reactive power for steady-state studiesand to be constant impedance for dynamic simulations. The steady-state power flow isdone in order to determine initial conditions necessary for dynamic simulations.
3.2 DISTRIBUTED ENERGY RESOURCES DEPLOYMENT SCENARIOS
Three di�erent scenarios of penetration for each base case system are described andPV generation was considered to be the only DER present in this grid. These tests aimsto assess the impact of di�erent levels of DERs’ installed capacity in the studied system.PV panels are connected to Buses number 634, 675 and 680. In order to provide a morerealistic approach, the PV connected to the low voltage bus 634 is completely unbalanced,while the ones connected to medium voltage buses 675 and 680 have a balanced profile. Inorder to describe the PV installed power capacity for each bus in all cases, Table 3 wascreated. Other parameters for PV generation are described in Appendix B.
Table 3 – Case with 30% Penetration.
Case with 30 % PenetrationBus Phase A [kW] Phase B [kW] Phase C [kW]634 105.42 23.19 134.55675 135 135 135680 135 135 135
Case with 40 % PenetrationBus Phase A [kW] Phase B [kW] Phase C [kW]634 140.56 30.92 179.40675 180 180 180680 180 180 180
Case with 50 % PenetrationBus Phase A [kW] Phase B [kW] Phase C [kW]634 175.70 38.65 224.25675 225 225 225680 225 225 225
The diagram used in these simulations is basically the same used for base casescenarios and it is depicted in Figure 3. The di�erence lies in the representation of
22
distributed PV generation in buses 634, 675 and 680.
HybridInterface
Three-PhaseModelling
1
2
3 4
56
7
8
9
10
11
12
13
14
G1
G2
C1
C2
C3
650
632
646 645 633 634
611 684 692 675
671
680652
PV634
PV675
PV680
Figure 3 – Diagram for the studied system with distributed PV generation.
3.3 STUDIED CONTINGENCY
In order to perform the study of a dynamic phenomena, a fault located near to Bus 4is proposed. The fault has an impedance of jX = 0.5 pu and the duration of 100 ms, when
23
the line between buses 2 and 4 opens and the fault ceases its e�ects. Table 4 summarizesparameters used in simulation. All cases are tested under this same contingency.
Table 4 – Contingency data.
Parameter Value/DescriptionFault Spot Bus 4
Fault Impedance jX = 0.5 pu
Fault Instant 12 s
Fault Duration 100 ms
Line Opened Line 4 ≠ 2
3.4 RADIANCE AND TEMPERATURE CONDITIONS
Since there are PV panels in the DER deployment scenarios, the variation ofradiance or temperature would influence the panel’s output power engendering anotherinteresting dynamic phenomena. Room temperature is di�cult to change significantlywithin seconds, and thus only radiance is chosen to vary during the simulation. Figures4 and 5 show that temperature is kept constant in Te = 22.5 [¶C] while the radiancehas a linear variation within 2s (starting in t = 11s and ending in t = 13s) fromHa = 744.0 [W/m
2] to Ha = 297.6 [W/m2] (60% reduction).
0 2 4 6 8 10 12 14 16 18 2021
22
23
24
Time [s]
Tempe
rature
[�C]
Figure 4 – Room temperature during simulation of all DER deployment cases.
24
0 2 4 6 8 10 12 14 16 18 20200
400
600
800
Time [s]
Rad
ianc
e[W
/m2 ]
Figure 5 – Radiance during simulation of all DER deployment cases.
3.5 MACHINE MODELS
The simulations performed in this study, aims to assess the dynamic performanceof this system, under di�erent conditions. Therefore, machine and its automatic voltageregulator models have critical importance in the present work. In this study, G standsfor a synchronous generator (or infinite bus for G1 in the SG-connected cases) and Cstands for synchronous condenser. In the studied system, the model used for synchronousmachine is the Order VI available in OpenIPSL and modeled in [36]. The set of di�erentialequations 3.1 defines the synchronous machine model. Data used in this study for allmachines (G1, G2, C1, C2 and C3) are presented in Appendix B.
Y_________________]
_________________[
” = 2fif · (Ê ≠ 1),
Ê = 12H
[Pm ≠ Pe ≠ D · (Ê ≠ 1)] ,
eÕq
= 1T
Õd
·3
≠eÕq
≠5xd ≠ x
Õd
≠ TÕÕd
TÕd
xÕÕd
xÕd(xd ≠ x
Õd)6
id +3
1 ≠ TaaT
Õd
4vf
4,
eÕÕq
= 1T
ÕÕd
·3
≠eÕÕq
+ eÕq
≠5x
Õd
≠ xÕÕd
+ TÕÕd
TÕd
xÕÕd
xÕd(xd ≠ x
Õd)6
id + TaaT
Õd
vf
4,
eÕd
= 1T Õ
q·
3≠e
Õd
≠5xq ≠ x
Õq
≠ TÕÕq
T Õq
xÕÕq
xÕq(xq ≠ x
Õq)6
iq
4,
eÕÕd
= 1T ÕÕ
q·
3≠e
ÕÕd
+ eÕd
+5x
Õq
≠ xÕÕq
+ TÕÕq
T Õq
xÕÕq
xÕq(xq ≠ x
Õq)6
iq
4
(3.1)
where the rotor angle ”, rotor speed Ê, q-axis transient voltage eÕq, q-axis subtransient
voltage eÕÕq, d-axis transient voltage e
Õd, and d-axis subtransient voltage e
ÕÕd
are the statevariables, and
Y____]
____[
Pe = (vq + raiq)iq + (vd + raid)id,
0 = vq + raiq ≠ eÕÕq
+ (xÕÕd
≠ xl)id,
0 = vd + raid ≠ eÕÕd
+ (xÕÕq
≠ xl)iq
(3.2)
Besides, Table 5 summarizes the definition for all parameters used in the set ofequations (3.1) and (3.2).
25
Table 5 – Machine parameter description. Adapted from [36].
Parameter Description Unitf Frequency rating HzH Inertia constant kWs/kVAPm Mechanical power p.u.Pe Electrical power p.u.D Damping coe�cient -T
Õd
d-axis open circuit transient time constant sT
ÕÕd
d-axis open circuit subtransient time constant sTaa d-axis additional leakage time constant sT
Õq
q-axis open circuit transient time constant sT
ÕÕq
q-axis open circuit subtransient time constant sxd d-axis synchronous reactance p.u.x
Õd
d-axis transient reactance p.u.x
ÕÕd
d-axis subtransient reactance p.u.xq q-axis synchronous reactance p.u.x
Õq
q-axis transient reactance p.u.x
ÕÕq
q-axis subtransient reactance p.u.xl Leakage reactance p.u.ra Armature resistance p.u.vf Field voltage p.u.vd d-axis terminal voltage p.u.vq q-axis terminal voltage p.u.id d-axis current p.u.iq q-axis current p.u.
3.6 EXCITATION SYSTEM MODEL
The model for the direct current commutator exciter used in this study is presentedin Figure 6. It is an automatic voltage regulator (AVR) based on the Type DC2A from[37]. The inputs of this models are the output from the transducer of terminal voltage vtr,which is subtracted from the reference Vref. In addition, the stabilizing feedback is alsosubtracted in order to produce an error signal which is amplified in the regulator. Thisregulators often are una�ected by short transients om synchronous machines. Moreover,the data for all parameters’ values used in this study are presented in Appendix B.
26
11+sTr
Vtr
Vref
+
≠
≠ q 1+sTC1+sTB
KA1+sTA
VRmax
VRmin
≠
+ q 1sTE
KE
sKF1+sTF
Efd
Figure 6 – AVR block diagram.
3.7 SIMULATION DATA
The Modelica-based software used for simulations in this study is the OpenMod-elica[38]. Start time of simulation is 0s and the stop time is 20s and 4000 intervals areanalyzed. Therefore, each interval has 0.005s. The dassl integration method was adopted,with tolerance of 0.0001 and coloredNumerical Jacobians. The entire set-up is summarizedin Table 6.
Table 6 – Summary of simulation data.
Parameter Value/DescriptionTime Simulated 20 s
Interval 0.0001 s
Tolerance 0.0001Method dasslJacobian coloredNumerical
27
4 METHODS
This chapter presents the modeling of three-phase and hybrid elements along withsingle-phase PV generation, which are necessary for studying the system described inChapter 3, System Overview. The first section, Modeling Elements, describes the modelingof the hybrid interface element, three-phase components and of a single-phase solar PVgeneration. These models are used to assemble the three-phase modeled distribution systemalong with the connection to the transmission network. The second section, ValidationTest Systems, describes three tests to validate the models developed in this chapter. Theresults from the validation tests are presented in Chapter 5, Section 5.1, Validation Results.
4.1 MODELING ELEMENTS
4.1.1 Hybrid Single-Phase◊Three-Phase Element
This subsection presents the mathematical modeling of a hybrid single-phase◊three-phase element. This element is crucial to this work, since it makes the interface betweentransmission systems modeled in single-phase (positive sequence only) with distributionsystems modeled in three-phase. The hybrid formulation in which the model is based onis proposed and described in [15, 16].
The modeling of this hybrid element is divided in the next three subsubsections.The first subsubsection is dedicated to describe the single phase element modeling, whilethe second subsubsection is dedicated to the three phase modeling of a passive fi element.Finally, the third and last section shows how the hybrid element was modeled. It should benoted that all subsubsections are based in modeling passive fi elements as it is suggestedin [15, 16]. Transformers and transmission lines which have fi-equivalent models can havetheir own hybrid three-phase ◊ single-phase versions.
4.1.1.1 Single-Phase Modeling
In single phase modeling only the positive sequence is considered while negative andzero sequence components are ignored. This type of model is mostly used on simulation ofsystems that can be considered completely balanced, such as transmission systems.
On a fi element, two terminals, k and m, are connected using three components. Onthis model, each component is represented by a positive sequence equivalent admittance.One component is connected in series between the two terminals (y+
ser), while the other
two are connected as shunt elements. The first one is connected on terminal k (y+shtk
) whilethe second one is on m (y+
shtm). The graphic representation of this element is shown in
Figure 7.
28
≠
+
v+k
i+k
y+ser
≠
+
v+m
i+m
y+shtk
y+shtm
Figure 7 – Diagram for a single-phase fi element.
Considering the circuit in Figure 7, it is possible to write the following set ofequations:
Y_]
_[
i+k
= y+shtk
v+k
+ y+ser
(v+k
≠ v+m
),
i+m
= ≠y+ser
(v+k
≠ v+m
) + y+shtm
v+m
(4.1)
That can be simplified to:
Y_]
_[
i+k
= (y+ser
+ y+shtk
)v+k
≠ y+ser
v+m
,
i+m
= ≠y+ser
v+k
+ (y+ser
+ y+shtm
)v+m
(4.2)
Using the equations above, it is possible to write the system of equations on amatrix form that can be seen below. This is the basic equation of a single-phase fi element.
S
U i+k
i+m
T
V =S
Uy+ser
+ y+shtk
≠y+ser
≠y+ser
y+ser
+ y+shtm
T
V
S
U v+k
v+m
T
V (4.3)
4.1.1.2 Three-Phase Modeling
This model has a representation for each of the phase components, a, b and c.Therefore it is mostly used on power systems that are completely unbalanced, such asdistribution systems. The graphic representation is analog to the one that was shown inFigure 7. The main di�erence is that components and values are not scalar, but arraysand matrices in this model. Hence, capital letters were used to represent these values.The diagram for a in Figure 8.
Although the values are now matrices and arrays, the equations that can be derivedfrom the three-phase model above are similar to those derived from the single-phase.
29
≠
+
Vabc
k
Iabc
k
Yabc
ser
≠
+
Vabc
m
Iabc
m
Yabc
shtkY
abc
shtm
Figure 8 – Diagram for a three-phase fi element.
Therefore:
Y_]
_[
Iabc
k= Y
abc
serV
abc
k+ Y
abc
shtk(V abc
k≠ V
abc
m),
Iabc
m= ≠Y
abc
ser(V abc
k≠ V
abc
m) + Y
abc
shtmV
abc
m
(4.4)
Organizing the system of matrix equations above, it is possible to write:
Y_]
_[
Iabc
k= (Y abc
ser+ Y
abc
shtk)V abc
k≠ Y
abc
shtkV
abc
m,
Iabc
m= ≠Y
abc
serV
abc
k+ (Y abc
ser+ Y
abc
shtm)V abc
m
(4.5)
Finally, it is possible to combine system (4.5) above in a matrix equation (4.6)below. This is the basic model equation for a three-phase fi element.
S
U Iabc
k
Iabc
m
T
V
6◊1
=S
UYabc
ser+ Y
abc
shtk≠Y
abc
shtk
≠Yabc
serY
abc
ser+ Y
abc
shtm
T
V
6◊6
S
U Vabc
k
Vabc
m
T
V
6◊1
(4.6)
4.1.1.3 Hybrid Modeling
In order to develop the hybrid model it is necessary to separate equation (4.6)into two equations. Equation (4.7) describes phase currents injected on terminal k, whileequation (4.8) describes the ones injected on terminal m.
ËI
abc
k
È
3◊1=
ËY
abc
ser+ Y
abc
shtk
È
3◊3
ËV
abc
k
È
3◊1+
Ë≠Y
abc
ser
È
3◊3
ËV
abc
m
È
3◊1(4.7)
ËI
abc
m
È
3◊1=
Ë≠Y
abc
ser
È
3◊3
ËV
abc
k
È
3◊1+
ËY
abc
ser+ Y
abc
shtm
È
3◊3
ËV
abc
m
È
3◊1(4.8)
Systems which are modeled as single-phase equivalent have only positive sequencesources and are considered balanced. Therefore, if such system is connected to terminal
30
k, assumptions shown in equations (4.9) and (4.10) are valid. Besides, it means thatevery voltage and current unbalance has its origins in terminal m, which is modeled asthree-phase.
i≠k
= i0k
= 0 (4.9)
v≠k
= v0k
= 0 (4.10)
In addition, positive sequence power is going to flow through the hybrid element.On the other hand, negative and zero sequence power are going to flow through equivalentnorton admittances y
≠nrtk
, and y0nrtk
, located on terminal k [16]. The diagram for allsequences is shown in Figure 9.
i+k
y+ser
≠
+
v+m
i+m
y+shtk
y+shtm
Positive Sequence
y≠nrtk
y≠ser
≠
+
v≠m
i≠m
y≠shtk
y≠shtm
Negative Sequence
y0nrtk
y0ser
≠
+
v0m
i0m
y0shtk
y0shtm
Zero Sequence
Figure 9 – Representation of a fi element on each sequence for symmetrical components.
31
Consider now equation (4.11) for converting terminal k phase voltages in symmet-rical components, and (4.12) for converting the terminal’s current.
Vabc
k=
S
WWWU
1 1 11 –
2–
1 – –2
T
XXXV · V0+≠
k=
S
WWWU
1 1 11 –
2–
1 – –2
T
XXXV ·
S
WWWU
v0k
v+k
v≠k
T
XXXV (4.11)
I0+≠k
=
S
WWWU
i0k
i+k
i≠k
T
XXXV = 13
S
WWWU
1 1 11 – –
2
1 –2
–
T
XXXV Iabc
k(4.12)
where – = ej
2fi3 .
If equations (4.9) and (4.10) are valid, then it is possible to summarize equations(4.11) and (4.12) into the set of equations (4.13) below.
Y________]
________[
Vabc
k= T1 · v
+k
=
S
WWWWU
1
–2
–
T
XXXXVv
+k
i+k
= T2 · Iabc
k= 1
3
51 – –
26
Iabc
k
(4.13)
Consider that the values for negative and zero sequence norton equivalent ad-mittances on terminal k, y
≠nrt and y
0nrt
are not known, or that the voltage unbalanceon terminal k is negligible. Therefore, the assumption presented in equation (4.14) isreasonable [15]. The resulting model will provide approximated results and, therefore, itwill be considered an approximated model.
y≠nrt
= y0nrt
= Π(4.14)
Hence, if equations (4.7) and (4.8) have their values for Vabc
kand i
+k
replaced forthe values from (4.13), it will be possible to write equations (4.15) and (4.16) below.
i+k
= T2 ·ËY
abc
ser+ Y
abc
shtk
È
3◊3· T1 · v
+k
+ T2 ·Ë≠Y
abc
ser
È
3◊3
ËV
abc
m
È
3◊1(4.15)
ËI
abc
m
È
3◊1=
Ë≠Y
abc
ser
È
3◊3· T1 · v
+k
+ËY
abc
ser+ Y
abc
shtm
È
3◊3
ËV
abc
m
È
3◊1(4.16)
Finally, it is possible to write the equation that is used on the hybrid single-phase◊ three-phase approximate model. The diagram that represents such model is shown in
32
Figure 10.
S
U i+k
Iabc
m
T
V
4◊1
=S
UT2 ·1Y
abc
ser+ Y
abc
shtk
2· T1 T2 ·
1≠Y
abc
ser
2
1≠Y
abc
ser
2· T1 Y
abc
ser+ Y
abc
shtm
T
V
4◊4
S
U v+k
Vabc
m
T
V
4◊1
(4.17)
≠
+
v+k
i+k
Yabc
ser
≠
+
Vabc
m
Iabc
m
Yabc
shtkY
abc
shtm
Figure 10 – Representation of a hybrid approximate model, where negative and zero sequenceadmittances are considered infinite.
However, if the negative and zero sequence equivalent admittances on terminal k
have known finite values, their e�ects must be included on the fi element. Precision ofresults depends on the inclusion of these elements. The analysis made below is describedin [15] and [16].
The first step to include negative and zero admittances is to the calculate nortonequivalent admittance matrix on phase components. Equation (4.18) presents the calcula-tion needed to obtain Y
abc
nrt. This step is made considering that there is only one hybrid
element used to interface the single-phase modeled system with the three-phase one. Iftwo or more hybrid components are used in order to connect the same two subsystem,another analysis must be made. This analysis is presented in [16] and it is not made inthis work.
Yabc
nrt=
S
WWWU
1 1 11 –
2–
1 – –2
T
XXXV ·
S
WWWU
y0nrt
y+nrt
y≠nrt
T
XXXV · 13 ·
S
WWWU
1 1 11 – –
2
1 –2
–
T
XXXV (4.18)
Now, consider the two following matrices that will work as a positive sequencefilter and a zero and negative sequece filter, respectively.
F+ =
S
WWWU
1 1 11 –
2–
1 – –2
T
XXXV ·
S
WWWU
01
0
T
XXXV · 13 ·
S
WWWU
1 1 11 – –
2
1 1 –
T
XXXV (4.19)
33
F≠0 =
S
WWWU
1 1 11 –
2–
1 – –2
T
XXXV ·
S
WWWU
10
1
T
XXXV · 13 ·
S
WWWU
1 1 11 – –
2
1 1 –
T
XXXV (4.20)
Equations (4.19) and (4.20) can still be simplified to the following equations:
F+ = 1
3
S
WWWU
1 – –2
–2 1 –
– –2 1
T
XXXV (4.21)
F0≠ = 1
3
S
WWWU
2 ≠– ≠–2
≠–2 2 ≠–
≠– ≠–2 2
T
XXXV (4.22)
The filters will help matrix Yabc
nrtbe added to the model. In the model, terminal
k is only modeled in positive sequence and it is considered to be completely groundedfor zero and negative sequence when it is seen from terminal m. Therefore, the e�ectsof negative and zero admittance must be take into account in terminal m. In order todo so, it is necessary to filter series admittance Y
abc
ser, multiplying it by F
+ and allowingonly positive sequence current to flow trough the branch between k and m. Furthermore,the negative and zero sequence currents that would flow through the fi element musthave their e�ects considered. In order to do so, a parallel association between Y
abc
nrtand
Yabc
shtkmust be done, which must be series associated with Y
abc
ser. Since the whole system is
already modeled for positive sequence, the matrix combination is filtered for the othersymmetrical components, being multiplied by F
0≠ and resulting in Y0≠
sht. Finally, it is
connected as a shunt element in terminal m.
The following routine was made in order to simplify the steps made to include thee�ects of finite negative and zero sequence admittances:
1. In terminal m, Yabc
shtmand the following matrix Y
0≠sht
must be added in parallel.
Y0≠
sht= F
0≠51
Yabc
nrt+ Y
abc
shtk
2≠1+
1Y
abc
ser
2≠16≠1(4.23)
2. Multiply the matrix Yabc
serby the positive sequence filter matrix.
Ynew
ser= F
+ · Yabc
ser(4.24)
Note that, in order to complete the first step, Yabc
sermust be an invertible matrix.
Therefore, this equation is not valid for a transformer that introduces a phase displacement.To solve this problem [15] suggests to relocate the hybrid interface element to an adjacentbranch, that does not contain an element introducing phase displacement.
34
Finally, the diagram for the hybrid single-phase ◊ three-phase complete model isdepicted in Figure 11. Its matrix model is described in equation (4.25) below.
S
U i+k
Iabc
m
T
V =S
UT2 ·1Y
new
ser+ Y
abc
shtk
2· T1 T2 · (≠Y
new
ser)
(≠Ynew
ser) · T1 Y
new
ser+
1Y
abc
shtm+ Y
0≠sht
2
T
V
S
U v+k
Vabc
m
T
V (4.25)
≠
+
v+k
i+k
Ynew
ser
≠
+
Vabc
m
Iabc
m
Yabc
shtk(Y abc
shtm+ Y
0≠sht
)
Figure 11 – Complete hybrid model with negative and zero sequence admittances considered.
4.1.2 Three-Phase Transmission Line
This subsection presents the mathematical modeling for a three-phase transmissionline. Such element is modeled as a fi-equivalent model such as suggested by [39]. However,instead of considering the series element as an impedance, it is considered as an admittance.This minor change was made in order to standardize all elements in the equation asadmittance elements.
The fi-equivalent model for a three-phase transmission line is similar to the onedescribed in subsubsection Three-Phase Modeling. Its diagram is shown in Figure 12. Notethat in this model, Y
abc
serwas separated in G
abc
ser+ jB
abc
serand that Y
abc
shtis considered to have
only an imaginary part, jBabc
sht. Their values are presented on equations (4.26), (4.27) and
(4.28) below.
Gabc
ser=
S
WWWU
Gaa
serG
ab
serG
ac
ser
Gba
serG
bb
serG
bc
ser
Gca
serG
cb
serG
cc
ser
T
XXXV =
S
WWWU
Gaa
serG
ab
serG
ac
ser
Gab
serG
bb
serG
bc
ser
Gac
serG
bc
serG
cc
ser
T
XXXV (4.26)
Babc
ser=
S
WWWU
Baa
serB
ab
serB
ac
ser
Bba
serB
bb
serB
bc
ser
Bca
serB
cb
serB
cc
ser
T
XXXV =
S
WWWU
Baa
serB
ab
serB
ac
ser
Bab
serB
bb
serB
bc
ser
Bac
serB
bc
serB
cc
ser
T
XXXV (4.27)
35
Babc
shtk= B
abc
shtm= B
abc
sht=
S
WWWU
Baa
shtB
ab
shtB
ac
sht
Bba
shtB
bb
shtB
bc
sht
Bca
shtB
cb
shtB
cc
sht
T
XXXV =
S
WWWU
Baa
shtB
ab
shtB
ac
sht
Bab
shtB
bb
shtB
bc
sht
Bac
shtB
bc
shtB
cc
sht
T
XXXV (4.28)
≠
+
Vabc
k
Iabc
k
Gabc
ser+ jB
abc
ser
≠
+
Vabc
m
Iabc
m
jBabc
shtkjB
abc
shtm
Figure 12 – Three-phase transmission line fi-equivalent model.
Note that this model allows the representation of both balanced and unbalancedlines [16]. Besides, it is important to observe that all conductance and susceptance matricesare symmetric.
Considering the equation for a fi element and the model presented in Figure 12 thefinal equation for a three-phase transmission line model is presented below.
S
U Iabc
k
Iabc
m
T
V =S
UGabc
ser+ j(Babc
ser+ B
abc
sht) ≠G
abc
ser≠ jB
abc
ser
≠Gabc
ser≠ jB
abc
serG
abc
ser+ j(Babc
ser+ B
abc
sht)
T
V
S
U Vabc
k
Vabc
m
T
V (4.29)
4.1.3 Three-Phase Transformers
This subsection presents the mathematical modeling for three-phase transformersin di�erent types of connections. This element and its conventional connections, suchas �-Wye, are well documented and studied. Therefore, there are plenty of models inliterature [16, 39, 40, 42, 41]. In general, the two magnetic coupled coils are representedas a di�erent type of fi element as shown in Figure 13. The di�erence consists in twotypes of series admittance between primary side (terminal k) and secondary side (terminalm)[39, 16]. Besides, each transformer connection presents a di�erent set of admittancematrices that need to be studied and di�erentiated.
36
≠
+
Vabc
k
Iabc
k
Yabc
ser,k|Y abc
ser,m
≠
+
Vabc
m
Iabc
m
Yabc
shtkY
abc
shtm
Figure 13 – Representation of a transformer modeled as fi element.
Consider a transformer defined by its primary-secondary dispersion resistance rt
and reactance xt. Then, it is possible to find its primary-secondary dispersion admittanceyt in equation (4.30):
yt = rt
(rt)2 + (xt)2 ≠ jxt
(rt)2 + (xt)2 (4.30)
Now, it is reasonable to define three matrices Y1, Y2 and Y3 that appear frequentlyin calculation of self and mutual transformer admittances [16].
Y1 =
S
WWWU
yt
yt
yt
T
XXXV , Y2 =
S
WWWU
2yt ≠yt ≠yt
≠yt 2yt ≠yt
≠yt ≠yt 2yt
T
XXXV , Y3 =
S
WWWU
≠yt yt
≠yt yt
yt ≠yt
T
XXXV (4.31)
Furthermore, Table 7 summarize all possible sets of self and mutual admittancesmatrices, based on [39, 40, 16] using variables Y1, Y2 and Y3 defined in (4.31).
Table 7 – Self and mutual admittance matrices for conventional three-phase transformer models.
Connection Self-Admittance Mutual-AdmittanceTerminal k Terminal m Ykk Ymm Ymk Ykm
Grounded Wye Grounded Wye Y1 Y1 ≠Y1 ≠Y1Grounded Wye Wye 1
3Y213Y2 ≠1
3Y2 ≠13Y2
Grounded Wye � Y1 Y2 Y3 YT
3Wye Wye 1
3Y213Y2 ≠1
3Y2 ≠13Y2
Wye � 13Y2 Y2 Y3 Y
T
3� � Y2 Y2 ≠Y2 ≠Y2
In addition, if a transformer has a – : — tap configuration, it is necessary to includeit in the equations. In order to do so, Ykk must be divided by –
2, Ymm by —2, Ykm by –—
37
and Ymk by —– [16]. Moreover, it is important to remember that a � connection has ainherent tap of
Ô3 which must be considered. Therefore, if a transformer is represented
by Figure 13, the set of equations (4.32) must be considered.
Y________]
________[
Vabc
ser,k= ≠ 1
–—Ykm
Yabc
ser,m= ≠ 1
—–Ymk
Yabc
shtk= 1
–2 Ykk + 1–—
Ykm
Yabc
shtm= 1
—2 Ymm + 1—–
Ymk
(4.32)
Finally, the matrix equation for a fi model of a transformer is:S
U Iabc
k
Iabc
m
T
V =S
UYabc
ser,p+ Y
abc
shtk≠Y
abc
ser,p
≠Yabc
ser,sY
abc
ser,s+ Y
abc
shtm
T
V
S
U Vabc
k
Vabc
m
T
V (4.33)
4.1.4 Three-Phase Loads
This subsection describes how wye-connected and delta-connected loads are modeled.Normally, three-phase load models have individual specifications for active and reactivepower consumed in each phase (wye-connected loads) or between phases (delta-connectedloads). Furthermore, active and reactive power can be considered as having constantvalues or they can be modeled as functions of terminal voltage [16]. For instance, a ZIPmodel allows the description of a load as a composition of its constant power, current andimpedance parts which are functions of terminal voltage. In order to provide models withdi�erent applications, each load is modeled as constant power and as ZIP model in thiswork. Besides, two-phase and single-phase load models are also provided.
4.1.4.1 Wye-Connected Loads
The diagram shown in Figure 14 represents a three-phase wye-connected load.Equations for this model could be written to represent each phase independently[43]. It ispossible to write equation (4.34), considering current phasors ia, ib and ic as having thedirections as depicted in Figure 14.
Y____]
____[
Sa = Pa + jQa = van · iúa
Sb = Pb + jQb = vbn · iúb
Sc = Pc + jQc = vcn · iúc
(4.34)
38
where superscript (ú) over x stands for the complex conjugate of variable x, and
Y____]
____[
van = va ≠ vn
vbn = vb ≠ vn
vcn = vc ≠ vn
(4.35)
ia
Pa + jQ
a
Pc + jQ
cP
b + jQb
ib
ic
vn
va
vb
vb
Figure 14 – Diagram for wye-connected load.
Equations for active and reactive power are independent for each phase and thereforethey can be described as one simple matrix equation, presented in (4.36). Note that thissame equation can be used for two-phase and single-phase loads.
S
WWWU
Pa + jQa
Pb + jQb
Pc + jQc
T
XXXV =
S
WWWU
van
vbn
vcn
T
XXXV
S
WWWU
iúa
iúb
iúc
T
XXXV (4.36)
Equations for loads using ZIP model describes how active and reactive powers varyaccording to terminal voltages. Two models are well documented in literature, exponentialand polynomial. In this work only the latter model is explored and single-phase equationsfor it are shown in (4.37) below.
Y_]
_[
Px(vxn) = Pp + Pi ·1
v0vxn
2+ Pz ·
1v0
vxn
22
Qx(vxn) = Qp + Qi ·1
v0vxn
2+ Qz ·
1v0
vxn
22 (4.37)
where subscript (p), (i) and (z) stands for constant power, current and impedance respec-tively and v0 represents the voltage when active and reactive power Pi, Qi, Pz, Qz aremeasured. However, it is possible to assign percentages – of total load P
0x
+ jQ0x
to eachdi�erent type of load represented by subscripts (p), (i) and (z) [43]. Besides, consider that
39
all loads are measured on voltage v0 = 1 pu. Hence, it is possible to rewrite equation(4.37) as (4.38) below.
Y__]
__[
Px(vxn) = P0x
5–
p
x+ –
i
x
11
vxn
2+ –
z
x
11
vxn
226
Qx(vxn) = Q0x
5–
p
x+ –
i
x
11
vxn
2+ –
z
x
11
vxn
226 (4.38)
where
–p
x+ –
i
x+ –
z
x= 1 (4.39)
Therefore, matrix equation (4.40) is written for a wye-connected three-phase ZIPmodel. Note that values for active and reactive power are influenced by each phase terminalvoltage.
S
WWWU
(Pa + jQa) “a
(Pb + jQb) “b
(Pc + jQc) “c
T
XXXV =
S
WWWU
van
vbn
vcn
T
XXXV
S
WWWU
iúa
iúb
iúc
T
XXXV (4.40)
where
Y____]
____[
“a = –p
a+ –
i
a
11
van
2+ –
z
a
11
van
22
“b = –p
b+ –
i
b
11
vbn
2+ –
z
b
11
vbn
22
“c = –p
c+ –
i
c
11
vcn
2+ –
z
c
11
vcn
22(4.41)
4.1.4.2 Delta-Connected Loads
A diagram for a delta-connected load is shown in Figure 15 below. Analyzing thediagram, it is possible to write an equation for active and reactive power between eachpair of phases [43, 44]. This set of equations is written on equation (4.42) below.
Y____]
____[
Sab = Pab + jQab = vab · iúab
Sbc = Pbc + jQbc = vbc · iúbc
Sca = Pca + jQca = vca · iúca
(4.42)
where Y____]
____[
vab = va ≠ vb
vbc = vb ≠ vc
vca = vc ≠ va
(4.43)
40
and Y____]
____[
ia = iab ≠ ica
ib = ibc ≠ iab
ic = ica ≠ ibc
(4.44)
ia
ib
ic
va
vb
vc
Pab + jQ
ab Pca + jQ
ca
Pbc + jQ
bc
Figure 15 – Diagram for delta-connected load.
However, it may be desirable to know phase currents ia, ib and ic. Therefore,considering equations (4.44), and (4.42) it is possible to write (4.44) below.
Y____]
____[
ia =1
Pab+jQabvab
2ú≠
1Pca+jQca
vca
2ú
ib =1
Pbc+jQbcvbc
2ú≠
1Pab+jQab
vab
2ú
ic =1
Pca+jQca
vca
2ú≠
1Pbc+jQbc
vbc
2ú
(4.45)
Therefore, considering constant active and reactive loads, (4.45) can be re-writtenin a matrix form as (4.46) below.
Iabc =
S
WWWU
ia
ib
ic
T
XXXV =
S
WWWU
1Pab+jQab
vab
2ú≠
1Pca+jQca
vca
2ú
1Pbc+jQbc
vbc
2ú≠
1Pab+jQab
vab
2ú
1Pca+jQca
vca
2ú≠
1Pbc+jQbc
vbc
2ú
T
XXXV (4.46)
For a ZIP model, it is necessary to use equations (4.38) and (4.39) in order torepresent the partition of loads Pab +jQab, Pbc +jQbc and Pca +jQca into parts representingconstant power, current and impedance parts. Making the same assumption that reference
41
voltage v0 = 1 pu,it is possible to write the matrix equation (4.47) below.
Iabc =
S
WWWU
ia
ib
ic
T
XXXV =
S
WWWU
1Pab+jQab
vab
2ú· “ab ≠
1Pca+jQca
vca
2ú· “ca1
Pbc+jQbcvbc
2ú· “bc ≠
1Pab+jQab
vab
2ú· “ab1
Pca+jQca
vca
2ú· “ca ≠
1Pbc+jQbc
vbc
2ú· “bc
T
XXXV (4.47)
where
Y____]
____[
“ab = –p
ab+ –
i
ab
11
vab
2+ –
z
ab
11
vab
22
“bc = –p
bc+ –
i
bc
11
vbc
2+ –
z
bc
11
vbc
22
“ca = –p
ca+ –
i
ca
11
vca
2+ –
z
ca
11
vca
22(4.48)
Note that, for both constant PQ and ZIP model, if there is a load only betweentwo phases, say a and b, it is necessary to consider all the other active and reactive powersexcept for Pab + jQab are equals to zero. The final equation for this case is (4.49) forconstant PQ model and (4.50) for ZIP model.
Iabc =
S
WWWU
ia
ib
ic
T
XXXV =
S
WWWU
1Pab+jQab
vab
2ú≠
10
vca
2ú
10
vbc
2ú≠
1Pab+jQab
vab
2ú
10
vca
2ú≠
10
vbc
2ú
T
XXXV =
S
WWWU
1Pab+jQab
vab
2ú
≠1
Pab+jQabvab
2ú
0
T
XXXV (4.49)
Iabc =
S
WWWU
ia
ib
ic
T
XXXV =
S
WWWU
1Pab+jQab
vab
2ú· “ab ≠
10
vca
2ú· “ca1
0vbc
2ú· “bc ≠
1Pab+jQab
vab
2ú· “ab1
0vca
2ú· “ca ≠
10
vbc
2ú· “bc
T
XXXV =
S
WWWU
1Pab+jQab
vab
2ú· “ab
≠1
Pab+jQabvab
2ú· “ab
0
T
XXXV (4.50)
4.1.5 Three-Phase Shunt Capacitor Bank
This subsection presents the modeling of three-phase capacitor banks. Theseelements are commonly used in power systems to help voltage regulation as well as toprovide reactive power support for the grid, as it is stated in [43]. Although capacitorbanks are modeled as shunt elements, such as loads, they must be considered as constantimpedance. Hence, it is necessary to consider –
p
x= –
i
x= 0 and –
z
x= 1. Moreover, since
capacitor banks can be wye-connected or delta connected, it is necessary to write equationsfor both cases.
For a three-phase grounded wye-connected capacitor bank, it is necessary to useequation (4.40). Making the active power for each phase equal to zero and admittingthat reactive power Qa, Qb and Qc are delivered when phase voltage is equal to 1 pu it ispossible to write matrix equation (4.51). Figure 16 shows the diagram for the model of a
42
wye-connected capacitor bank. Note that the same equation could be used for single-phaseor two-phase grounded wye-connected capacitor banks.
S
WWWWU
jQa
11
van
22
jQb
11
vbn
22
jQc
11
vcn
22
T
XXXXV=
S
WWWU
van
vbn
vcn
T
XXXV
S
WWWU
iúa
iúb
iúc
T
XXXV (4.51)
ia
jQa
jQc
jQb
ib
ic
vn
va
vb
vb
Figure 16 – Diagram for wye-connected load.
On the other hand, for three-phase delta-connected capacitor banks, it is necessaryto use equation (4.47). Similarly to the grounded wye-connection, constant impedanceloads are used in order to express the equation for the capacitor bank. Hence, it is possibleto write the matrix equation (4.52) below. Figure 17 shows the diagram for the model ofa delta-connected capacitor bank. Moreover, note that this same equation can be used forone capacitor bank connected only between two phases. In this case, the final equationwould be similar to (4.50) described for loads connected between two phases.
Iabc =
S
WWWU
ia
ib
ic
T
XXXV =
S
WWWWU
1jQabvab
2ú·
11
vab
22≠
1jQca
vca
2ú·
11
vca
22
1jQbcvbc
2ú·
11
vbc
22≠
1jQabvab
2ú·
11
vab
22
1jQca
vca
2ú·
11
vca
22≠
1jQbcvbc
2ú·
11
vbc
22
T
XXXXV(4.52)
43
ia
ib
ic
va
vb
vc
jQab jQ
ca
jQbc
Figure 17 – Diagram for delta-connected capacitor bank.
4.1.6 Three-Phase Photovoltaic Generation
The three-phase PV generation model is based on the connection of three single-phase models. The single-phase model’s main structure is depicted in Figure 18. It wasdesigned to be a simple and dynamic model, and therefore not all control dynamics areincluded. In fact, the only represented strategy is the DC link voltage control.
Te
HaPV
CELL DC
DCCdc
DC
ACVac
Ppv Pdc Pac
Figure 18 – PV generation diagram.
Temperature Te and radiance Ha have direct influence in the maximum point ofpower output from PV cell Ppv. This behavior can be written as following equation (4.53)[45].
Ppv = Parray · Ha
Href
· [1 + “ · (Tc ≠ Tref )] (4.53)
where Parray is the installed power capacity, “ is the thermal coe�cient for maximumpower given by manufacturers, Href is the reference radiation in standard test conditions(STC), Tref is the cell reference temperature and
Tc = Te + Ha
800 · (Tnoct ≠ Tref ) (4.54)
44
where Tnoct is the normal operation cell temperature. In addition it is possible to relatethe power that is transmitted to the DC link Pdc with the power output of the solar cellPpv. The relation between both power values is basically the e�ciency of the maximumpower point tracking (MPPT) algorithm ÷mppt. Therefore, it is possible to write equation(4.55) below.
Pdc = ÷mppt · Ppv (4.55)
Besides, it is possible to write the relation between AC power and DC link poweras follows [46, 47]:
Pac = Pdc ≠ Pcap = Pdc ≠ vdcCdc
d
dt(vdc) (4.56)
where
Pcap = vdc · icap = vdcCdc
d
dt(vdc) (4.57)
However, if AC power can be written as the product of phase voltage vg and currentig we can rewrite equation (4.56) as (4.58) below.
vdcCdc
d
dt(vdc) = ÷mppt · Ppv ≠ vgig (4.58)
Applying Laplace transformation, it is possible to obtain equation (4.59) as follows.
sCdc(V 2dc
) = ÷mppt · Ppv ≠ VgIg (4.59)
Figure 19 shows the DC link voltage controller which is based on the dynamicbehavior state on equation (4.59) and on (4.57) for capacitors. Note that a proportional-integral (PI) controller block is present in order to calculate the grid current Ig from thedi�erence of measured V
2dc
and reference (V 2dc
)ú. The "≠1" gain block is placed in order tocompensate the power direction adopted for all calculations.
(V 2dc
)ú
≠
+ q≠1 Kp + Ki
s◊
Vg
q≠ +
÷mpptPpv
÷ 1sCDC
◊V
2dc
Figure 19 – Block diagram for DC link voltage control.
45
In order to provide equations for determining the parameters of PI controller block,it is necessary to write the block diagram transfer function. This equation is presented in(4.60) below.
V2
dc
(V 2dc
)ú =s
≠KpVg
Cdc+ ≠KiVg
Cdc
s2 + s≠KpVg
Cdc+ ≠KiVg
Cdc
(4.60)
Equation (4.60) is compared to the canonical form (4.61) of a second order transferfunction. Besides, consider that grid voltage Vg does not have large variations from itsoriginal value of 1 p.u.. Hence, it is possible to set the PI parameters as (4.62)
H(s) = 2Ên›s + Ê2n
s2 + s2Ên› + Ê2n
(4.61)
Y_]
_[
Kp = ≠2›ÊnCdc
Ki = ≠Ê2nCdc
(4.62)
4.2 VALIDATION TEST SYSTEMS
This section presents three test systems used to provide validation of the modelsdeveloped in Section 4.2, Modeling Elements. Two tests, IEEE 4-Node Test Systemand IEEE 13-Node Test System, consists in a steady state analysis while the other one,Modified IEEE 14-Bus Test System, uses a dynamic phenomena.
4.2.1 IEEE 4-Node Test System
This test feeder is a simple four bus system which is described on [48] and it isrepresented in the Figure 20 below. The system presents two transmission lines and atransformer in its configuration. The first line is located between buses 1 and 2 andis modeled in single-phase. The second line is located between buses 3 and 4 and it ismodeled as three-phase. The transformer is located between buses 2 and 3 is modeled asan interface hybrid element. Besides, the load connected to bus 4 is modeled as constantPQ. This information is summarized in Table 8.
1 32 4
GL12
T23
L34
Load
Figure 20 – IEEE 4 node test feeder diagram.
46
Table 8 – Type of modeling for each component.
Element Type Model UsedG Generator Infinite Bus
L12 Transmission Line Single-PhaseT23 Step-Down Transformer HybridL34 Transmission Line Three-Phase
Load Unbalanced Loads Constant PQ | Grounded Y
The power flow is performed in order to validate the hybrid single-phase◊three-phase interface for steady state conditions. Therefore, two tests are made: one for theapproximate model, in which negative and zero admittances are considered to have infinitevalues; and one for the complete model, in which negative and zero admittances haveknown finite values. In addition, the first test uses a �-Grounded Wye connection inorder to allow the approximate model to be used. The second test uses the GroundedWye - Grounded Wye connection in order to aggravate the unbalances, demanding thecomplete hybrid model to be used. Both tests are made in two softwares, OpenModelicaand Simulight, in order to perform a software-to-software validation.
4.2.2 IEEE 13-Node Test System
This test feeder is described in [48] and consists of a three-phase highly loadeddistribution system. This system has thirteen buses, as shown in Figure 21, along withunbalanced lines and loads which are interesting features for assessing three-phase models’behavior under unbalanced conditions. Therefore, a power flow is performed aiming thevalidation of three-phase components developed in the last section for steady state studies.The only adaptations made in this test was to eliminate the line voltage regulator and toconsider Bus 650 as having the regulator output voltage. All data for performing this testis presented in Appendix A.
47
646 645 633 634
611 684 692 675
650
632
671
680652
G
Figure 21 – IEEE 13 node test feeder diagram.
4.2.3 Modified IEEE 14-Bus Test System
The tested grid consists in a modification of the consolidated IEEE 14-bus testsystem presented in [49] and it represents a small transmission network connected to twothree-phase buses as it is depicted in Figure 22. The transient stability study was madein order to provide a software-to-software validation of the hybrid single-phase◊three-phase element dynamic behavior. Hence, the test was performed in both OpenModelicaand Simulight for the same contingency. The data for the whole system is presented inAppendix A, while the data for the contingency data is presented in in Table 9. Simulationparameters are shown in Table 10.
Table 9 – Contingency data for validation test.
Parameter Value/DescriptionFault Spot Bus 4
Fault Impedance jX = 0.6 pu
Fault Instant 12 s
Fault Duration 100 ms
48
HybridInterface
Three-PhaseModelling
1
2
3 4
56
7
8
9
10
11
12
13
14
G1
G2
C1
C2
C3
650
632
Figure 22 – Modified IEEE 14-bus test feeder diagram.
Table 10 – Summary of simulation data for both softwares.
Parameter OpenModelica SimulightTime Simulated 20 s 20 s
Interval 0.001 s 0.0001 s
Tolerance 0.0001 0.0001
49
5 RESULTS AND DISCUSSIONS
The first section of this chapter, Validation Results, are related to the tests describedin Chapter 4, Section 4.2. It shows comparison between simulation outcomes obtainedusing the developed models and consolidated softwares. Comparison for the IEEE 13-nodetest feeder results is made with the provided o�cial results. The second section of thischapter, Integrated System Results, presents the outcomes obtained in tests performedin the studied system, described in Chapter 3. Comparison between base case scenariosand di�erent penetration levels is provided in order to assess the impact of distributedgeneration in transmission and distribution system. Furthermore, analysis of results isalso presented.
5.1 VALIDATION RESULTS
This section presents the results from tests described in Chapter 4, Section 4.2,Validation Test Systems.
5.1.1 IEEE 4-Node Test System
In the first case, the connection between primary and secondary bus in the trans-former was chosen to be � - Grounded Wye. Therefore, there is a zero sequence currentblocking on the primary side of the transformer by the � configuration, and it is reasonableto consider that y
0nrt
and y≠nrt have infinite values. The results obtained with Simulight R•
and with OpenModelica are shown in Table 11 below. Comparing the result from bothsoftwares shows that there are small variations between them. However, note that thelargest variation encountered is of ≠0.17% in bus 4 phase C voltage magnitude. In addition,results for voltage angles are shown to have small errors as well.
Table 11 – Results for IEEE 4-node first case.
Simulight OpenModelicaBus Voltage [p.u.] Angle [¶] Voltage [p.u.] Angle [¶]
1 1.000 0 1.000 02 0.9891 -0.36 0.9891 -0.36
3A 0.9537 27.49 0.9529 27.493B 0.9401 -93.79 0.9393 -93.793C 0.9238 145.23 0.9229 145.234A 0.8976 25.69 0.8968 25.684B 0.8038 -97.08 0.8028 -97.094C 0.7736 133.53 0.7723 133.49
50
The second case consists on the substitution of the transformer connection from� ≠ GrY to Grounded Wye - Grounded Wye connection. In this case, there is going to bea zero sequence current flowing on the primary side of the transformer and, therefore y
0nrt
and y≠nrt cannot be considered infinite, demanding the usage of hybrid model’s complete
formulation. Results can be seen on Table 12 below. Both softwares present really similarresults, since the largest variation encountered in this test was of ≠0.10%, in bus 3 phaseC voltage magnitude.
Table 12 – Results for IEEE 4-node second case.
Simulight OpenModelicaBus Voltage [p.u.] Angle [¶] Voltage [p.u.] Angle [¶]
1 1.000 0 1.000 02 0.9891 -0.4 0.9890 -0.4
3A 0.9625 -2.3 0.9619 -2.33B 0.9383 -123.6 0.9374 -123.63C 0.9179 114.8 0.9173 114.84A 0.9083 -4.1 0.9078 -4.14B 0.8015 -126.8 0.8015 -126.84C 0.7636 102.9 0.7635 102.9
5.1.2 IEEE 13-Node Test System
The power flow test was performed in this IEEE 13-node test system and theresults are presented in Table 13. The comparison for validation is made between theresults obtained with OpenModelica and the o�cial results provided by the IEEE test case.Note that results for both voltage magnitudes and angles are very similar and supportthe claim that three-phase models have a good performance for steady state studies. Thelargest error encountered was of 0.01% in voltage magnitude of bus 692’ phase C.
51
Table 13 – Results for the 13-node test case.
O�cial Results OpenModelicaBus Phase Voltage [p.u.] Angle [¶] Voltage [p.u.] Angle [¶]
A 1.0625 0 1.0625 0650 B 1.0500 -120.0 1.0500 -120
C 1.0687 120.0 1.0687 120A 1.0210 -2.5 1.0210 -2.5
632 B 1.0420 -121.7 1.0420 121.7C 1.0174 117.8 1.0175 117.8A 1.0180 -2.6 1.0180 -2.5
633 B 1.0401 -121.8 1.0401 -121.8C 1.0148 117.8 1.0148 117.8A 0.9940 -3.2 0.9940 -3.2
634 B 1.0218 -122.2 1.0218 -122.2C 0.9960 117.3 0.9960 117.3
645 B 1.0329 -121.9 1.0328 -121.9C 1.0155 117.9 1.0155 117.9
646 B 1.0311 -122.0 1.0311 -122.0C 1.0134 117.9 1.0134 117.9A 0.9900 -5.3 0.9900 -5.3
671 B 1.0529 -122.3 1.0529 -122.3C 0.9778 116.0 0.9777 116.0A 0.9900 -5.3 0.9900 -5.3
680 B 1.0529 -122.3 1.0529 -122.3C 0.9778 116.0 0.9777 116.0
684 A 0.9881 -5.3 0.9881 -5.3C 0.9758 115.9 0.9757 115.9
611 C 0.9738 115.8 0.9738 115.8652 A 0.9825 -5.2 0.9825 -5.2
A 0.9900 -5.3 0.9900 -5.3692 B 1.0529 -122.3 1.0529 -122.3
C 0.9778 116.0 0.9777 116.0A 0.9835 -5.6 0.9835 -5.5
675 B 1.0553 -122.5 1.0553 -122.5C 0.9758 116.0 0.9758 116.0
5.1.3 Modified IEEE-14 Bus Test System
Since this test is a dynamic study, the results are presented as charts and graphics,showing the voltage variation during time. Figure 23 shows the voltage magnitude variation
52
for bus 1. The result is separated into the first ten seconds and into the last ten secondsof simulation.
0 2 4 6 8 10 12 14 16 18 201.02
1.04
1.06
1.08
1.1
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(a) Complete simulation.
10 11 12 13 14 15 16 17 18 19 201.02
1.04
1.06
1.08
1.1
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(b) Last 10 seconds of simulation.
0 1 2 3 4 5 6 7 8 9 101.05
1.05
1.06
1.06
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(c) First 10s of simulation.
Figure 23 – Bus 1 voltage magnitude.
This was made in order to highlight that di�erences between results obtained withSimulight and OpenModelica appears in the first ten seconds of simulation. However, thesedi�erences occur due to di�erent initialization process used in each software. This smallissue was also noted in [27] when the results obtained with Modelica-based software Dymolawere compared with Eurostag. In this reference, it is suggested to start the simulation ofthe Modelica model with three seconds in order to avoid this oscillations. However, it ispossible to observe that both simulations goes to the same solution over time. Besides,
53
it is important to note that the last ten seconds of simulation shows that both modelspresent the same dynamic behavior.
Figure 24 shows the results for two single-phase modeled buses, 6 and 11. Thesimulation outcome is similar to the one presented in Figure 23: there are di�erences inthe first seconds of simulation due to di�erent AVR and machine parameter initializationon both softwares. However, once more, it is possible to observe that both results showsthe same dynamic behavior.
0 2 4 6 8 10 12 14 16 18 20
0.98
1
1.02
1.04
1.06
1.08
1.1
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(a) Bus 6 voltage magnitude.
0 2 4 6 8 10 12 14 16 18 200.95
1
1.05
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(b) Bus 11 voltage magnitude
Figure 24 – Complete results for other single-phase modeled buses.
Results presented in Figures 25 and 26 shows the dynamic behavior of all threephases in buses 650 and 632, respectively. The first ten seconds of simulation were omittedin order to allow the focus to be on during-fault and after-fault behavior of voltagemagnitudes. Although the first ten seconds were not presented, the same issue regardinginitialization could be spotted in the complete twenty seconds simulation. However, oncemore, the oscillations stopped as the results from both softwares tended to the same valuebefore the fault, as it is represented by the results from ten to twelve seconds. In addition,it is possible to make important claims about these results.
Firstly and foremost, the outcomes show that both simulations resulted in thesame behavior for each phase voltage. Resulting curves of Simulight overlaps the curves
54
obtained with OpenModelica during the presented last 10 seconds of simulation. This factcorroborates that the Modelica hybrid single-phase◊three-phase element was correctlymodeled.
10 11 12 13 14 15 16 17 18 19 200.9
0.95
1
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(a) Phase A voltage magnitude.
10 11 12 13 14 15 16 17 18 19 20
0.95
1
1.05
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(b) Phase B voltage magnitude.
10 11 12 13 14 15 16 17 18 19 20
0.92
0.94
0.96
0.98
1
1.02
1.04
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(c) Phase C voltage magnitude.
Figure 25 – Results for three-phase bus 650.
Secondly, the hybrid interface allowed the dynamic of a completely unbalancedthree-phase network to be represented. While the transmission system was representedonly with its positive sequence in a single-phase equivalent network, buses 650 and 632formed a small three-phase-modeled distribution feeder. Therefore, the results presentedin this subsection along with the ones presented in 5.1.1 (IEEE 4-Node Test System)
55
validates the hybrid formulation proposed in [15]. As a matter of fact, the outcomes alsoshows that the proposed hybrid formulation is an interesting tool.
10 11 12 13 14 15 16 17 18 19 20
0.88
0.9
0.92
0.94
0.96
0.98
1
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(a) Phase A voltage magnitude.
10 11 12 13 14 15 16 17 18 19 20
0.95
1
1.05
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(b) Phase B voltage magnitude.
10 11 12 13 14 15 16 17 18 19 200.86
0.88
0.9
0.92
0.94
0.96
0.98
Time [s]
Voltage
[pu]
OpenModelicaSimulight
(c) Phase C voltage magnitude.
Figure 26 – Results for three-phase bus 632.
5.2 INTEGRATED SYSTEM RESULTS
This section summarizes the main results obtained through the simulation of thestudied system described in Chapter 3, System Overview, using OpenModelica. Thestudied contingency, as describe in Table 4, consists in a fault at bus 4, happening att = 12s with duration of ”t = 100ms. After this time, the line between buses 4 and 2 is
56
opened. All scenarios presents this exact same sequence of events. Results are presentedas charts in order to facilitate subsequent analysis.
5.2.1 Machines Frequency
The coherence between machines is never lost in any of the cases analyzed, whichsuggests that the PV penetration level suggested in this work is not enough to harm thesystem angular stability. However, some considerations need to be made on the frequencyof machines operating under di�erent scenarios.
0 2 4 6 8 10 12 14 16 18 2059.8
59.9
60
60.1
60.2
Time [s]
Freque
ncy[H
z]
Base Case50% PV Penetration
Figure 27 – Frequency for synchronous condenser C2 for SG-connected cases.
Considering the cases where the whole system is connected to a strong grid which isrepresented by an infinite bus (SG-connected cases), there is a small frequency oscillationduring the fault. But the frequency is reestablished to the original value of 60 Hz in lessthan 2 seconds. This result is shown in Figure 27 for synchronous condenser C2. Note thatthe result is the same for both the base case and for the 50% penetration scenario. Thisstrongly suggests that the PV presented installed capacity has no e�ect on the systemfrequency if the network is considered to be connected to a strong grid.
0 2 4 6 8 10 12 14 16 18 2059.8
59.9
60
60.1
60.2
Time [s]
Freque
ncy[H
z]
Base Case50% PV Penetration
Figure 28 – Frequency for synchronous condenser C2 for isolated cases.
57
On the other hand, when the analysis is made over the cases in which the systemis considered to be isolated a small influence is noted. The results presented in Figure28 suggests that higher values of PV penetration allows frequency to stabilize during theperiod after-fault in values that are closer to the original frequency of 60 Hz. However,the PV penetration has no clear e�ect on the oscillations in the period during-fault, sinceboth cases have very similar values during the first three oscillation cycles.
5.2.2 Bus Voltages
5.2.2.1 Transmission System
The comparison between both base case scenarios is crucial for the understandingwhat to expect from the scenarios with di�erent PV penetration levels. This comparisonis presented in Figure 29 and shows that both base cases had di�erent behaviors asexpected. The SG-connected scenario had little oscillations and a voltage sag that reachedV11 = 0.9649 pu, while the isolated scenario had more oscillations, a voltage sag that reachedV11 = 0.9422 pu which is only 2% lower, and an overshoot that reached V11 = 1.0706 pu.Although the dynamic behavior is shown to be completely di�erent, it is interesting tonote that both cases reach the same steady-state value.
0 2 4 6 8 10 12 14 16 18 20
0.95
1
1.05
1.1
Time [s]
Voltage
Magnitude
[pu]
Isolated Base Case ScenarioSG Connected Base Case Scenario
Figure 29 – Comparison between voltage magnitude in bus 11 for both base cases.
Results obtained for the transmission network for all cases shows that PV deploy-ment level has little influence over a transmission system dealing with much higher powerlevels. This claim is supported by Figures 30 and 31, that shows the voltage magnitudefor bus 11 over SG-connected scenarios and the voltage magnitude for bus 6 over isolatedscenarios, respectively.
58
0 2 4 6 8 10 12 14 16 18 200.95
1
1.05
Time [s]
Voltage
Magnitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
Figure 30 – Voltage magnitude in bus 11 for SG-connected scenarios.
Although Figures 30 and 31 shows that all cases have overlapping curves asoutcomes for di�erent PV scenarios, it is important to consider that the power producedby the DERs is very low if compared with the power of one synchronous machine in thesystem. Generator G2, for instance, has a power outcome of 40 [MW], while the totalsolar production in the higher penetration scenario is of 1789 [kW], less than 4.5% of thepower delivered by the system’s smaller synchronous generator.
0 2 4 6 8 10 12 14 16 18 200.95
1
1.05
1.1
Time [s]
Voltage
Mag
nitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
Figure 31 – Voltage magnitude in bus 6 for isolated scenarios.
Figure 32 shows the bus 11 voltage for all isolated scenarios. In addition, thementioned figure also presents a zoom showing di�erent voltage levels for the scenarios induring-fault period. However, the di�erence between voltage sags is less than 0.0004 pu,and therefore it can be neglected. Bus 11’s voltage results show that for this penetrationlevel of DERs, there is no considerable e�ect over voltage’s dynamic behavior nor steady-state values in the transmission side of the system. In fact, bus 11 was chosen because itis the transmission bus in which the distribution feeder with DERs is connected.
59
0 2 4 6 8 10 12 14 16 18 20
0.95
1
1.05
Time [s]
Voltage
Magnitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
(a) Complete simulation.
12.02 12.04 12.06 12.08 12.1 12.12 12.14 12.160.94
0.94
0.94
0.95
0.95
0.95
Time [s]
Volta
geMag
nitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
(b) Representation of di�erent voltage sag levels during the fault.
Figure 32 – Voltage magnitude in bus 11 for isolated scenarios.
5.2.2.2 Distribution System
The first step taken in the analysis of the distribution system is also a comparisonbetween base case scenarios, and in this case the three-phase bus 632 is analyzed. Figure33 shows the di�erence between base cases for each phase. Note that the voltages areunbalanced but their dynamics resembles the behavior observed for positive sequencevoltage of single-phase bus 11 presented in Figure 29, as expected. It is also importantto note that even though the oscillations on during-fault and after-fault are di�erent,steady-state values are similar if not the same in base case scenarios.
In addition, voltage sags are greater in the isolated case, in which there is anvoltage overshoot in the after-fault period, just like what was observed in the voltage ofbus 11. There is another interesting fact regarding the voltage sag in the analysis of basecase scenarios. Voltage sag is approximately 11% for all phases in the isolated base casescenario, while it is 8.7% for all phases in the SG-connected base case scenario. Hence, itis correct to state that the voltage sag proportion is maintained for each phase.
60
0 2 4 6 8 10 12 14 16 18 200.85
0.9
0.95
1
1.05
Time [s]
Voltage
Mag
nitude
[pu]
Isolated Base Case ScenarioSG Connected Base Case Scenario
(a) Phase A voltage magnitude.
0 2 4 6 8 10 12 14 16 18 200.9
0.95
1
1.05
1.1
Time [s]
Voltage
Mag
nitude
[pu]
Isolated Base Case ScenarioSG Connected Base Case Scenario
(b) Phase B voltage magnitude.
0 2 4 6 8 10 12 14 16 18 200.85
0.9
0.95
1
1.05
Time [s]
Voltage
Mag
nitude
[pu]
Isolated Base Case ScenarioSG Connected Base Case Scenario
(c) Phase C voltage magnitude.
Figure 33 – Comparison of base cases scenarios for three-phase bus 632.
The comparison of 50% PV penetration for isolated and SG-connected scenarios ismade in Figure 34. The analysis of such figure shows that the steady-state voltage valuesare the same, even for both 50% PV penetration scenarios. Therefore, it is reasonableto assume that steady state values are not influenced by the isolated or SG-connectioncondition of the transmission system. In fact, both curves show the same voltage valueseven in the period of time when the radiance slowly decreased, between t = 11s andt = 12s.
61
0 2 4 6 8 10 12 14 16 18 200.85
0.9
0.95
1
1.05
Time [s]
Voltage
Mag
nitude
[pu]
Isolated 50% PV ScenarioSG Connected 50% PV Scenario
(a) Phase A voltage magnitude.
0 2 4 6 8 10 12 14 16 18 200.9
0.95
1
1.05
1.1
Time [s]
Voltage
Mag
nitude
[pu]
Isolated 50% PV ScenarioSG Connected 50% PV Scenario
(b) Phase B voltage magnitude.
0 2 4 6 8 10 12 14 16 18 200.85
0.9
0.95
1
1.05
Time [s]
Voltage
Mag
nitude
[pu]
Isolated 50% PV ScenarioSG Connected 50% PV Scenario
(c) Phase C voltage magnitude.
Figure 34 – Comparison of 50% PV penetration in isolated and SG-connected scenarios forthree-phase bus 632.
Figure 35 shows bus 671’s three-phase voltages for all isolated scenarios. Note thatphase B is the one which shows the smallest voltage values variation among all di�erentpenetration scenarios. This is due to the fact that phase B contains the smallest PVinstalled power capacity. By analogy, note that phase C has the largest voltage valuesvariation among all di�erent penetration scenarios, since it is the one with the largest PVinstalled power capacity. Note that the largest PV installed power capacity scenario, thehigher the bus phase voltage is. Besides, it is important to observe that the voltage sagslowly diminish its proportional value. For instance, if we analyze phase A voltage sagging
62
during the fault, it accounts for 10, 8% in the base case scenario, and for 10, 65% in the50% PV penetration scenario.
0 2 4 6 8 10 12 14 16 18 20
0.85
0.9
0.95
1
Time [s]
Voltage
Magnitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
(a) Phase A voltage magnitude.
0 2 4 6 8 10 12 14 16 18 200.9
0.95
1
1.05
1.1
Time [s]
Voltage
Mag
nitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
(b) Phase B voltage magnitude.
0 2 4 6 8 10 12 14 16 18 20
0.85
0.9
0.95
1
Time [s]
Voltage
Magnitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
(c) Phase C voltage magnitude.
Figure 35 – Comparison of isolated scenarios for three-phase bus 671.
Moreover, Figure 36 depicts an interesting phenomena that must be analyzed. Itshows how the bus voltage changes due to the variation of the radiance during that time.This claim is clearly supported by the fact that the base case scenario does not have itsvoltage altered for any phase. Another evidence of such influence is that the higher PVinstalled power capacity, the largest the voltage sag is. In phase C, the voltage dropsaround 1% of its value in just one second, without the occurrence of any contingency.It is important to consider that the radiance have diminished approximately 30% of its
63
value during that time. Therefore, distribution feeders with high PV penetration canface voltage fluctuations during periods when the PV is injecting active power on thedistribution network.
11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 120.94
0.96
0.98
1
Time [s]
Voltage
Mag
nitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
(a) Phase A voltage magnitude.
11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 120.99
1
1.01
1.02
Time [s]
Voltage
Mag
nitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
(b) Phase B voltage magnitude.
11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 120.94
0.96
0.98
1
Time [s]
Voltage
Mag
nitude
[pu]
Base Case30% PV Penetration40% PV Penetration50% PV Penetration
(c) Phase C voltage magnitude.
Figure 36 – Comparison of voltage fluctuation on isolated scenarios for three-phase bus 634.
5.2.3 Currents Injected
During the fault, there is a fast dropping in all system’s voltage level. Since the PVgeneration has its power output determined only by conditions such as temperature andradiance, a current peak is expected to appear in during-fault period. In fact, Figure 37shows that current increases as fast as the voltage drop during the fault. In this study, the
64
radiance starts to drop one second before the start of the fault, and ceases its dropping onesecond after the short-circuit started. If the radiance had not diminished in this period,the current peak would represent a threat to the safe operation of certain equipments inthe distribution network, such as the inverter itself.
0 2 4 6 8 10 12 14 16 18 20
1
2
3
4
5·10�3
Time [s]
Current
Mag
nitude
[pu]
30% PV Penetration40% PV Penetration50% PV Penetration
(a) Phase A current magnitude.
0 2 4 6 8 10 12 14 16 18 20
1
2
3
4
5·10�3
Time [s]
Current
Mag
nitude
[pu]
30% PV Penetration40% PV Penetration50% PV Penetration
(b) Phase B current magnitude.
0 2 4 6 8 10 12 14 16 18 20
1
2
3
4
5·10�3
Time [s]
Current
Mag
nitude
[pu]
30% PV Penetration40% PV Penetration50% PV Penetration
(c) Phase C current magnitude.
Figure 37 – Comparison of SG-connected scenarios current peak for three-phase bus 680.
This behavior is also noted on the current that is drained from the bus 11 by thetransformer that feeds the distribution network. However, in this case the e�ect is theopposite. While the reduction of radiance would increase the current that is needed bythe feeder, the short-circuit causes a sudden reduction followed by an also sudden increaseof the drained current. Figure 38 shows this very occurrence.
65
0 2 4 6 8 10 12 14 16 18 20
1
2
3
4
5·10�2
Time [s]
Current
Mag
nitude
[pu]
30% PV Penetration40% PV Penetration50% PV Penetration
Figure 38 – Current magnitude drained from bus 11 by the transformer connected to bus 650.
66
6 CONCLUSIONS
Models for three-phase components along with the model for a hybrid element thatcan interface single-phase grids with three-phase systems have been presented in this work.Results obtained by experiments performed in IEEE 4-Node Test System, IEEE 13-NodeTest System, and Modified IEEE 14-Bus Test System presents small errors if comparedwith o�cial results or other softwares’ results. This fact supports that the validation ofmodels three-phase and hybrid models was successfully performed.
Three-phase and hybrid single-phase/three-phase models are now part of OpenIPSLover the name of ThreePhase. ThreePhase package allows the study of three-phase powernetworks. Besides, the connection between single-phase and three-phase systems issupported in ThreePhase by the presence of hybrid transformers and transmission lines.
The impacts of distributed generation with integrated distribution in power systemsand transmission networks have been assessed in the present study. A power system whichconsists of a transmission grid, based on IEEE 14-Bus test system, connected to adistribution feeder, based on IEEE 13-node test system, with DERs has been successfullyassembled. Simulation results have been presented.
Results regarding the system’s frequency variation strongly suggests that smallinstalled power capacities show very little influence over frequency. Specially whenconsidering the transmission system connected to a strong grid. In addition, PV distributedgeneration allow the system frequency to have a lower steady state value after the shortcircuit, if compared to the case without any distributed generation.
Results regarding voltages magnitudes shows that the PV generation shows muchhigher influence over the distribution feeder buses. While the voltage levels from thetransmission grid have not shown any significant changes over di�erent PV penetrationscenarios, the voltage magnitude from the distribution grid have shown higher levels forhigher PV penetrations. Moreover, distribution feeders are shown to have their voltagelevels dependent on radiance and temperature values.
Results regarding the current injection by the distributed PV generation shows thatevents occurring in the transmission system can damage equipments in the distributionnetwork. In fact, the current is shown to have a sudden increase, leading to concerns aboutthe safe operation of distribution-connected apparatus. The assessment of integrateddistribution and transmission system are shown to be important for studies regardingprotection of equipments connected to the distribution grid.
All systems have been successfully assembled using Modelica-based libraries ina open-source software. Modelica language have been shown as an interesting tool forassessing dynamic phenomena in modern power system analysis.
67
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71
APPENDIX A – Validation Test Systems Data
This appendix presents the data for tested systems.
A.1 Data for IEEE 4 Node Test System
Table 14 – Branches configuration for tests.
Bus k Bus m Element Length (miles)1 2 L12 0.37882 3 T23 —3 4 L34 0.4735
Line parameters for L12 model are:
z≠12 = z
+12 = 0.306 + j0.6272 �/mile
z012 = 0.5919 + j2.9855 �/mile
(A.1)
Line parameters for L34 model are:
Zabc
34 =
Q
ccca
0.4013 + j1.4133 0.0953 + j0.8515 0.0953 + j0.72660.0953 + j0.8515 0.4013 + j1.4133 0.0953 + j0.78020.0953 + j0.7266 0.0953 + j0.7802 0.4013 + j1.4133
R
dddb �/mile (A.2)
Data for transformer T23 is presented below:
Table 15 – Step-Down transformer configuration.
Power Base (MVA) LL-Volt.-high(kV) LL-Volt.-low(kV) R (pu) X (pu)100 12.47 4.16 0.16667 1.00
Load configuration for tests is:
Table 16 – Unbalanced load data.
Phase P(kW) Q(kVAr) Power FactorA 1275 790.174 0.85 laggingB 1800 871.780 0.9 laggingC 2375 780.605 0.95 lagging
72
A.2 Data for IEEE 13-Node Test System
This appendix has all data necessary for IEEE 13-Node test system.
Table 17 – Branch data.
Bus k Bus m Length (miles) Line Configuration632 645 0.0947 603632 633 0.0947 602633 634 — XFM-1645 646 0.0568 603650 632 0.3788 601684 652 0.1515 607632 671 0.3788 601671 684 0.0568 604671 680 0.1894 601671 692 — Switch684 611 0.0568 605692 675 0.0947 606
Table 18 – XFM-1 transformer configuration.
Power Base (kVA) High(kV) Low(kV) R (pu) X (pu)500 4.16 0.48 0.011 0.02
Table 19 – Capacitor data.
Node Phase A(kVAr) Phase B(kVAr) Phase C(kVAr)675 200 200 200611 — — 100
Table 20 – Delta-connected load data.
Node Model PAB QAB PBC QBC PCA QCA
(kW) (kVAr) (kW) (kVAr) (kW) (kVAr)646 Const-Z 0 0 230 132 0 0671 Const-PQ 385 220 385 220 385 220692 Const-I 0 0 0 0 170 151
73
Table 21 – Wye-connected load data.
Node Model PA QA PB QB PC QC
(kW) (kVAr) (kW) (kVAr) (kW) (kVAr)632 Const-PQ 8.5 5 33 19 58.5 34634 Const-PQ 160 110 120 90 120 90645 Const-PQ 0 0 170 125 0 0652 Const-Z 128 86 0 0 0 0671 Const-PQ 8.5 5 33 19 58.5 34675 Const-PQ 485 190 68 60 290 212
Line configuration 601:
Z601 =
Q
ccca
0.3465 + j1.0179 0.1560 + j0.5017 0.1580 + j0.42360.1560 + j0.5017 0.3375 + j1.0478 0.1535 + j0.38490.1580 + j0.4236 0.1535 + j0.3849 0.3414 + j1.0348
R
dddb �/mile (A.3)
Line configuration 602:
Z602 =
Q
ccca
0.7526 + j1.1814 0.1580 + j0.4236 0.1560 + j0.50170.1580 + j0.4236 0.7475 + j1.1983 0.1535 + j0.38490.1560 + j0.5017 0.1535 + j0.3849 0.7436 + j1.2112
R
dddb �/mile (A.4)
Line configuration 603:
Z603 =
Q
ccca
0.0000 + j0.0000 0.0000 + j0.0000 0.0000 + j0.00000.0000 + j0.0000 1.3294 + j1.3471 0.2066 + j0.45910.0000 + j0.0000 0.2066 + j0.4591 1.3238 + j1.3569
R
dddb �/mile (A.5)
Line configuration 604:
Z604 =
Q
ccca
1.3238 + j1.3569 0.0000 + j0.0000 0.2066 + j0.45910.0000 + j0.0000 0.0000 + j0.0000 0.0000 + j0.00000.2066 + j0.4591 0.0000 + j0.0000 1.3294 + j1.3471
R
dddb �/mile (A.6)
Line configuration 605:
Z605 =
Q
ccca
0.0000 + j0.0000 0.0000 + j0.0000 0.0000 + j0.00000.0000 + j0.0000 0.0000 + j0.0000 0.0000 + j0.00000.0000 + j0.0000 0.0000 + j0.0000 1.3292 + j1.3475
R
dddb �/mile (A.7)
Line configuration 606:
Z606 =
Q
ccca
0.7982 + j0.4463 0.3192 + j0.0328 0.2849 ≠ j0.01430.3192 + j0.0328 0.7891 + j0.4041 0.3192 + j0.03280.2849 ≠ j0.0143 0.3192 + j0.03289 0.7982 + j0.4463
R
dddb �/mile (A.8)
74
Line configuration 607:
Z607 =
Q
ccca
1.3425 + j0.5124 0.0000 + j0.0000 0.0000 + j0.00000.0000 + j0.0000 0.0000 + j0.0000 0.0000 + j0.00000.0000 + j0.0000 0.0000 + j0.0000 0.0000 + j0.0000
R
dddb �/mile (A.9)
Shunt susceptance jBsht is considered to be zero for all line configurations.
A.3 Data for Modified IEEE 14-Bus Test System
This section has all data necessary for modified IEEE 14-bus test system forSbase = 100 MVA.
Table 22 – Branch data.
Bus k Bus m R (p.u.) X (p.u.) B (p.u.) Tap1 2 0.01938 0.05917 0.0528 —1 5 0.05403 0.22304 0.0492 —2 3 0.04699 0.19797 0.0438 —2 4 0.05811 0.17632 0.0340 —2 5 0.05695 0.17388 0.0346 —3 4 0.06701 0.17103 0.0128 —4 5 0.01335 0.04211 0.0000 —4 7 0.00000 0.20912 0.0000 0.9784 9 0.00000 0.55618 0.0000 0.9695 6 0.00000 0.25202 0.0000 0.9326 11 0.09498 0.19890 0.0000 —6 12 0.12291 0.25581 0.0000 —6 13 0.06615 0.13027 0.0000 —7 8 0.00000 0.17615 0.0000 1.0007 9 0.00000 0.11001 0.0000 1.0009 10 0.03181 0.08450 0.0000 —9 14 0.12711 0.27038 0.0000 —10 11 0.08205 0.19207 0.0000 —12 13 0.22092 0.19988 0.0000 —13 14 0.17093 0.34802 0.0000 —
75
Table 23 – Bus data.
Bus Pgen Qgen Pload Qload Shunt ControlledVoltage
(p.u.) (p.u.) (p.u.) (p.u.) (p.u.) (p.u.)1 2.324 -0.169 0.000 0.000 — 1.0602 0.400 0.424 0.217 0.127 — 1.0453 0.000 0.234 0.942 0.190 — 1.0104 — — 0.478 -0.390 — —5 — — 0.076 0.016 — —6 0.000 0.122 0.112 0.075 — 1.0707 — — 0.000 0.000 — —8 0.000 0.174 0.000 0.000 — 1.0909 — — 0.295 0.166 0.190 —10 — — 0.090 0.058 — —11 — — — — — —12 — — 0.061 0.016 — —13 — — 0.135 0.058 — —14 — — 0.149 0.050 — —
Data for hybrid transformer:
Table 24 – Transformer parameters.
Power Base (kVA) R (pu) X (pu) Connection (High-Low)5000 0.01 0.08 �≠GrWye
Data for three-phase transmission line between buses 650 and 632:
Yabc
ser=
S
WWWU
0.1982 ≠ j0.5712 ≠0.0841j0.2112 ≠0.046 + j0.1579≠0.0841 + j0.2112 0.1735 ≠ j0.5413 ≠0.0219 + j0.1206≠0.046 + j0.1579 ≠0.0219 + j0.1206 0.1535 ≠ j0.5106
T
XXXV p.u. (A.10)
Babc
shtk= B
abc
shtm= 0 p.u. (A.11)
Data for three-phase load:
Table 25 – Three-phase load data.
Phase P(kW) Q(kVAr)A 1318.05 677.6B 937.61 347.43C 1429.46 677.85
76
A.4 Data for IEEE 14-Bus Plus IEEE 13-Node Tests Systems
The transmission system branch parameters are described in Table 22. Single phaseloads are described in Table 23. The parameters for the hybrid transformer connectedbetween buses 11 and 650 are described in Table 24. The distribution system branchparameters are described in Tables ?? and ??. Three-phase loads for this system aredescribed in Tables ?? and ??. The capacitor bank connected in the distribution systemis described in Table ??.
77
APPENDIX B – Machine, PV and Regulator Data
B.1 Data for Machines
The parameters used for all machines are:
Table 26 – Data for synchronous generators and condensers.
Parameter G1 G2 C1 C2 C3
Sbase (MVA) 615 60 60 25 25xl (p.u.) 0.2396 0.0000 0.0000 0.0000 0.0000ra (p.u.) 0.0000 0.0031 0.0031 0.0041 0.0041xd (p.u.) 0.8979 1.0500 1.0500 1.2500 1.2500x
Õd
(p.u.) 0.2995 0.1850 0.1850 0.2320 0.2320x
ÕÕd
(p.u.) 0.2300 0.1300 0.1300 0.1200 0.1200T
Õd
7.40 6.10 6.10 4.75 4.75T
ÕÕd
0.03 0.04 0.04 0.06 0.06Taa 0.00 0.00 0.00 0.00 0.00
xq (p.u.) 0.6460 0.9800 0.9800 1.2200 1.2200x
Õq
(p.u.) 0.6460 0.3600 0.3600 0.7150 0.7150x
ÕÕq
(p.u.) 0.4000 0.1300 0.1300 0.1200 0.1200T
Õq
0.0001 0.30 0.30 1.50 1.50T
ÕÕq
0.033 0.099 0.099 0.210 0.210H 5.148 6.540 6.540 5.060 5.060D 2.00 2.00 2.00 2.00 2.00
B.2 Data for Solar PV
Parameters used for PV panels and controllers are:
Table 27 – PV parameters.
Parameter PV 634 PV 675 PV 680Cdc (mF) 0.829 0.829 0.829
Kp 3.27276 3.27276 3.27276Ki 0.20835 0.20835 0.20835
Ên (Hz) 10 10 10› 2.00 2.00 2.00
Href (W/m2) 1000 1000 1000
Tref (K) 298.15 298.15 298.15Tnoct (K) 320.15 320.15 320.15“ (W/K) -0.005 -0.005 -0.005
÷mppt 0.99 0.99 0.99
78
B.3 Data for AVRs
The parameters used for all static exciters are:
Table 28 – Data for static exciter of each synchronous machine.
Parameter G1 G2 C1 C2 C3
Tr 0.001 0.001 0.001 0.001 0.001KA 200 20 20 20 20TA 0.02 0.02 0.02 0.02 0.02TB 0.00 0.00 0.00 0.00 0.00TC 0.00 0.00 0.00 0.00 0.00
VRmax 7.32 4.38 4.38 6.81 6.81VRmin 0.00 0.00 0.00 1.395 1.395KE 1.00 1.00 1.00 1.00 1.00TE 0.19 1.98 1.98 0.70 0.70KF 0.0012 0.0010 0.0010 0.0010 0.0010TF 1.00 1.00 1.00 1.00 1.00
