Boost Matrix Converter Applied to Wind Energy Conversion Systems

Transcription

Boost Matrix Converter Applied to Wind Energy Conversion Systems
João Pedro Costa e Castro
Thesis to obtain the Master of Science Degree in
Electrical and Computer Engineering
Supervisors: Prof. Sónia Maria Nunes dos Santos Paulo Ferreira Pinto and
Prof. José Fernando Alves da Silva
Examination Committee
Chairperson: Prof. Maria Eduarda de Sampaio Pinto de Almeida Pedro
Supervisor: Prof. Sónia Maria Nunes dos Santos Paulo Ferreira Pinto
Member of the Committee: Prof. Joaquim José Rodrigues Monteiro
October 2014
Agradecimentos
A entrega da dissertação representa o final de um capítulo importante da minha vida, por isso
resta-me deixar algumas palavras de agradecimento às pessoas que me acompanharam neste
percurso e que, de uma maneira ou de outra, contribuíram para que fosse possível concluir
com sucesso esta etapa.
Em primeiro lugar, gostaria de agradecer à Professora Sónia Pinto e ao Professor Fernando
Silva pela confiança que depositaram em mim ao aceitarem orientar a minha dissertação. À
Professora Sónia Pinto um agradecimento muito especial pela dedicação, paciência e entusiasmo inexcedíveis e pela total disponibilidade para esclarecer as inúmeras dúvidas e problemas que foram surgindo no decurso deste trabalho; ao Professor Fernando Silva um agradecimento pelo excelente trabalho de coorientação, com sugestões e conselhos que contribuíram
para melhorar o produto final.
À minha mãe, ao meu pai e ao meu irmão, pelo apoio prestado nas horas mais difíceis e por
terem proporcionado as condições psicológicas e materiais indispensáveis à concretização
deste objetivo. Ao meu pai, em particular, agradeço os comentários, revisões e sugestões que
acrescentaram valor ao trabalho.
À Catarina Albuquerque, pelo carinho, compreensão e inspiração e por me ter ajudado a ultrapassar os momentos mais complicados com um sorriso.
Aos meus grandes amigos André Duarte, Gonçalo Saraiva, Gonçalo Mendes, Ricardo Pires e
Flávio Lopes, pela amizade, camaradagem e espírito de grupo.
Aos grandes amigos que conheci neste percurso, agradeço os momentos de estudo e convívio,
em especial ao Pedro Carlos, ao João Maurício, ao Francisco Pires, ao Francisco Marques, ao
Gonçalo Silva e ao Pedro Antunes.
Aos meus professores, pela contribuição para a minha formação académica e por terem despertado o meu interesse nestes temas.
i
Resumo
A presente dissertação apresenta uma nova contribuição para o estudo dos conversores matriciais trifásicos, garantido que o conversor matricial tem características de elevador de tensão.
Simultaneamente, o conversor matricial permite a adaptação de frequências entre as grandezas de entrada e de saída, fator de potência regulável no ponto de ligação à rede elétrica e
bidirecionalidade no trânsito de energia. Este é o Conversor Matricial Elevador.
Neste trabalho, é desenvolvido um modelo completo e detalhado, incluindo uma abordagem
inovadora ao processo de modulação, que combina a Modulação por Vetores Espaciais e a
Modulação por Largura de Impulso com técnicas clássicas de projeto de controladores.
A validação do processo de modulação é efetuada com sucesso, primeiro para uma carga RL
genérica, e depois com a ligação aos terminais do conversor matricial de um gerador de magnetos permanentes acoplado a uma turbina eólica, usando, para esse efeito, modelos do sistema global.
Os resultados obtidos com recurso ao Matlab/Simulink®, que podem ser posteriormente complementados com testes em ambiente de laboratório, antecipam o potencial do Conversor
Matricial Elevador nas mais variadas aplicações de engenharia, como transmissão em corrente
contínua, reguladores ativos de tensão, reguladores de trânsito de energia em redes de transporte e acionamentos elétricos.
Palavras-chave: Conversor Matricial Trifásico, Conversor Matricial Elevador, Modulação por
Vetores Espaciais, Modulação por Largura de Impulso, Projeto de Controladores, Gerador Síncrono de Magnetos Permanentes, Turbina Eólica
ii
Abstract
The present thesis offers a novel contribution to the study of three-phase matrix converters,
ensuring that the matrix converter presents voltage step-up characteristics. Simultaneously,
the matrix converter allows an input/output frequency adaptation, adjustable power factor in
the point of common coupling and bidirectional power flow. This is the Boost Matrix Converter
(Boost MC).
A full detailed model is developed, including an innovative approach to the modulation strategy, which uses Space Vector Modulation and Pulse-Width Modulation combined with classical
techniques of controller design.
The validation of the modulation process is successfully performed, firstly with a generic RL
load, and then with a set permanent magnet synchronous generator + wind turbine connected
to the matrix converter terminals.
The obtained results in Matlab/Simulink®, which can be later complemented with tests in laboratory environment, anticipate the Boost MC potential in several engineering applications,
like High-Voltage Direct Current, Dynamic Voltage Restorer, Unified Power Flow Controller and
electrical drives.
Keywords: Three-phase Matrix Converter, Boost Matrix Converter, Space Vector Modulation,
Pulse-Width Modulation, Controller Design, Permanent Magnet Synchronous Generator, Wind
Turbine
iii
Table of Contents
Agradecimentos ............................................................................................................................. i
Resumo.......................................................................................................................................... ii
Abstract ........................................................................................................................................ iii
Table of Contents ..........................................................................................................................iv
List of Figures ................................................................................................................................vi
List of Tables ................................................................................................................................ viii
Acronyms.......................................................................................................................................ix
1.
2.
Introduction .......................................................................................................................... 2
1.1.
Context and Motivation ................................................................................................ 2
1.2.
Objectives ...................................................................................................................... 5
1.3.
State-of-the-art ............................................................................................................. 6
1.4.
Contents ...................................................................................................................... 10
Matrix Converter ................................................................................................................. 14
2.1.
Matrix Converter Basics .............................................................................................. 14
2.2.
Modulation Strategy ................................................................................................... 17
2.2.1.
Space Vector Representation .............................................................................. 17
2.2.2.
Indirect Modulation ............................................................................................ 22
2.2.3.
Space Vector Modulation – Application to the Boost Converter........................ 26
2.2.4.
Indirect Modulation – Application to the Boost Converter ................................ 33
2.3.
2.3.1.
Voltage Regulator ................................................................................................ 35
2.3.2.
Current Regulator ................................................................................................ 40
2.3.3.
Reference Values Setting .................................................................................... 44
2.4.
3.
Regulators Design........................................................................................................ 35
Filters Sizing ................................................................................................................. 46
2.4.1.
Load Filter ............................................................................................................ 46
2.4.2.
Grid Filter............................................................................................................. 50
Wind Turbine Generator ..................................................................................................... 54
3.1.
Wind Turbine ............................................................................................................... 54
3.1.1.
Structure and Main Components ........................................................................ 54
3.1.2.
Power in the Wind............................................................................................... 56
iv
3.1.3.
Turbine Model ..................................................................................................... 56
3.1.4.
Generator Power Curve....................................................................................... 57
3.1.5.
Torque Control .................................................................................................... 58
3.2.
4.
3.2.1.
Description .......................................................................................................... 62
3.2.2.
Machine’s Model ................................................................................................. 65
3.2.3.
Field-Oriented Control ........................................................................................ 70
Validation Results and Discussion ...................................................................................... 74
4.1.
Step 1 – Boost matrix converter feeding a generic RL load ........................................ 74
4.1.1.
Case-study 1 ........................................................................................................ 75
4.1.2.
Case-Study 2 ........................................................................................................ 77
4.1.3.
Case-Study 3 ........................................................................................................ 79
4.1.4.
Case Study 4 ........................................................................................................ 80
4.2.
5.
Permanent Magnet Synchronous Generator .............................................................. 62
Step 2 – Boost matrix converter feeding a wind conversion system .......................... 82
Conclusions ......................................................................................................................... 88
Bibliography ................................................................................................................................ 91
v
List of Figures
Figure 1.1: Single-line diagram of a generic system with an AC/AC converter............................. 2
Figure 1.2: Indirect converter........................................................................................................ 2
Figure 1.3: Direct converter .......................................................................................................... 3
Figure 1.4: Three-phase matrix converter topology ..................................................................... 3
Figure 1.5: Buck Matrix Converter (top) and Boost Matrix Converter (down) ............................. 5
Figure 1.6: Single-line diagram of the system studied .................................................................. 5
Figure 1.7: DVR single-line diagram (Alcaria, 2012) ...................................................................... 6
Figure 1.8: UPFC single-line diagram (Monteiro, Silva, Pinto, & Palma, 2011) ............................. 7
Figure 1.9: Matrix converters and high frequency transformer in traction substations (Mendes,
2013) ............................................................................................................................................. 8
Figure 1.10: Wind turbine driven by DFIG with matrix converter (Afonso, 2011) ........................ 9
Figure 1.11: Wind turbine driven by PMSG with matrix converter (Fernandes, 2013) ................ 9
Figure 2.1: Generic matrix converter with nxm phases .............................................................. 14
Figure 2.2: Three-phase matrix converter (Pinto S. F., 2003) ..................................................... 15
Figure 2.3: Space vector representation (groups II e III) ............................................................. 18
Figure 2.4: Representation of the twelve location zones of input voltages ............................... 20
Figure 2.5: Output voltage vectors to Zone 1 ............................................................................. 21
Figure 2.6: Representation of the twelve location zones of output currents ............................. 21
Figure 2.7: Input Current Vectors (Zone 1) ................................................................................. 22
Figure 2.8: Equivalent model of a rectifier-inverter association (Pinto S. F., 2003) ................... 22
Figure 2.9: Buck Matrix Converter single-line diagram ............................................................... 24
Figure 2.10: Boost Matrix Converter single-line diagram ........................................................... 26
Figure 2.11: Spatial location of the vectors (I1-I9) needed to control the output current (Pinto
S. F., 2003) ................................................................................................................................... 27
Figure 2.12: Example of synthesis of
in sector 0 (Pinto S. F., 2003) ........................ 28
Figure 2.13: Spatial location of the vectors (V0-V7) needed to control the input voltage (Pinto
S. F., 2003) ................................................................................................................................... 30
in sector 0 (Pinto S. F., 2003).......................... 31
Figure 2.15: PWM modulation process used to select the time interval during the appropriate
vectors are applied ...................................................................................................................... 34
Figure 2.16: Selection scheme for the SVM vectors ................................................................... 35
Figure 2.17: Single-line diagram of the whole system – voltage regulator focus ....................... 36
Figure 2.18: Single-phase equivalent used do extract the system equations............................. 36
Figure 2.19: Voltage regulator block diagram ............................................................................. 38
Figure 2.20: Simplified voltage regulator block diagram ............................................................ 38
Figure 2.21: Single-line diagram of the whole system – current regulator focus ....................... 40
Figure 2.22: Current regulator block diagram............................................................................. 42
Figure 2.23: Simplified current regulator block diagram ............................................................ 42
Figure 2.24: Load filter in the global system ............................................................................... 47
vi
Figure 2.25: Load filter single-phase equivalent ......................................................................... 47
Figure 2.26: Grid filter in the global system ................................................................................ 50
Figure 2.27: Grid filter single-phase equivalent .......................................................................... 50
Figure 3.1: Wind turbine structure ............................................................................................. 54
Figure 3.2: Single-line diagram of the set turbine + generator ................................................... 55
Figure 3.3: Power curve of a 2 MW generator (Castro, 2012) .................................................... 57
Figure 3.4: Cp variation with and ......................................................................................... 59
Figure 3.5: Cp variation with ( = ) ...................................................................................... 60
Figure 3.6: Cross section of a typical PMSG (adapted from (Fernandes, 2013)) ........................ 63
Figure 3.7: Armature winding arrangement (adapted from (Fernandes, 2013))........................ 63
Figure 3.8: Graphical view of the application of Concordia transformation (Fernandes, 2013) 67
Figure 3.9: Graphical view of the application of Park’s transformation (adapted from
(Fernandes, 2013)) ...................................................................................................................... 68
Figure 3.10: Field-Oriented Control – graphical view ................................................................. 72
Figure 4.1: Boost matrix converter feeding a RL load ................................................................. 74
Figure 4.2: Load voltage and grid voltage – case study 1 ........................................................... 76
Figure 4.3: Voltage and current in the grid side- case study 1.................................................... 76
Figure 4.4: d component of the voltage regulator error............................................................. 77
Figure 4.5: q component of the voltage regulator error............................................................. 77
Figure 4.6: d component of the current regulator error............................................................. 77
Figure 4.7: q component of the current regulator error............................................................. 77
Figure 4.8: Load current vs grid current – frequency adaptation – case study 2 ....................... 78
Figure 4.9: d component of the current regulator error............................................................. 78
Figure 4.10: q component of the current regulator error........................................................... 78
Figure 4.11: Grid voltage and grid current – unitary power factor – case study 3 ..................... 79
Figure 4.12: Grid voltage and grid current – capacitive power factor – case study 3 ................ 80
Figure 4.13: Load voltage and grid voltage – case-study 4 ......................................................... 81
Figure 4.14: Load current and grid current – case-study 4 ......................................................... 82
Figure 4.15: Boost matrix converter feeding a wind turbine generator ..................................... 83
Figure 4.16: Piece of a wind profile............................................................................................. 83
Figure 4.17: Torque at the Maximum Power Point (equal to the reference torque) ................. 84
Figure 4.18: Single-phase voltage imposed by the Boost MC to the PMSG’s terminals ............. 84
Figure 4.19: d component of the voltage regulator error........................................................... 84
Figure 4.20: q component of the voltage regulator error........................................................... 84
vii
List of Tables
Table 2-1: All possible combinations for the switch state combinations with the instantaneous
values of output voltages and input currents ............................................................................. 16
Table 2-2: Output voltage vectors and input current vectors resulting from the application of
Concordia’s transformation ........................................................................................................ 19
Table 2-3: All possible switch state combinations of the rectifier-inverter association ............. 25
Table 2-4: State vectors generated by the rectifier for all the possible combinations ............... 27
Table 2-5: State vectors generated by the inverter for all the possible combinations............... 30
Table 2-6: Matrix converter’s vectors used in the modulation of the input voltages and output
currents ....................................................................................................................................... 34
Table 2-7: Voltage controller gains and associated constants .................................................... 40
Table 2-8: Rated values ............................................................................................................... 46
Table 2-9: Load filter parameters and the associated constants ................................................ 50
Table 2-10: Grid filter parameters and the associated constants............................................... 51
Table 3-1: PMSG’s parameters .................................................................................................... 69
Table 4-1: Simulation conditions to case study 1........................................................................ 75
Table 4-3: Simulation conditions to case study 3 (unitary power factor) ................................... 79
Table 4-4: Simulation conditions to case study 3 (capacitive power factor) .............................. 80
viii
Acronyms
AC – Alternate Current
BJT – Bipolar Junction Transistor
DC – Direct Current
DFIG – Doubly Fed Induction Generator
DVR – Dynamic Voltage Restorer
FOC – Field-Oriented Control
GTO – Gate Turn-Off Thyristor
HVDC – High-Voltage Direct Current
IGBT – Insulated Gate Bipolar Transistor
ITAE – Integral of Time and Absolute Error
KCL – Kirchhoff’s Current Law
KVL – Kirchhoff’s Voltage Law
MC – Matrix Converter
MCT – MOS Controlled Thyristor
MPPT – Maximum Power Point Tracking
PCC – Point of Common Coupling
PI – Proportional-Integral
PMSG – Permanent Magnet Synchronous Generator
PWM – Pulse-Width Modulation
RB-IGBT – Reverse Blocking Insulated Gate Bipolar Transistor
RMS – Root Mean Square
SVM – Space Vector Modulation
TSO – Transmission System Operator
UPFC – Unified Power Flow Controller
THD – Total Harmonic Distortion
TSR – Tip Speed Ratio
VSI – Voltage Source Inverter
ix
CHAPTER 1
Introduction
Abstract
Chapter 1 presents the context and the motivation to develop the thesis theme, as
well as the main objectives to be achieved and the structure of the thesis. A summary
of the most important limitations of the classic matrix converter (Buck MC) utilization
in some engineering applications is made and a different configuration of the matrix
converter (Boost MC) is presented, in order to overcome those limitations.
1
1. Introduction
1.1. Context and Motivation
In 1980, Venturini and Alesina (Venturini & Alesina, 1980) proposed the first algorithm capable
of synthetizing output sinusoidal voltages from a three-phase voltage source connected to the
matrix converter input terminals. Since then, this field has been subject to intense research
and development.
In power electronics, an AC/AC converter is a power electronic device, normally fed by a sinusoidal voltage system (input) and composed by power semiconductors (GTO, TJB, IGBT, MCT or
RB-IGBT), which displays output sinusoidal quantities with different characteristics of the input
(RMS value, frequency and/or power factor). Figure 1.1 presents a single-line diagram of a
generic system with an AC/AC converter.
Figure 1.1: Single-line diagram of a generic system with an AC/AC converter
There are two types of AC/AC converters:
Indirect converter: association of two independent converters - an AC/DC converter and a
DC/AC converter connected in cascade – and a DC-link with energy storage components (usually a bank of capacitors), as represented in Figure 1.2.
Vin1
Vin 2
Vin 3
Vinn
AC
Vout 1
Vout 2
Vout 3
DC
AC
DC
Figure 1.2: Indirect converter
2
Voutm
Direct converter: single converter that performs directly the AC/AC conversion without the use
of any energy storage elements, as depicted in Figure 1.3.
Vin1
Vin 2
Vin 3
Vinn
AC
AC
Vout 1
Vout 2
Vout 3
Voutm
Figure 1.3: Direct converter
Indirect converters operation is not the scope of this thesis, direct converters being the focus
here. Therefore, we will pay special attention to the main type of direct converters, the threephase matrix converters.
A three-phase matrix converter has the generic topology as depicted in Figure 1.4.
Vina
Vinb
Vinc
Figure 1.4: Three-phase matrix converter topology
Up to now, the conventional configuration of the matrix converter presents the characteristics
of a step down converter and will be designated here as Buck Matrix Converter (Buck MC),
because the output RMS voltage is lower than the input RMS voltage.
3
The main advantages of the Buck MC are as follows:
•
•
•
•
•
High efficiency.
Capability to control the fundamental frequency and the RMS value of the output voltages and also the input power factor as seen from the generator.
Lower volume, with the correspondent power density increase.
Bidirectional power flow.
Input current waveforms nearly sinusoidal.
On the opposite, there are also several disadvantages:
•
•
•
•
Large number of semiconductors.
Complexity of the control system.
Output RMS voltage limited to, at most, √3/2 of the input.
Higher probability of disturbances due to the high frequency operation of semiconductors.
In recent years, matrix converters are being increasingly used in several applications: Dynamic
Voltage Restorer (DVR) (Alcaria, 2012), (Wang & Venkataramanan, 2009), (Pandey &
Rajlakshmi, 2013), (Gamboa, Silva, Pinto, & Margato, 2009), Unified Power Flow Controller
(UPFC) (Monteiro, Silva, Pinto, & Palma, 2014), (Monteiro, Silva, Pinto, & Palma, 2011), traction substations in railway network (Mendes, 2013), (Drabek, Peroutka, Pittermann, & Cédl,
2011), electrical drives with ⁄ command, renewable energy applications (control of Doubly
Fed Induction Generator (DFIG) and Permanent Magnet Synchronous Generator (PMSG))
(Afonso, 2011), (Djeriri, Meroufel, Massoum, & Boudjema, 2014), (Fernandes, 2013).
In the applications described above, the matrix converter is operated as a Buck MC. This
means that the voltage displayed by the conventional matrix converter is reduced, at least, by
a factor of √3⁄2 in relation to the input voltage. In some applications, this can be a serious
drawback. It could be interesting to have a device that could increase the voltage, instead of
reducing it, because this type of feature is required by some applications.
In order to achieve the purpose of increasing the output voltage with respect to the input voltage, a different configuration of the matrix converter is studied in this thesis – the Boost Matrix Converter (Boost MC).
Figure 1.5 shows schematically the difference between the two converters.
4
Figure 1.5: Buck Matrix Converter (top) and Boost Matrix Converter (down)
In this thesis, the Boost MC will be used in association with a wind conversion system with the
aim of controlling the voltage at the wind generator terminals and the current injected into the
grid, as depicted in Figure 1.6.
Figure 1.6: Single-line diagram of the system studied
1.2. Objectives
This thesis’ main goal is to give a novel contribution to the study of the Boost MC, as this type
of matrix converter is poorly assessed in the available literature. In order to achieve such goal,
the following partial objectives are to be accomplished:
1. Development of a full model, including modulation strategy, regulators design and filters sizing, to enable the study of the Boost matrix converter.
5
2. Validation of the modulation process, with the capability to control, over a wide range,
the input/output quantities in terms of RMS value, frequency and power factor.
3. Simulation of a real application, namely a wind turbine driven by a permanent magnet
synchronous generator controlled by a Boost matrix converter.
These are quite innovative aspects, portraying original contributions of this thesis, because a
review of the available literature showed that Boost MC have been insufficiently addressed, up
to now.
1.3. State-of-the-art
There are several applications where Buck MC is being used. A list of the main applications of
Buck MC follows.
Dynamic Voltage Restorer (DVR)
A DVR is a power electronic topology that aims at protecting sensitive loads (mainly in Low
Voltage (LV) distribution grids) from disturbances in the power supply. Usually, DVR ensures a
low time response, therefore allowing the distribution grid to become nearly immune to voltage sags and voltage swells. This goal is achieved by inserting a voltage in series with the LV
distribution grid, being that compensation voltage granted by a matrix converter (Alcaria,
2012), (Wang & Venkataramanan, 2009), (Pandey & Rajlakshmi, 2013), (Gamboa, Silva, Pinto,
& Margato, 2009), as illustrated in Figure 1.7.
Figure 1.7: DVR single-line diagram (Alcaria, 2012)
6
Unified Power Flow Controller (UPFC)
An UPFC is a power electronic device capable of regulating the power flow in transmission
grids. Selecting an appropriate matrix converter switching state, it is possible to control the
active and reactive power that flows through some branches of the network. Recent approaches, based on sliding mode control techniques, also guarantee a decoupled control of
active and reactive power (Monteiro, Silva, Pinto, & Palma, 2014), (Monteiro, Silva, Pinto, &
Palma, 2011). Figure 1.8 shows a transmission network with a UPFC implemented with a matrix converter.
Figure 1.8: UPFC single-line diagram (Monteiro, Silva, Pinto, & Palma, 2011)
Traction substations in railway network
Traction substations are located along a railway network and its function is to receive the electrical power from the transmission grid and convert it to an adequate voltage to supply the
locomotive’s traction systems. However, there are numerous locomotives crossing several
countries, supplied by different traction substations, so locomotives’ traction systems are
equipped with some electronic and mechanical devices that adjust the type of supply. To avoid
the use of these devices, it is proposed that the traction substation on its own is able to adapt
the supply to the characteristics of the locomotive crossing its zone. This can be accomplished
using matrix converters and a high frequency transformer (Mendes, 2013), (Drabek, Peroutka,
Pittermann, & Cédl, 2011). In Figure 1.9 an example of a traction substation with the configuration described above is presented.
7
Figure 1.9: Matrix converters and high frequency transformer in traction substations (Mendes, 2013)
Electrical drives with ⁄ command
⁄ command is widely used in electrical drives, namely in applications that involve asynchronous motors. The torque developed by the motor is nearly proportional to the ratio of voltage
amplitude and frequency of the supply, so it is possible, by actuating in the ⁄ ratio, to keep
the torque constant throughout the speed range (Dente, 2011). A matrix converter, due to its
capability to change output voltage amplitude and frequency, is being gradually adopted in
this type of applications.
Renewable energy applications
Matrix converters are replacing indirect converters in some electrical generators used in renewable energy applications, like DFIG (Castro, 2012) or PMSG. When a DFIG is used, the matrix converter is controlled with the aim of extracting the maximum available power from the
wind and to control the power factor in the point of common coupling (PCC) (see Figure 1.10),
(Afonso, 2011), (Djeriri, Meroufel, Massoum, & Boudjema, 2014); when a PMSG is used, the
matrix converter is controlled with the double objective of extracting the maximum available
power from the wind and adapting the variable frequency of the stator quantities to the constant frequency of the grid (see Figure 1.11), (Fernandes, 2013).
8
Figure 1.10: Wind turbine driven by DFIG with matrix converter (Afonso, 2011)
Figure 1.11: Wind turbine driven by PMSG with matrix converter (Fernandes, 2013)
We will see now a few exemplificative situations wherein a larger range of output voltage variation is required, therefore justifying the use of a Boost MC.
9
DVR
In distribution networks, but also in transmission networks, the use of DVR helps in keeping
the voltage profile under control, namely when voltage swells or voltage sags occur. When
voltage sags are to be addressed, an increase in the output voltage with respect to the input
voltage would be welcome.
UPFC
To enhance active and reactive power control in transmission grids, a wider range of voltage
variation would be a plus, since those quantities depend upon the voltage.
Electrical drives with ⁄ command
Sometimes, in electrical drives with ⁄ command, it is necessary to increase the frequency of
the supply , to achieve a desired machine speed. To maintain the magnetization level and, at
the same time, to keep the torque nearly constant, the voltage must be increased. As so, a
matrix converter capable of increasing the voltage amplitude can be useful.
High-Voltage Direct Current (HVDC)
An HVDC is a long distance transmission system. It is composed by an AC/DC converter station
at the emission, a DC transmission line and a DC/AC converter station at the receiving end. In
order to guarantee an adequate DC transmission voltage, the input AC voltage must be step up
using a transformer. To perform this function, a Boost MC maybe used instead.
1.4. Contents
The thesis is organized into five chapters and a list of references is provided at the end.
Chapter 1 presents the context and the motivation to develop the thesis theme, as well as the
main objectives to be achieved and the structure of the thesis. A summary of the most important limitations of the classic matrix converter (Buck MC) utilization in some engineering
applications is made and a different configuration of the matrix converter (Boost MC) is presented, in order to overcome those limitations.
10
Chapter 2 provides a full description of the innovative approach adopted to control the Boost
MC. The changes performed in the conventional modulation process, as well as the details
about the regulators design and the filters sizing, are assessed.
In Chapter 3 it is described, in some detail, the dynamic models of the wind turbine and of the
permanent magnet synchronous generator. This wind application is connected to the matrix
converter terminals.
Chapter 4 presents the results that validate the modulation process in the context of a wind
application and also the respective discussion.
Chapter 5 finalizes the thesis by presenting a set of conclusions that can be drawn and giving
some suggestions for future work.
11
CHAPTER 2
Matrix Converter
Abstract
Chapter 2 provides a full description of the innovative approach adopted to control
the Boost matrix converter, implying an adequate modification in the conventional
modulation process. Furthermore, details about the regulators design and the filters
sizing are also in the scope of this chapter.
13
2. Matrix Converter
2.1. Matrix Converter Basics
A generic matrix converter is composed by a set of
bidirectional switches, which allows
the interconnection between two distinct systems, with and phases respectively. In Figure
2.1 a generic matrix converter with
phases is depicted.
Vin1
Vin 2
Iin1
Iin 2
Vin 3 Iin 3
Vinn Iinn
Vout 1
Vout 2
Vout 3
Voutm
Iout 1
Iout 2
Iout 3
Ioutm
Figure 2.1: Generic matrix converter with nxm phases
This type of converters is normally represented with an input with voltage source characteristics and an output with current source characteristics.
Theoretically, as each switch has two possible states (turned on and turned off), this would
allow the existence of 2 × possible switch state combinations. However, neither short-circuit
voltage sources (feeder) nor open current sources (load) are desirable, as so the topological
.
constraints reduce the number of possible combinations to
14
Assuming that the semiconductors that compose the bidirectional switches are ideal, each
switch can be represented by a variable ! , described as follows:
!
1, %&'()ℎ(,- ./0
="
0, %&'()ℎ(,- ./0
', 2 ∈ 41,2,35
(2.1)
The three-phase matrix converter ( = 3 and = 3) referred to in (2.1) is a particular case of
the generic matrix converter and is composed by nine bidirectional switches that allow a total
of 36 = 27 possible combinations. Three-phase matrix converters allow the connection of
each one of the three output phases to any one of the three input phases.
Figure 2.2 represents a typical three-phase matrix converter and Table 2-1 contains all possible
combinations for the switch state combinations (numbered from 1 to 27), with the instantaneous values of output voltages 89: and input currents ';<= .
Figure 2.2: Three-phase matrix converter (Pinto S. F., 2003)
The switch states can be compacted in a 3 3 matrix :
=>
??
@?
6?
?@
@@
6@
?6
@6 A
(2.2)
66
15
in which;
∑6!C?
!
', 2 ∈ 41,2,35
=1
(2.3)
Table 2-1: All possible combinations for the switch state combinations with the instantaneous values of output
voltages and input currents
1
DEE DEF DEG DFE DFF DFG DGE DGF DGG
1
0
0
0
1
0
0
0
1
2
0
1
0
0
0
1
1
0
0
3
0
0
1
1
0
0
0
1
0
4
1
0
0
0
0
1
0
1
0
5
0
1
0
1
0
0
0
0
1
6
0
0
1
0
1
0
1
0
0
7
1
0
0
0
1
0
0
1
0
8
0
1
0
1
0
0
1
0
0
9
0
1
0
0
0
1
0
0
1
10
0
0
1
0
1
0
0
1
0
11
0
0
1
1
0
0
1
0
0
12
1
0
0
0
0
1
0
0
1
13
0
1
0
1
0
0
0
1
0
14
1
0
0
0
1
0
1
0
0
15
0
0
1
0
1
0
0
0
1
16
0
1
0
0
0
1
0
1
0
17
1
0
0
0
0
1
1
0
0
18
0
0
1
1
0
0
0
0
1
19
0
1
0
0
1
0
1
0
0
20
1
0
0
1
0
0
0
1
0
21
0
0
1
0
0
1
0
1
0
22
0
1
0
0
1
0
0
0
1
23
1
0
0
1
0
0
0
0
1
24
0
0
1
0
0
1
1
0
0
25
1
0
0
1
0
0
1
0
0
26
0
1
0
0
1
0
0
1
0
27
0
0
1
0
0
1
0
0
1
N.º
H (J)
; (()
< (()
< (()
< (()
= (()
= (()
; (()
< (()
; (()
= (()
< (()
; (()
= (()
< (()
; (()
= (()
< (()
; (()
= (()
; (()
< (()
= (()
': (()
= (()
< (()
< (()
< (()
= (()
; (()
= (()
; (()
NO (J)
; (()
; (()
< (()
M (J)
= (()
= (()
; (()
16
L (J)
= (()
< (()
; (()
= (()
< (()
; (()
= (()
; (()
< (()
< (()
= (()
= (()
; (()
< (()
= (()
= (()
< (()
; (()
= (()
; (()
< (()
= (()
< (()
; (()
< (()
'8 (()
'9 (()
'8 (()
': (()
'9 (()
'9 (()
': (()
= (()
< (()
−'8 (()
'8 (()
= (()
; (()
−'9 (()
'9 (()
−': (()
< (()
0
; (()
; (()
= (()
'8 (()
'8 (()
0
0
'9 (()
−'9 (()
0
−'9 (()
0
−': (()
= (()
= (()
'9 (()
0
0
< (()
0
−'8 (()
0
−': (()
−': (()
0
0
': (()
< (()
': (()
−'8 (()
−'9 (()
; (()
'9 (()
'8 (()
0
0
< (()
'8 (()
−'9 (()
−'9 (()
= (()
'8 (()
'9 (()
; (()
'8 (()
'9 (()
'8 (()
−'8 (()
; (()
': (()
'9 (()
0
0
NQ (J)
'8 (()
−'8 (()
−'8 (()
< (()
': (()
'8 (()
; (()
= (()
NP (J)
': (()
'9 (()
'9 (()
0
0
': (()
−': (()
0
0
−': (()
0
0
0
0
0
0
': (()
0
': (()
': (()
0
It is worth to point out that matrix relates line-to-neutral output voltages 89: with line-toneutral input voltages ;<= . Input currents ';<= are converted in output currents '89: through
matrix ( S ).
>
8
9A
:
= >
';
>'< A =
'=
S
;
<A
(2.4)
'8
>'9 A
':
(2.5)
=
Furthermore, matrix = establishes a relationship between line-to-line output voltages (
9: and :8 ) with line-to-neutral input voltages.
−
> 9: A = > @? −
:8
6? −
89
??
@?
6?
??
−
@@ −
6@ −
?@
@@
6@
?@
−
@6 −
66 −
?6
@6
;
66 A > < A
=
?6
=
=
>
;
<A
=
89 ,
(2.6)
Matrices and = will play an important role in establishing the converter’s control strategy, as
will be seen, further in this chapter.
2.2. Modulation Strategy
2.2.1. Space Vector Representation
Two modulation strategies can be adopted to control the matrix converter: the Venturini approach and the Space Vector Modulation (SVM) approach. In the scope of this work, SVM approach was chosen over Venturini approach, because it ensures better input-output transfer
relationships and also minimum harmonic distortion (Pinto S. F., 2003).
The main principle of the SVM approach is to consider that, at each time instant, the output
voltages and the input currents can be represented as vectors in the TU plan. This polar repre-
17
sentation can be achieved by applying the Concordia’s transformation1 to each switching state
combination of Table 2-1.
Table 2-2 presents the amplitude and the angle for each output voltage/ input current vector,
resulting from the application of Concordia’s transformation to each state combination of Table 2-1.
Consulting Table 2-2, it is possible to divide the resulting vectors in three groups:
•
Rotating vectors (group I): constant amplitude, but variable angle.
•
Pulsating vectors (group II): constant angle, but variable amplitude and signal
along the time.
•
Null vectors (group III): null amplitude and angle.
Rotating vectors were not considered in the modulation process, because the rotation in TU
plan increases the process complexity (Pinto S. F., 2003), but both pulsating and null vectors
will be used. The representation in TU plan of all considered vectors is shown in Figure 2.3.
Figure 2.3: Space vector representation (groups II e III)
Concordia’s transformation, which performs the coordinate’s change VW) → TU0, is given by:
?
0
_1
√@b
?a
√6
@^ ?
YZ[ = \ ^−
@
@
√@a
6
^ ?
? à
√6
]− @ − @ √@
1
18
Table 2-2: Output voltage vectors and input current vectors resulting from the application of Concordia’s transformation
Output Voltage
Group
N.º
Name
1
1g
2
2g
3
3g
4
4g
5
5g
6
6g
7
+1
8
-1
9
+2
10
-2
11
+3
12
-3
13
+4
14
-4
15
+5
16
-5
17
+6
18
-6
19
+7
20
-7
21
+8
22
-8
23
+9
24
-9
25
za
−k2⁄3
26
zb
27
zc
I
II
III
Input Current
Amplitude
Angle
Angle
−√3c =
d (()
Amplitude
−d (()
+ 4j⁄3
d (()
+ 4j⁄3
√3ef
−gf (
√3c =
−d (()
√3ef
−gf (
+ 2j⁄3
−√3c =
−d (()
+ 2j⁄3
√3ef
gf (
+ 4j⁄3
√3c =
−√3c =
√3c =
k2⁄3
;< (()
−k2⁄3
k2⁄3
−k2⁄3
k2⁄3
−k2⁄3
k2⁄3
−k2⁄3
k2⁄3
−k2⁄3
k2⁄3
−k2⁄3
k2⁄3
−k2⁄3
k2⁄3
−k2⁄3
k2⁄3
;< (()
<= (()
<= (()
=; (()
=; (()
;< (()
;< (()
<= (()
<= (()
=; (()
=; (()
;< (()
;< (()
<= (()
<= (()
=; (()
d (()
+ 2j⁄3
0
0
0
0
0
0
2j⁄3
√3ef
√3ef
√3ef
√2'8 (()
−√2'8 (()
√2'8 (()
−√2'8 (()
√2'8 (()
−√2'8 (()
4j⁄3
4j⁄3
4j⁄3
4j⁄3
j ⁄2
j ⁄2
7j⁄6
7j⁄6
√2'9 (()
−√2'9 (()
−j⁄6
j ⁄2
7j⁄6
7j⁄6
√2': (()
−j⁄6
−√2': (()
j ⁄2
−√2': (()
√2': (()
−j⁄6
j ⁄2
√2': (()
7j⁄6
-
-
−√2': (()
0
7j⁄6
0
-
0
-
0
-
0
-
0
=; (()
19
4j⁄3
−j⁄6
j ⁄2
√2'9 (()
4j⁄3
−j⁄6
−√2'9 (()
2j⁄3
2j⁄3
−gf (
+ 4j⁄3
−j⁄6
−√2'9 (()
2j⁄3
gf (
+ 2j⁄3
√2'9 (()
2j⁄3
2j⁄3
gf (
Since the variable amplitude of pulsating vectors is dependent of the instantaneous values of
input voltages ;<= or output currents '89: (see Table 2-2), spatial location of output voltage
vectors depends on time location of input voltages and spatial location of input current vectors
depends on time location of output currents. The time location of input voltages is performed,
as depicted in Figure 2.4, dividing the waveform in twelve zones.
1
2
3
4
5
6
7
8
9
10
11
12
Figure 2.4: Representation of the twelve location zones of input voltages
The criterion to perform the waveform division is to choose some notable points where a significant change in the relative position of the variables can occur. For example, for Zone 1, the
output voltage vectors according to input voltage time location are as shown in Figure 2.5.
From Figure 2.4, it is possible to identify, in Zone 1, the voltage which has the highest value, in
this case is (c=; ). In Table 2, and considering only the vectors with null angle, ±41,2,35, we can
see that the vectors which depend from this voltage are ±3; these are the vectors that have
the highest amplitude, as can be seen in Figure 2.5. A similar logic is adopted to complete the
remaining polar representation to Zone 1. Furthermore, as far as the other eleven zones are
concerned, the same procedure is followed.
Also, the same technique is applied in what concerns the time location of output currents. The
waveform division in twelve zones for the output currents is presented in Figure 2.6 and an
example of the input current vectors representation to Zone 1 is displayed in Figure 2.7.
20
Figure 2.5: Output voltage vectors to Zone 1
1
2
3
4
5
6
7
8
9
10
11
12
Figure 2.6: Representation of the twelve location zones of output currents
21
Figure 2.7: Input Current Vectors (Zone 1)
2.2.2. Indirect Modulation
SVM approach requires a Pulse-Width Modulation (PWM) modulator, whose main function is
to synthetize output voltages from input voltages and input currents from output currents. To
allow the application of the well-known PWM modulation techniques, the matrix converter is
represented by a rectifier-inverter association without DC link, in a process known as indirect
modulation. An equivalent model is shown in Figure 2.8.
Figure 2.8: Equivalent model of a rectifier-inverter association (Pinto S. F., 2003)
Under these conditions, the goal is to control input currents ';<= based on the intermediate
stage current en: and to control output voltages 89: based in the intermediate stage voltage
22
cn: . However, as there is no DC link, cn: and en: are not real constant quantities, as so they
suffer a time variation.
This rectifier-inverter association leads to several switch state combinations. Each switch is
represented by a variable ( o ! or p ! , for the rectifier and inverter, respectively) that assumes
the value ‘1’ when the switch is turned on and ‘0’ when the switch is turned off.
In order to avoid short-circuits in the feeder of the rectifier, it is necessary that the following
condition is satisfied:
∑6!C?
o !
= 1 ' ∈ 41,25
(2.7)
With respect to the inverter case, in order to avoid open-circuits in the load, the following restriction must be followed:
∑@C?
p !
= 1 2 ∈ 41,2,35
(2.8)
Based in (2.7) and (2.8), all the possible switch state combinations of the rectifier-inverter association can be established (see Table 2-3 below). One must keep in mind that input currents
';<= are directly related to en: through the modulation function Y o [ – as in (2.9) – and output
voltages 89: are directly related to cn: through the modulation function Y p [ – as in (2.10).
';
>'< A = q
'= >
89
9: A
:8 o ??
o @?
o @@ r s
o ?@
o ?6
=q
o @6
p ??
p ?@
en:
t=Y
−en:
en:
o [ s−e t
n:
(2.9)
cn:
c
t = Y p [ s n: t
−cn:
−cn:
p @@ r s
p @?
p 6?
p 6@
(2.10)
Intermediate stage voltage cn: is imposed by the rectifier and is directly dependent on input
voltages ;<= , as can be seen in (2.11):
cn: = Y
o ??
−
o @?
o ?@
−
o @@
o ?6
−
23
;
o @6 [ > < A
=
(2.11)
On the other hand, intermediate stage current en: is imposed by the inverter and is directly
dependent on output currents '89: , as displayed in (2.12):
en: = Y
−
p ??
p ?@
p @?
−
p @@
The resulting modulation matrix Y
op
= Y p [Y
o
[S
=q
+
o ?? +
o ?? +
p ?? o ??
p @?
p 6?
op [
p 6?
'8
p 6@ [ >'9 A
': −
is the product of Y
p ?@ o @?
p @@ o @?
p 6@ o @?
+
o ?@ +
o ?@ +
p ?? o ?@
p @?
p 6?
(2.12)
o [ and
p ?@ o @@
p @@ o @@
p 6@ o @@
Y p [, as follows:
+
o ?6 +
o ?6 +
p ?? o ?6
p @?
p 6?
p ?@ o @6
p @@ o @6 r(2.13)
p 6@ o @6
For each pair of rectifier-inverter possible combinations (Table 2-3), there is a correspondent
configuration of switches of the matrix converter (Table 2-1) and thus an associated vector
(Table 2-2), excluding rotating vectors.
As discussed previously2, the output voltage displayed by the conventional matrix converter is
limited to, at most,
√6
@
u 1 of the input voltage. This means that this converter reduces the
voltage displayed in the output, as compared to the input voltage, so it is here called Buck
Matrix Converter (Buck MC). A single-line diagram of the converter is shown in Figure 2.9.
Figure 2.9: Buck Matrix Converter single-line diagram
2
See Chapter 1 – Introduction.
24
Table 2-3: All possible switch state combinations of the rectifier-inverter association
Rectifier-Inverter Association
S R 11
1
0
0
0
0
1
1
0
0
S R 12
0
1
1
0
0
0
0
1
0
S R 13
0
0
0
1
1
0
0
0
1
S R 21
0
0
1
1
0
0
1
0
0
S R 22
0
0
0
0
1
1
0
1
0
S R 23
1
1
0
0
0
0
0
0
1
V DC
-v ca
v bc
-v ab
v ca
-v bc
v ab
0
0
0
Matrix
S I 11
S I 12
S I 21
S I 22
S I 31
S I 32
I DC
S 11
S 12
S 13
S 21
S 21
S 23
S 31
S 32
S 33
Vector
1
0
0
1
0
1
iA
1
0
0
0
0
1
0
0
1
-3
1
0
1
0
0
1
-i C
1
0
0
1
0
0
0
0
1
9
0
1
1
0
0
1
iB
0
0
1
1
0
0
0
0
1
-6
0
1
1
0
1
0
-i A
0
0
1
1
0
0
1
0
0
3
0
1
0
1
1
0
iC
0
0
1
0
0
1
1
0
0
-9
1
0
0
1
1
0
-i B
1
0
0
0
0
1
1
0
0
6
1
0
0
1
1
0
0
1
1
0
0
1
0
0
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
za
zc
1
0
0
1
0
1
iA
0
1
0
0
0
1
0
0
1
2
1
0
1
0
0
1
-i C
0
1
0
0
1
0
0
0
1
-8
0
1
1
0
0
1
iB
0
0
1
0
1
0
0
0
1
5
0
1
1
0
1
0
-i A
0
0
1
0
1
0
0
1
0
-2
0
1
0
1
1
0
iC
0
0
1
0
0
1
0
1
0
8
1
0
0
1
1
0
-i B
0
1
0
0
0
1
0
1
0
-5
1
0
0
1
1
0
0
1
1
0
0
1
0
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
zb
zc
1
0
0
1
0
1
iA
0
1
0
1
0
0
1
0
0
-1
1
0
1
0
0
1
-i C
0
1
0
0
1
0
1
0
0
7
0
1
1
0
0
1
iB
1
0
0
0
1
0
1
0
0
-4
0
1
1
0
1
0
-i A
1
0
0
0
1
0
0
1
0
1
0
1
0
1
1
0
iC
1
0
0
1
0
0
0
1
0
-7
1
0
0
1
1
0
-i B
0
1
0
1
0
0
0
1
0
4
1
0
0
1
1
0
0
1
1
0
0
1
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
zb
za
1
0
0
1
0
1
iA
0
0
1
1
0
0
1
0
0
3
1
0
1
0
0
1
-i C
0
0
1
0
0
1
1
0
0
-9
0
1
1
0
0
1
iB
1
0
0
0
0
1
1
0
0
6
0
1
1
0
1
0
-i A
1
0
0
0
0
1
0
0
1
-3
0
1
0
1
1
0
iC
1
0
0
1
0
0
0
0
1
9
1
0
0
1
1
0
-i B
0
0
1
1
0
0
0
0
1
-6
1
0
0
1
1
0
0
1
1
0
0
1
0
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
zc
za
1
0
0
1
0
1
iA
0
0
1
0
1
0
0
1
0
-2
1
0
1
0
0
1
-i C
0
0
1
0
0
1
0
1
0
8
0
1
1
0
0
1
iB
0
1
0
0
0
1
0
1
0
-5
0
1
1
0
1
0
-i A
0
1
0
0
0
1
0
0
1
2
0
1
0
1
1
0
iC
0
1
0
0
1
0
0
0
1
-8
1
0
0
1
1
0
-i B
0
0
1
0
1
0
0
0
1
5
1
0
0
1
1
0
0
1
1
0
0
1
0
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
zc
zb
1
0
0
1
0
1
iA
1
0
0
0
1
0
0
1
0
1
1
0
1
0
0
1
-i C
1
0
0
1
0
0
0
1
0
-7
0
1
1
0
0
1
iB
0
1
0
1
0
0
0
1
0
4
0
1
1
0
1
0
-i A
0
1
0
1
0
0
1
0
0
-1
0
1
0
1
1
0
iC
0
1
0
0
1
0
1
0
0
7
1
0
0
1
1
0
-i B
1
0
0
0
1
0
1
0
0
-4
1
0
0
1
1
0
0
1
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
za
zb
1
0
0
1
0
1
iA
1
0
0
1
0
0
1
0
0
za
1
0
1
0
0
1
-i C
1
0
0
1
0
0
1
0
0
za
0
1
1
0
0
1
iB
1
0
0
1
0
0
1
0
0
za
0
1
1
0
1
0
-i A
1
0
0
1
0
0
1
0
0
za
0
1
0
1
1
0
iC
1
0
0
1
0
0
1
0
0
za
1
0
0
1
1
0
-i B
1
0
0
1
0
0
1
0
0
za
1
0
0
1
1
0
0
1
1
0
0
1
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
za
za
1
0
0
1
0
1
iA
0
1
0
0
1
0
0
1
0
zb
1
0
1
0
0
1
-i C
0
1
0
0
1
0
0
1
0
zb
0
1
1
0
0
1
iB
0
1
0
0
1
0
0
1
0
zb
0
1
1
0
1
0
-i A
0
1
0
0
1
0
0
1
0
zb
0
1
0
1
1
0
iC
0
1
0
0
1
0
0
1
0
zb
1
0
0
1
1
0
-i B
0
1
0
0
1
0
0
1
0
zb
1
0
0
1
1
0
0
1
1
0
0
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
zb
zb
1
0
0
1
0
1
iA
0
0
1
0
0
1
0
0
1
zc
1
0
1
0
0
1
-i C
0
0
1
0
0
1
0
0
1
zc
0
1
1
0
0
1
iB
0
0
1
0
0
1
0
0
1
zc
0
1
1
0
1
0
-i A
0
0
1
0
0
1
0
0
1
zc
0
1
0
1
1
0
iC
0
0
1
0
0
1
0
0
1
zc
1
0
0
1
1
0
-i B
0
0
1
0
0
1
0
0
1
zc
1
0
0
1
1
0
0
1
1
0
0
1
0
0
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
zc
zc
25
In this case, the quantities to be controlled are the output (load) voltage and the input (feeder)
currents, so the PWM modulator receives the reference values (cvwx yz{ and e yz{ ) and also
the information about the quantities (cn: and en: ) used to generate the controlled quantities.
In the matrix converter developed in the scope of this thesis, it is intended that the voltage
displayed by the converter is higher than the input voltage, so it is named Boost Matrix Converter (Boost MC). A single-line diagram of the new converter is presented in Figure 2.10.
Figure 2.10: Boost Matrix Converter single-line diagram
In the Boost MC, the quantities to be generated by the PWM modulator are the input voltage
and the output current, but now these quantities are treated as command quantities, being
the control process of the load voltages and the feeder currents performed by two independent regulators. The design of these regulators will be detailed in the next section.
The Boost MC vector modulation in the rectifier and in the inverter will be dealt with next.
2.2.3. Space Vector Modulation – Application to the Boost Converter
Rectifier
The rectifier has two functions:
•
•
To guarantee that the output current (a command current) follows its reference.
To generate the intermediate stage voltage cn: .
When applying Concordia’s transformation, the nine possible switch state combinations of the
rectifier (see Table 2-3) result in nine spatial vectors, as shown in Table 2-4 and Figure 2.11.
26
Table 2-4: State vectors generated by the rectifier for all the possible combinations
D|EE
D|EF
D|EG
D|FE
D|FF
D|FG
1
0
0
0
0
1
en:
0
1
0
0
0
1
0
0
1
0
1
0
0
−en:
0
0
1
1
0
0
−en:
0
0
1
0
1
0
0
1
0
0
0
1
0
e…
en:
1
0
0
1
0
0
0
e‡
0
1
0
0
1
0
0
0
1
0
0
1
Vector
e?
e@
e6
e‚
eƒ
e„
e†
2
NO
NP
0
en:
en:
NQ
}~N • }
€N
√2en:
j
2
−en:
√2en:
0
√2en:
−en:
en:
j
6
5j
6
5j
6
√2en:
−
0
√2en:
−
0
0
0
-
0
0
0
0
-
0
0
0
0
-
0
−en:
−en:
en:
√2en:
−
j
2
j
6
1
0
3
Ioutref αβ
4
5
Figure 2.11: Spatial location of the vectors (I1-I9) needed to control the output current (Pinto S. F., 2003)
Knowing the desired location for the output current in the TU plan, it is possible to synthetize
it using the vectors adjacent to the respective sector. For example, if the reference vector
evwx yz{ˆ‰ is located in sector 0, it can be generated by applying vectors e? , e„ and one of the
27
null vectors e… ,e† ore‡ during an appropriate amount of time. This process is illustrated in Figure 2.12.
I1
Ioutref αβ
I6
in sector 0 (Pinto S. F., 2003)
where eŠ‹ are generic adjacent vectors, /Š‹Œ are the appropriate duty cycles and • is the angle
of the current reference vector in the respective sector.
Considering that the converter operates with a switching frequency much higher than the output frequency ( Ž ≫ vwx ), it can be ensured that, in each switching period, the reference vector evwx yz{ˆ‰ is given nearly by:
evwx yz{ˆ‰ ≅ e‹ /‹ + eŠ /Š + eŒ /Œ
(2.14)
Based on a trigonometric analysis of the diagram depicted in Figure 2.12, the duty-cycles
/‹ ,/Š and /Œ are given by the following expressions:
/‹ =
•
= sin( 6
−•)
‘ /Š = = sin(• )
/Œ = 1 − /‹ − /Š
wherein
=
(2.15)
is the current modulation index.
28
relates the amplitude of the reference vector evwx yz{ˆ‰ with the intermediate
=
stage current en: as follows:
The constant
=
=
p–—˜ ™š›
pœ•
wherein evwx
(2.16)
;ž
is the amplitude of the reference vector evwx yz{ˆ‰ .
It should be noted that the intermediate stage current en: is imposed by the inverter.
The intermediate stage voltage cn: is the rectifier output and can be calculated considering
that the active power is invariant along the system:
Ÿn: = Ÿvwx
(2.17)
wherein Ÿn: is the power in the DC intermediate stage and Ÿvwx is the output power.
Developing (2.17), we get:
cn: en: = @ evwx
6
wherein cvwx
;ž
c
;ž vwx
;ž
cos(¢vwx )
(2.18)
is the amplitude of the output voltage and ¢vwx is the angle between the
output voltage and the output current.
From (2.18), it results:
cn: = @
6 p–—˜ ™š›
cvwx
pœ•
;ž
cos(¢vwx ) = @
6
= cvwx
;ž
cos(¢vwx )
Inverter
The inverter has two functions:
•
•
To guarantee that the input voltage (a command voltage) follows its reference.
To generate the intermediate stage current en: .
29
(2.19)
Once again, when applying Concordia’s transformation, the eight possible switch state combinations of the inverter (see Table 2-3) result in eight spatial vectors. The related process is
presented in Table 2-5 and Figure 2.13.
Table 2-5: State vectors generated by the inverter for all the possible combinations
Vector
c?
c@
c6
c‚
cƒ
c„
cŒ
c…
D~EE
D~EF
D~FE
D~FF
D~GE
D~GF
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
1
0
cn
1
0
1
0
1
0
0
0
0
0
-
0
1
0
1
0
1
0
0
0
0
-
2
H
L
cn
M
√2cn:
j
2
c:
√2cn:
cn
c:
√2cn:
cn
c:
cn
cn
c:
cn
c:
€
c:
cn
c:
}£N • }
c:
c:
cn
j
6
5j
6
5j
6
√2cn:
−
√2cn:
−
√2cn:
−
1
0
3
Vinref αβ
4
5
Figure 2.13: Spatial location of the vectors (V0-V7) needed to control the input voltage (Pinto S. F., 2003)
30
j
2
j
6
The technique is similar to the one presented for output currents. In this case, for example, if
the reference vector c yz{ˆ‰ is located in sector 0, it can be generated by applying vectors c?,
c„ and one of the null vectors cŒ orc… during an appropriate amount of time. This process is
illustrated in Figure 2.14.
V1
Vinref αβ
V6
in sector 0 (Pinto S. F., 2003)
where cˆ‰ are generic adjacent vectors, /ˆ‰Œ are the appropriate duty cycles and •¤ is the
angle of the voltage reference vector in the respective sector.
Again, we are going to assume that the converter operates with a switching frequency much
higher than the input frequency ( Ž ≫ ). As so, it can be ensured that, in each switching
period, the reference vector c yz{ˆ‰ is given by:
c
yz{ˆ‰
≅ cˆ /ˆ + c‰ /‰ + cŒ /Œ
(2.20)
Trigonometric analysis applied to the diagram displayed in Figure 2.14, allows to conclude that
the duty-cycles /ˆ ,/‰ and /Œ are given by the following expressions:
/ˆ = ¤ sin( 6 − •¤ )
¥ /‰ = ¤ sin(•¤ )
/Œ = 1 − /ˆ − /‰
•
(2.21)
31
¤
wherein
is the voltage modulation index.
Now, the constant ¤ creates the relationship between the amplitude of the reference vector
c yz{ˆ‰ with the intermediate stage voltage cn: as follows:
¤
=
¦§¨ ™š›
¦œ•
wherein c
(2.22)
;ž
is the amplitude of the reference vector c
yz{ˆ‰
.
In this situation, the intermediate stage voltage cn: is imposed by the rectifier.
The intermediate stage current en: is the inverter output and, once again, can be estimated
considering that the active power is invariant along the system:
Ÿn: = Ÿ
(2.23)
where Ÿ is the input power.
Developing (2.23), we obtain:
cn: en: = e
@
6
wherein e
;ž
;ž
c
;ž
cos(¢ )
(2.24)
is the amplitude of the input current and ¢
is the angle between the input
voltage and the input current.
From (2.24), it results:
en: = @
6 ¦§¨ ™š›
e
¦œ•
;ž
cos(¢ ) = @
6
¤e
;ž
cos(¢ )
32
(2.25)
2.2.4. Indirect Modulation – Application to the Boost Converter
Considering switching frequencies high enough when compared to the input and output frequencies ( Ž ≫
and Ž ≫ vwx ), it can be assumed that, during a switching period, the average values of cn: and en: are constant. Based on this assumption, the input voltage modulation and the output current modulation can be applied to the rectifier-inverter association,
considering that the modulation function requires five state vectors: two nonzero vectors,
which are used to perform the input voltage modulation, another two nonzero vectors, with
the aim of performing the ouput current modulation, and a null vector.
The operating times of each one of the chosen vectors is obtained by multiplying the rectifier
and inverter related duty-cycles. The result of these operations is given in (2.26).
¬
ª
ª
/‹ /ˆ =
/‹ /‰ =
=
=
•
¤ sin( 6
− • )sin( 6 − •¤ )
•
¤ sin( 6
•
− • )sin(•¤ )
/Š /ˆ = = ¤ sin(• )sin( 6 − •¤ )
«
/Š /‰ = = ¤ sin(• )sin(•¤ )
ª
ª
/
=
© Œ 1 − /‹ /ˆ − /‹ /‰ − /Š /ˆ − /Š /‰
For a matter of simplicity,
•
=
(2.26)
is defined as 1.
Once the duty-cycles are defined, it is mandatory to determine the vectors that participate in
the modulation process, as well as the order in which they are selected3. The selection of the
vector in each instant depends not only on the sector location of the input reference voltage
(£N- ®), but also on the sector location of the output reference current (~¯°J ®). The following
look-up table (Table 2-6) was built to help in the selection process.
To fully understand how Table 2-6 was built, it is useful to have a look at an example. If both
the input voltage and output current references are located in sector 0 (c Ž = 0 and
evwx Ž = 0), the vectors used to perform the modulation process are V1, V6 (see Figure 2.14), I1,
I6 (see Figure 2.12) and a null vector. According to the same figures, V6 corresponds to /ˆ , V1
to /‰ , I1 to /Š and I6 to /‹ , therefore the vector pairs I6-V6, I6-V1, I1-V6 and I1-V1 will correspond
the duty-cycles /‹ /ˆ , /‹ /‰ , /Š /ˆ and /Š /‰ , respectively. Consulting first Table 2-4 and Table
2-5, and then Table 2-3, we note that, for example, the pair I6-V6 corresponds to the matrix
vector −4, as stated in Table 2-6.
3
This degree of freedom could be used to minimize the harmonic distortion of the currents and/or to minimize the
number of switching commutations.
33
Table 2-6: Matrix converter’s vectors used in the modulation of the input voltages and output currents
£N- ®
0
1
2
~¯°J ®
0
±² ±•
-4
±² ±
+1
±€ ±•
+6
±€ ±
1
+6
-3
-5
+2
2
-5
+2
+4
-1
3
+4
-1
-6
+3
4
-6
+3
+5
5
+5
-2
0
+1
1
2
3
£N- ®
~¯°J ®
0
±² ±•
+4
±² ±
-1
±€ ±•
-6
±€ ±
1
-6
+3
+5
-2
2
+5
-2
-4
+1
3
-4
+1
+6
-3
-2
4
+6
-3
-5
+2
-4
+1
5
-5
+2
+4
-1
-7
-3
+9
0
-1
+7
+3
-9
-3
+9
+2
-8
1
+3
-9
-2
+8
+2
-8
-1
+7
2
-2
+8
+1
-7
-1
+7
+3
-9
3
+1
-7
-3
+9
4
+3
-9
-2
+8
4
-3
+9
+2
-8
5
-2
+8
+1
-7
5
+2
-8
-1
+7
0
-7
+4
+9
-6
0
+7
-4
-9
+6
1
+9
-6
-8
+5
1
-9
+6
+8
-5
2
-8
+5
+7
-4
3
+7
-4
-9
+6
4
-9
+6
+8
5
+8
-5
-7
-3
3
4
+3
2
+8
-5
-7
+4
3
-7
+4
+9
-6
-5
4
+9
-6
-8
+5
+4
5
-8
+5
+7
-4
5
To know the time interval during which the corresponding vectors are applied to the converter, a sawtooth high frequency carrier waveform is compared to the duty cycles stated in (2.26).
This process can be seen in Figure 2.15.
d γ dα
+ d γ d β + dδ dα + d δ d β
d γ dα
+ d γ d β + dδ dα
d γ dα
+ dγ d β
d γ dα
Ts
0
Ts d γ dα
Ts d 0
Ts dδ dα
Ts d γ d β
Ts d δ d β
2Ts
Ts d γ dα
Ts d0
Ts dδ dα
Ts d γ d β
Ts dδ d β
Figure 2.15: PWM modulation process used to select the time interval during the appropriate vectors are applied
34
To summarize the whole process, it will be assumed that, at a particular time instant, both the
input voltage and output current references are located in sector 0 (c Ž = 0 and evwx Ž = 0).
Consulting Table 2-6, we can see that there are a set of vectors that can be applied to the converter (−4,+1,+6 and −3), being each one of them applied during an amount of time defined
by the process illustrated in Figure 2.15. For example, during a switching period, when the
components /‹ /ˆ are selected, the vector −4 is applied during ³Ž /‹ /ˆ ; when the components
/‹ /‰ are selected, the vector +1 is applied during ³Ž /‹ /‰ , etc. Figure 2.16 illustrates the
whole selection scheme, with an exemplificative situation.
dγ dα
0
Ts
2Ts
Vins = 0
Iouts = 0
Figure 2.16: Selection scheme for the SVM vectors
2.3. Regulators Design
This section presents the detailed project of the two regulators in the system. So, recovering
Figure 2.10, from this point on, the load will be treated as a PMSG coupled to a wind turbine
and the feeder will be treated as a grid connection.
One of the regulators controls the voltage at the PMSG’s terminals and the other one controls
the current injected in the grid. These regulators will generate the references to the command
quantities used by the modulation process described in the previous section, therefore controlling the PMSG’s voltages and the current injected in the grid.
2.3.1. Voltage Regulator
The voltage regulator is designed to ensure that the voltage at the PMSG’s terminals effectively follows a certain reference. So, the controlled voltage is the capacitor voltage c= , which is
35
almost equal to the generator voltage4 c´z , as can be seen in the single-line diagram of Figure
2.17.
Figure 2.17: Single-line diagram of the whole system – voltage regulator focus
In the design of this regulator, the generator current e´z is treated as a disturbance of the
system. The current e ;xy ž represents the current that flows through the matrix converter
and that is controlled by the current regulator, whose design will be further discussed next.
The single-phase equivalent used to extract the equations that describe the voltage regulator
operation is represented in Figure 2.18.
Ic
Igen
C
Vc
Imatrix
Figure 2.18: Single-phase equivalent used do extract the system equations
Applying the KCL (Kirchhoff’s Current Law) to the node and considering an alpha-beta representation, we have the following system of equations:
‘
4
Z
Z
µ¦¶ ·
µx
µ¦¶ ¸
µx
= '´z
= '´z
ˆ
‰
−'
−'
;xy ž ˆ
(2.27)
;xy ž ‰
For this purpose, the inductance of the external filter of the PMSG can be neglected.
36
where c= ˆ‰ are the capacitor voltages, '
converter and '´z
ˆ‰
;xy ž ˆ‰
are the generator currents, all in alfa-beta coordinates. Z is the capaciare the currents that flow through the matrix
tance value of the capacitor.
Now, using Park’s transformation5 and assuming that ¢ = g´z ( (¢ - transformation angle,
g´z – angular frequency related to the generator operation), we set the dq representation of
the system, in the canonical form:
µ¦¶ ¹
µx
‘µ¦
¶¾
µx
=
=
º»¨ ¹
:
º»¨ ¾
:
−
−
™š˜¼§› ¹
:
™š˜¼§› ¾
:
+ g´z c= ½
− g´z c= µ
(2.28)
where the additional terms g´z c= µ½ are the cross terms that result from the application of
Park’s transformation; these terms represent the interaction between the two components of
the transformation.
Rewriting the equations above as functions of e= µ½ , the command currents that allow the voltage control, we get the following system:
µ¦¶ ¹
µx
‘ µ¦
¶¾
µx
=
=
º»¨ ¹
:
º»¨ ¾
:
+ : e= µ
?
?
+ : e= ½
(2.29)
Thus,
e= µ = −'
¿
e= ½ = −'
where '
'
À
;xy ž µ½
À
;xy ž µ
À
;xy ž ½
+ Zg´z c= ½
− Zg´z c= µ
À
;xy ž µ½ is
(2.30)
an image of the current '
differs from '
;xy ž µ½ by
;xy ž µ½
that flows through the converter.
a delay introduced by the converter operation.
These equations can be used to build the voltage regulator block diagram, which is presented
in Figure 2.19.
Park’s transformation, which performs the coordinate’s change TU → /Á, is given by:
)0%¢ −%' ¢
YŸ[ = s
t
%' ¢ )0%¢
5
37
αv
Igend
Vrefd
+
αv
−
K pv +
Icd
K iv
s
'
matrixd
I matrix d
−
− Imatrixq'
I matrix q
I
−
+
Vmeasd
+
ωgenC
ωgenC
Vrefq
+
−
αv
K pv
Ic q
K
+ iv
s
−
−
Vmeasq
+
Igenq
αv
Figure 2.19: Voltage regulator block diagram
Neglecting the cross terms and simplifying the notation, we obtain the compact block diagram
of the voltage regulator as in Figure 2.20.
Vrefdq
αv
+
−
Ic dq
K pv +
K iv
s
1
αi
Imatrixdq
Igendq
−
+
sTdv + 1
1
sC
Vmeasdq
αv
Figure 2.20: Simplified voltage regulator block diagram
The difference between the reference voltage cyz{¹¾ and the measured capacitor voltage
c
z;Ž¹¾ ,
the voltage error, is applied to a Proportional-Integral (PI) Controller, which generates
the reference current e= µ½ used by the modulation process. The block
?⁄ˆ§
,
ŽS¹Â Ã?
a first order
transfer function with a delay ³µ¤ , represents the matrix converter controlled by current and
the modulation process; the constants T¤ and T are the sensor voltage gain and sensor current gain, respectively6.
With respect to the PI controller, it normally ensures a null static error and a reasonable rise
time. Parameters ÄÅ¤ and Ä ¤ can be calculated by deriving the transfer function that repre6
This is a consequence of we are assuming a real situation, wherein the global system is tested in laboratory environment.
38
sents the capacitor voltage response to the disturbance introduced by the generator current.
Considering the voltage regulator block diagram above, the desired transfer function, in the
canonical form, is given by:
¦™»šÆ¹¾
pº»¨
¹¾
(Ž)
=
Æ
(ŽS¹Â Ã?)
•Ç¹Â
É
·Â ËÌÂ ·Â Ë§Â
Æ
ŽÈ Ã
ÊŽ
Ê
Ç¹Â
•·§ Ç¹Â •·§ Ç¹Â
(2.31)
To determine the PI controller parameters, the denominator is compared, term by term, with
the third order ITAE (Integral of Time and Absolute Error) polynomial:
Ÿ6 (%) = % 6 + 1.75&Œ % @ + 2.15&Œ@ % + &Œ6
(2.32)
The result is:
1
¬ 1.75&Œ = Î³µ¤
ª ˆÂ ÏÌÂ
−
= 2.15&Œ@
ˆ§
« :S¹Â
ª − ˆÂ Ï§Â = & 6
Œ
©
:S ˆ
(2.33)
¹Â §
After some algebraic manipulation, the proportional gain ÄÅ¤ and the integral gain Ä ¤ of the
voltage controller can be obtained as:
‘
ÄÅ¤ = − ˆ
Ä¤ = −ˆ
@.?ƒ:ˆ§
É
Â S¹Â ?.…ƒ
:ˆ§
É
È
S
Â ¹Â ?.…ƒ
(2.34)
In what concerns the average delay of the system ³µ¤ , the dynamic of the capacitor voltage is
considerably slower than the dynamic of the controlled grid current. As so, it assumes a relatively larger value, in this case, one-tenth of the grid period. As the grid period is ³ý µ =
0.02%, ³µ¤ = 2 %.
Table 2-7 presents the values of the voltage controller gains and also the associated constants.
Details about the capacitor sizing can be found in the section devoted to filters sizing.
39
Table 2-7: Voltage controller gains and associated constants
MYÐÑ[
•
•N
Ò± YÓ®[
ÔÕ
ÔN
253.9
0.01
0.001
2
0.0089
1.1844
2.3.2. Current Regulator
The current regulator aims at imposing the grid injected eý µ current within some predetermined values. Figure 2.21 shows the single-line diagram of the system, emphasizing the
current regulator focus.
Figure 2.21: Single-line diagram of the whole system – current regulator focus
The operating principle of the current regulator is related with the following equations, which
result from the application of the KVL (Kirchhoff’s Voltage Law) to the grid mesh, in abc coordinates:
µ º¼§¹
š
µx
¬
ªµ
= − Ø Ö§×˜ 'ý µ −
o
Ö§×˜
;
š
ØÖ§×˜
¦º¼§¹
+Ø
µx
o
wherein 'ý µ
Ö§×˜
;<=
<
Ü
ØÖ§×˜
+Ø
?
Ö§×˜
cÙÚÛ ;
cÙÚÛ <
«
¦º¼§¹
?
ª µ º¼§¹ ¶ = − oÖ§×˜ '
− Ø ¶ + Ø cÙÚÛ =
ØÖ§×˜ ý µ =
© µx
Ö§×˜
Ö§×˜
º¼§¹ Ü
= − Ø Ö§×˜ 'ý µ −
¦º¼§¹
?
Ö§×˜
are the RMS grid injected currents, cý µ
(2.35)
;<=
cÙÚÛ ;<= are the voltages displayed by the matrix converter, Ý{ Þx is the resistance of the grid
filter and ß{ Þx is the inductance of the grid filter.
40
are the RMS grid voltages,
Applying Park’s transformation to the equations above and choosing a reference frame synchronous with the grid voltage (¢ = gý µ (, ¢ - transformation angle, gý µ – angular frequency of the grid), we obtain the description of the system in dq coordinates as follows:
µ º¼§¹
¹
µx
¥ µ º¼§¹
µx
¾
=−
'
ØÖ§×˜ ý µ µ
oÖ§×˜
= −Ø
oÖ§×˜
Ö§×˜
−
'ý µ −
½
¦º¼§¹
¹
ØÖ§×˜
¦º¼§¹
ØÖ§×˜
¾
+
?
c
ØÖ§×˜ ÙÚÛ µ
+Ø
?
Ö§×˜
+ gý µ 'ý µ
cÙÚÛ ½ − gý µ 'ý µ
wherein the additional terms gý µ 'ý µ
µ½
½
µ
(2.36)
are the cross terms that result from the application
of Park’s transformation; these terms represent the interaction between the two components
of the transformation.
Now, equations (2.36) can be rewritten in order to be expressed as a function of à= µ½ voltages, the command voltages used by the modulation process that ensure grid currents follow
their references:
µ º¼§¹
¹
µx
¥ µ º¼§¹
µx
¾
= −Ø
oÖ§×˜
Ö§×˜
'ý µ −
µ
= − Ø Ö§×˜ 'ý µ −
o
Ö§×˜
½
¦º¼§¹
¹
ØÖ§×˜
¦º¼§¹
¾
ØÖ§×˜
+Ø
+Ø
?
Ö§×˜
?
Ö§×˜
à= µ
à= ½
(2.37)
As so,
à=¹ = cÙÚÛ Àµ + gý µ ß{ Þx 'ý µ
½
¿
À
à=¾ = cÙÚÛ ½ − gý µ ß{ Þx 'ý µ
(2.38)
µ
wherein cÙÚÛ Àµ½ is an image of the voltage cÙÚÛ µ½ displayed by the converter. cÙÚÛ Àµ½ differs
from cÙÚÛ µ½ by a delay introduced by the converter operation.
Based on these equations, the current regulator block diagram obtained is represented in Figure 2.22:
41
αi
Igridrefd
+
αi
−
1 + sTz
sTp
Ucd
+
−
VPWM d'
VPWM d
Imeasd
'
+ VPWM q
VPWM q
Imeasq
ωgrid Lfilt
ωgrid Lfilt
Igridrefq
αi
Ucq
1 + sTz
sTp
+
−
+
αi
Figure 2.22: Current regulator block diagram
As the weight of the cross terms is not significant, they can be neglected at this stage and the
block diagram above can be compacted as follows:
Igridrefdq
αi
+
−
1 + sTz
sTp
Ucdq
K
sTdi + 1
VPWMdq
Vgriddq
+
−
1
sLfilt + Rfilt
Imeasdq
αi
Figure 2.23: Simplified current regulator block diagram
The reference current eý µyz{¹¾ and the measured current e
age à= µ½ used by the SVM. Moreover, the block
Ï
,
ŽS¹§ Ã?
z;Ž¹¾ are
subtracted and the dif-
ference is applied to a Proportional-Integral (PI) Controller, which returns the command volta first order transfer function with a
delay ³µ , models the matrix converter and the modulation process. The constant T , that affects the reference current and the measured current, is a sensor current gain, as was referred
previously. Ä is an incremental gain that aims at representing the gain of the converter, relating the matrix converter output voltage with the command voltage.
In what concerns the PI controller, parameters ³á and ³Å can be estimated by deriving the
transfer function that relates the measured grid current with the reference grid current.
For simplicity in the controller design, the contribution of the disturbance, the grid voltage
cý µ , is accounted with a fictitious resistance Ýý µ , which is added to the filter resistance
Ý{ Þx . As so, the resultant resistance Ýx is given by:
42
Ýx = Ý{ Þx + Ýý µ
where Ýý µ =
(2.39)
¦º¼§¹
pº¼§¹
Then, to obtain ³á , the normal procedure is considering that the zero of the PI Controller cancels the lowest frequency pole, introduced by the filter (Pinto S. , Silva, Silva, & Frade, 2011).
Thus, ³á is given by:
³á =
ØÖ§×˜
o˜
(2.40)
So, in the canonical form, the desired transfer function is given by:
p™»šÆ¹¾ (Ž)
pº¼§¹¼»Ö¹¾ (Ž)
=
Ë·§
ÇÌ Ç¹§ â˜
Ë·§
ã
ŽÉ ÃŽ Ã
Ç¹§ ÇÌ Ç¹§ â˜
(2.41)
Now, the denominator can be compared to the denominator of a second order transfer function written in the canonical form, as follows:
ä(%) =
åÉ̈
ŽÉ Ã@æå¨ ŽÃåÉ̈
(2.42)
where g is the natural frequency of the system and ç is the damping factor.
Equating the proper terms, we get:
‘
2çg = S
g =
@
?
¹§
Ïˆ§
SÌ S¹§ o˜
(2.43)
and after some algebraic manipulation, considering a typical damping factor ç =
tain:
43
√@
,
@
we ob-
³Å =
@Ïˆ§ S¹§
o˜
(2.44)
Normally, the PI controller is presented with the following formulation:
ZÙp (%) = ÄÅ +
Ï§§
Ž
(2.45)
where:
‘
ÄÅ = è
S
Ä =
S
Ì
?
SÌ
(2.46)
So, replacing (2.40) and (2.44) in (2.46), we obtain:
‘
ÄÅ = @Ïˆ S
Ä =
ØÖ§×˜
§ ¹§
o˜
@Ïˆ§ S¹§
(2.47)
To choose the appropriate value of the system average delay ³µ , the most common criterion
is considering a value with the same order of magnitude of the matrix converter switching
period (Pinto S. , Silva, Silva, & Frade, 2011). This is reasonable, since the maximum response
time of the converter with regard to oscillations of the modulating waveform is a switching
period (³Ž = 200é%). As so, we consider ³µ as an half of the switching period, ³µ = 100é%.
As the incremental gain Ä and the parameter Ýx change with the operating conditions, ÄÅ
and Ä (which depend on Ä) are not constant, also depending on the operating conditions.
2.3.3. Reference Values Setting
One crucial aspect in the regulators design is the interaction between the two regulators, as
the reference current in the grid side is directly related to the reference voltage in the load
side.
In dq coordinates, active power and reactive power are given by:
44
Ÿ=
"
ê=
µ 'µ
+
'
µ ½−
½ '½
½ 'µ
(2.48)
Referring to the load side, we get:
ë
ŸÞv;µ =
êÞv;µ =
µ Þv;µ 'µ Þv;µ
µ Þv;µ '½ Þv;µ
+
−
½ Þv;µ '½ Þv;µ
½ Þv;µ 'µ Þv;µ
(2.49)
Referring to the grid side, we obtain:
ë
Ÿý µ =
µ ý µ 'µ ý µ
êý µ =
µ ý µ '½ ý µ
+
−
½ ý µ '½ ý µ
½ ý µ 'µ ý µ
(2.50)
Assuming that the reactive power flow in the grid is null,
Ÿý µ =
½ ý µ
= 0 and '½ ý µ = 0, so:
µ ý µ 'µ ý µ
(2.51)
Therefore,
'µ ý µ = ¤
Ùº¼§¹
¹ º¼§¹
(2.52)
Neglecting losses, Ÿý µ ≅ ŸÞv;µ , (2.52) can be approximated by:
eý µyz{ = ¦ ×–š¹
Ù
µ
º¼§¹ ¹
(2.53)
wherein eý µyz{ is the reference current to the current regulator (d component) and ŸÞv;µ
µ
and cý µ are well defined quantities that can be measured instantaneously.
µ
45
The q component of eý µyz{ is a degree of freedom that can be adjusted to control the reactive power injected in the grid.
Assuming that the power extracted from the wind (turbine) changes along the time and that
this power has to equal in every moment the power delivered to the grid (the losses inherent
to the converter operation are negligible), the voltage/current references given to both controllers must follow these power fluctuations. So, the transferred active power is measured
instantaneously and the references are continuously updated. This point will be further discussed in the next chapter.
2.4. Filters Sizing
This section provides the methodology used to calculate the load and grid filter parameters.
Table 2-8 contains the rated values assumed.
Table 2-8: Rated values
ìGíî Yïð[
5
£î Y£[
2000
√2
2.4.1. Load Filter
The switching process of the semiconductors introduces high-frequency harmonics in the converter currents and voltages, which may cause additional losses and excite electrical resonance. To overcome this problem, a second order low pass filter is installed between the load
and the converter (Figure 2.24), in order to minimize the harmonic content of the quantities
associated to the load. Besides that, the correct load filter sizing assumes a significant role in
the control process, as the controlled voltage is the capacitor voltage7.
7
See the previous section Regulators Design on this chapter.
46
Figure 2.24: Load filter in the global system
In this case, a parallel arrangement of the damping resistor Ý{ and inductance ß{ was adopted
(Silva, Input Filter Design for Power Converters, 2013), as can be seen in the single-phase
equivalent of Figure 2.25.
In Figure 2.25, - is an incremental negative resistance that models the converter. eñ is the
current that flows through the fictitious resistance and à= is the applied voltage. So, - is given
by the following expression:
- = µp ¶
µñ
ò
(2.54)
Lf
Igen Imatrix
Vgen
Rf
IU
Ic
Cf
Vc
ri
UC
Figure 2.25: Load filter single-phase equivalent
The single phase power ŸÛ: that flows in the matrix converter is given by:
ŸÛ: = à= eñ
(2.55)
Ÿv is the single-phase delivered power and it is a fraction of ŸÛ: , as follows:
47
Ÿv = óŸÛ: = óà= eñ
(2.56)
wherein ó is the converter efficiency.
So, à: is given by:
à: = ôp–
Ù
ò
(2.57)
Replacing (2.57) in (2.54) and developing the equation, we get:
- = µp õôp– ö = − ôp–É = −ó Ù• = −óÝv ¦•É
µ
Ù
ò
ò
Ù
ò
ñÉ
–
ñÉ
–
(2.58)
wherein Ýv and cv are the equivalent resistance and voltage related to the delivered power Ÿv ,
respectively. Ýv has the following expression:
Ýv =
¦–â÷ø É
Ù–
(2.59)
wherein cvoÛù is the RMS value of the voltage related to the delivered power Ÿv .
The ratio ñ– is nearly the matrix converter voltage transfer ratio (√3⁄2), so (2.58) simplifies in:
¦
•
- = −ó 6 Ýv
‚
(2.60)
The incremental resistance - assumes a negative value only when the converter operates at
constant power (Silva, Input Filter Design for Power Converters, 2013).
Now, the auxiliary parameter úÅ and its variation range are introduced.
48
1 ≤ úÅ ≤ É
@æ
?
(2.61)
wherein ç is the damping factor of the filter, which, in general, can be selected in the range
0.5 ≤ ç ≤ 0.7. However, simulation results showed that better filter performance is achieved
with ç =
√@
,
@
so úÅ should follows the condition:
úÅ ≤ 1
(2.62)
This auxiliary parameter allows the filter parameters to be adjusted through the filter characteristic impedance ü{ , as follows:
ü{ =
@æ É ýÌ Ê?
æýÌ
-
(2.63)
At this time, it is possible to present the expressions that allow the calculation of the filter parameters Z{ , ß{ and Ý{ (see Figure 2.25):
¬ Z{ = þÖåÖ
ª
þ
ß{ = åÖ
Ö
«
ªÝ = y§ þÖ
© { @æy§ ÊþÖ
?
(2.64)
wherein g{ is the filter cut-off angular frequency.
The filter cut off-frequency { (g{ = 2j { ) should be at least one decade above the grid frequency ( ý µ = 50ú ) and one decade bellow the switching frequency ( Ž = 5000ú ) (Pinto
& Silva, 2001), so it is set to { = 500ú .
As a final remark, it should be noted that the capacitance value obtained Z{ must be divided by
3, due to the delta connection performed in load capacitors (see Figure 2.24). So, the actual
capacitance value Z is given by:
Z=
:Ö
6
(2.65)
49
The load filter parameters and the associated constants are presented in Table 2-9.
£¯|ïD Y£[
0.985
cy
√3
2
Table 2-9: Load filter parameters and the associated constants
ì¯ Y ð[
Ÿ6 y
3
|¯ Y [
0.9
Õ
0.8
√2
2
Y [
0.4179
Y
[
500
MYµ [
253.9
Yµ [
133.02
| Y [
0.2364
2.4.2. Grid Filter
In the PCC (see Figure 2.26), the RL filter aims to reduce the harmonic content present in the
grid injected currents.
Figure 2.26: Grid filter in the global system
The grid filter single-phase equivalent is presented in Figure 2.27.
Figure 2.27: Grid filter single-phase equivalent
To calculate the value of the inductance ß{ Þx , it is considered that the single-phase equivalent
of the global system can be approximated by the Voltage Source Inverter (VSI). In this type of
configuration, an empirical expression to estimate ß{ Þx is given by:
50
ß{ Þx =
„
ñœ• SÆ
pº¼§¹
(2.66)
wherein àn: is the DC link voltage, ³Ž is the switching period and Δeý µ is the grid current
ripple.
Adapting (2.66) to our case, we get:
ß{ Þx =
¦¶ SÆ
„ pº¼§¹
(2.67)
wherein c= is the capacitor voltage.
As the dynamic of the capacitor voltage is much slower than the grid current dynamic, this
approximation doesn’t introduce a significant error. In the calculations, c= = 2000c (maximum value of the rated voltage) is considered, together with a 10% grid current ripple of the
rated grid current.
To estimate Ý{ Þx , the first concern is to ensure that power losses ŸÞvŽŽ dissipated in the filter
resistance do not exceed 0.5% of the rated power. So:
Ý{ Þx = 6p
Ù×–ÆÆ
º¼§¹â÷ø
É
(2.68)
wherein eý µoÛù is calculated from the rated conditions.
Table 2-10 presents the grid filter parameters and the associated constants.
Table 2-10: Grid filter parameters and the associated constants
£Q Y£[
2000
~
îN± YH[
341.5975
ì ¯®® Y ð[
25
51
|
N J YÓ
1.4
[
N J Yµ
195.16
[
CHAPTER 3
Wind Turbine Generator
Abstract
This chapter is organized in two parts: in the first part, the study of the wind turbine is performed, including a succinct description of its structure and main components, together with the presentation of the model used and the control strategy adopted; in the second one, a brief overview of the permanent magnets synchronous generator’s characteristics is given alongside with the presentation of the
model that describes its dynamics.
53
3. Wind Turbine Generator
3.1. Wind Turbine
3.1.1. Structure and Main Components
The wind turbine is the driving machine that provides mechanical energy to the synchronous
generator. A typical wind turbine has the structure as depicted in Figure 3.1.
Figure 3.1: Wind turbine structure
The wind turbine most significant components are:
•
Blades: are used to extract kinetic energy from the wind. Also, they execute the power
control either by aerodynamic design or by changing the pitch. Power control is required to prevent wind generator’s nominal power to be exceeded. Indeed, the turbine’s classification is made according to the type of control performed by the blades:
stall turbines (blades designed to enter in aerodynamic loss when the wind speed
reaches a certain value) and pitch turbines (blades change the angle between the
blade and the turbine longitudinal shaft, the pitch angle, therefore, decreasing the
conversion efficiency). Power control provided by pitch turbines is more effective.
54
•
Rotor: is composed by the blades and the hub where the blades are fixed.
•
Gear box: the low-speed turbine shaft is coupled to the high-speed generator shaft
through a gear box that adapts the turbine rotor frequency to the generator frequency.
•
Nacelle: is a compartment in the top of the tower, where the shafts, the generator,
the gear box, the brake and some other support devices are located.
•
Tower: supports the nacelle and allows that the rotor is mounted at a high enough
height, so that the wind is more intense and suffers less perturbations.
Now, it is important to present a single-line diagram of the set turbine + generator, for a better
understanding of the system’s dynamic and the quantities involved.
u, Pavai
d
in
W
ωT
ωG
TT
Pm
TG
Pm
Pe
ωT
Figure 3.2: Single-line diagram of the set turbine + generator
Some considerations can be made about the single-line diagram represented in Figure 3.2:
•
•
Assuming negligible losses in the gearbox and shaft, the mechanical power extracted
from the turbine shaft equals the mechanical power delivered to the generator.
The mechanical power Ÿ is then given by:
Ÿ = ³S g S = ³ g
(3.1)
wherein ³S and ³ are the turbine torque and the generator torque, respectively, and g S and
g are the turbine angular speed and the generator angular speed, respectively.
55
•
The gear box is represented by a simple multiplicative gain, so, in order to respect the
power conservation:
ä=
SÇ
S
=
å
åÇ
(3.2)
3.1.2. Power in the Wind
, associated to an air
The kinetic energy available to a wind turbine is the kinetic energy,
volume with mass (in kg), moving with constant speed , (in m/s) in the direction (in m):
=@
?
,@ = @ (
?
),@
(3.3)
wherein is the air density ( = 1.225 / 6, according with the International Standard
Atmosphere), is the flat cross-section (in m2) and is the thickness of the air volume (in m).
So, in absence of the turbine, the available power in the wind Ÿ;¤; (in W) is given by:
Ÿ;¤; =
µ
§¨
µx
= @õ
?
µž
ö ,@
µx
=@
?
,6
(3.4)
Eq. (3.4) emphasizes the cubic dependence of the wind speed ,.
3.1.3. Turbine Model
As the wind has to leave the blades plane with nonzero speed, the available power in the wind
can’t be fully converted in mechanical power in the turbine shaft. The maximum conversion
rate – Betz’s law – is 59.3%, but this is a theoretical value, with modern wind turbines achieving only 50% at most of wind-mechanical energy conversion. This means that, in normal conditions, only a maximum of 50% of the available power in the wind is converted into mechanical
power in the turbine shaft.
The quantity that measures the total conversion efficiency is the power coefficient ZÙ , given by
the following expression:
56
ZÙ (,) =
Ù»
ÙšÂš§
(3.5)
where Ÿz is the electrical power available at the generator’s terminals. Eq. (5) considers the
efficiencies of both turbine and generator and is wind speed dependent.
So, we can write:
Ÿz = Ÿ;¤; ZÙ (,) = ZÙ (,)
?
@
,6
(3.6)
3.1.4. Generator Power Curve
In wind applications, the generators are designed to provide the maximum electrical power
(nominal power) to a certain wind speed (rated wind speed). In Figure 3.3, a power curve
Ÿz (,) of a typical wind generator of 2 MW is shown:
ucut −in
ur
ucut −out
Figure 3.3: Power curve of a 2 MW generator (Castro, 2012)
Four operation zones can be distinguished in the power curve of Figure 3.3:
57
•
Zone 1: as it is not economically feasible to extract energy for wind speeds lower than
the cut-in wind speed (,=wxÊ ), the generator is not connected to the grid in this zone.
Zone 2: the zone between the cut-in wind speed and the rated wind speed (,y ) corresponds to the maximum power extraction. In this zone, the electrical power has an approximate cubic behaviour, because the power coefficient depends on the wind speed,
as emphasized by (3.6).
•
•
Zone 3: for values higher than the rated wind speed, it is not economically feasible to
increase the electrical power (there is no return on the investment because this speed
range occurs only a few times during a year), so, in these cases, the generator is regulated to run at constant power.
•
Zone 4: For safety reasons, the generator is turned off for wind speeds higher than the
cut-out wind speed (,=wxÊvwx ).
The control process that allows the generator operation in the constant power zone (Zone 3 in
Figure 3.3) is outside the scope of this thesis. As so, it is considered that the generator runs
always in Zone 2, meaning that a Maximum Power Point Tracking (MPPT) is implemented. The
MPPT follows a control strategy, known as torque control, which imposes the generator
torque to follow a reference value ³o . The necessary assumptions will be detailed next.
3.1.5. Torque Control
As far as the power coefficient ZÙ is concerned, one of the most referred expressions to describe its behaviour as a function of the tip speed ratio and the pitch angle U is (Castro,
2012):
ZÙ = 0.22(
??„
§
− 0.4U − 5)exp(−
)
?@.ƒ
§
(3.7)
where
=
?
ã
%.%È'
Ê
#$%.%&¸ ¸È $ã
(3.8)
58
U is the pitch angle8 and is the Tip Speed Ratio (TSR), the relation between the linear speed
at the blade tip and the wind speed, which is given in (3.9):
=
åÇ o
w
(3.9)
where Ý is the radius of the circle described by the blades rotation (in m), , is the wind speed
(in m/s) and g S is the turbine angular speed (in rad/s). To clarify some points, it could be necessary to confer Figure 3.2.
The representation of ZÙ as function of , to several values of U, is depicted in Figure 3.4.
Cp - Power Coefficient
0.5
0.45
β =0°
β =10°
0.4
β =20°
β =30°
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
λ - Tip Speed Ratio
Figure 3.4: Cp variation with
and
As it can be seen in Figure 3.4, maximum efficiency is achieved for a pitch angle equal to zero.
Therefore, U = 0 will be considered in this work.
Under this assumption, (3.7) becomes:
8
See section Structure and Main Components on this chapter.
59
ZÙ = 0.22(
??„
§
− 5)exp(−
)
?@.ƒ
§
(3.10)
where
=ã
#
?
ÊŒ.Œ6ƒ
(3.11)
Figure 3.5 depicts the variation of the power coefficient with the tip speed ratio. It can be observed that there is a particular value of the tip speed ratio that maximizes the power coefficient. In Figure 5, this value is denoted as vÅx .
Figure 3.5: Cp variation with
( = )
One of the assumptions made by the torque control process is that the power coefficient ZÙ is
maximum (optimal conversion efficiency), so the value of that maximizes ZÙ (named vÅx ) is
obtained by:
µ:(
µ
= 0}
C –Ì˜
(3.12)
60
from where results,
vÅx
= 6.32497
(3.13)
Replacing (3.13) in (3.9) and rewriting the equation, we get:
g SvÅx =
„.6@‚‡…w
o
(3.14)
Recovering equation (3.6) and replacing the adequate equations for the power coefficient
((3.10) and (3.11)), one obtains:
Ÿz = jÝ @ ,6 0.22(
@
?
??„
ã
ã
*%.%È'
#
− 5)exp(−
?@.ƒ
ã
ã
*%.%È'
#
)
(3.15)
Knowing that the generator torque ³ = » and remembering that g = äg S ,
å
Ù
³ =
?
?
åÇ–Ì˜ @
jÝ @ ,6 0.22(
??„
ã
ã
*%.%È'
#
− 5)exp(−
?@.ƒ
ã
ã
*%.%È'
#
)
(3.16)
The torque at the Maximum Power Point is obtained by replacing the value of the optimal TSR
((3.13) in (3.16)).
³ÛÙÙS =
?
?
åÇ–Ì˜ @
jÝ @ ,6 0.22(
??„
ã
ã
*%.%È'
+.ÈÉ,-.
− 5)exp(−
?@.ƒ
ã
ã
*%.%È'
+.ÈÉ,-.
Replacing numerical values, one gets the reference torque as:
61
)
(3.17)
³o
= ³ÛÙÙS =
Œ.†‚6@?6oÉ wÈ
åÇ–Ì˜
=
Œ.†‚6@?6oÈ wÉ
„.6@‚‡…
(3.18)
For implementation purposes, the constant values of (3.18) are Ý = 56.5
and ä = 77.
The implementation of a torque control assumes that there is no speed control, so the turbine
speed (or the generator speed, since they are directly related through the gear box gain ä) is
always the optimal speed for a certain value of ,, as given in (3.14); whenever a variation of ,
occurs, the turbine/generator speed is automatically updated accordingly to (3.14), as to maintain = )0 %(V ( = vÅx and, consequently, to maximize ZÙ . The reference torque ³o ,
which is chosen to guarantee the maximum power extraction, is a function of the wind speed
, (Eq. (3.18)). This subject is related to the Field-Oriented Control, as will be discussed in the
next section.
3.2. Permanent Magnet Synchronous Generator
3.2.1. Description
A synchronous generator is an electromechanical converter that receives the mechanical energy from a driving machine (a wind turbine, for example) and delivers the electrical energy to
the grid, with a very high efficiency. The term synchronous is related to the fact that this kind
of rotating machine, in steady state, runs at constant frequency and speed, synchronously with
the other grid connected synchronous machines.
A classic synchronous generator is constituted by a fixed metallic piece – stator –the armature
winding wherein is located, and by a rotating metallic piece – rotor – wherein the inductor coil
is wounded. An auxiliary source injects a DC current in the inductor coil, which creates a magnetic excitation field that closes itself through the stator. As the rotor is rotating with constant
speed, a rotating magnetic flux is created, which, according to Faraday Law, induces an electromotive force at the armature winding terminals.
A permanent magnet synchronous generator (PMSG) is a variant of the conventional synchronous generator using neither auxiliary source nor inductor coil in the rotor. This type of generator, increasingly used in wind applications, has the same operating principle described above,
with the difference that the excitation field is provided by a set of permanent magnets coupled
to the rotor. As there is no electrical circuit in the rotor, Joule’s losses are minimal and a PMSG
usually presents higher efficiency than a classic one. Besides that, another advantage of the
permanent magnets is that they dispense the use of brushes and slip rings9, hence reducing
9
The brushes and the slip rings are used to make the connection between the auxiliary source and the inductor coil
(rotor).
62
the maintenance cost. So, for these reasons, a PMSG connected to a wind turbine was chosen
to feed the matrix converter developed in this thesis.
The typical architecture of a PMSG is presented in Figure 3.6:
Figure 3.6: Cross section of a typical PMSG (adapted from (Fernandes, 2013))
As it can be seen, the armature winding in the stator is composed by three distinct windings.
These windings are spatially lagged by 120º (see Figure 3.7) in order that electromotive forces
temporally lagged by 120º are produced, as so forming a balanced symmetric three-phased
system.
Figure 3.7: Armature winding arrangement (adapted from (Fernandes, 2013))
63
The frequency (in Hz) of the electromotive forces induced at the armature winding terminals
is proportional to the rotor speed y (in rpm). To estimate this proportionality constant, first of
all, it is necessary to distinguish mechanical angles from electrical angles, which are related as
follows:
•z = /•
(3.19)
wherein •z is the electrical angle (in rad), • is the mechanical angle, also called angular position of the rotor (in rad), and / is the number of pairs of poles.
The angular frequency of the electromotive forces, gz (in rad/s), is given by:
gz =
µ0»
µx
(3.20)
Replacing (3.19) in (3.20), one obtains:
gz = /
µ0™
µx
= /gy
(3.21)
wherein gy is the angular speed of the rotor (in rad/s).
Knowing that:
gz = 2j
(3.22)
and that:
gy = 2j
y
= 2j „Œ¼
(3.23)
64
the electromotive force frequency
pression:
and the rotor speed
y
are related by the following ex-
= / „Œ¼
(3.24)
3.2.2. Machine’s Model
Machine’s model in abc coordinates
The PMSG is well documented in the literature. A brief description of the main principles of a
standard PMSG model follows.
Assuming that the external circuit of the generator is closed (output stator current different
from zero), the voltages at the PMSG’s terminals c;<= are described by the following matrix
equation:
Yc;<= [ = YÝŽ [Y';<= [ +
µY1šÜ¶ [
µx
(3.25)
wherein ÝŽ is the resistance of the armature winding, ';<= is the vector of the currents and
2;<= is the vector of the linkage fluxes, which are dependent from the machine’s inductances.
This relation can be seen in the matrix presented below:
2;
ß? + ß@ )0%2•;
>2< A = >3? + 3@ )0%2•<
2=
3? + 3@ )0%2•=
3? + 3@ )0%2•<
ß? + ß@ )0%2•=
3? + 3@ )0%2•;
3? + 3@ )0%2•= ';
3? + 3@ )0%2•< A >'< A
ß? + ß@ )0%2•; '=
(3.26)
wherein L coefficients are self-inductances and M coefficients are mutual inductances.
In abc representation of the machine, both mutual and self-inductances change with the cosine of the rotor’s angular position (• ), which, in turn, is a function of the time, as can be
seen in:
• = g( + 4v
(3.27)
65
where 4v is a generic initial angle.
This is a serious drawback, as this set of equations is too difficult to solve and analyse. As so, it
is common practice to apply first Concordia’s transformation and then Park’s transformation,
in order to eliminate the dependences described above. These calculations will be briefly detailed next.
Machine’s model in • coordinates
Concordia’s transformation, which performs the coordinate’s change VW) → TU0, is given by:
0
_1
@^ ?
√6
YZ[ = \ ^− @
@
6
^ ?
√6
]− @ − @
T
V
and: 5W6 = YZ[ 5U 6;
)
0
?
√@ b
?a
√@ a
? à
√@
(3.28)
T
V
5U 6 = YZ[S 5W 6
)
0
The application of Concordia’s transformation to (3.26) allows the representation of the machine’s dynamics in a two-phase equivalent system, as can be seen in (3.29)
2ˆ
ß + ßvŽ )0%2•
s2 t = s ;¤
ßvŽ %. 2•
‰
'ˆ
ßvŽ %. 2•
ts t
ß;¤ − ßvŽ )0%2• '‰
(3.29)
where ß;¤ and ßvŽ are fictitious inductances that result from the application of the transformation.
A graphical illustration of what has been done is depicted in Figure 3.8.
66
Figure 3.8: Graphical view of the application of Concordia transformation (Fernandes, 2013)
Some simplification has been achieved, but still not the desired one, as the dependence of the
rotor’s angular position has not been eliminated. To obtain further simplification, the application of Park’s transformation is required.
Machine’s model in dq coordinates
Park’s transformation, which performs the coordinate’s change TU → /Á, is given by:
)0%¢
YŸ[ = s
%' ¢
−%' ¢
t
)0%¢
T
/
and: 7U 8 = YŸ[ s t;
Á
(3.30)
T
/
s t = YŸ[S 7U 8
Á
In this transformation, the armature windings are seen from a rotating frame, called dq referential, which rotates at rotor speed and has the same origin than TU referential, but lagged
from ¢ degrees. The d-axis (direct axis) is aligned with the angular position of the rotor and the
q-axis (quadrature axis) is located at an angle of 90 degrees from the d-axis. For a better perception, a graphical representation is presented in Figure 3.9.
67
Figure 3.9: Graphical view of the application of Park’s transformation (adapted from (Fernandes, 2013))
Applying Park’s transformation to (3.29), we get the following set of equations, in dq coordinates:
‘
µŽ
½Ž
= ÝŽ 'µŽ +
= ÝŽ '½Ž +
µ1¹Æ
µx
µ1¾Æ
µx
− gz 2½Ž
+ gz 2µŽ
(3.31)
where µ½Ž are the voltages at the PMSG’s terminals, 'µ½Ž are the PMSG’s stator currents and
2µ½Ž are the linkage fluxes, (all in dq components). gz is the angular frequency of the PMSG’s
voltages.
The linkage fluxes 2µ½Ž are given by:
2µŽ = 2{Œ + ßµŽ 'µŽ
(3.32)
2½Ž = ß½Ž '½Ž
wherein ßµ½Ž are the machine’s inductances in dq coordinates and 2{Œ is the constant flux
generated by the permanent magnets.
As required, the inductances are now constant and the analysis is easier to perform.
68
As far as the mechanical part of the dynamic model is concerned, it is first necessary to establish the expression of the electromagnetic torque displayed by the generator, ³z . By definition, in dq coordinates, the electromagnetic torque is:
³z
= /(2µŽ '½Ž − 2½Ž 'µŽ )
(3.33)
Replacing (3.32) in (3.33), we get:
³z
= /(2{Œ + (ßµŽ − ß½Ž )'µŽ )'½Ž
(3.34)
The swing equation describes the mechanical dynamic of the system and is given by:
³z − ³Þ = 9
µå¼
µx
(3.35)
wherein 9 is the generator’s moment of inertia (in kgm2) and ³Þ is the load torque (in Nm), provided by the mechanical shaft.
As a final note, it is important to notice that this model assumes a motor convention in the
equations’ establishment. However, as the synchronous machine works as generator, both the
electromagnetic torque ³z and the load torque ³Þ must be affected by a negative sign, in
order to maintain the coherence.
The overall PMSG dynamic model is defined by equations (3.31) e (3.35).
Table 3-1 contains the PMSG’s parameters used in this work.
Table 3-1: PMSG’s parameters
: YðP[
0.865
±® YÓ
0.09
[
;® YÓ
0.09
69
[ |® YÓ<[ =Y
2
ÓF [ Õ
1000
4
3.2.3. Field-Oriented Control
Field-Oriented Control (FOC) is a well-known strategy control, widely used in the framework of
electrical machines. The main principle of this strategy is to consider that any electrical machine is a system that produces a torque from a reference torque and a reference flux (Djeriri,
Meroufel, Massoum, & Boudjema, 2014), (Marques, 2007).
The application of FOC to the PMSG aims to control the power angle d (angle between the
rotor linkage flux and the stator linkage flux – see Figure 3.10 bellow for better understanding)
under certain conditions:
1. Constant stator linkage flux 2Ž .
2. Minimal (null) reactive power, with decoupled control between active and reactive
power.
In this case, 2Ž is oriented along the d-axis, meaning that:
2µŽ = 2Ž
(3.36)
2½Ž = 0
(3.37)
In dq coordinates, the active and reactive powers that flow through the stator are given by:
ŸŽ = µŽ 'µŽ +
"
êŽ = − ½Ž 'µŽ +
½Ž '½Ž
µŽ '½Ž
(3.38)
To achieve the goal stated in point 2 (êŽ = 0), it is necessary to set 'µŽ = 0. This condition
affects equation (3.32), as can be seen in (3.39):
2µŽ = 2{Œ = 2Ž = )0 %(V (
(3.39)
Equation (3.39) confirms point 1.
70
Knowing that
µ1¹Æ
µx
= 0 and also 'µŽ = 0 and 2½Ž = 0, as was seen previously, equation (3.31)
simplifies as follows:
µŽ
=0
(3.40)
what means êŽ = 0, as intended in point 2.
Equation (3.33) is also affected by the considerations made above, leading to:
³z
= /2µŽ '½Ž = /2{Œ '½Ž
(3.41)
Rewriting equation (3.41),
'½Ž = Å1»™
S
Ö%
(3.42)
The currents 'µŽ and '½Ž are to be incorporated in a controller (designed with the Symmetrical
Optimum criterion (Silva, Electrónica Industrial: Semicondutores e Conversores de Potência,
2013)) that will generate the reference values to the voltages at PMSG’s terminals and to the
grid currents. In order to extract the maximum power available in the wind, the electromagnetic torque, ³z in (3.42), should equal the torque at the Maximum Power Point, ³ÛÙÙS (or
³o ) (refer to the “Wind Turbine” above section for more details). So, one gets the reference
currents:
'µŽ yz{ = 0
¿
Sâ>?
'½Ž yz{ = Å1
(3.43)
Ö%
Figure 3.10 illustrates the FOC application to the PMSG.
71
β
β
q
q
d
d
ψr
ψr
ψs
δ
ψ qs
δ
ψ ds
θ
ψ qs = 0
α
Figure 3.10: Field-Oriented Control – graphical view
72
ψ s = ψ ds
θ
α
CHAPTER 4
Validation Results and
Discussion
Abstract
In this Chapter, the models developed in the previous chapters are implemented in
Matlab/Simulink® environment and the piece of software produced is used as a validation tool. The effectiveness of the models and of its implementation is demonstrated, as the results closely follow the expectations.
73
4. Validation Results and
Discussion
The simulation and the respective results record are performed in Matlab/Simulink® and include two main steps:
1. Validation of the adopted modulation process, with a few tests performed with a generic RL load connected to the Boost matrix converter terminals.
2. Simulation of a real application, with a test performed with a set composed by a wind
turbine + PMSG connected to the Boost matrix converter terminals.
4.1. Step 1 – Boost matrix converter feeding a generic
RL load
For validation purposes, a generic RL load (ÝÞv;µ = 1Ω and ßÞv;µ = 133.02éú) is connected
to the matrix converter terminals, as depicted in Figure 4.1.
Rload
Lload
Figure 4.1: Boost matrix converter feeding a RL load
Bearing in mind the approach followed in this thesis in terms of modulation strategy, regulators design and filter sizing, the Boost matrix converter must be able:
•
•
To increase the RMS value of the voltage displayed at its own terminals.
To adapt the frequency of the quantities associated to the load to the grid frequency.
74
•
To control the power factor as seen from the grid (variable ¢ý µ ).
The tests are conducted with the aim of showing the above mentioned Boost matrix converter
features.
4.1.1. Case-study 1
In this case-study, the Boost MC capability of displaying a voltage with an amplitude value
(cÞv;µ ) higher than the grid voltage amplitude value (cý µ ), for the same frequency conditions, will be evaluated. In this case, the objective is to set a load voltage equal to 2000 V (maximum value) from a feeder voltage equal to 976 V (maximum value). The simulation conditions
are presented in Table 4-1.
Table 4-1: Simulation conditions to case study 1
Load side
£ ¯O± îÓ® Y £[
¯O± Y
A[
£
1.4
50
îN± îÓ® Y£[
îN± Y
í
A[
îN± Y°[
Grid side
690
50
0
Figure 4.2 shows both the time evolutions of the load voltage displayed by the Boost MC and
the grid voltage.
75
Figure 4.2: Load voltage and grid voltage – case study 1
From Figure 4.2, it can be observed the increase of the voltage amplitude between the grid
side and the load side of the converter, which was the purpose of this test. The step-up load
voltage is imposed by the Boost MC.
Figure 4.3 depicts the voltage and the current in the PCC.
Figure 4.3: Voltage and current in the grid side- case study 1
In Figure 4.3, it can be seen that the Boost MC imposes a unitary power factor in the interconnection point with the grid, as required (see Table 4-1). Besides that, the harmonic analysis
performed results in a THD (Total Harmonic Distortion) nearly equals to 4%, which complies
whit international standards.
The correct controllers’ performance is verified in Figure 4.4, Figure 4.5, Figure 4.6 and Figure
4.7, wherein are represented the dq components of the voltage and current regulators errors,
along an extended time.
76
Figure 4.4: d component of the voltage regulator error
Figure 4.5: q component of the voltage regulator error
Figure 4.6: d component of the current regulator error
Figure 4.7: q component of the current regulator error
4.1.2. Case-Study 2
In this case, a frequency test is made in order to proof that the Boost MC is able to adapt the
load frequency to the grid frequency. For this purpose, a step in the load frequency is imposed,
wherein the frequency associated to the load quantities jumps from an initial value ( =
50ú ) to a final value ( { = 100ú ) at 0.8 seconds. Table 4-2 presents the simulation conditions.
Load side
£ ¯O± îÓ® Y £[
¯O± Y
A[
£
1.4
50→100
îN± îÓ® Y£[
îN± Y
í
77
A[
îN± Y°[
Grid side
690
50
0
In order to demonstrate that a successful frequency adaptation is performed by the Boost MC,
a simulation was carried on and the corresponding results referring to the time evolution of
both the load current and the grid current are depicted in Figure 4.8.
Figure 4.8: Load current vs grid current – frequency adaptation – case study 2
As can be seen from Figure 4.8, when a frequency fluctuation occurs in the load side of the
Boost MC, the frequency of the grid current remains unchanged, as required. It should be noted that to the boost effect in the load voltage corresponds a decrease of the load current so
that the active power remains constant.
The correct controller’ performance is confirmed by Figure 4.9 and Figure 4.10, wherein are
represented the dq components of the current regulator error.
Figure 4.9: d component of the current regulator error
Figure 4.10: q component of the current regulator error
78
4.1.3. Case-Study 3
In this test, the Boost MC’s capability to control the power factor at the interconnection point
with the grid is to be verified. Firstly, the simulation conditions required to ensure unitary grid
power factor are presented in Table 4-3.
Table 4-3: Simulation conditions to case study 3 (unitary power factor)
Load side
£ ¯O± îÓ® Y £[
¯O± Y
A[
£
1.4
50
îN± îÓ® Y£[
îN± Y
í
Grid side
A[
îN± Y°[
690
50
0
Figure 4.11 shows the voltage and the current in the grid side.
Figure 4.11: Grid voltage and grid current – unitary power factor – case study 3
Figure 4.11 allows the conclusion that, despite the inductive characteristics of the load, the
power factor in the point of common coupling is unitary. This shows the effectiveness of the
Boost MC controllers.
79
Now, we set the simulation conditions to achieve a capacitive power factor in the grid (see
Table 4-4).
Table 4-4: Simulation conditions to case study 3 (capacitive power factor)
Load side
£ ¯O± îÓ® Y £[
¯O± Y
A[
£
1.4
50
îN± îÓ® Y£[
îN± Y
í
Grid side
A[
îN± Y°[
690
50
-20
The grid voltage and the grid current are depicted in Figure 4.12.
Figure 4.12: Grid voltage and grid current – capacitive power factor – case study 3
Analysing the results presented in Figure 4.12, one come to the conclusion that, although the
load inductive characteristics, the current injected into the grid can be controlled so that it
presents a capacitive power factor. It should be noticed that this procedure is required, in
many cases, by the Transmission System Operator (TSO), to help voltage control.
4.1.4. Case Study 4
As far as case-study 4 is concerned, a double change in the simulation conditions will be performed: at 0.6 seconds, the frequency of the load voltage jumps from 50ú to 100ú and, at
80
the same time, the amplitude of the load voltage jumps from 1500c to 2000c, as stated in
Table 4-5.
Load side
£ ¯O± îÓ® Y £[
¯O± Y
A[
£
1.1→1.4
50→100
îN± îÓ® Y£[
îN± Y
í
Grid side
A[
îN± Y°[
690
50
0
Regarding this case-study, the most interesting results are shown in Figure 4.13 and Figure
4.14. In the first one, both the grid and load voltage are shown, whereas in the last one, the
grid current are the load current are depicted.
Figure 4.13: Load voltage and grid voltage – case-study 4
81
Figure 4.14: Load current and grid current – case-study 4
Concerning Figure 4.13, until 0.6 seconds, the Boost MC successful elevates the voltage to
1500c, with the desired frequency of 50ú . After this critical point, when the simulation conditions are modified, a short transient can be observed. However, the converter’s response is
very fast and the desired values of voltage amplitude (cÞv;µ = 2000c) and frequency
( Þv;µ = 100ú ) are reached in a short amount of time.
With respect to Figure 4.14, the main conclusion is that, despite the existence of a current
peak at the critical point, the current injected in the grid is not affected by the change in the
load conditions (cÞv;µ = 2000c and Þv;µ = 100ú ): the frequency remains invariant and
equal to 50ú and the inevitable increase in the current amplitude in order to accommodate
the boost in load voltage amplitude can be observed.
4.2. Step 2 – Boost matrix converter feeding a wind
conversion system
In this step, the simulation of a real situation is performed, with a wind turbine generator connected to the Boost matrix converter terminals, as shown in Figure 4.15.
82
Figure 4.15: Boost matrix converter feeding a wind turbine generator
It is assumed that a MPPT is already implemented, meaning that the generator torque follows,
at each instant, the reference torque that guarantees the maximum power extraction from the
wind. In this test, the Boost MC function is to display at its own terminals the voltage value
correspondent to maximum power extraction, following the reference value provided by the
FOC strategy combined with an auxiliary controller. For further details, please refer to Chapter
3.
To illustrate some important aspects, a simple wind variation profile is considered, as depicted
in Figure 4.16
Figure 4.16: Piece of a wind profile
The most relevant results are presented in Figure 4.17, Figure 4.18, Figure 4.19 and Figure
4.20, wherein the reference torque, the Boost MC voltage and the two components of the
voltage regulator errors are depicted, respectively.
83
Figure 4.17: Torque at the Maximum Power Point (equal to the reference torque)
Figure 4.18: Single-phase voltage imposed by the Boost MC to the PMSG’s terminals
Figure 4.19: d component of the voltage regulator
error
Figure 4.20: q component of the voltage regulator error
84
Figure 4.17 shows the time evolution of the torque at the Maximum Power Point (equal to the
reference torque), which, as can be seen, is an image of the wind speed represented in Figure
4.16. This torque is defined by the MPPT and its instantaneous value is used to generate the
reference voltages at PMSG’s terminals that correspond to the maximum power extraction.
In Figure 4.18, it is emphasized the boost effect over the voltage at PMSG’s terminals; despite
the connection of the wind turbine generator, the converter is still able to display voltages
with a higher amplitude than the grid voltage. However, in this case, the reference voltages
depend on the wind profile, so the generated voltages follow the oscillations registered in the
torque at the Maximum Power Point, to allow the maximum power extraction.
Figure 4.19 and Figure 4.20 show the d component and the q component of the voltage regulator error, respectively. It is apparent that both the voltage error components are almost zero,
thus confirming the good Boost MC dynamic performance in wind applications.
85
CHAPTER 5
Conclusions
Abstract
The final chapter presents the main conclusions of the work completed in the
scope of this Master Thesis, together with some suggestions for future developments in this field.
87
5. Conclusions
Matrix converters are advanced power electronics converters that have been subject to intense research and development over the past years. Still, up to now, matrix converters have
been used as Buck matrix converters, meaning that they operate as AC/AC electronic transformers, in which the output voltage is lower than the input voltage. However, its operation as
Boost matrix converters, i.e. electronic AC/AC transformers that increase the output voltage
with respect to the input voltage, can be envisaged. This was the main objective of this Thesis:
to propose a novel operation of the matrix converter with voltage boost characteristics.
The work performed in the scope of this Thesis allowed the development of a fully detailed
model able to describe the behaviour of the Boost matrix converter. The used methodology
combined Space Vector Modulation, Pulse-Width Modulation and classic control techniques in
an innovative approach to the conventional modulation strategies.
After the completion of the Thesis, it was theoretically established and shown via simulation
that this converter is able to:
•
•
•
Increase the RMS value of the voltage displayed at its own terminals.
Perform a frequency adaptation between the input and output voltages and currents.
Control the power factor in the point of common coupling.
The features listed above were successfully implemented in appropriate Matlab/Simulink®
models. Validation tests were carried out in two simulation conditions:
•
•
Connection of a generic RL load to the matrix converter terminals.
Connection of a set wind turbine + permanent magnet synchronous generator to the
matrix converter terminals.
The full integrated theoretical model was established after the development of adequate
components models:
•
•
•
For the Boost matrix converter, which combined Space Vector Modulation, Pulse
Width Modulation and Indirect Modulation; furthermore, the respective regulators
were properly designed and the filters were duly sized.
For the wind turbine, using wind power theory concepts, like power coefficient and tip
speed ratio, thus enabling a torque control to be devised.
For the Permanent Magnet Synchronous Generator, based on the application of Concordia’s and Park’s transformations, in order to obtain an easier to use transient model.
88
From the tasks listed above, the most significant one is undoubtedly the innovative Boost matrix converter modulation strategy and its validation through simulation results. This fills a gap
in the matrix converter area, since the conventional matrix converter – the Buck matrix converter – presents some relevant limitations that restrain its application field. In engineering
applications that require a larger range of output voltage, like High-Voltage Direct Current,
Dynamic Voltage Restorer, Unified Power Flow Controller or electrical drives with / control,
an increase in the output voltage with respect to the input voltage will be most welcome,
hence the interest of the Boost matrix converter.
Nevertheless this contribution, the research in Boost matrix converters is widely open. Hereafter, some topics for future research follow:
•
•
•
Implementation in laboratory environment of the proposed model to describe the
Boost matrix converter.
Research about the model response when used in association with other applications
(DVR, HVDC, UPFC, etc.), in Matlab/Simulink® simulation and laboratory environment.
Optimization of the modulation process, in order to minimize harmonic distortion and
the semiconductors switching frequency, using, for example, a triangular carrier waveform instead of a sawtooth.
89
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Boost Matrix Converter Applied to Wind Energy Conversion Systems

Transcription

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