Boost Matrix Converter Applied to Wind Energy Conversion … · gy, which uses Space Vector...

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

i

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 entusi-

asmo inexcedíveis e pela total disponibilidade para esclarecer as inúmeras dúvidas e proble-

mas que foram surgindo no decurso deste trabalho; ao Professor Fernando Silva um agradeci-

mento 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 ultra-

passar 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 des-

pertado o meu interesse nestes temas.

ii

Resumo

A presente dissertação apresenta uma nova contribuição para o estudo dos conversores matri-

ciais trifásicos, garantido que o conversor matricial tem características de elevador de tensão.

Simultaneamente, o conversor matricial permite a adaptação de frequências entre as grande-

zas 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 mag-

netos permanentes acoplado a uma turbina eólica, usando, para esse efeito, modelos do sis-

tema global.

Os resultados obtidos com recurso ao Matlab/Simulink®, que podem ser posteriormente com-

plementados 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 trans-

porte e acionamentos elétricos.

Palavras-chave: Conversor Matricial Trifásico, Conversor Matricial Elevador, Modulação por

Vetores Espaciais, Modulação por Largura de Impulso, Projeto de Controladores, Gerador Sín-

crono de Magnetos Permanentes, Turbina Eólica

iii

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 strate-

gy, 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 la-

boratory 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

iv

Table of Contents

Agradecimentos ............................................................................................................................. i

Resumo .......................................................................................................................................... ii

Abstract ........................................................................................................................................ iii

Table of Contents ..........................................................................................................................iv

List of Figures ................................................................................................................................vi

List of Tables ................................................................................................................................ viii

Acronyms ....................................................................................................................................... ix

1. Introduction .......................................................................................................................... 2

1.1. Context and Motivation ................................................................................................ 2

1.2. Objectives ...................................................................................................................... 5

1.3. State-of-the-art ............................................................................................................. 6

1.4. Contents ...................................................................................................................... 10

2. 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. Regulators Design ........................................................................................................ 35

2.3.1. Voltage Regulator ................................................................................................ 35

2.3.2. Current Regulator ................................................................................................ 40

2.3.3. Reference Values Setting .................................................................................... 44

2.4. Filters Sizing ................................................................................................................. 46

2.4.1. Load Filter ............................................................................................................ 46

2.4.2. Grid Filter ............................................................................................................. 50

3. 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

v

3.1.3. Turbine Model ..................................................................................................... 56

3.1.4. Generator Power Curve....................................................................................... 57

3.1.5. Torque Control .................................................................................................... 58

3.2. Permanent Magnet Synchronous Generator .............................................................. 62

3.2.1. Description .......................................................................................................... 62

3.2.2. Machine’s Model ................................................................................................. 65

3.2.3. Field-Oriented Control ........................................................................................ 70

4. 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. Step 2 – Boost matrix converter feeding a wind conversion system .......................... 82

5. Conclusions ......................................................................................................................... 88

Bibliography ................................................................................................................................ 91

vi

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

Figure 2.14: Example of synthesis of �� 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

vii

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

viii

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


ix

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

1

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.

2

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 sinus-

oidal 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 (usu-

ally a bank of capacitors), as represented in Figure 1.2.

AC

DC AC

DC1inV

2inV

3inV

innV

1outV

2outV

3outV

outmV

Figure 1.2: Indirect converter

3

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.

AC

AC

1inV

2inV

3inV

innV

1outV

2outV

3outV

outmV

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 three-

phase matrix converters.

A three-phase matrix converter has the generic topology as depicted in Figure 1.4.

ainV

binV

cinV

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.

4

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 volt-

ages 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 semiconduc-

tors.

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), trac-

tion 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 volt-

age, a different configuration of the matrix converter is studied in this thesis – the Boost Ma-

trix Converter (Boost MC).

Figure 1.5 shows schematically the difference between the two converters.

5

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 fil-

ters sizing, to enable the study of the Boost matrix converter.

6

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 volt-

age 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)

7

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 ap-

proaches, 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 ma-

trix 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 elec-

trical 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 configu-

ration described above is presented.

8

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 asynchro-

nous 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 re-

newable energy applications, like DFIG (Castro, 2012) or PMSG. When a DFIG is used, the ma-

trix 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 con-

stant frequency of the grid (see Figure 1.11), (Fernandes, 2013).

9

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 vari-

ation is required, therefore justifying the use of a Boost MC.

10

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 im-

portant 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 pre-

sented, in order to overcome those limitations.

11

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.

13

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.

14

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.

1inV

2inV

3inV

innV

1outV 2outV 3outV outmV

1inI

2inI

3inI

innI

1outI 2outI 3outI outmI

Figure 2.1: Generic matrix converter with nxm phases

This type of converters is normally represented with an input with voltage source characteris-

tics 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 ��.

15

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 instantane-

ous 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@ �66A (2.2)

16

in which;

∑ � ! = 16!C? ', 2 ∈ 41,2,35 (2.3)

Table 2-1: All possible combinations for the switch state combinations with the instantaneous values of output

voltages and input currents

N.º DEE DEF DEG DFE DFF DFG DGE DGF DGG �H(J) �L(J) �M(J) NO(J) NP(J) NQ(J)

1 1 0 0 0 1 0 0 0 1 �;(() �<(() �=(() '8(() '9(() ':(()

2 0 1 0 0 0 1 1 0 0 �<(() �=(() �;(() '9(() ':(() '8(()

3 0 0 1 1 0 0 0 1 0 �=(() �;(() �<(() ':(() '8(() '9(()

4 1 0 0 0 0 1 0 1 0 �;(() �=(() �<(() '8(() ':(() '9(()

5 0 1 0 1 0 0 0 0 1 �<(() �;(() �=(() '9(() '8(() ':(()

6 0 0 1 0 1 0 1 0 0 �=(() �<(() �;(() ':(() '9(() '8(()

7 1 0 0 0 1 0 0 1 0 �;(() �<(() �<(() '8(() −'8(() 0

8 0 1 0 1 0 0 1 0 0 �<(() �;(() �;(() −'8(() '8(() 0

9 0 1 0 0 0 1 0 0 1 �<(() �=(() �=(() 0 '8(() −'8(()

10 0 0 1 0 1 0 0 1 0 �=(() �<(() �<(() 0 −'8(() '8(()

11 0 0 1 1 0 0 1 0 0 �=(() �;(() �;(() −'8(() 0 '8(()

12 1 0 0 0 0 1 0 0 1 �;(() �=(() �=(() '8(() 0 −'8(()

13 0 1 0 1 0 0 0 1 0 �<(() �;(() �<(() '9(() −'9(() 0

14 1 0 0 0 1 0 1 0 0 �;(() �<(() �;(() −'9(() '9(() 0

15 0 0 1 0 1 0 0 0 1 �=(() �<(() �=(() 0 '9(() −'9(()

16 0 1 0 0 0 1 0 1 0 �<(() �=(() �<(() 0 −'9(() '9(()

17 1 0 0 0 0 1 1 0 0 �;(() �=(() �;(() −'9(() 0 '9(()

18 0 0 1 1 0 0 0 0 1 �=(() �;(() �=(() '9(() 0 −'9(()

19 0 1 0 0 1 0 1 0 0 �<(() �<(() �;(() ':(() −':(() 0

20 1 0 0 1 0 0 0 1 0 �;(() �=(() �<(() −':(() ':(() 0

21 0 0 1 0 0 1 0 1 0 �=(() �=(() �<(() 0 ':(() −':(()

22 0 1 0 0 1 0 0 0 1 �<(() �<(() �=(() 0 −':(() ':(()

23 1 0 0 1 0 0 0 0 1 �;(() �;(() �=(() −':(() 0 ':(()

24 0 0 1 0 0 1 1 0 0 �=(() �=(() �;(() ':(() 0 −':(()

25 1 0 0 1 0 0 1 0 0 �;(() �;(() �;(() 0 0 0

26 0 1 0 0 1 0 0 1 0 �<(() �<(() �<(() 0 0 0

27 0 0 1 0 0 1 0 0 1 �=(() �=(() �=(() 0 0 0

17

It is worth to point out that matrix � relates line-to-neutral output voltages �89: with line-to-

neutral input voltages �;<=. Input currents ';<= are converted in output currents '89: through

matrix (�S).

>�8�9�:A = � >�;�<�=A (2.4)

>';'<'= A = �S >'8'9':A (2.5)

Furthermore, matrix �= establishes a relationship between line-to-line output voltages (�89, �9: and �:8) with line-to-neutral input voltages.

>�89�9:�:8A = >�?? − �@? �?@ − �@@ �?6 − �@6�@? − �6? �@@ − �6@ �@6 − �66�6? − �?? �6@ − �?@ �66 − �?6A >�;�<�= A = �= >�;�<�= A (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 ap-

proach and the Space Vector Modulation (SVM) approach. In the scope of this work, SVM ap-

proach 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-

18

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 Ta-

ble 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)

1 Concordia’s transformation, which performs the coordinate’s change VW) → TU0, is given by:

YZ[ = \@6 ]_ 1 0 ?√@− ?@ √6@ ?√@− ?@ − √6@ ?√@a

aab

19

Table 2-2: Output voltage vectors and input current vectors resulting from the application of Concordia’s trans-

formation

Output Voltage Input Current

Group N.º Name Amplitude Angle Amplitude Angle

I

1 1g √3c = d (() √3ef gf(

2 2g −√3c = −d (()+ 4j 3⁄ √3ef −gf(

3 3g −√3c = −d (() √3ef −gf(+ 2j 3⁄

4 4g √3c = d (()+ 4j 3⁄ √3ef

gf(+ 2j 3⁄

5 5g √3c = d (()+ 2j 3⁄ √3ef

gf(+ 4j 3⁄

6 6g −√3c = −d (()+ 2j 3⁄ √3ef

−gf(+ 4j 3⁄

II

7 +1 k2 3⁄ �;<(() 0 √2'8(() −j 6⁄

8 -1 −k2 3⁄ �;<(() 0 −√2'8(() −j 6⁄

9 +2 k2 3⁄ �<=(() 0 √2'8(() j 2⁄

10 -2 −k2 3⁄ �<=(() 0 −√2'8(() j 2⁄

11 +3 k2 3⁄ �=;(() 0 √2'8(() 7j 6⁄

12 -3 −k2 3⁄ �=;(() 0 −√2'8(() 7j 6⁄

13 +4 k2 3⁄ �;<(() 2j 3⁄ √2'9(() −j 6⁄

14 -4 −k2 3⁄ �;<(() 2j 3⁄ −√2'9(() −j 6⁄

15 +5 k2 3⁄ �<=(() 2j 3⁄ √2'9(() j 2⁄

16 -5 −k2 3⁄ �<=(() 2j 3⁄ −√2'9(() j 2⁄

17 +6 k2 3⁄ �=;(() 2j 3⁄ √2'9(() 7j 6⁄

18 -6 −k2 3⁄ �=;(() 2j 3⁄ −√2'9(() 7j 6⁄

19 +7 k2 3⁄ �;<(() 4j 3⁄ √2':(() −j 6⁄

20 -7 −k2 3⁄ �;<(() 4j 3⁄ −√2':(() −j 6⁄

21 +8 k2 3⁄ �<=(() 4j 3⁄ √2':(() j 2⁄

22 -8 −k2 3⁄ �<=(() 4j 3⁄ −√2':(() j 2⁄

23 +9 k2 3⁄ �=;(() 4j 3⁄ √2':(() 7j 6⁄

24 -9 −k2 3⁄ �=;(() 4j 3⁄ −√2':(() 7j 6⁄

III

25 za 0 - 0 -

26 zb 0 - 0 -

27 zc 0 - 0 -

20

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 sig-

nificant 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.

21

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

22

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

23

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:

∑ �o ! = 16!C? ' ∈ 41,25 (2.7)

With respect to the inverter case, in order to avoid open-circuits in the load, the following re-

striction must be followed:

∑ �p ! = 1@ C? 2 ∈ 41,2,35 (2.8)

Based in (2.7) and (2.8), all the possible switch state combinations of the rectifier-inverter as-

sociation 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�o?? �o@?�o?@ �o@@�o?6 �o@6r s en:−en:t = Y�o[ s en:−en:t (2.9)

>�89�9: �:8A = q�p?? �p?@�p@? �p@@�p6? �p6@r s cn:−cn:t = Y�p[ s cn:−cn:t (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 −�o@6[ >�;�<�= A (2.11)

24

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@@ �p6? −�p6@[ > '8'9': A (2.12)

The resulting modulation matrix Y�op[ is the product of Y�o[ and Y�p[, as follows:

�op = Y�p[Y�o[S = q�p??�o?? + �p?@�o@? �p??�o?@ + �p?@�o@@ �p??�o?6 + �p?@�o@6�p@?�o?? + �p@@�o@? �p@?�o?@ + �p@@�o@@ �p@?�o?6 + �p@@�o@6�p6?�o?? + �p6@�o@? �p6?�o?@ + �p6@�o@@ �p6?�o?6 + �p6@�o@6r(2.13)

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.

25

Table 2-3: All possible switch state combinations of the rectifier-inverter association

Matrix

S R 11 S R 12 S R 13 S R 21 S R 22 S R 23 V DC 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 i A 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 i B 0 0 1 1 0 0 0 0 1 -6

1 0 0 0 0 1 -v ca 0 1 1 0 1 0 -i A 0 0 1 1 0 0 1 0 0 3

0 1 0 1 1 0 i C 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 1 0 1 0 0 1 0 0 1 0 0 1 0 0 za

0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 zc

1 0 0 1 0 1 i A 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 i B 0 0 1 0 1 0 0 0 1 5

0 1 0 0 0 1 v bc 0 1 1 0 1 0 -i A 0 0 1 0 1 0 0 1 0 -2

0 1 0 1 1 0 i C 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 1 0 1 0 0 0 1 0 0 1 0 0 1 0 zb

0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 zc

1 0 0 1 0 1 i A 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 i B 1 0 0 0 1 0 1 0 0 -4

0 1 0 1 0 0 -v ab 0 1 1 0 1 0 -i A 1 0 0 0 1 0 0 1 0 1

0 1 0 1 1 0 i C 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 1 0 1 0 0 0 1 0 0 1 0 0 1 0 zb

0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0 za

1 0 0 1 0 1 i A 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 i B 1 0 0 0 0 1 1 0 0 6

0 0 1 1 0 0 v ca 0 1 1 0 1 0 -i A 1 0 0 0 0 1 0 0 1 -3

0 1 0 1 1 0 i C 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 1 0 1 0 0 0 0 1 0 0 1 0 0 1 zc

0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0 za

1 0 0 1 0 1 i A 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 i B 0 1 0 0 0 1 0 1 0 -5

0 0 1 0 1 0 -v bc 0 1 1 0 1 0 -i A 0 1 0 0 0 1 0 0 1 2

0 1 0 1 1 0 i C 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 1 0 1 0 0 0 0 1 0 0 1 0 0 1 zc

0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0 zb

1 0 0 1 0 1 i A 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 i B 0 1 0 1 0 0 0 1 0 4

1 0 0 0 1 0 v ab 0 1 1 0 1 0 -i A 0 1 0 1 0 0 1 0 0 -1

0 1 0 1 1 0 i C 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 1 0 1 0 0 1 0 0 1 0 0 1 0 0 za

0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0 zb

1 0 0 1 0 1 i A 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 i B 1 0 0 1 0 0 1 0 0 za

1 0 0 1 0 0 0 0 1 1 0 1 0 -i A 1 0 0 1 0 0 1 0 0 za

0 1 0 1 1 0 i C 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 1 0 1 0 0 1 0 0 1 0 0 1 0 0 za

0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0 za

1 0 0 1 0 1 i A 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 i B 0 1 0 0 1 0 0 1 0 zb

0 1 0 0 1 0 0 0 1 1 0 1 0 -i A 0 1 0 0 1 0 0 1 0 zb

0 1 0 1 1 0 i C 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 1 0 1 0 0 0 1 0 0 1 0 0 1 0 zb

0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0 zb

1 0 0 1 0 1 i A 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 i B 0 0 1 0 0 1 0 0 1 zc

0 0 1 0 0 1 0 0 1 1 0 1 0 -i A 0 0 1 0 0 1 0 0 1 zc

0 1 0 1 1 0 i C 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 1 0 1 0 0 0 0 1 0 0 1 0 0 1 zc

0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 zc

Rectifier-Inverter Association

26

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 (cvwxyz{ 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 Con-

verter (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 independ-

ent 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.

27

Table 2-4: State vectors generated by the rectifier for all the possible combinations

Vector D|EE D|EF D|EG D|FE D|FF D|FG NO NP NQ }~N��} �N e? 1 0 0 0 0 1 en: 0 −en: √2en:

j6

e@ 0 1 0 0 0 1 0 en: −en: √2en: j2

e6 0 1 0 1 0 0 −en: en: 0 √2en: 5j6

e� 0 0 1 1 0 0 −en: 0 en: √2en: − 5j6

e� 0 0 1 0 1 0 0 −en: en: √2en: − j2

e� 1 0 0 0 1 0 en: −en: 0 √2en: − j6

e� 1 0 0 1 0 0 0 0 0 0 -

e� 0 1 0 0 1 0 0 0 0 0 -

e� 0 0 1 0 0 1 0 0 0 0 -

0

12

3

4 5

refoutIαβ

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 evwxyz{��is located in sector 0, it can be generated by applying vectors e?, e� and one of the

28

null vectors e�,e� ore� during an appropriate amount of time. This process is illustrated in Figu-

re 2.12.

1I

6I

refoutIαβ

Figure 2.12: Example of synthesis of �� in sector 0 (Pinto S. F., 2003)

where e��are generic adjacent vectors, /�� 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 out-

put frequency (�� ≫ �vwx), it can be ensured that, in each switching period, the reference vec-

tor evwxyz{�� is given nearly by:

evwxyz{�� ≅ e�/� + e�/� + e�/� (2.14)

Based on a trigonometric analysis of the diagram depicted in Figure 2.12, the duty-cycles /�,/� and /� are given by the following expressions:

�/� = �=sin(�6 − � )/� = �=sin(� )/� = 1 − /� − /� (2.15)

wherein �= is the current modulation index.

29

The constant �= relates the amplitude of the reference vector evwxyz{��with the intermediate

stage current en: as follows:

�= = p��p�� (2.16)

wherein evwx�;� is the amplitude of the reference vector evwxyz{��.

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:

�n: = �vwx (2.17)

wherein �n: is the power in the DC intermediate stage and �vwx is the output power.

Developing (2.17), we get:

cn:en: = 6@ evwx�;�cvwx�;�cos(¢vwx) (2.18)

wherein cvwx�;� 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��p�� cvwx�;� cos(¢vwx) = 6@ �=cvwx�;� cos(¢vwx) (2.19)

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:.

30

Once again, when applying Concordia’s transformation, the eight possible switch state combi-

nations 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 D~EE D~EF D~FE D~FF D~GE D~GF �H �L �M }£N��} ��

c? 1 0 0 1 0 1 cn c: c: √2cn: j6

c@ 1 0 1 0 0 1 cn cn c: √2cn: j2

c6 0 1 1 0 0 1 c: cn c: √2cn: 5j6

c� 0 1 1 0 1 0 c: cn cn √2cn: − 5j6

c� 0 1 0 1 1 0 c: c: cn √2cn: − j2

c� 1 0 0 1 1 0 cn c: cn √2cn: − j6

c� 1 0 1 0 1 0 0 0 0 0 -

c� 0 1 0 1 0 1 0 0 0 0 -

12

3

4 5

0

refinVαβ

Figure 2.13: Spatial location of the vectors (V0-V7) needed to control the input voltage (Pinto S. F., 2003)

31

The technique is similar to the one presented for output currents. In this case, for example, if

the reference vector c �yz{�� is located in sector 0, it can be generated by applying vectors c?, c� and one of the null vectors c� orc� during an appropriate amount of time. This process is

illustrated in Figure 2.14.

refinVαβ

1V

6V

Figure 2.14: Example of synthesis of �� in sector 0 (Pinto S. F., 2003)

where c�� are generic adjacent vectors, /�� 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 (�� ≫ � �). As so, it can be ensured that, in each switching

period, the reference vector c �yz{�� is given by:

c �yz{�� ≅ c�/� + c�/� + c�/� (2.20)

Trigonometric analysis applied to the diagram displayed in Figure 2.14, allows to conclude that

the duty-cycles /�,/� and /� are given by the following expressions:

¥/� = �¤sin(�6 − �¤)/� = �¤sin(�¤)/� = 1 − /� − /� (2.21)

32

wherein �¤ is the voltage modulation index.

Now, the constant �¤ creates the relationship between the amplitude of the reference vector c �yz{�� with the intermediate stage voltage cn: as follows:

�¤ = ¦§¨��¦�� (2.22)

wherein c ��;� is the amplitude of the reference vector c �yz{��.

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:

�n: = � � (2.23)

where � � is the input power.

Developing (2.23), we obtain:

cn:en: = 6@ e ��;�c ��;�cos(¢ �) (2.24)

wherein e ��;� 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@ ¦§¨��¦�� e ��;� cos(¢ �) = 6@ �¤e ��;� cos(¢ �) (2.25)

33

2.2.4. Indirect Modulation – Application to the Boost Converter

Considering switching frequencies high enough when compared to the input and output fre-

quencies (�� ≫ � � and �� ≫ �vwx), it can be assumed that, during a switching period, the aver-

age values of cn: and en: are constant. Based on this assumption, the input voltage modula-

tion 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).

©ªª«ªª¬ /�/� = �=�¤sin(�6 − � )sin(�6 − �¤)/�/� = �=�¤sin(�6 − � )sin(�¤)/�/� = �=�¤sin(� )sin(�6 − �¤)/�/� = �=�¤sin(� )sin(�¤)/� = 1 − /�/� − /�/� − /�/� − /�/�

(2.26)

For a matter of simplicity, �= 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 �� = 0 and evwx� = 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 /�, V1

to /�, I1 to /� and I6 to /�, therefore the vector pairs I6-V6, I6-V1, I1-V6 and I1-V1 will correspond

the duty-cycles /�/�, /�/�, /�/� and /�/�, 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.

34

Table 2-6: Matrix converter’s vectors used in the modulation of the input voltages and output currents

£N® ~¯°J® ±²±� ±²±� ±�±� ±�±� £N® ~¯°J® ±²±� ±²±� ±�±� ±�±�

0

0 -4 +1 +6 -3

3

0 +4 -1 -6 +3

1 +6 -3 -5 +2 1 -6 +3 +5 -2

2 -5 +2 +4 -1 2 +5 -2 -4 +1

3 +4 -1 -6 +3 3 -4 +1 +6 -3

4 -6 +3 +5 -2 4 +6 -3 -5 +2

5 +5 -2 -4 +1 5 -5 +2 +4 -1

1

0 +1 -7 -3 +9

4

0 -1 +7 +3 -9

1 -3 +9 +2 -8 1 +3 -9 -2 +8

2 +2 -8 -1 +7 2 -2 +8 +1 -7

3 -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

2

0 -7 +4 +9 -6

5

0 +7 -4 -9 +6

1 +9 -6 -8 +5 1 -9 +6 +8 -5

2 -8 +5 +7 -4 2 +8 -5 -7 +4

3 +7 -4 -9 +6 3 -7 +4 +9 -6

4 -9 +6 +8 -5 4 +9 -6 -8 +5

5 +8 -5 -7 +4 5 -8 +5 +7 -4

To know the time interval during which the corresponding vectors are applied to the convert-

er, 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.

sT d dγ α sT d dδ α

sT d dγ β

d dγ α

d d d dγ α γ β+

d d d d d dγ α γ β δ α+ +

d d d d d d d dγ α γ β δ α δ β+ + +

sT d dδ β

0sT d

sT0 2 sT

sT d dγ α sT d dδ α

sT d dγ β sT d dδ β

0sT d

Figure 2.15: PWM modulation process used to select the time interval during the appropriate vectors are applied

35

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 �� = 0 and evwx� = 0).

Consulting Table 2-6, we can see that there are a set of vectors that can be applied to the con-

verter (−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 /�/�are selected, the vector −4 is applied during ³�/�/�; when the components /�/� are selected, the vector +1 is applied during ³�/�/�, etc. Figure 2.16 illustrates the

whole selection scheme, with an exemplificative situation.

sT0 2 sT

d dγ α

0sinV = 0

soutI =

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 con-

trolling 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 effective-

ly follows a certain reference. So, the controlled voltage is the capacitor voltage c=, which is

36

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 � 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.

genImatrixI

cI

CcV

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 repre-

sentation, we have the following system of equations:

�Z µ¦¶·µx = '´z�� − '�;xy ��Z µ¦¶¸µx = '´z�� − '�;xy �� (2.27)

4 For this purpose, the inductance of the external filter of the PMSG can be neglected.

37

where c=�� are the capacitor voltages, '�;xy �� are the currents that flow through the matrix

converter and '´z�� are the generator currents, all in alfa-beta coordinates. Z is the capaci-

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¦¶¹µx = º»¨¹: − ��¼§�¹: + g´z�c=½µ¦¶¾µx = º»¨¾: − ��¼§�¾: − g´z�c=µ (2.28)

where the additional terms g´z�c=µ½are 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 volt-

age control, we get the following system:

�µ¦¶¹µx = º»¨¹: + ?: e=µµ¦¶¾µx = º»¨¾: + ?: e=½ (2.29)

Thus,

¿e=µ = −'�;xy �µÀ + Zg´z�c=½e=½ = −'�;xy �½À − Zg´z�c=µ (2.30)

where '�;xy �µ½À is an image of the current '�;xy �µ½ that flows through the converter. '�;xy �µ½À differs from '�;xy �µ½by 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.

5 Park’s transformation, which performs the coordinate’s change TU → /Á, is given by: Y�[ = s)0%¢ −%'�¢%'�¢ )0%¢ t

38

ivpv

KK

s+vα + −drefV

ivpv

KK

s+vα +

−qrefV

vα

dcI

qcI −

+−

−

vα

ωgenC

ωgenC

'matrixdI

'matrixqI

matrixdI

matrixqI

dmeasV

qmeasV

gendI

genqI

−+

+−

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.

dqcI

ivpv

KK

s+

1

1i

dvsTα

+

dqmatrixI−

dqgenI

+ 1sCvα

vα

+−

dqmeasVdqrefV

Figure 2.20: Simplified voltage regulator block diagram

The difference between the reference voltage cyz{¹¾ and the measured capacitor voltage c�z;�¹¾, 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 ? �§⁄�S¹ÂÃ?, 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 cur-

rent 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 repre-

6 This is a consequence of we are assuming a real situation, wherein the global system is tested in laboratory envi-

ronment.

39

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:

¦�»�Æ¹¾pº»¨¹¾(�) = Æ�Ç¹Â(�S¹ÂÃ?)�ÈÃ ÆÉÇ¹ÂÊ�·ÂËÌÂ�·§Ç¹ÂÊ ·ÂË§Â�·§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:

�6(%) = %6 + 1.75&�%@ + 2.15&�@% + &�6 (2.32)

The result is:

©ª«ª¬ 1.75&� = 1 ³µ¤Î− �ÂÏÌÂ:S¹Â�§ = 2.15&�@− �ÂÏ§Â:S¹Â�§ = &�6

(2.33)

After some algebraic manipulation, the proportional gain ÄÅ¤ and the integral gain Ä ¤ of the

voltage controller can be obtained as:

�ÄÅ¤ = − @.?�:�§�ÂS¹Â?.��ÉÄ ¤ = − :�§�ÂS¹ÂÉ?.��È (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 rela-

tively larger value, in this case, one-tenth of the grid period. As the grid period is ³ y 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.

40

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 e y µ current within some pre-

determined 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 coor-

dinates:

©ª«ª¬µ º¼§¹�µx = − oÖ§×�ØÖ§×� 'ý µ; − ¦º¼§¹�ØÖ§×� + ?ØÖ§×� cÙÚÛ;µ º¼§¹Üµx = − oÖ§×�ØÖ§×� 'ý µ< − ¦º¼§¹ÜØÖ§×� + ?ØÖ§×� cÙÚÛ<µ º¼§¹¶µx = − oÖ§×�ØÖ§×� 'ý µ= − ¦º¼§¹¶ØÖ§×� + ?ØÖ§×� cÙÚÛ=

(2.35)

wherein 'ý µ;<= are the RMS grid injected currents, c y µ;<=are the RMS grid voltages, 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.

41

Applying Park’s transformation to the equations above and choosing a reference frame syn-

chronous with the grid voltage (¢ = gý µ(, ¢ - transformation angle, gý µ – angular fre-

quency of the grid), we obtain the description of the system in dq coordinates as follows:

¥µ º¼§¹¹µx = − oÖ§×�ØÖ§×� 'ý µµ − ¦º¼§¹¹ØÖ§×� + ?ØÖ§×� cÙÚÛµ + gý µ'ý µ½µ º¼§¹¾µx = − oÖ§×�ØÖ§×� 'ý µ½ − ¦º¼§¹¾ØÖ§×� + ?ØÖ§×� cÙÚÛ½ − gý µ'ý µµ (2.36)

wherein the additional terms gý µ'ý µµ½are 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 voltag-

es, the command voltages used by the modulation process that ensure grid currents follow

their references:

¥µ º¼§¹¹µx = − oÖ§×�ØÖ§×� 'ý µµ − ¦º¼§¹¹ØÖ§×� + ?ØÖ§×� à=µµ º¼§¹¾µx = − oÖ§×�ØÖ§×� 'ý µ½ − ¦º¼§¹¾ØÖ§×� + ?ØÖ§×� à=½ (2.37)

As so,

¿à=¹ = cÙÚÛµÀ + gý µß{ Þx'ý µ½à=¾ = cÙÚÛ½À − gý µß{ Þx'ý µµ (2.38)

wherein cÙÚÛµ½À is an image of the voltage cÙÚÛµ½displayed by the converter. cÙÚÛµ½À differs

from cÙÚÛµ½by a delay introduced by the converter operation.

Based on these equations, the current regulator block diagram obtained is represented in Fig-

ure 2.22:

42

+1 z

p

sTsT

α i+ −dgridrefI

+1 z

p

sTsT

α i+−

qgridrefI

α i

dcU

qcU+

−+

+

α i

grid filtLω

grid filtLω

'PWMdV

'PWMqV

PWMdV

PWMqV

dmeasI

qmeasI

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:

dqcU+1 z

p

sTsT 1di

KsT +

dqPWMV+

dqgridV

−+1

filt filtsL Rα i

α i

+−

dqmeasIdqgridrefI

Figure 2.23: Simplified current regulator block diagram

The reference current e y µyz{¹¾and the measured current e�z;�¹¾are subtracted and the dif-

ference is applied to a Proportional-Integral (PI) Controller, which returns the command volt-

age à=µ½used by the SVM. Moreover, the block Ï�S¹§Ã?, a first order transfer function with a

delay ³µ , models the matrix converter and the modulation process. The constant T , that af-

fects 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, relat-

ing 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 c y µ, is accounted with a fictitious resistance Ý´y µ, which is added to the filter resistance Ý{ Þx. As so, the resultant resistance Ýx is given by:

43

Ýx = Ý{ Þx + Ý´y µ (2.39)

where Ý´y µ = ¦º¼§¹pº¼§¹

Then, to obtain ³á, the normal procedure is considering that the zero of the PI Controller can-

cels 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�»�Æ¹¾(�)pº¼§¹¼»Ö¹¾(�) = Ë·§ÇÌÇ¹§â��ÉÃ� ãÇ¹§Ã Ë·§ÇÌÇ¹§â� (2.41)

Now, the denominator can be compared to the denominator of a second order transfer func-

tion written in the canonical form, as follows:

ä(%) = åÉ�ÉÃ@æå¨�ÃåÉ (2.42)

where g� is the natural frequency of the system and ç is the damping factor.

Equating the proper terms, we get:

� 2çg� = ?S¹§g�@ = Ï�§SÌS¹§o� (2.43)

and after some algebraic manipulation, considering a typical damping factor ç = √@@ , we ob-

tain:

44

³Å = @Ï�§S¹§o� (2.44)

Normally, the PI controller is presented with the following formulation:

ZÙp(%) = ÄÅ + Ï§§� (2.45)

where:

�ÄÅ = SèSÌÄ = ?SÌ (2.46)

So, replacing (2.40) and (2.44) in (2.46), we obtain:

�ÄÅ = ØÖ§×�@Ï�§S¹§Ä = o�@Ï�§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 (³� = 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:

45

"� = �µ'µ + �½'½ê = �µ'½ − �½'µ (2.48)

Referring to the load side, we get:

ë�Þv;µ = �µÞv;µ'µÞv;µ + �½Þv;µ'½Þv;µêÞv;µ = �µÞv;µ'½Þv;µ − �½Þv;µ'µÞv;µ (2.49)

Referring to the grid side, we obtain:

ë� y µ = �µý µ'µý µ + �½ý µ'½ý µêý µ = �µý µ'½ý µ − �½ý µ'µý µ (2.50)

Assuming that the reactive power flow in the grid is null, �½ý µ = 0 and '½ý µ = 0, so:

� y µ = �µý µ'µý µ (2.51)

Therefore,

'µý µ = Ùº¼§¹¤¹º¼§¹ (2.52)

Neglecting losses, � y µ ≅ �Þv;µ, (2.52) can be approximated by:

e y µyz{µ = Ù×��¹¦º¼§¹¹ (2.53)

wherein e y µyz{µ is the reference current to the current regulator (d component) and �Þv;µ

and c y µµare well defined quantities that can be measured instantaneously.

46

The q component of e y µyz{ is a degree of freedom that can be adjusted to control the reac-

tive 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 con-

trollers must follow these power fluctuations. So, the transferred active power is measured

instantaneously and the references are continuously updated. This point will be further dis-

cussed 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

ìGíîYïð[ £îY£[ 5

2000√2

2.4.1. Load Filter

The switching process of the semiconductors introduces high-frequency harmonics in the con-

verter currents and voltages, which may cause additional losses and excite electrical reso-

nance. 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.

47

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:

- = µñ¶µpò (2.54)

cI

fCgenV cV

fR

fL

genImatrixI UI

CUir

Figure 2.25: Load filter single-phase equivalent

The single phase power �Û: that flows in the matrix converter is given by:

�Û: = à=eñ (2.55)

�v is the single-phase delivered power and it is a fraction of �Û:, as follows:

48

�v = ó�Û: = óà=eñ (2.56)

wherein ó is the converter efficiency.

So, à: is given by:

à: = Ù�ôpò (2.57)

Replacing (2.57) in (2.54) and developing the equation, we get:

- = µµpò õ Ù�ôpòö = − Ù�ôpòÉ = −ó ñ�ÉÙ� = −óÝv ñ�É¦�É (2.58)

wherein Ýv and cv are the equivalent resistance and voltage related to the delivered power �v,

respectively. Ýv has the following expression:

Ýv = ¦�â÷øÉÙ� (2.59)

wherein cvoÛù is the RMS value of the voltage related to the delivered power �v.

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.

49

1 ≤ úÅ ≤ ?@æÉ (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 charac-

teristic impedance ü{, as follows:

ü{ = @æÉýÌÊ?æýÌ - (2.63)

At this time, it is possible to present the expressions that allow the calculation of the filter pa-

rameters Z{, ß{ and Ý{ (see Figure 2.25):

©ª«ª¬ Z{ = ?þÖåÖß{ = þÖåÖÝ{ = 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 fre-

quency (� y µ = 50ú�) and one decade bellow the switching frequency (�� = 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 = :Ö6 (2.65)

50

The load filter parameters and the associated constants are presented in Table 2-9.

Table 2-9: Load filter parameters and the associated constants

� £¯|ïDY£[ ì¯Y�ð[ |¯Y�[ �Õ � ��Y�[ ��Y�[ MYµµµµ[ ��Yµµµµ�[ |�Y�[ 0.985 cy √32

�6�y3 0.9 0.8 √22 0.4179 500 253.9 133.02 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:

51

ß{ Þx = ñ��SÆ� pº¼§¹ (2.66)

wherein àn: is the DC link voltage, ³� is the switching period and Δe y 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 (maxi-

mum 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 �Þv�� dissipated in the filter

resistance do not exceed 0.5% of the rated power. So:

Ý{ Þx = Ù×�ÆÆ6pº¼§¹â÷øÉ (2.68)

wherein e y 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

£QY£[ �~�îN±YH[ ì�¯®®Y�ð[ |�N�JYÓ�[ ��N�JYµµµµ�[ 2000 341.5975 25 1.4 195.16

53

CHAPTER 3 Wind Turbine Generator

Abstract

This chapter is organized in two parts: in the first part, the study of the wind tur-

bine is performed, including a succinct description of its structure and main com-

ponents, together with the presentation of the model used and the control strate-

gy adopted; in the second one, a brief overview of the permanent magnets syn-

chronous generator’s characteristics is given alongside with the presentation of the

model that describes its dynamics.

54

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 re-

quired to prevent wind generator’s nominal power to be exceeded. Indeed, the tur-

bine’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.

55

• 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 frequen-

cy.

• 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.

ePmP

Wind

, avaiu P

Tω

TTTω

mPGTGω

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 �� is then given by:

�� = ³SgS = ³�g� (3.1)

wherein ³S and ³� are the turbine torque and the generator torque, respectively, and gS and g� are the turbine angular speed and the generator angular speed, respectively.

56

• The gear box is represented by a simple multiplicative gain, so, in order to respect the

power conservation:

ä = SÇS� = å�åÇ (3.2)

3.1.2. Power in the Wind

The kinetic energy available to a wind turbine is the kinetic energy, �� , associated to an air

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 �;¤; (in W) is given by:

�;¤; = µ��§¨µx = ?@ õ�� µ�µxö ,@ = ?@��,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 achiev-

ing only 50% at most of wind-mechanical energy conversion. This means that, in normal condi-

tions, 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:

57

ZÙ(,) = Ù»Ù�Â�§ (3.5)

where �z 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:

�z = �;¤; 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 �z(,) of a typical wind generator of 2 MW is shown:

cut inu − rucut outu −

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:

58

• 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) corre-

sponds to the maximum power extraction. In this zone, the electrical power has an ap-

proximate 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 regu-

lated 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 de-

scribe 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(− ?@.�

�§ ) (3.7)

where

� = ?ã#$%.%&¸Ê%.%È'¸È$ã (3.8)

59

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):

� = åÇow (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 gS is the turbine angular speed (in rad/s). To clarify some points, it could be nec-

essary to confer Figure 3.2.

The representation of ZÙ as function of �, to several values of U, is depicted in Figure 3.4.

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.

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

λ - Tip Speed Ratio

Cp

- P

ower

Coe

ffic

ient

β=0°β=10°β=20°β=30°

60

ZÙ = 0.22(??��§ − 5)exp(− ?@.�

�§ ) (3.10)

where

� = ?ã#Ê�.�6� (3.11)

Figure 3.5 depicts the variation of the power coefficient with the tip speed ratio. It can be ob-

served that there is a particular value of the tip speed ratio that maximizes the power coeffi-

cient. 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)

61

from where results,

�vÅx = 6.32497 (3.13)

Replacing (3.13) in (3.9) and rewriting the equation, we get:

gSvÅx = �.6@��wo (3.14)

Recovering equation (3.6) and replacing the adequate equations for the power coefficient

((3.10) and (3.11)), one obtains:

�z = ?@�jÝ@,60.22( ??�ãã#*%.%È' − 5)exp(− ?@.�ãã#*%.%È') (3.15)

Knowing that the generator torque ³� = Ù»å� and remembering that g� = ägS,

³� = ?�åÇ�Ì� ?@�jÝ@,60.22( ??�ãã#*%.%È' − 5)exp(− ?@.�ãã#*%.%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@,60.22( ??�ãã+.ÈÉ,-.*%.%È' − 5)exp(− ?@.�ãã+.ÈÉ,-.*%.%È') (3.17)

Replacing numerical values, one gets the reference torque as:

62

³o�� = ³ÛÙÙS = �.��6@?6oÉwÈ�åÇ�Ì� = �.��6@?6oÈwÉ�.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 main-

tain � = )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 ener-

gy 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 mag-

netic 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 elec-

tromotive force at the armature winding terminals.

A permanent magnet synchronous generator (PMSG) is a variant of the conventional synchro-

nous generator using neither auxiliary source nor inductor coil in the rotor. This type of gener-

ator, 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).

63

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))

64

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 posi-

tion 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 �¼�� (3.23)

65

the electromotive force frequency � and the rotor speed �y are related by the following ex-

pression:

� = / �¼�� (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;<=[ = YÝ�[Y';<=[ + µY1�Ü¶[µx (3.25)

wherein Ý� 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;2<2= A = > ß? + ß@)0%2�; 3? + 3@)0%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�; ß? + ß@)0%2�; A >';'<'=A (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 co-

sine of the rotor’s angular position (��), which, in turn, is a function of the time, as can be

seen in:

�� = g( + 4v (3.27)

66

where 4vis 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 de-

tailed next.

Machine’s model in �� coordinates

Concordia’s transformation, which performs the coordinate’s change VW) → TU0, is given by:

YZ[ = \@6 ]_ 1 0 ?√@− ?@ √6@ ?√@− ?@ − √6@ ?√@a

aab (3.28)

and: 5VW)6 = YZ[ 5TU06; 5TU06 = YZ[S 5VW)6

The application of Concordia’s transformation to (3.26) allows the representation of the ma-

chine’s dynamics in a two-phase equivalent system, as can be seen in (3.29)

s2�2�t = sß;¤ + ßv�)0%2� ßv�%.�2�ßv�%.�2� ß;¤ − ßv�)0%2�t s'�'�t (3.29)

where ß;¤ and ßv� are fictitious inductances that result from the application of the transfor-

mation.

A graphical illustration of what has been done is depicted in Figure 3.8.

67

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 applica-

tion 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:

Y�[ = s)0%¢ −%'�¢%'�¢ )0%¢ t (3.30)

and: 7TU8 = Y�[ s/Át; s/Át = Y�[S 7TU8

In this transformation, the armature windings are seen from a rotating frame, called dq refer-

ential, 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 per-

ception, a graphical representation is presented in Figure 3.9.

68

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 coordi-

nates:

��µ� = Ý�'µ� + µ1¹Æµx − gz2½��½� = Ý�'½� + µ1¾Æµx + gz2µ� (3.31)

where �µ½� are the voltages at the PMSG’s terminals, 'µ½� are the PMSG’s stator currents and

2µ½� are the linkage fluxes, (all in dq components). gz is the angular frequency of the PMSG’s

voltages.

The linkage fluxes 2µ½� are given by:

2µ� = 2{� + ßµ�'µ� (3.32)

2½� = ß½�'½�

wherein ßµ½� are the machine’s inductances in dq coordinates and 2{� is the constant flux

generated by the permanent magnets.

As required, the inductances are now constant and the analysis is easier to perform.

69

As far as the mechanical part of the dynamic model is concerned, it is first necessary to estab-

lish the expression of the electromagnetic torque displayed by the generator, ³z�. By defini-

tion, in dq coordinates, the electromagnetic torque is:

³z� =/(2µ�'½� − 2½�'µ�) (3.33)

Replacing (3.32) in (3.33), we get:

³z� = /(2{� + (ßµ� − ß½�)'µ�)'½� (3.34)

The swing equation describes the mechanical dynamic of the system and is given by:

³z� − ³Þ = 9 µå¼µx (3.35)

wherein 9 is the generator’s moment of inertia (in kgm2) and ³Þ is the load torque (in Nm), pro-

vided 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[ �±®YÓ�[ �;®YÓ�[ |®YÓ<[ =Y��ÓF[ Õ

0.865 0.09 0.09 2 1000 4

70

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 ma-

chine 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 2�.

2. Minimal (null) reactive power, with decoupled control between active and reactive

power.

In this case, 2� is oriented along the d-axis, meaning that:

2µ� = 2� (3.36)

2½� = 0 (3.37)

In dq coordinates, the active and reactive powers that flow through the stator are given by:

" �� = �µ�'µ� + �½�'½�ê� = −�½�'µ� + �µ�'½� (3.38)

To achieve the goal stated in point 2 (ê� = 0), it is necessary to set 'µ� = 0. This condition

affects equation (3.32), as can be seen in (3.39):

2µ� = 2{� = 2� = )0�%(V�( (3.39)

Equation (3.39) confirms point 1.

71

Knowing that µ1¹Æµx = 0 and also 'µ� = 0 and 2½� = 0, as was seen previously, equation (3.31)

simplifies as follows:

�µ� = 0 (3.40)

what means ê� = 0, as intended in point 2.

Equation (3.33) is also affected by the considerations made above, leading to:

³z� =/2µ�'½� = /2{�'½� (3.41)

Rewriting equation (3.41),

'½� = S»�Å1Ö% (3.42)

The currents 'µ� and '½� 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 electromag-

netic 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:

¿ 'µ�yz{ = 0'½�yz{ = Sâ>?Å1Ö% (3.43)

Figure 3.10 illustrates the FOC application to the PMSG.

72

α

β

dq

sψdsψ

qsψ

δ

θ

rψ

α

β

dq

s dsψ ψ=

0qsψ =

δ

θ

rψ

Figure 3.10: Field-Oriented Control – graphical view

73

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 val-

idation tool. The effectiveness of the models and of its implementation is demon-

strated, as the results closely follow the expectations.

74

4. Validation Results and

Discussion

The simulation and the respective results record are performed in Matlab/Simulink® and in-

clude two main steps:

1. Validation of the adopted modulation process, with a few tests performed with a ge-

neric 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.

loadR loadL

Figure 4.1: Boost matrix converter feeding a RL load

Bearing in mind the approach followed in this thesis in terms of modulation strategy, regula-

tors 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.

75

• To control the power factor as seen from the grid (variable ¢´y 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 (c y µ), for the same frequency condi-

tions, will be evaluated. In this case, the objective is to set a load voltage equal to 2000 V (max-

imum 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 Grid side

£�¯O±îÓ®Y�£[ 1.4 £�îN±îÓ®Y£[ 690

��¯O±Y�A[ 50 ��îN±Y�A[ 50

í�îN±Y°[ 0

Figure 4.2 shows both the time evolutions of the load voltage displayed by the Boost MC and

the grid voltage.

76

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 intercon-

nection 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.

77

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 condi-

tions.


Load side Grid side

£�¯O±îÓ®Y�£[ 1.4 £�îN±îÓ®Y£[ 690

��¯O±Y�A[ 50→100 ��îN±Y�A[ 50

í�îN±Y°[ 0

78

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 not-

ed 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

79

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 Grid side

£�¯O±îÓ®Y�£[ 1.4 £�îN±îÓ®Y£[ 690

��¯O±Y�A[ 50 ��îN±Y�A[ 50

í�îN±Y°[ 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.

80

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 Grid side

£�¯O±îÓ®Y�£[ 1.4 £�îN±îÓ®Y£[ 690

��¯O±Y�A[ 50 ��îN±Y�A[ 50

í�îN±Y°[ -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 per-

formed: at 0.6 seconds, the frequency of the load voltage jumps from 50ú� to 100ú� and, at

81

the same time, the amplitude of the load voltage jumps from 1500c to 2000c, as stated in

Table 4-5.


Load side Grid side

£�¯O±îÓ®Y�£[ 1.1→1.4 £�îN±îÓ®Y£[ 690

��¯O±Y�A[ 50→100 ��îN±Y�A[ 50

í�îN±Y°[ 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

82

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 con-

ditions 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 con-

nected to the Boost matrix converter terminals, as shown in Figure 4.15.

83

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.

84

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

85

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 regula-

tor 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.

87

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 develop-

ments in this field.

88

5. Conclusions

Matrix converters are advanced power electronics converters that have been subject to in-

tense 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 trans-

formers, 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 Con-

cordia’s and Park’s transformations, in order to obtain an easier to use transient mod-

el.

89

From the tasks listed above, the most significant one is undoubtedly the innovative Boost ma-

trix 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 con-

verter – 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. Hereaf-

ter, 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 wave-

form instead of a sawtooth.

91

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