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Performance of DFIG and PMSG
Wind Turbines
Due to environmental pollution and climate change, the use of renewable energy
sources as an alternative means of power generation is on the rise globally. This is
because of their clean nature, which makes them eco-friendly with little or no pollu-
tion compared to traditional fossil fuel power-generated power plants.
Among the various renewable energy sources, wind energy is one of the most
widely employed, due to its promising technology. Wind turbine technologies could
be classified into two groups as follows: Fixed Speed Wind Turbines (FSWTs) and
Variable Speed Wind Turbines (VSWTs). There have been tremendous improvement
in wind turbine technology over the years, from FSWTs to VSWTs, as a result of fast
innovations and advanced developments in power electronics. Thus, VSWTs have
better wind energy capture and conversion efficiencies, less acoustic noise and
mechanical stress, and better power quality in power grids without support from
external reactive power compensators due to the stochastic nature of wind energy.
The two most widely employed VSWTs in wind farm development are the Doubly
Fed Induction Generator (DFIG) and the Permanent Magnet Synchronous Generator
(PMSG) wind turbines. In order to solve transient stability intricacies during power
grid faults, this book proposes different control strategies for the DFIG and PMSG
wind turbines.
Performance of DFIG and
PMSG Wind Turbines
Kenneth E. Okedu
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press

4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Kenneth E. Okedu
Reasonable efforts have been made to publish reliable data and information, but the author and publisher
cannot assume responsibility for the validity of all materials or the consequences of their use. The authors
and publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
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transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter
invented, including photocopying, microfilming, and recording, or in any information storage or retrieval
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or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-
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Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used
only for identification and explanation without intent to infringe.
ISBN: 978-1-032-39507-4 (hbk)

ISBN: 978-1-032-39690-3 (pbk)
ISBN: 978-1-003-35091-0 (ebk)
DOI: 10.1201/9781003350910
Typeset in Times
by SPi Technologies India Pvt Ltd (Straive)
Contents
Preface��xi
Acknowledgments..................................................................................................... xv
Author.....................................................................................................................xvii
Chapter 1 Overview of Wind Energy Installations and Wind Turbine Technologies..... 1

1.1 Overview of Global Wind Energy Installations�� 1
1.2 Classification of Wind Energy Conversion System�� 6
1.3 Overview of Wind Turbine Technologies�� 6
1.3.1 Fixed Speed Wind Turbines�� 6
1.3.2 Variable Speed Wind Turbines�� 6
1.4 Overview of DFIG and PMSG Wind Turbines�� 7
1.5 DFIG Wind Turbine Modeling�� 9
1.6 PMSG Wind Turbine Modeling�� 11
1.7 Chapter Conclusion�� 14
References�� 14
Chapter 2 DFIG with Different Inverter Schemes............................................... 17

2.1 Chapter Introduction�� 17
2.2 Model System of Study�� 18
2.3 Variable Speed Drive Control�� 19
2.4 DFIG Neutral Point Clamped Multilevel
Converter Topology�� 22
2.5 DFIG Parallel Interleaved Multilevel Converter Topology�� 23
2.6 Simulation Results and Discussions�� 28
2.6.1 SDBR Switching and Position�� 28
2.6.2 2-Level Interleaved Inverter and SDBR�� 30
2.6.3 3-Level Inverter, 2-Level Interleaved Inverter
and SDBR�� 32
2.6.4 Proposed PLL Control Strategy and the Various
Schemes�� 34
References�� 37
Chapter 3 DFIG Performance and Excitation Parameters................................... 39

3.2 Model System of Study�� 39
3.3 DFIG Variable Speed Drive Control�� 40
3.4 Insulated Gate Bipolar Transistor Excitation Parameters�� 42
3.5 Fault Ride Through Requirements for Wind Farms�� 43
3.6 Evaluation of the System Performance�� 43
v
viContents
3.6.1 Effects of the Insulated Gate Bipolar Excitation

Parameters on the DFIG Wind Turbine�� 43
3.7 Proposed PLL Control Strategy and
SDBR Scheme�� 48
References�� 50
Chapter 4 PMSG Performance and Excitation Parameters.................................. 51

4.2 Modeling of the PMSG Wind Turbine�� 53
4.3 Control Strategy for the PMSG Wind Turbine�� 53
4.4 Excitation Parameters of the Insulated Gate Bipolar
Transistors of the PMSG Wind Turbine�� 53
4.5 Evaluation of System Performance�� 57
Parameters on the PMSG without Overvoltage
Protection Scheme (OVPS)�� 57
Parameters on the PMSG Considering
Overvoltage Protection Scheme (OVPS)��59
4.6 Analysis of the DFIG Wind Turbine Considering the
Power Converters Excitation Parameters��61
References�� 63
Chapter 5 DFIG and PMSG Machine Parameters............................................... 67

5.2 Turbine Characteristics and Model of DFIG Wind
Turbine�� 69
5.3 Turbine Characteristics and Model of PMSG Wind
Turbine�� 71
5.4.1 DFIG Wind Turbine Machine Parameters’
Evaluation�� 72
5.4.2 PMSG Wind Turbine Machine Parameters’
Evaluation�� 77
References�� 82
Chapter 6 PMSG in Different Grid Strengths...................................................... 87

6.2 Modeling of the PMSG Wind Turbine�� 88
6.3 The Model System and Control Strategy of the PMSG�� 88
Contents vii
6.4 Mathematical Dynamics of Connecting the SDBR in the

PMSG Wind Turbine�� 89
6.5 Dynamics of Weak Grids and Voltage Source Inverters�� 93
6.6 Results and Discussions�� 94
6.6.1 The Placement and Effective Sizing of the
SDBR in PMSG-Based Wind Turbine�� 94
6.6.2 Improving the Performance of PMSG Wind
Turbine in Weak Grids Considering the Effective
Size of SDBR�� 97
6.6.3 Analysis of the Proposed Scheme Considering
Asymmetrical Faults�� 99
6.6.3.1 Double-Line-to-Grid Fault
(2 LG Fault)�� 99
6.6.3.2 Line-to-Line Grid Fault (LL Fault)�� 100
6.6.3.3 Line-to-Ground Grid Fault
(1 LG Fault)�� 102
6.6.3.4 Comparison of the Various Asymmetric
Fault Conditions�� 103
References�� 104
Chapter 7 DFIG and PMSG in Weak and Strong Grids..................................... 107

7.2 Modeling and Control�� 108
7.2.1 Wind Turbine Characteristics�� 108
7.2.2 DFIG Model and Control�� 108
7.2.3 PMSG Model and Control�� 110
7.3 Mathematical Dynamics of SDBR in DFIG Wind
Turbine�� 111
7.4 Mathematical Dynamics of SDBR in PMSG Wind
Turbine�� 112
7.5 Results and Discussions�� 115
7.5.1 Operation of the DFIG and PMSG Wind
Turbines at Different Grid Strengths�� 115
7.5.2 Improving the Performance of DFIG and
PMSG Wind Turbines in Weak Grids
Considering the Effective Sizing of SDBR�� 118
7.5.3 Improving the Performance of DFIG and
PMSG Wind Turbines in Weak Grids
Considering Overvoltage Protection System
(OVPS)�� 120
7.5.4 Improving the Performance of DFIG and PMSG
Wind Turbines in Weak Grids Considering
SDBR and Overvoltage Protection System�� 120
viiiContents
7.5.5 Improving the Performance of DFIG and PMSG

Wind Turbines in Weak Grids Considering 75
and 50% of the Effective SDBR and Overvoltage
Protection System�� 121
7.5.6 Investigating the Performance of DFIG
and PMSG Wind Turbines in Weak Grids
Considering the Effective SDBR and
Overvoltage Protection System in a
Single-Line-to-Ground Fault�� 123
Chapter 8 DFIG Wind Turbines and Supercapacitor Scheme............................ 127

8.2.1 Wind Turbine Characteristics and DFIG
Control�� 128
8.3 The DFIG Model with Supercapacitor System�� 128
8.3.1 The Dynamics of the Traditional Supercapacitor
System�� 128
8.3.2 The Dynamics of the Supercapacitor System
in DFIG Wind Turbines�� 130
8.4 The Model System of Study and Parameters�� 132
8.5 Simulation Results and Discussions�� 133
8.5.1 Evaluation of the Proposed DFIG Supercapacitor
Scheme�� 133
Scheme and Parallel DFIG Capacitor Scheme�� 136
System during Asymmetrical Faults at Super-
synchronous and Sub-synchronous Speeds�� 139
8.5.4 Performance of the Proposed Scheme under
Zero-Voltage Condition at the Terminal
of the Machine�� 141
Chapter 9 PMSG Wind Turbine with Series and Bridge Fault

Current Limiters................................................................................ 145
9.2 Overview of Fault Current Limiter Topologies�� 145
9.3 Modeling of the PMSG Wind Turbine with
Back-to-Back Converters�� 146
Contents ix
9.4 The Power Converter Control of the PMSG Wind

Turbine�� 148
9.5 The PMSG Model System with SDBR and BFCL�� 149
Chapter 10 PMSG with Capacitive Bridge Fault Current Limiters..................... 161

10.2 The PMSG Model System with the Different Fault
Current Limiters�� 162
10.2.1 PMSG Wind Turbine with SDBR�� 162
10.2.2 PMSG Wind Turbine with BFCL�� 166
10.2.3 PMSG Wind Turbine with the CBFCL�� 167
10.3 Control Strategy of the PMSG Wind Turbine��169
Chapter 11 Comparative Study of DFIG and PMSG with Different Fault

Current Limiters................................................................................ 177
11.3 DFIG and PMSG Model Systems with the Fault
Current Limiters�� 180
11.4 Mathematical Dynamics of SDBR in DFIG and PMSG
wind turbines�� 182
11.4.1 SDBR in DFIG Wind Turbines�� 182
11.4.2 SDBR in PMSG Wind Turbines�� 184
11.5 Dynamics of BFCL and CBFL on DFIG and PMSG
Wind Turbines�� 187
11.5.1 DFIG and PMSG Wind Turbines with BFCL�� 187
11.5.2 DFIG and PMSG Wind Turbines with CBFCL�� 188
11.6.1 Performance of the DFIG and PMSG Wind
Turbines Considering SDBR and BFCL�� 190
11.6.2 Performance of the DFIG Wind Turbine
Considering SDBR, BFCL and CBFCL�� 192
11.6.3 Performance of the PMSG Wind Turbine
Considering SDBR, BFCL and CBFCL�� 194
xContents
Chapter 12 DFIG and PMSG Wind Turbines Life Cycle Cost Analysis............. 201
12.2 Fundamental and Theoretical Background�� 201
12.2.1 Reliability, Availability and Downtime�� 201
12.2.2 Operation and Maintenance Cost, Failure Mode
and Effects Analysis�� 203
12.3 Life Cycle Cost Analysis of the DFIG and PMSG Wind
Turbines�� 203
12.3.1 Production Losses of the DFIG and PMSG
Wind Turbines�� 203
12.3.2 Preventive Maintenance of the DFIG and
PMSG Wind Turbines�� 204
12.3.3 Corrective Maintenance of DFIG and
PMSG Wind Turbines�� 204
12.3.4 Reliability-centered Maintenance��204
12.4 Typical Levelized Cost of Energy for Wind
Farm Projects�� 206
Perspective�� 210
Index�� 213
Preface
With the recent proliferation and penetration of wind farms into existing power
grids, it is paramount to conduct numerous studies to counter grid disturbances
based on operational grid codes. Electric power from wind energy could be extracted
by employing the promising technologies of the Doubly Fed Induction Generator
(DFIG) and Permanent Magnet Synchronous Generators (PMSG) wind turbines. In
order to solve transient stability intricacies posed by the stochastic nature of wind
energy during grid faults, this book would propose different control strategies for
DFIG and PMSG wind turbines. The control strategies would be based on Fault
Current Limiters (FCLs). The Series Dynamic Braking Resistor (SDBR) would be
the first FCL to be investigated in this book. The best location to place the SDBR on
the machine side and grid side converters of both DFIG and PMSG wind turbines
would be investigated, considering different switching strategies. Efforts would also
be made to determine the suitable sizing of the SDBR. The performance of the SDBR
would be investigated at various network strengths in weak and strong grids for both
wind turbines. Both severe three-line-to-ground faults and asymmetrical faults of
line-to-line, double-line-to-ground and one-line-to-ground would be used to test the
robustness and rigidity of the controllers of the wind generators.
In weak grids, the challenges of network stability are a result of wind energy pen-
etration. The Voltage Source Inverter (VSI) based on Pulse Width Modulation (PWM)
is employed widely in interfacing sources with regard to renewable energy and the
grid. The utilization of these inverters would cause stability issues in the power grid.
The studies carried out in the literature show that VSI control could affect the stabil-
ity of power grids. In addition, the stability of grid-connected VSI can be affected by
a weak grid. A grid that has a low Short Circuit Ratio (SCR) is said to be weak. In
other words, a grid is said to be weak if it has an impedance that is high and low
inertia constant. In this book, the performance of both wind turbines would be tested
in weak, normal and strong grids, in addition to the SDBR implementation on both
wind turbines. The recently stipulated grid codes require that wind generators re-
initiate normal power production after grid voltage sag. This book will also present a
comparative performance of two commonly employed variable speed wind turbines
in today’s electricity market: the DFIG and the PMSG wind turbines. The evaluation
of both wind turbines was done for weak, normal and strong grids, considering the
same machine ratings of the wind turbines. Because of the critical situations of the
wind turbines during faulty conditions in the weak grids, an analysis was done con-
sidering the use of effective SDBR for both wind turbines. The grid voltage variable
was employed as the signal for switching the SDBR in both wind turbines during
transient state. Also, an overvoltage protection system was considered for both wind
turbines using the DC chopper in the DC-link excitation circuitry of both wind tur-
bines. Furthermore, a combination of the SDBR and DC chopper was employed in
both wind turbines at weak grid condition in order to improve the performance of the
variable speed wind turbines.
xi
xiiPreface
The performance of other FCLs would be investigated in the DFIG and PMSG
wind turbines in this book. The FCLs are the Bridge FCL and the Capacitive FCL.
Controlling variable speed wind turbines during transient state is challenging. The
use of variable speed wind turbines based on PMSG is on the rise due to some of the
features of the wind turbine. According to the grid code requirements, grid-connected
wind turbine systems should achieve active power control and provide Low Voltage
Ride Through (LVRT) capability. Thus, the primary target of the wind turbine control
system is to keep the turbine connected to the grid during grid disturbances or fail-
ures. In this book, the SDBR and the Bridge Fault Current Limiter (BFCL) were used
to improve the LVRT of PMSG wind turbines. The topology of the PMSG grid side
voltage source converter, with the SDBR and BFCL, was modeled during steady and
transient states. The performance of both schemes on the PMSG was analyzed and
compared during a severe balanced fault scenario. In addition, a scenario with any of
the schemes was also considered. For fair comparison, the PMSG wind turbine was
operating at its rated speed during the low voltage and the same conditions of opera-
tion were used for all the considered scenarios.
This book proposes a supercapacitor strategy for improving the capability of grid-
connected Doubly Fed Induction Generator (DFIG) wind turbines during fault sce-
narios. Supercapacitors are one of the important components in sustainable energy
systems that are commonly used to store energy. In DFIGs, the super-capacitor is
used to compensate for voltage dips and damping oscillations. In this book, a new
topology of supercapacitor system was investigated in a DFIG wind turbine during
transient state. The model system employed was a DFIG connected to the earlier
wind turbine technology of a fixed speed squirrel cage induction generator. Efforts
were made to determine the effective parameters and switching strategies of the
supercapacitor by considering different scenarios, in order to improve the transient
state of the wind generator. The results obtained under severe grid fault were com-
pared considering the different parameters of the resistance, inductance and capaci-
tance of the supercapacitor. The DC-link voltage and grid voltage switching strategies
of the supercapacitor were investigated. Furthermore, the results of the proposed
DFIG supercapacitor were compared with the traditional parallel capacitor scheme
for DFIG system. For fair comparison between the DFIG supercapacitor and parallel
capacitor-based solution, the capacitance value considered was the same to buffer the
transient energy.
This book also investigates the transient performance of the two DFIG and the
PMSG wind turbines. The machine parameters of both wind turbines were varied
considering different scenarios, while keeping other parameters constant, in order to
study the behavior of the wind generators. The study was carried out using the same
operating conditions of rated wind speed, based on the wind turbine characteristics
of both wind turbine technologies. The wind turbines were subjected to a severe
three-phase-to-ground bolted fault, in order to test the robustness of their controllers
during grid fault conditions. Efforts were made to carry out an extensive comparative
study to investigate the machine parameters that have more influence on the stability
of the different wind turbines considered in this study. Effective machine parameter
selection could help solve Fault Ride Through (FRT) problems in a cost-effective
Preface xiii
way for both VSWTs, without considering external circuitry and changing of the
original architecture of the wind turbines.
The system performance carried out in this book was evaluated using Power
System Computer Design and Electromagnetic Transient Including DC (PSCAD/
EMTDC) platform. The same conditions of operation were used in investigating the
various scenarios considered in this study, for effective comparison.
Kenneth E. Okedu,
Visiting Professor, Department of Electrical and
Communication Engineering,
National University of Science and Technology,
Muscat, Sultanate of Oman.
Adjunct Professor, Department of Electrical
and Electronic Engineering,
Nisantasi University,
Istanbul, Turkey.
Acknowledgments
The author of the book would like to thank God Almighty, his wife (Imonina Blessing
Oked-Kenneth), daughter (Imonisa Ambless Okedu-Kenneth), parents (Sir and Mrs.
Simon Williams Okedu) and mentors (Prof. Junji Tamura and Prof. S. M. Muyeen)
for their support, love and care.
xv
Author
Kenneth E. Okedu was a research fellow in
the Department of Electrical and Computer
Engineering, Massachusetts Institute of Technology
(MIT), Boston, USA, in 2013. He obtained his PhD
from the Department of Electrical and Electronic
Engineering, Kitami Institute of Technology, Japan,
in 2012. He received his BSc and MEng in Electrical
and Electronic Engineering from the University of
Port Harcourt, Nigeria, in 2003 and 2007, respec-
tively, where he was retained as a faulty member
from 2005 until the present day. He has also been
a visiting faculty member at the Abu Dhabi National Oil Company (ADNOC)
Petroleum Institute. He was also a visiting faculty member at the Caledonian College
of Engineering, Oman (Glasgow Caledonian University, UK). He is presently a visit-
ing professor in the Department of Electrical and Computer Engineering, National
University of Science and Technology (NUST), Oman, and an adjunct professor
in the Department of Electrical and Electronic Engineering, Nisantasi University,
Turkey. He was recognized as a top 1% peer reviewer in Engineering by Publons in
2018 and 2019 and was the editor’s pick in the Journal of Renewable and Sustainable
Energy in 2018. Dr. Okedu has published several books and journals/transactions in
the field of renewable energy. He is an editor for including Frontiers in Renewable
Energy Research (Smart Grids), Energies (MDPI), International Journal of Smart
Grids, International Journal of Electrical Engineering, Mathematical Problems in
Engineering and Trends in Renewable Energy. His research interests include power
system stability, renewable energy systems, stabilization of wind farms, stabil-
ity analysis of Doubly Fed Induction Generators (DFIGs) and Permanent Magnet
Synchronous Generators (PMSG) variable speed wind turbines, augmentation and
integration of renewable energy into power systems, grid frequency dynamics, wind
energy penetration, FACTS devices and power electronics, renewable energy storage
systems and hydrogen and fuel cells. Dr. Okedu was listed in the global Stanford
University list of the top 2% of the world most academically cited researchers in
the Scopus Worldwide Database. He also won the Outstanding Publication Award
for publishing most Scopus-indexed papers in the year 2021-2022 at the National
University.
xvii
1 Overview of Wind
Energy Installations
and Wind Turbine
Technologies
1.1 OVERVIEW OF GLOBAL WIND ENERGY INSTALLATIONS

The global cumulative wind power capacity is around 743 Gigawatts (GW), as a
result of the recent installation of 93 GW, in 2020 [1]. There was a 59% increment
in the wind onshore market in 2020, accounting for 86.9 GW more installations,
compared to 2019. The two countries taking the lead for onshore wind markets glob-
ally are China and the USA, with new onshore additions and increased market share
ranging from 15 to 76%. Also, in the year 2020, on the regional level, onshore wind
installations in Asia Pacific, North America and Latin America were on the rise. The
total new wind installations in these three regions was 74 GW of new onshore wind
capacity or 76% more than in 2019. As a result of the slow recovery of onshore
installations in Germany in recent times, Europe experienced only 0.6% new onshore
installations, while in Africa and the Middle East only 8.2 GW wind onshore installa-
tions were observed, with little or no considerable increment with the previous year.
Globally, in the offshore market, 6.1 GW was added in 2020, which is one of the
highest wind offshore installations, with China taking the lead in installing half of the
new offshore wind capacity. Europe experienced steady growth, with the Netherlands,
Belgium, the UK, Germany and Portugal, taking leads, accordingly. Besides, in
2020, the United States and South Korea shared the remaining new offshore wind
installations, making the cumulative offshore wind power capacity more than 35 GW
or 4.8% of global cumulative wind power capacity [1]. Though, the global wind
market growth would likely slow down in the near future because of an anticipated
drop in onshore wind installations in China and in the United States as a result of
incentive schemes that would expire [2, 3]. However, the wind market outlook still
remains promising, with an expectation of over 469 GW of new wind onshore and
offshore wind capacity in the next five years (accounting for about 94 GW of yearly
new installations until 2025), considering present and new policies [1, 4].
Also, there was a milestone commitment to carbon neutrality in 2020, with the
European Union, Japan, South Korea, Canada and South Africa each pledging to
achieve net zero by the year 2050. In addition to China’s net zero by 2060 target and
the United States by 2050, the current net zero targets is 2/3 of the global economy,
representing 63% of global emissions [5, 6]. Based on these facts, it is obvious that
the era of fossil fuels is over and quickly taken over by the global energy transition.
DOI: 10.1201/9781003350910-1 1
2 Performance of DFIG and PMSG Wind Turbines
FIGURE 1.1 Global Annual Wind Installations to achieve Net Zero scenario by 2050.
Source: GWEC Market Intelligence; IEA World Energy Outlook (2020), volume in 2022–
2024 and 2026–2029 are estimates.
There is no doubt that the wind industry has demonstrated incredible resilience,
recently. The proliferation of wind farms and renewable energy is required to help
limit global warming to below 2 °C, based on the Paris Agreement. Figure 1.1 shows
the total new wind installations required by International Energy Agency (IEA), to
achieve net zero by 2050 [7]. Figure 1.1 reflects that the annual wind installations
must increase steadily in order to achieve net zero by 2050, by the addition of new
global wind installations in (GW), ranging from 60 GW in 2019 to 280 GW in 2030,
with an increased Compound Annual Growth Rate (CAGR) value of 17% at the end
of 2025 and 12% at the end of 2030 [1, 7].
Figure 1.2 shows that in the year 2020, the new installations in the onshore wind
market got up to 86.9 GW, while the offshore wind market was 6.1 GW, reflecting a
considerable high onshore and offshore wind power installations around the globe.
This was driven basically by China, Asia Pacific, where wind power continues to take
the lead in global wind power developments, increasing its global market region by
8.5% in 2020 [1]. Another driving force for global wind power installation is the
United States, and North America with a global market share of 18.4%, replacing
Europe with 15.9%, as the second-largest regional market for new wind power instal-
lations. Latin America holds the fourth-largest regional market wind power installa-
tions with 5.0% in 2020, followed by Africa and the Middle East with regional wind
power installations of 0.9%, in the same year. Figure 1.3 shows the new wind power
capacity by region in 2020, while Figure 1.4 shows the global top five markets in the
year 2020 for new wind power installations, where China, the United States, Brazil,
Netherlands and Germany are taking the lead with a total of 80.6% of global wind
power installations. Based on the cumulative wind power installations, as of the end
of the year 2020, the top five markets are the same. The markets are China, the
United States, Germany, India and Spain, accounting for 73% of global wind power
installations.
Wind Energy Installations and Wind Turbine Technologies 3
FIGURE 1.2 Global new onshore and offshore wind installations [1, 7].
FIGURE 1.3 New wind power capacity installations by region [1].
The global offshore wind industry recorded over 6 GW of new installation in

2020, despite the effect of COVID-19. China took the lead in global new offshore
wind installations for consecutive three years with over 3 GW of new offshore wind
capacity recorded in 2020, as shown in Figure 1.4. In Europe, steady growth in off-
shore wind power installations was achieved, with the Netherlands taking the lead of
nearly 1.5 GW new offshore wind power, then Belgium with 706 MW, the UK with
483 MW and Germany with 237 MW, respectively. Apart from China and Europe,
FIGURE 1.4 New wind power capacity for the top five markets [1].
FIGURE 1.5 New global offshore wind power installations in 2020 [1, 7].
South Korea with 60 MW and the United States with 12 MW were the two other
countries that recorded high new offshore wind installations in the year 2020, given
in Figure 1.5 [7, 8]. More so, in the same year, in Portugal two new floating offshore
wind turbines, were commissioned, making a total of 16.8 MW. A comparison of
both onshore and offshore wind power capacity changes between 2019 and 2020 is
given in Figure 1.6, where all regions increased installations except Europe, Africa
and the Middle East. Figure 1.7 shows the major wind turbine manufacturers, where
Vestas and General Electric are taking the lead.
FIGURE 1.6 Changes in new wind power installations 2019–2020 [1, 2].
FIGURE 1.7 Wind turbine manufacturers.

1.2 CLASSIFICATION OF WIND ENERGY CONVERSION SYSTEM

Wind energy conversion systems can be classified as follows.
(1) They are two broad classifications based on the axis of the machine:
(a) Horizontal Axis Machines: The axis of rotation is horizontal and the
aero turbine plane is vertical, facing the wind.
(b) Vertical Axis Machines: The axis of rotation is vertical. The sails or
blades may also be vertical.
(2) They may be classified according to size:
(a) Small-scale machines (up to 2 kW): Used in low-power applications.
(b) Medium-size machines (2–100 kW)
(c) Large-scale or large-size machines (100 kW and up): Used to generate
power for distribution in central power grids.
(3) Classification as per type of output power
(a) DC output: DC generators, Alternator rectifiers
(b) AC output: Variable frequency variable or constant voltage, constant
frequency variable or constant voltage
1.3 OVERVIEW OF WIND TURBINE TECHNOLOGIES

Wind turbine technologies could be classified into two groups as follows: Fixed
Speed Wind Turbines (FSWTs) and Variable Speed Wind Turbines (VSWTs). There
has been tremendous improvement in wind energy technology over the years, as a
result of the fast innovations and developments in power electronics [9, 10]. This has
resulted in the replacement of FSWTs with VSWTs. The following are some of the
features of both wind turbine technologies.
1.3.1 Fixed Speed Wind Turbines

• This class of wind turbine has a limited range of power capture because it
operates using fixed speed.
• This class of wind turbine lacks voltage and frequency control capability.
• This class of wind turbine is rugged in construction, has low running cost, is
maintenance-free, with operational simplicity and possesses superior brush-
less features.
• This class of wind turbine needs large reactive power compensation during
the transient state, in order to recover the air gap flux.
• This class of wind turbine system is expensive because of the installa-
tion of external reactive power compensation devices, such as Flexible AC
Transmission Systems (FACTS) that could be Static Synchronous Compensators
(STATCOM), Energy Capacitor Systems (ECS) or Superconducting Magnetic
Energy Storage System (SMES), to provide reactive power.
1.3.2 Variable Speed Wind Turbines

• This class of wind turbine has high energy conversion efficiency, during low
and high winds because it has variable speed operation.
DFIG
IG
(a) (b)
Filter
PMSG
(c)
FIGURE 1.8 Wind turbine technologies. (a) Fixed speed induction generator wind turbine,
(b) doubly fed induction generator variable speed wind turbine and (c) permanent magnet
synchronous generator variable speed wind turbine.
• This class of wind turbine has less acoustic noise and mechanical stress.
• The class of wind turbine has better power quality in power grids without
support from external reactive power compensators like the FACTS devices.
• This class of wind turbine employs power converters for secondary exci-
tation, between 20 and 30% for a DFIG system and 100% for a PMSG
system.
• This class of wind turbine has a lower cost of operation because it can gen-
erate power to the grid and the same time help in providing reactive power
support to achieve stability of the network.
FSWTs are based on Squirrel Cage Induction Generator (SCIG), while VSWTs
are based on Doubly Fed Induction Generator or Permanent Magnet Synchronous
Generator (PMSG), as shown in Figure 1.8 (a–c) for FSWTs and VSWTs, respec-
tively. A brief distinction between the three types of wind turbine-driven generators
is given below.
1.4 OVERVIEW OF DFIG AND PMSG WIND TURBINES

Variable speed turbines are becoming the norm for new wind farm installations,
because of high energy capture efficiency, coupled with reduced drive train stresses
[11]. The PMSG VSWT also known as the direct-drive synchronous generator with
permanent magnet excitation and the DFIG VSWT, with doubly fed back-to-back
power converter type technologies, have become the two generator alternatives. The
former has the disadvantage of cost mainly due to a fully rated power converter
of 100% for energy capture. Although in the latter, a gearbox is needed, the DFIG
requires a converter of only 20–30% of the generator’s rating for an operating speed
range of 0.7–1.3 per unit (p.u), resulting in a lower cost. This book would be focus-
ing on the DFIG and PMSG VSWTs, since modern wind farms are built using both
wind turbines.
Although the DFIG is not as rugged and robust as the squirrel-cage wind tur-
bine type, its brushes have little wear and sparking when compared to DC machines
and it is the only acceptable option for alternative energy conversion in the mega-
watts power range. With the help of modern power electronic devices, it is possible
to recover the slip power dissipated in resistances [12]. The DFIG wind turbine
uses a back-to-back power inverter system connected between the rotor side and
the grid side of the machine, while the stator is directly connected to the grid. The
DFIG can effectively operate at a wide range of speeds, based on the available
wind speed and other specific operational requirements. Thus, it allows for better
capture of wind energy [13, 14], and dynamic slip and pitch control may contribute
to rebuilding the voltage at the wind turbine terminals and, at the same time, main-
taining the power system stability after clearance of an external short-circuit fault
[15]. Besides, DFIG wind turbine has shown better behavior regarding system
stability during short-circuit faults in comparison to SCIG, because of its ability to
decouple the control of active and reactive power output. The superior dynamic
performance of the DFIG is achieved from the frequency or power converters
which typically operate with sampling and switching frequencies of above 2 kHz
[16]. At lower voltages down to 0%, the IGBTs (Insulated Gate Bipolar Transistors)
of the DFIG are switched off and the system remains in standby mode [17–21]. If
the voltages are above a certain cutoff or threshold value during grid disturbances,
the DFIG wind turbine system is very quickly synchronized and is back in opera-
tion again.
The VSWT PMSG is connected through a back-to-back converter to the grid. The
PMSG wind turbine is tied via the Grid Side Converter (GSC), control systems and
Machine Side Converter (MSC) to the power network. This provides maximum flex-
ibility, enabling full real and reactive power control and fault ride through capability
during voltage dips, as compared to the VSWT DFIG technology. However, the use
of this wind turbine technology is limited when compared to the DFIG technology
due to high cost. Compared to the widely used DFIG wind turbine, the PMSG-based
VSWT has a more feasible technology that is promising in wind generation because
it is self-excited, hence, is possible to operate with higher efficiency and power fac-
tor. Besides, there is no gearbox system because of its low rotational speed.
Therefore, no careful and regular maintenance is required in this wind turbine topol-
ogy, unlike the DFIG-based wind turbine [22]. In addition, the converters have room
for flexible control of active and reactive dissipation of power in normal and tran-
sient states [23, 24]. However, some drawbacks of the PMSG wind turbine are com-
plex construction and controller control strategy, compared to the traditional FSWT
based on SCIGs.
1.5 DFIG WIND TURBINE MODELING

A wind turbine is an electromechanical energy conversion device that captures kinetic
energy from the wind and turns it into electrical energy. The primary components of
a wind turbine for modeling purposes consist of the turbine rotor or prime mover,
a shaft and a gearbox unit (a speed changer) [25]. The dynamics interaction involv-
ing forces from the wind and the response of wind turbine determines the amount
of kinetic energy that can be extracted. The aerodynamic torque and the mechanical
power of a wind turbine are given as follows [26–28].
R 3
TM V w 2 Ct NM (1.1)
2
R 2
PM V w3 Cp W (1.2)
2
where ρ is the air density, R is the radius of the turbine, Vw is the wind speed, Cp ,
is the power coefficient given by

Cp , 0.5 0.02 2 5.6 e 0.17 (1.3)
The relationship between C t and C p is
Cp
Ct (1.4)

rR
(1.5)
Vw
R 3600
In (1.3), and in (1.5), λ is the tip speed ratio.
1609
The wind turbine characteristics [29] for both IGs and DFIGs are shown in Figures
1.9 and 1.10, respectively. In Figure 1.10, the power capture characteristics of the
0.5
0.4
β= 0
Power coefficient
0.3
β= 8
0.2
Cp
β = 16
0.1 β = 24
0.0
0 2 4 6 8 10 12 14
Tip Speed Ratioλ
FIGURE 1.9 CP- λ curves for different pitch angles (for FSWT).
Pmax
1.4
Turbine Input Power [pu]

1.2 Pref
1.0
0.8
13m/s
0.6
12m/s
0.4
11m/s
0.2 10m/s
9m/s
6m/s 7m/s 8m/s
0.0
0.4 0.6 0.8 1.0 1.2 1.4 1.6
Turbine Speed [pu]
FIGURE 1.10 Turbine characteristic with maximum power point tracking (for DFIG
VSWT).
Section A
Pref 2 0.2147V w 1.668
1
1
0 +
Pref 1 0.1571V w 1.035 Pref
CTRL + 0
+
Comparator
Vw 11 1
Vw 11 0
+
Vw ω wt_opt 0.0775Vw 5
Section B
-
.
ω wt Ti 0.1 + 0
-
+ +
0.7 Section C
-100
5
FIGURE 1.11 Control block to determine active power reference Pref.
turbine with respect to the rotor speed are shown. The dotted lines show the locus of
the maximum power point of the turbine, which is used to determine the reference of
active power output Pref . Equations (1.6) and (1.7) are used to calculate the reference
of the active power output Pref , as shown in Section A of Figure 1.11. The optimum
rotor speed ω ropt is given in Equation (1.8). The operation range of the rotor of DFIG
is chosen from 0.7 to 1.3 pu, as shown in the turbine characteristics in Figure 1.10.
Pref 1 0.1571Vw 1.035 pu (1.6)
Pref 2 0.2147Vw 1.668 pu (1.7)
opt 0.0775Vw pu (1.8)
The power extracted from the wind can be limited by pitching the rotor blades.
The angle control is usually done with a Proportional Integral (PI) controller in such
d
G dt
ω wt PI
1 ST

d
dt
100

KP = 350 G = 1.0 Max. 10[o/s]
Ti = 0.05 T = 5.0 Min. 10[o/s]
Upper limit 100 Upper limit 1.0
Lower limit 1.2 Lower limit 0.0 1.3
FIGURE 1.12 Pitch controller for DFIG VSWT.
d
+ G dt
P-IG PI
- 1 ST
1.0 KP = 100 Upper limit 1000 G = 1.0 Max. 10[o/s]
Ti = 0.01 Lower limit 0.0 T = 5.0 Min. 10[o/s]
FIGURE 1.13 Pitch controller for SCIG FSWT.
a way that the pitch controller shown in Figure 1.12 controls the angle when the rotor
speed exceeds 1.3 pu for the case of DFIG that operates in variable speed mode.
Figure 1.13 shows the pitch controller for the fixed speed Wind Turbine Generator
System (WTGS). In order to get a realistic response in the pitch angle control system,
the servomechanism accounts for a servo time constant, Tservo and a limitation of both
the pitch angle and its rate of change, as shown in Figures 1.12 and 1.13, respectively.
The rate of change limitation is very important during grid fault, because it decides
how fast the aerodynamic power can be reduced in order to prevent over-speeding
during fault [30, 31]. Considering the realistic scenario for a heavy mechanical sys-
tem, the rate limiter must be incorporated to simulate the pitch controller. Therefore
the pitch rate limiter of ±10 deg./sec. is used for both pitch controllers in this text.
1.6 PMSG WIND TURBINE MODELING

The PMSG turbine is made up of a generator blade, system controller, components of
power electronics and transformer [32]. The wind turbine is tied to the power grid via
its back-to-back full-power converters that are capable of converting wind into elec-
trical energy. To realize wind energy maximum power tracking [33, 34], the motor
speed or torque of the wind generator is controlled by the MSC. The stabilization of
the voltage of the DC-link and regulation of the power factor and quality of the wind
generator is done basically by the GSC.
The mechanical power extracted by the wind generator can be expressed as [35]:
1
Pw R 2Vw3Cp , (1.9)
2
From Equation (1.9), Pw is the wind power that is captured, expressed in (W), the
air density is ρ , expressed in (kg / m 3 ), the radius R is expressed in (m) and the wind
speed Vw is expressed in (m/s). The wind generator’s power coefficient is Cp and is

related to the ratio of the tip speed (λ ) and angle of the pitch (β ), respectively, as
expressed in Equation (1.91) [36].
c5
c
Cp , c1 2 c3 c4 e i c6 (1.10)

i
where
1 1 0.035
3 (1.11)
i 0.08 1
In Equation (1.10), c1 to c6 are the characteristic coefficients of the wind turbine.

In the PMSG wind turbine, the Maximum Power Point Tracking (MPPT) is based on
the rotor speed and the maximum power could be obtained by [37]:
3
1 R
PMPPT R 2 r cpopt (1.12)
2 opt
where λopt is the optimal value of λ , cpopt is the optimal power coefficient and ωr is
the rotor speed of the wind generator. The wind generator characteristics relating to
the turbine output power and the rotor speed for varying wind speeds are shown in
Figure 1.14. The maximum obtainable power output is 1.0 pu at 12 m/s occurring at
FIGURE 1.14 Turbine characteristic with maximum power point tracking (for PMSG
VSWT).
1.0 pu rotational speed. It should be noted that the reference power Pref of the wind
turbine is limited to the wind generator rated power.
The rotating frame based on d-q reference for the dynamic model in the PMSG
wind turbine is expressed as [38]:
d sd
Vsd Rs I sd e sq (1.13)
dt
d sq
Vsq Rs I sq e sd (1.14)
dt
From Equations (1.13) and (1.14),
sd Lsd Lmd I sd m (1.15)
sq Lsq Lmq I sq (1.16)
where Vsd and Vsq are the voltages of the stator circuit, Rs is the winding resistance of
the stator, I sd and I sq are the currents in the stator d and q reference frames, ωe is the
rotational speed of the wind generator, csd and csq are the flux linkages of the stator
circuit, Lsd and Lsq are the stator wind leakage inductances, Lmd and Lmq are the mag-
netizing inductances and cm is the linkage flux of the machine’s permanent magnet.
Putting Equations (1.15), (1.16) into (1.13), (1.14), the PMSG differential equa-
tions could be obtained as:
dI sd
Ld Vsd Rs I sd e Lq I sq (1.17)
dt
dI sq
Lq Vsq Rs I sq e Ld I sd e m (1.18)
dt
With
Ld Lsd Lmd (1.19)
Lq Lsq Lmq (1.20)
The PMSG active and reactive powers are given as:
Ps Vsd I sd Vsq I sq (1.21)
Qs Vsq I sd Vsd I sq (1.22)
The wind generator electrical torque (Te ) with number of pole pairs is given as:

Te 0.5 p m I sq Ld Lq I sd I sq (1.23)
1.7 CHAPTER CONCLUSION
Recently, the penetration of wind energy into power grids is on the rise globally.
This is evident from the wind energy installation statistics presented in this chapter,
considering the various regions in the world and the top markets for wind energy.
China and Asia Pacific continue to take the lead in global wind power developments,
increasing its global market region by 8.5% in recent years. The USA and North
America have a global market share of 18.4% and have recently replaced Europe
with 15.9% as the second-largest regional market for new wind power installations.
Latin America with 5.0% and Africa and the Middle East region with 0.9% hold the
fourth and fifth positions in global wind installations, respectively. China, the USA,
Brazil, the Netherlands and Germany have 80.6% of global wind power installa-
tions. The recent top five countries in wind power installations are China, the USA,
Germany, India and Spain, possessing 73% of global wind power installations.
The VSWT technology is superior to the fixed speed with turbine technology in
wind energy conversion system due to their high wind energy capture efficiency. The
DFIG and the PMSG are the two main types of VSWTs employed in modern wind
farms. This chapter presented the basic modeling of both wind turbines considering
their MPPT characteristics.
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2 DFIG with Different
Inverter Schemes
2.1 CHAPTER INTRODUCTION
In wind energy applications, the Doubly Fed Induction Generator (DFIG) has one
main merit of utilizing only 20–30% of the wind generator rating for the power
converters linking the rotor side and the grid side [1, 2]. Lately, the use of Insulated
Gate Bipolar Transistors (IGBTs) is on the rise for high-power applications among
semiconductor devices. This is because the current capability of the IGBT switches
can be increased by configuring them in parallel connection [3]. Because the IGBT
switches have high power and are required to be subjected to high voltage and cur-
rent, their transient operations during periods of several microseconds are vital [4, 5].
Therefore, the ability for variable speed wind turbine IGBT based on withstanding
abnormal conditions is strictly paramount to achieve the lifetime specifications [6,
7] of the wind turbine and at the same time to fulfill the requirement of grid codes.
As the use of multilevel converters is becoming popular in wind energy conver-
sion systems, because of their robustness during transient conditions, this chapter
tends to improve the performance of the six-step 2-level IGBT inverter by proposing
a coordinated hybrid control of a new Phase Lock Loop (PLL) configuration and a
Series Dynamic Braking Resistor (SDBR). The high voltage usually experienced
during grid disturbances is shared by the small inserted resistance because of the
series connection strategy of the braking resistor employed in this study. Thus, the
loss of the converter control system is not experienced in this topology due to the
effects of induced overvoltage. In addition, the series-connected braking resistor
strategy significantly reduces the very high current in the rotor circuitry of the wind
generator to lower values. Consequently, the overvoltage of the DC-link that was
supposed to be dangerous to the power converters of the wind generator is avoided
because of the low DC-link capacitor charging current [8–10]. Although many stud-
ies in the literature considered the use of fault current limiter for enhancing power
quality and limiting fault current of DFIG wind turbines in wind farms [11–13], in
this chapter, the preferred position of the braking resistor in the wind generator sys-
tem was analyzed considering different switching signals. The optimal braking resis-
tor position and switching strategy were used for further analysis of the proposed
wind generator schemes used in this study. In addition, a comparative study using the
proposed scheme for the 2-level IGBT inverter was carried out with the schemes hav-
ing parallel interleaved IGBT inverter and 3-level IGBT inverter. Simulations were
run in PSCAD/EMTDC [14]. The proposed hybrid scheme could help to increase the
current capability and post-fault recovery of the wind turbine. In addition, the space
vector modulation of the inverter schemes resulted in maximum value change in
Common Mode (CM) voltage, using the proposed hybrid control strategy of the PLL
DOI: 10.1201/9781003350910-2 17
and SDBR scheme [15]. Consequently, there is improved switched output voltage of
the converter leg of the voltage source converter. The results show that the hybrid
scheme in the various inverter topologies considered in the study can enhance the
performance of the wind generator variables during severe three-phase-to-grid fault.
2.2 MODEL SYSTEM OF STUDY

The model system of study is shown in Figure 2.1. The DFIG wind turbine model-
ing can be referred to in Chapter 1 of this book. In Figure 2.1, the rotor side (A),
the grid side (B) and the stator side (C) of the wind generator show the insertion of
the proposed braking resistor scheme in a series connection, respectively. A three-
phase-to-ground fault was used to investigate the optimal position of the braking
resistor considering the model system of study in Figure 2.1, which is connected to
an infinite bus. Figure 2.2 shows the various control signals, DC-link voltage, current
of the rotor for the wind generator and terminal voltage of the grid, along with the
conditions of operation of the braking resistor. The optimal position and switching
strategy of the braking resistor in the wind generator were used for further analysis
of the inverter schemes proposed in this chapter.
Wind
DFIG
C
SDBR
0.1+j0.6
A B
Infinite bus
SDBR SDBR
j0.2
β RSC DC GSC 0.1+j0.6
Fault
Pitch Angle
Control
DFIG-Side Grid-Side
Control Control
FIGURE 2.1 Model system.
Edc <1.5Edc or Ir <2Ir or Vg > 0.9: 1 Normal condition

Edc >1.5Edc or Ir >2Ir or Vg < 0.9: 0 Fault condition
Bypass switch Control
Resistor
FIGURE 2.2 SDBR control strategy.

DFIG with Different Inverter Schemes 19
2.3 VARIABLE SPEED DRIVE CONTROL

Figure 2.3 shows the control strategy of the DFIG-based variable speed wind turbine.
The cost of the crowbar protection scheme used in a conventional DFIG system is
more than the other schemes like the braking resistor or DC chopper. During grid
fault, the crowbar makes the variable speed wind generator based on DFIG act like
a fixed speed wind generator. This is done by disconnecting the wind generator’s
rotor side converter. In this chapter, the DC-link (chopper) scheme is the alternative
employed in place of the traditional crowbar switch, as shown in Figure 2.3. The
coordinated controls of the active and reactive power of the DFIG system via abc
to dq and dq to abc conversion to generate pulses for the switching of the PWM are
shown in Figure 2.3.
The new PLL control strategy used in this study has a delay element incorporated
to improve the performance of the DFIG system. The PLL scheme is designed based
Gearbox
Wind Turbine
DFIG Grid
𝜔 Vg
Va,b,c
Pitch Control
Ig(a,b,c)
Variable Voltage DC-link
and Frequency
Ir(a,b,c) RSC GSC Transformer
- >
Fixed Voltage
and Frequency
<
Carrier signal + DC-link
PMPPT Carrier signal
+ *
Ira,b,c Voltage Source Converters
- PI
>
PDFIG <
abc 𝜃 + PLL
PI dq -
+ PI
𝜔 න𝑑 𝑡
+ -
𝜋Τ
1.0pu
- + 𝜃 abc + -
- *
Vg >0.9: 1 Normal operation dq Iga,b,c
QDFIG >
< Vg <0.9: 0 Fault condition +
PI PI
Qgsc
Vg 0.9pu -
*
- Qref =0
+ Edc (pu)
*
Edc =1.0pu
FIGURE 2.3 DFIG control strategy.

FIGURE 2.4 Conventional PLL scheme.
on a frequency of 50 Hz and is rated line-to-line voltage of 0.69 kV. Unlike the con-
ventional PLL scheme, the proposed PLL scheme in this chapter integrates the Sine
and Cosine function angles for the three phases via a multiplier before the insertion
of a delay element, in order to boost its synchronizing strength with the grid for better
performance during transient state.
Figure 2.4 shows the conventional three-phase PLL scheme, which is basically an
error signal feedback system based on the principle of a synchronous rotating frame,
with low pass filters and voltage-controlled oscillator. The working principle is based
on the conversion of the measured voltage of a three-phase system to d-q component
via conversion coordinate and set DC voltage reference of q-axis vref. Figure 2.5
shows the vector partition diagram for the PLL. From Figure 2.5, the d-axis compo-
nent is fully co-phased with the vector voltage when the q-axis component is zero,
despite the values of the voltages on the various q-axes. The Proportional Integral
(PI) controller in Figure 2.4 helps in obtaining the frequency of the system. With the
grid voltage having only positive sequence fundamental components, the d-q steady
value coordinate is DC current and its phase and frequency can be locked by
q
U
U d
Ud
Uq
FIGURE 2.5 PLL vector partition.

DFIG with Different Inverter Schemes 21
controlling the q-axis component to zero. Frequency detection when the grid voltage
is balanced is achieved for the conventional three-phase PLL based on tracking the
grid voltage positive sequence fundamental components, as the inner of the PLL is
closed loop controller. However, during the transient state, there will be a sudden
change giving rise to instantaneous negative sequence and zero sequence fundamen-
tal components, which leads to the oscillation of the PLL output angle.
In the course of the grid voltage being unbalanced, there exist positive, negative
and zero sequence fundamental components. For a typical three-phase system with-
out a neutral point, the zero sequence is not usually considered. Thus, the grid volt-
age can be expressed as [16]:
cos t

cos t 0

2 2
vabc vabc vabc V cos t V cos t 0 (2.1)
3 3

cos t 2 2
cos t
0
3 3
From Equation (2.1), V+, V− gives the voltage amplitude separately for the positive
and negative sequences, respectively. 0 gives the relative phase angle of the initial
voltage negative sequence. The voltage of the output is achieved after a 3/2 conver-
sion in αβ static coordinates and expressed as [17, 18]:
v
v T vabc (2.2)
v
1 1
1
2 2 2
T (2.3)
3 3 3
0
2 2
In the static αβ static coordinates of the grid voltage, the fundamental positive and
negative sequence component is given as:
cos wt
cos( wt o

v v v V V
(2.4)
sin wt sin(wt o
After d-q transformation to the synchronous coordinate system, the following

equation is obtained:

vdq Tdq v Tdq v

Tdq v

(2.5)
cos sin
Tdq (2.6)
sin cos
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230. Pascual de Gayangos. Cartas y Relaciones de Hernán Cortés al
Emperador Carlos V. París. 1866. Pág. 196.
231. Op. cit., pág. 166.
232. Op. cit. pág. 137.
233. Op. cit. pág. 255.
234. A. H. Keane. Central and South America. 1878. Sayce. Science of
Language. 1886. A. H. Keane. Indians Americans. 1903.
235. P. Sagot. Vocabulaire Francais—Arrouge. Paris. 1882.
236. Hernhutes de Zittau.—Vocabulaire Arrouge—Allemand. Paris. 1882.
237. Th. Schulz. Grammaire Arrouge. París. 1882.
238. R. Breton. Op. cit.
239. Raymond Breton. Op. cit.
240. Voyage du Pere Labat aux isles de l’Amerique. La Haye. 1724. En el t. 2°
p. 123 dice: “Ils reconnoissent du moins confusement deux principes, l’un bon et
l’autre manvais.” J. Ballet. Les Caraibes. Congrés intern. des Americanistes. Nancy.
1875 t. 1° pág. 433.
241. Revue des Deux Mondes. S. Champlain.—Voyages et Descouvertry. París.
1620.
242. La ou francesa tiene sonido de ú española.
243. Ihering.—Prehist. de los indo-europeos. Madrid. 1896.
244. Schleicher.—Jahrbücher fur Nationalokonomie.
245. Diario de bitácora de Cristóbal Colón del primer viaje.—Anotación del
Domingo 16 de Diciembre de 1492.
246. Edición de Viena, tipografía imperial y real de la Corte.—1868.
247. Pedro Mártir de Anglería. Década II. cap. IV.
248. Véanse estas palabras indo-antillanas en el Vocabulario.
249. Guayama—Gua-yara-ma: gua, este; yara, sitio; y ma, grande. Este sitio
grande.
250. Canuy—Canua-ní: canua, canoa; ní, agua. Es decir, canoa y agua. Hoy
sintetizamos la idea en el español pasaje.
251. Guanajibo—Gua-sabana-ní-abo: gua, he aquí, na, por sabana, llano; ji,
por ní, agua; bo por abo, lugar. He aquí un lugar llano con agua. Como si di
dijéramos: Buen sitio de labranza. Una de las ideas principales del boriqueño era
buscar en la isla buenos sitios donde sembrar sus yucubías y sus ajes y batatas. Así
como elegir los lugares de agua abundante para sus baños, después del juego de
pelota.
252. Según el moderno viajero doctor Crévaux, también los tarumas, trios,
rucuyús, apalais y carijonas llaman al agua tuna. Y según Segarra y Juliá (Costa
Rica. 1907. pág. 585) los indios guatusos la llaman tí y las otras tribus indígenas de
Costa Rica, dí.
253. Arístides Rojas.—El padre nuestro en lenguas venezolanas.—Caracas.
1878.
254. Fray Matías Ruíz Blanco.—Conversión en Piritú (Colombia) de los indios
cumanagotos y palenques. Nueva edición matritense. 1892, pág. 162.
255. R. Breton.—Ob. cit., p. 35.
256. Lucien Adam. Matériaux pour servir à l’établissement d’une grammaire
comparée des dialectes de la famille Caribe. París. 1893, pág. 32.
257. Cayetano Coll y Toste. Rep. hist. de Puerto Rico. 1896. pág. 17.
258. Arístides Rojas.—Ob. cit.
259. Pbro. Rafael Celedón.—Gramática de la lengua koggaba. París. 1886.
260. Don Pedro Tomás de Córdova guarda silencio sobre este particular en sus
Memorias, porque era un empleado muy adicto á la Monarquía absoluta. Carlos
Espinosa fué un general, gobernador de Cádiz y capitán general de Andalucía, muy
adicto á los principios liberales. Se distinguió en la guerra de la Independencia y
mandó el ejército constitucional de Navarra en 1822. Murió de avanzada edad, en
1850. Juan Díaz Porlier, fué uno de los mártires de la libertad española. Nació en
Cartajena de Indias, en 1775. Se halló de guardia marina en Trafalgar. Fué mariscal
de campo en la guerra de la Independencia. Combatió el despotismo de Fernando
VII de abolir la Constitución. Fué preso. Se sublevó en Septiembre de 1815. Le
nombraron presidente de la junta revolucionaria de Galicia. Cayó prisionero y fué
ahorcado en la Coruña el 3 de Octubre del mismo año. La tremenda reacción
borbónica hizo desaparecer de la topografía de Puerto Rico los nombres de estos
tres ilustres generales españoles, salvándose Lacy en un anagrama y quedando el
nombre del general Espinosa en un barrio de Vega Alta. Del bravo Porlier no
queda recuerdo alguno.
TRANSCRIBER’S NOTESNOTAS
DEL TRANSCRIPTOR
Página Cambiado de Cambiado a
vi de Tierra Firme á de Tierra Firme á una
una franca paz, franca paz, como al
como al principio, principio, porque las
por que las expediciones
expediciones
vi Iñigo Abbad.— Íñigo Abbad.—
Facultades mentales Facultades mentales del
del aborigen.—La aborigen.—La vida en
vida en tribu ó tribu ó
7 una línea casi una línea casi regular,
regular, que sirva de que sirva de limite en
límite en este este
11 por que el terreno porque el terreno que
que corresponde á corresponde á este
este período período apareció muy
apareció muy netamente y con gran
netamente y con extensión en
gran extensión en Devonshire, Inglaterra.
Devonsgire, L. Figuier. La terre
Inglaterra. L. avant le déluge. París
Figuier. La terre
avant le déluge.
Paris
13 L. Figuier. La terre L. Figuier. La terre
avant le deluge. avant le déluge. París.
París. 1863 1863
16 contemporánea de contemporánea de
piedras se produce piedras se produce
ámpliamente ampliamente
16 estos acinamientos estos hacinamientos
calizos calizos
18 induce á creer, induce á creer, según la
según la ley ley geognóstica de Elie
geognósica de Elie de
de
18 J. B. Elie de J. B. Elie de Beaumont.
Beaumont. Notice Notice sur les systèmes
sur le systeme des de montagnes. París.
montagnes. París. 1852. Esta ley
1852. Esta ley
19 Dou Julio L. Don Julio L.
Vizcarrondo (Viaje á Vizcarrondo (Viaje á la
la isla de Puerto isla de Puerto Rico, el
Rico, el año de 1797, año de 1797, por Ledru
por Ledru y y
23 zonas. Los terrenos zonas. Los terrenos
compredidos en la comprendidos en la
zona N. y en zona N. y en
24 contíguos á la contiguos á la carretera
carretera central, en central, en los cortes de
los cortes de
25 granito granito desprendidos,
desprendidos, han han sido producidos por
sido producido por la dislocación
la dislocación
25 antidiluvianos por antediluvianos por estos
estos territorios. En territorios. En un
un
36 Hallado en Bayaney, Hallada en Bayaney,
Hatillo. Hatillo.
39 descubridor, en descubridor, en estado
estado nómade, á nómada, á semejanza
semejanza de las de las
51 americano. En un americano. En un
principio, creimos principio, creímos que
que
53 El triunfo de los El triunfo de los
invasores hnbiera invasores hubiera sido
sido seguro en seguro en
56 Monquin-Tandon Moquin-Tandon siguió
siguió á Cuvier. á Cuvier. Dumeril
Dumeriel
56 Malte Brun clasificó Malte-Brun clasificó al
al hombre en diez y hombre en diez y seis
seis
56 Cuvier. Reyne Cuvier. Règne animal,
animal, ed. 2ª t. 1º, ed. 2ª t. 1º, pág. 84.
pág. 84. París. 1829 París. 1829
56 Malte Brum. Malte-Brun.
Geographie, etc. Géographie, etc. París.
París. 1803–7 1803–7
56 Bory de Saint- Bory de Saint-Vicent.
Vicent. L’Homme. L’Homme. Essai
Essai zoologique sur zoologique sur le genre
le genre humaine. humaine. París
Paris
61 clasiflcación de clasificación de Ulloa,
Ulloa, que visto un que visto un indio
indio estaban estaban
62 llarmarse siboneyes, llamarse siboneyes,
haytianos, haytianos,
jamaiquinos y jamaiquinosboriqueños,
boriqueños, por que por que en el trascurso
en el trascurso del del tiempo
tiempo
63 caracteres especia es caracteres especiales
sostenidos y se sostenidos y se origina
origina la subraza la subraza
63 á ser dos razas á ser dos razas
fundament les y una fundamentales y una
tercera por tercera por
63 Todo e to es la Todo esto es la
influencia del medio influencia del medio
ambiente, con ambiente, con
65 en su Systemes of en su Systems of
Consanguinity and Consanguinity and
Affinity of the Affinity of the Human
Human Family, Family, sostiene la
sostiene la unidad unidad
70 en combatir y hacer en combatir y hacer
frente. Los indios frente. Los indios serían
seraín unos unos
70 españolas, seis de á españoles, seis de á
caballo y cien de á caballo y cien de á pie.
pie. La caballería La caballería
71 de igual modo que lo de igual modo que lo
hacían los indo hacían los indo-
antillanos antillanos
81 de S. A. Los Caribes de S. M. Los Caribes se
se los comen é los comen é hácenles
hácenles
82 rescibir á su recibir á su
conversasión á los conversasión á los
chrystianos, ni á los chrystianos, ni á los
predicadores predicadores
82 del dicho Golfo, está del dicho Golfo, está
otra provincia, que otra provincia, que se
se dise de los Oleros, dice de los Oleros, los
los quales quales
83 de más arriba hasta de más arriba hasta lo
lo demás abajo, que de más abajo, que no
no son declaradas son declaradas por de
por de
83 las dichas tierras é las dichas tierras é
provincias, guerra, provincias, guerra, ni
ni fuerza, ni fuerza, ni violencias, ni
violencias, ni extorsiones;
extorciones,
84 un sin número de un sinnúmero de
pueblos indígenas pueblos indígenas con
con distintas distintas
84 Archivos de Indias. Archivo de Indias. Doc.
Doc. inéd inéd
88 sobresalir. La sobresalir. La quijada es
quijada es antropológicamente
antropológimente ortognática [vertical]
ortognática mesognática [media] ó
[vertical]
mesognática [media]
ó
94 ni mucho menos; ni mucho menos; pero,
pero, si expresiones sí expresiones de
de aprecio aprecio
97 de Uravoán estaba de Urayoán estaba
junto al Guaorabo, junto al Guaorabo, en
en Yagüeca Yagüeca
97 la de Avmamón en la de Aymamón en las
las riberas del riberas del Coalibina
Coalibina
104 En ellos tendrián En ellos tendrían
también los también los caciques,
caciques, bohiques bohiques
106 magiiey, se le ha magüey, se le ha
agregado el agregado el retumbante
retumbante cuero cuero
107 Estrañará á alguno, Estrañará á algunos,
que hayamos que hayamos concedido
concedido
113 ofrendas. La ofrendas. La explicación
explicación de ésto de esto es bien sencilla:
es bien sencilla: el el
113 V. Durny. Ob. cit. V. Duruy. Ob. cit.
117 (Fitolatría) á la (Fitolatría) á la
bienhechora y bienhechora y
mistoriosa planta misteriosa planta
125 Pedro Mártir de Pedro Mártir de
Anglería. 1ª decada, Anglería. 1ª década, lib.
lib. IX., cap. IV. IX., cap. IV.
126 Para los indo- Para los indo-antillanos
antillanos no todo no todo terminaba
terrminaba
127 de continuo y ruje y de continuo y ruge y se
se encrespa, y los encrespa, y los ríos
ríos desde la desde la
131 por el ontrario, de por el contrario, de
peces en los ríos y peces en los ríos y
ensenadas ensenadas
139 están trabajadas están trabajadas
ligeramente ligeramente oblicuas;
oblícuas; pero con pero con una
una
147 fermentación, fermentación, habían de
habían de producirse
producirse necesariamente
necesamente
147 Puso Colón Puso Colón Fernandina
Fernandina á la isla á la isla que los indios
que los indios llamaban Yumaí
llamaban Yumai
148 entre ellos, según entre ellos, según como
como están más están más cerca ó
cercas ó
148 Lestrigones y los Lestrigones y los
Ciclopes. Y Ciclopes. Y Herodoto
Herodoto nos nos refiere, que fueron
refiere, que fueron canibales los Scitas,
canibales los Scitas, Germanos, Celtas,
Germanos, Celtas, Fenicios, Tártaros y
Fenicios, Tártaros y Etíopes. El hambre es
Etiopes. El hambre mala
es mal
148 los Scitas, los Scitas, Germanos,
Germanos, Celtas, Celtas, Fenicios,
Fenicios, Tártaros y Tártaros y Etíopes. El
Etiopes. El hambre hambre es mala
es mal consejera. No consejera. No
148 que tiene de extraño ¿qué tiene de extraño
que en la atrasada que en la atrasada
época de la bestia época de la bestia
humana lo fuéramos humana lo fuéramos
materialmente. materialmente?
150 de la Dominique de la Dominique «que
«que lors de la lors de la conquête des
conquête des êles, le eles, le chef caraibe
chef caraibe avait avait exterminé tous les
exterminé tous les
151 debemos la Avmara. debemos la Aymara. A
A los misioneros los misioneros Vega,
Vega, Valdivia Valdivia
151 Joseph de Anchieta. Joseph de Anchieta.—
—Arte de Arte de Grammatica da
Grammatica da lingua mais usada na
lingua mais usada costa do Brasil
nacosta do Brasil
152 que nosotos que nosotros
afirmamos, afirmamos,
apoyándonos en el apoyándonos en el
estudio estudio
152 mallorquín y él mallorquín y él dialecto
dialecto catalán, catalán, proceden de la
proceden de la lenua lengua
153 que caney procede que caney procede de
cana; maíz se origina cana; maíz se origina en
en mahizo mahizo
153 relación ó estracto relación ó extracto de
de una carta que una carta que escribió el
escribió el
158 los caños del Delta y los caños del Delta y en
en su desagiie en el su desagüe en el mar,
mar, en en
160 del escritor que lo del escritor que lo
anota, sufre también anota, sufre también
cierta variente cierta variante
166 El lenguaje indo El lenguaje indo-
antillano, por lo antillano, por lo poco
poco que que conservamos
conservamos
167 ésto lo hemos esto lo hemos
recopilado con recopilado con paciente
paciente labor. No labor. No
167 sabána, llano; ji, por sabana, llano; ji, por ní,
ní, agua; bo por abo, agua; bo por abo, lugar.
lugar. He aquí un He aquí un lugar llano
lugar llano con agua. con agua. Como si di
Como si di dijéramos: Buen sitio de
dijeramos: Buen labranza. Una de las
sitio de labranza. ideas principales del
Una de las ideas boriqueño era buscar en
principales del la isla buenos sitios
boriqueño era donde sembrar sus
buscar en la isla yucubías y sus ajes y
buenos sitios donde batatas. Así como elegir
sembrar sus
yucubías y sus ajes y
batatas. Así como
elejir
173 Espíritu benéfico.— Espíritu benéfico.—
Yukivu; Haytí, Yukiyu; Haytí, Yukajú;
Yukajú; Ci. Ci.
175 Generoso.—Matum. Generoso.—Matún
175 nim; Dk. tanka; DD. nim; Dk. tanka; DD.
tcho; Nabajo tcho; Navajo (apaches)
(apaches) cha cha
175 Hilo para canastos. Hilo para canastos.—
—Biiao. Bijao
187 que Montes de Oca que Montes de Oca
traduce padre, traduce padre,
adaptándo adaptando
187 Nosotros, siguiendo Nosotros, siguiendo á
á Lucien Adam, Lucien Adam,
traduciriamos traduciríamos
187 Lucien Adam. Lucien Adam.
Matériaux pour Matériaux pour servir à
servir á l’établissement d’une
l’établissement d’ grammaire comparée
une grammaire
comparée
189 Ki-umú-e titanvem Ki-umú-e titanyem
ubécuvum, santiket ubécuyum, santiket ála
ála evéti.—Nuestro eyéti.—Nuestro
190 dóminical en tupí- dominical en tupí-
guaraní, guaraní, sometiéndola á
sometiéndola á algunas
algunas
197 que nabos que nabos
comunmente.” El comunmente.” El
mismoa utor, en el mismo autor, en el
199 aves llamaban los aves llamaban los
españoles españoles alcatraces.”
alcatraces.” En árabe En arabe
199 para componer para componer areytos
arevtos ó ritmos.” ó ritmos.” Por orden de
Por orden de
202 dos primeras dos primeras palabras
palabras llevan llevan radicales indo-
radicales indo- antillanos
antillanas
203 Ateque.—Arbol de Ateque.—Arbol de
Cuba. (Cordia callo Cuba. (Cordia
cocca). callococca).
203 que el aborígen la que el aborigen la
cultivara. Oviedo cultivara. Oviedo (lib.
(lib. VII VII
203 pepo) con la pepo) con la candungo
candungo ó ó marimbo (cucurbita
marimbo (cucurbita lagenaria
lagenaira
203 (crecentia cujete). (crescentia cujete).
Probablemente, Probablemente,
después de después de importada
importada
207 se denomina punta se denomina punta
Maisí. Las Casas Maisí. Las Casas anota
anota Bavatiquiri. Bayatiquiri. Corrupción
Corrupción de de Bayatikeri.
Bavatikeri.
208 Bajaraque.—El Bajareque.—El bohío
bohío que tenía que tenía mucha
mucha extensión extensión
214 Terræ Novœ; y así Terræ Novæ; y así
aparece en las obras aparece en las obras de
de Oviedo Oviedo
219 buena como de lino, buena como de lino, é
é ésta llaman ésta llaman cabuya, la
cabuva, la penúltima penúltima
219 la voz cabuva viene la voz cabuya viene de
de cabo cabo
221 Hay también el Hay también el
Chrvsophvllum Chrysophyllum
oliviforme oliviforme
222 Caiaguala.—Vegetal Calaguala.—Vegetal
silvestre. Es el silvestre. Es el polipodio
polipodio
222 y el Presbítero Ponce y el Presbítero Ponce de
de León anotaron León anotaron Camuy
Camuv
222 tomando la e par tomando la e por una s,
una s, han hecho el han hecho el vocablo
vocablo
226 padre Nazario (Ob. padre Nazario (Ob. cit.)
cit.) á seguirles en á seguirles en esta
esta equivocacion equivocación
226 Sir Walter Ralegh, Sir Walter Raleigh,
desde la isla de desde la isla de Trinidad
Trinidad hasta hasta
229 Ciales.—No es Ciales.—No es palabra
palabra indígena. indígena. Nombre de un
Nombre de nu
234 cuyo monotono grito cuyo monótono grito
nocturno es coquí, nocturno es coquí,
coquí coquí
234 este cú ó kú de la este cú ó kú del radical
radical tu; pues tu- tu; pues tu-rey, era
rey, era
234 Cuaia.—Río de Santo Cuaja.—Río de Santo
Domingo, tributario Domingo, tributario del
del
236 El nombre indígena El nombre indígena era
era cabuva cabuya
237 Daiabón.—Lugar del Dajabón.—Lugar del
cacicazgo de Marien. cacicazgo de Marien.
Las Las
250 Guaraca del Guaraca del Guayaney,
Guavaney, y por y por último se quedó
último se quedó con con
251 Guavabacán.— Guayabacán.—Arbol.
Arbol. (Myrica (Myrica divaricata)
divaricata)
255 Haití.—Véase Havtí Haití.—Véase Haytí
262 su régulo Boiekio. su régulo Bojekio.
Comprendía á Comprendía á
Hanigagía, Yaquino Hanigagía, Yaquino
262 de los sucesos de la de los sucesos de la
conquista de la conquista de la Nueva
Nueva Espana España
265 haze el abrigo una haze el abrigo una
ysleta que tendrá de ysleta que tendrá de
amplido tres amplio tres
272 Manaca.—La palma Manaca.—La palma
real. (Oreodoxia real. (Oreodoxa regia)
regia)
274 Leónen Puerto Rico, León en Puerto Rico,
cuando visitó, la isla cuando visitó, la isla en
en 1508 1508
278 también esplica esta también explica esta
frase que hemos frase que hemos citado
citado
279 Nibajo.—Río Nibajo.—Río
dominican tributario dominicano tributario
del Yaque. del Yaque.
280 O.—Radical indo- O.—Radical indo-
antillana. Montaña antillano. Montaña
287 Semí.—La divinidad Semí.—La divinidad
tutelar del indo- tutelar del indo-
anttillano antillano
all Pedro Mártir (Dec. Pedro Mártir (Déc.
1. Errores tipográficos palpables corregidos

silenciosamente; retuvo ortografía y dialecto no
estándar.
2. Notas a pie de página reindexadas utilizando números y
recopiladas al final del último capítulo.
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Chapter 1 Overview of Wind Energy Installations and Wind Turbine Technologies..... 1

Chapter 2 DFIG with Different Inverter Schemes............................................... 17

Chapter 3 DFIG Performance and Excitation Parameters................................... 39

3.6.1 Effects of the Insulated Gate Bipolar Excitation

Chapter 4 PMSG Performance and Excitation Parameters.................................. 51

Chapter 5 DFIG and PMSG Machine Parameters............................................... 67

Chapter 6 PMSG in Different Grid Strengths...................................................... 87

6.4 Mathematical Dynamics of Connecting the SDBR in the

Chapter 7 DFIG and PMSG in Weak and Strong Grids..................................... 107

7.5.5 Improving the Performance of DFIG and PMSG

Chapter 8 DFIG Wind Turbines and Supercapacitor Scheme............................ 127

Chapter 9 PMSG Wind Turbine with Series and Bridge Fault

9.4 The Power Converter Control of the PMSG Wind

Chapter 10 PMSG with Capacitive Bridge Fault Current Limiters..................... 161

Chapter 11 Comparative Study of DFIG and PMSG with Different Fault

1.1 OVERVIEW OF GLOBAL WIND ENERGY INSTALLATIONS

FIGURE 1.3 New wind power capacity installations by region [1].

The global offshore wind industry recorded over 6 GW of new installation in

FIGURE 1.7 Wind turbine manufacturers.

1.2 CLASSIFICATION OF WIND ENERGY CONVERSION SYSTEM

1.3 OVERVIEW OF WIND TURBINE TECHNOLOGIES

1.3.1 Fixed Speed Wind Turbines

1.3.2 Variable Speed Wind Turbines

1.4 OVERVIEW OF DFIG AND PMSG WIND TURBINES

1.5 DFIG WIND TURBINE MODELING

The relationship between C t and C p is

Turbine Input Power [pu]

FIGURE 1.11 Control block to determine active power reference Pref.

Pref 1  0.1571Vw  1.035  pu  (1.6)

Pref 2  0.2147Vw  1.668  pu  (1.7)

opt  0.0775Vw  pu  (1.8)

FIGURE 1.12 Pitch controller for DFIG VSWT.

FIGURE 1.13 Pitch controller for SCIG FSWT.

1.6 PMSG WIND TURBINE MODELING

speed Vw is expressed in (m/s). The wind generator’s power coefficient is Cp and is

In Equation (1.10), c1 to c6 are the characteristic coefficients of the wind turbine.

From Equations (1.13) and (1.14),

 sd   Lsd  Lmd  I sd   m (1.15)

 sq   Lsq  Lmq  I sq (1.16)

Ld  Lsd  Lmd (1.19)

Lq  Lsq  Lmq (1.20)

The PMSG active and reactive powers are given as:

Ps  Vsd I sd  Vsq I sq (1.21)

1.1 OVERVIEW OF GLOBAL WIND ENERGY INSTALLATIONS

1.2 CLASSIFICATION OF WIND ENERGY CONVERSION SYSTEM

1.3 OVERVIEW OF WIND TURBINE TECHNOLOGIES

1.3.1 Fixed Speed Wind Turbines

1.3.2 Variable Speed Wind Turbines

1.4 OVERVIEW OF DFIG AND PMSG WIND TURBINES

1.5 DFIG WIND TURBINE MODELING

Pref 1 0.1571Vw 1.035 pu (1.6)

Pref 2 0.2147Vw 1.668 pu (1.7)

opt 0.0775Vw pu (1.8)

1.6 PMSG WIND TURBINE MODELING

sd Lsd Lmd I sd m (1.15)

sq Lsq Lmq I sq (1.16)

Ld Lsd Lmd (1.19)

Lq Lsq Lmq (1.20)

Ps Vsd I sd Vsq I sq (1.21)

Qs Vsq I sd Vsd I sq (1.22)

2.2 MODEL SYSTEM OF STUDY

2.3 VARIABLE SPEED DRIVE CONTROL

After d-q transformation to the synchronous coordinate system, the following