Modeling, Operation, and Analysis of DC Grids: From High Power DC Transmission to DC Microgrids

Modeling, Operation, and Analysis of DC Grids presents a unified vision of direct current grids with their core analysis techniques, uniting power electronics, power systems, and multiple scales of applications. Part one presents high power applications such as HVDC transmission for wind energy, faults and protections in HVDC lines, stability analysis and inertia emulation. The second part addresses current applications in low voltage such as microgrids, power trains and aircraft applications. All chapters are self-contained with numerical and experimental analysis.

• Provides a unified, coherent presentation of DC grid analysis based on modern research in power systems, power electronics, microgrids and MT-HVDC transmission
• Covers multiple scales of applications in one location, addressing DC grids in electric vehicles, microgrids, DC distribution, multi-terminal HVDC transmission and supergrids
• Supported by a unified set of MATLAB and Simulink test systems designed for application scenarios

• Language: English
• Publisher: Academic Press
• Release date: Jul 1, 2021
• ISBN: 9780128221020

Book preview

Front Cover for Modeling, Operation, and Analysis of DC Grids

Modeling, Operation, and Analysis of DC Grids

From High Power DC Transmission to DC Microgrids

First edition

Alejandro Garcés

publogo

Table of Contents

Cover image

Title page

Copyright

List of contributors

1: Introduction

Abstract

1.1. The battle of the currents

1.2. DC grids

1.3. Power electronics

1.4. High-power applications

1.5. Low-power applications

References

Part 1: High power applications

2: HVDC transmission for wind energy

Abstract

2.1. Wind energy

2.2. Slow-dynamics model of the wind turbine

2.3. HVDC transmission for wind farms

2.4. Stability of HVDC transmission lines

2.5. Summary

References

3: DC faults in HVDC

Abstract

3.1. Minimum requirements for the protection system of MTDC

3.2. Impact of DC faults in VSC

3.3. Analysis of the MTDC-HVDC during DC faults

3.4. Detection and identification strategies in MTDC

3.5. Clearance strategies for MTDC

3.6. HVDC circuit breakers

3.7. Fault current limiters

References

4: Eigenvalue-based analysis of small-signal dynamics and stability in DC grids

Abstract

4.1. Introduction

4.2. Introduction to state-space modeling of electrical systems

4.3. Synthesis of system-level state-space models of HVDC grids

4.4. Examples of sub-system modeling

4.5. Practical considerations for modular and automated generation of system-level small-signal state-space models

4.6. Example of small-signal analysis

4.7. Conclusion

References

5: Inertia emulation with HVDC transmission systems

Abstract

Acknowledgement

5.1. Introduction

5.2. Basis for a need of virtual inertia with VSC HVDC systems

5.3. VSC HVDC control approaches for inertia emulation

5.4. Fast frequency response service by VSC HVDC systems

5.5. Summary

References

6: Real-time simulation of a transient model for HVDC cables in SOC-FPGA

Abstract

6.1. Introduction

6.2. Frequency domain model formulation

6.3. Cable model with difference equations

6.4. VHDL conceptual design of the HVDC cable model

6.5. Integration and development of the HVDC cable in VHDL

6.6. Conclusions

References

7: Probabilistic analysis in DC grids

Abstract

7.1. Introduction

7.2. DC power grid model

7.3. Probabilistic power flow analysis in DC grids

7.4. Bayesian modeling of DC grids

7.5. Experimental validation

7.6. Conclusions

References

Part 2: Low power applications

8: Stationary-state analysis of low-voltage DC grids

Abstract

8.1. Introduction

8.2. Modeling the grid

8.3. Results

8.4. Conclusions

References

9: Stability analysis and hierarchical control of DC power networks

Abstract

9.1. Literature review and scope of the chapter

9.2. Power system and control system overview

9.3. Small-signal modeling of the DC microgrid

9.4. Case study and prototype description

9.5. Validation of the model predictive controller

9.6. Validation of the small-signal modeling approach

9.7. Conclusion

References

10: Digital control strategies of DC–DC converters in automotive hybrid powertrains

Abstract

Acknowledgements

10.1. Introduction

10.2. Analysis of the DC–DC power converters

10.3. Digital current control strategies

10.4. Simulation results

10.5. Summary

References

11: Adaptive control for second-order DC–DC converters: PBC approach

Abstract

Acknowledgements

11.1. Introduction

11.2. DC–DC converter modeling

11.3. Passivity-based control method

11.4. Control design for DC–DC converters

11.5. Simulation results

11.6. Conclusions

References

12: Advances in predictive control of DC microgrids

Abstract

Acknowledgement

12.1. Introduction

12.2. Predictive control of DC microgrids

12.3. Conclusion

References

13: Modeling and control of DC grids within more-electric aircraft

Abstract

13.1. Introduction to more-electric aircraft

13.2. Modeling of aircraft EPS

13.3. Control development

13.4. Summary

References

Index

Copyright

Academic Press is an imprint of Elsevier

125 London Wall, London EC2Y 5AS, United Kingdom

525 B Street, Suite 1650, San Diego, CA 92101, United States

50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States

The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom

Copyright © 2021 Elsevier Inc. All rights reserved.

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission.

The MathWorks does not warrant the accuracy of the text or exercises in this book.

This book's use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher's permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions.

This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

Library of Congress Cataloging-in-Publication Data

A catalog record for this book is available from the Library of Congress

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

ISBN: 978-0-12-822101-3

For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Brian Romer

Acquisitions Editor: Graham Nisbet

Editorial Project Manager: Ruby Gammell

Production Project Manager: Nirmala Arumugam

Designer: Mark Rogers

Typeset by VTeX

List of contributors

Jef Beerten

Department of Electrrical Engineering (ESAT-electa), KU Leuven, Heverlee, Belgium

EnergyVille, Genk, Belgium

Santiago Bustamante     Universidad Pontificia Bolivariana, Electrical Engineering Department, Medellín, Colombia

Hugo A. Cardona     Universidad Pontificia Bolivariana, Electrical Engineering Department, Medellín, Colombia

Salvatore D'Arco     SINTEF Energy Research, Trondheim, Norway

Gerardo Espinosa-Perez     Universidad Nacional Autónoma de México, Facultad de Ingeniería, CDMX, Mexico

Fei Gao     Department of Electrical Engineering, Key Laboratory of Power Transmission and Conversion, Shanghai Jiao Tong University, Shanghai, China

Alejandro Garcés     Universidad Tecnológica de Pereira, Department of electric power engineering, Pereira, Risaralda, Colombia

Walter Gil-González     Facultad de Ingeniería, Institución Universitaria Pascual Bravo, Medellín, Colombia

Jorge W. Gonzalez     Universidad Pontificia Bolivariana, Electrical Engineering Department, Medellín, Colombia

Catalina González-Castaño     Universitat Rovira i Virgili, Departament d'Enginyeria Electrònica, Elèctrica i Automàtica, Tarragona, Spain

Habibu Hussaini     Power Electronics, Machines and Control Group, Faculty of Engineering, The University of Nottingham, Nottingham, United Kingdom

Miguel Jiménez Carrizosa     Universidad Politecnica de Madrid, CEI & ETSIME, Madrid, Spain

Oscar Danilo Montoya

Universidad Distrital Francisco José de Caldas, Facultad de Ingeniería, Bogotá, D.C., Colombia

Universidad Tecnológica de Bolívar, Laboratorio Inteligente de Energía, Cartagena, Colombia

Javier Muñoz     Department of Electrical Engineering, Faculty of Engineering, Universidad de Talca, Campus Curicó, Chile

Milan Prodanovic     IMDEA Energy, Electrical Systems Unit, Móstoles, Madrid, Spain

Carlos Restrepo     Universidad de Talca, Department of Electromechanics and Energy Conversion, Curicó, Chile

Marco Rivera     Department of Electrical Engineering, Faculty of Engineering, Universidad de Talca, Campus Curicó, Chile

Alberto Rodríguez-Cabero     IMDEA Energy, Electrical Systems Unit, Móstoles, Madrid, Spain

Javier Roldán-Pérez     IMDEA Energy, Electrical Systems Unit, Móstoles, Madrid, Spain

Santiago Sanchez-Acevedo     SINTEF Energi AS, Department Energy Systems, Trondheim, Norway

Jon Are Suul

SINTEF Energy Research, Trondheim, Norway

Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway

Raymundo E. Torres-Olguin     SINTEF Energy, Department of energy systems, Trondheim, Norway

Ariel Villalón     Engineering Systems PhD. Program, Faculty of Engineering, Universidad de Talca, Campus Curicó, Chile

Cheng Wang     Power Electronics, Machines and Control Group, Faculty of Engineering, The University of Nottingham, Nottingham, United Kingdom

Tao Yang     Power Electronics, Machines and Control Group, Faculty of Engineering, The University of Nottingham, Nottingham, United Kingdom

Carlos D. Zuluaga R.     Institución Universitaria Pascual Bravo, Faculty of Engineering, Department of Electrical Engineering, Medellín, Colombia

1: Introduction

Alejandro Garcés    Universidad Tecnológica de Pereira, Department of electric power engineering, Pereira, Risaralda, Colombia

Abstract

Modern power systems include DC grids in different voltage levels, from high-power multiterminal HVDC systems to low-voltage DC distribution, microgrids, and electric transportation systems. This chapter presents an overview of these technologies with focus on modeling, operation, and analysis.

Keywords

Introduction; DC microgrids; HVDC transmission; DC converters; the battled of the currents

Chapter Points

•  A general overview of DC grids is presented, from high- to low-voltage applications.

•  Basic concepts of power electronics are presented.

•  Three main types of DC grids are described: multiterminal high-voltage direct current transmission, DC microgrids, and electric transportation systems.

1.1 The battle of the currents

At the end of the 19th century, there was a serious debate about the use of AC or DC for the emerging power systems. Modern historians call this episode as the battle of the currents [2]. Two major scientists were involved in this discussion, Tomas Alva Edison and Nicola Tesla. Edison stubbornly advocated the use of DC current arguing safety reasons, whereas Tesla advocated for the AC current due to the high efficiency of AC generators and the possibility to rise voltages to transmission levels. At the end, Tesla proved to be right at the point that the AC power was the rule during a century. In fact, AC transmission was the greatest engineering achievement of the twentieth century, according to the National Academy of Engineering [10]. However, the 21th century is witnessing a change due to new advances on power electronics. Modern power systems include portion of the grid operated in DC for both low- and high-power applications. This is motivated, in the first case, by an increasing penetration of renewable energies and electric vehicles under concepts such as DC-microgrids and DC-distribution [1], and in the second case, by advances in multiterminal HVDC systems for offshore applications [3] and supergrids [5]. Although some authors claim that Edison was right, modern power systems are hybrid with portions of the grid in DC and other parts in AC. Therefore we can say that the result of the battle of the currents is a technical draw between Edison and Tesla.

1.2 DC grids

DC technology appears in both high- and low-voltage applications. The most common application of DC technology is in high-voltage direct current transmission (hvdc), although modern power systems may include DC distribution, microgrids, and electric transportation systems as shown in Fig. 1.1.

Figure 1.1 DC technology in modern power systems.

This book overviews all these applications, starting from hvdc and multiterminal hvdc networks. These networks are characterized by a relatively small number of nodes, with power electronics consisting on voltage source converters and/or modular multilevel converters. Multiterminal systems for the integration of offshore wind farms are studied in Chapter 2 with emphasis on the control and operation of systems of this type. Modeling aspects of these systems are studied in detail in Chapter 3, including protection schemes. A small signal stability analysis is presented in Chapter 4. The interaction of the mt-hvdc grid with the AC grid is studied in Chapter 5, considering the concept of inertia emulation. The model of the DC-cable and its implementation for real-time simulation is presented in Chapter 6. Finally, a probabilistic analysis of the power flow is presented in Chapter 7.

The second part of the book includes low-voltage applications such as DC distribution, microgrids, and electric transportation systems. It starts with the stationary state analysis and power flow presented in Chapter 8. The stability and hierarchical control of DC grids is analyzed in Chapter 9. After that, the control of DC–DC converters is studied for electric transportation systems (Chapter 10) and for aircraft systems (Chapter 13). Passivity-based control and model predictive control are presented in Chapters 11 and Chapter 12, respectively.

1.3 Power electronics

Two components, the transformer and the synchronous machine, were key for the victory of AC current in the 20th century. On one hand, the transformer is the only way to increase voltage to transmission levels; recall that transmitted power is given by and hence, to transmit high power, it is necessary to have whether a high voltage or a high current; we prefer a high voltage since high currents lead to unacceptable conduction losses, since these increase quadratically as . On the other hand, the synchronous machine is the most efficient way to generate electrical power from a mechanical rotating source. DC generators require brushes and additional resistive components that reduce their electrical and mechanical efficiency.

These two components have achieved a high efficiency and will be part of the 21th century power grid. However, there is a third component that will play a key role, the power electronic converter. This device allows us to transform AC into DC current and vice versa. It also allows us to control voltages and currents on different components and, in some cases, compensate harmonics and reactive power.

There is a vast variety of power electronic converters being an active research area. However, two main technologies are used in most practical applications, the line-commutated converter and the force-commutated converter.

Line-commutated converters use a set of thyristor valves usually connected in a 6-pulse or a 12-pulse configuration as depicted in Fig. 1.2.

Figure 1.2 Schematic representation of a 6-pulse line-commutated converter and the line commutation concept.

In a thyristor the conduction process cannot be initiated without a proper polarity to the gate that is only able to control the thyristor turn-on. Once the conduction process has started, the valve continues to conduct until the current drops to zero, and the reverse voltage bias appears across the thyristor. This is the origin of the concept of line commutation since the turn-off action depends on the zero crossing of the line voltage (see [9] for more details). These types of converters have been widely used in high-voltage applications and motor drives [8].

The next generation of converters were force-commutated converters. This type devices use IGBTs (Insulated Gate Bipolar Transistor) or GTOs (Gate Turn-Off Thyristor), which allow us to control the turn-on and turn-off commutation at relatively high frequencies (higher than 1 kHz). In this way, it is possible to generate any desired voltage or current by a strategy called pulse-width modulation or PWM. Force-commutated converters are widely used in all kinds of applications, from high-voltage transmission to motor drivers and integration of renewable energies. In addition, force-commutation can be used in AC/DC, DC/DC, and AC/AC converters [9]. Although there are many configurations of AC/DC converters, the most common is the voltage source converter depicted in Fig. 1.3; this converter has the capacity of controlling nodal voltage with additional objectives such as reactive power compensation [4], harmonic filtering, and control of renewable energies [11].

Figure 1.3 Schematic representation of a voltage source converter and the pulse-width modulation. In this case the desired sinusoidal wave is embedded in the modulated signal.

A recent evolution of the voltage source converter for high-voltage applications is the modular-multilevel converter. This converter allows smooth and nearly ideal sinusoidal output voltage on the AC side with little or no filtering at all; it is able to operate at lower switching frequencies with high-efficiency and superior controllability compared to the voltage source converters. Modular-multilevel converters are specially designed for high-power applications and offshore wind farm integration.

Details about the modulation and control of each of these converters are beyond the scope of this book, since our objective is to analyze the grid by the functionality of the converters as will be described in Chapter 11. The interested reader can consult [11] and [9] for more details. Chapter 2 presents basic concepts about the operation of these power electronic converters.

1.4 High-power applications

Early applications of high-voltage direct current transmission (HVDC) were based on vacuum tubes. This technology was quickly replaced by line-commutated converters [7], which gave higher efficiency and reliability. Modern HVDC systems use voltage-source converters and modular-multilevel converters. However, the main drawback of HVDC transmission is the cost of the converters; therefore the technology seems to be only financially viable for long transmission lines and offshore applications.¹

HVDC transmission is evolving to the concept of multiterminal HVDC transmission (MT-HVDC), where different components are integrated into DC grids with radial or meshed configurations as depicted in Fig. 1.4. These multiterminal applications include offshore wind farms [12] and the massive integration of transmission systems in the form of supergrids [5]. Operation of these systems is challenging for several reasons. First, there is not a global variable such as the frequency in AC grids, which allows us to determine stability conditions without communications. Second, the renewable energy sources introduce high variability with a stochastic nature. Third, if we use the operation of AC grids, then we require a change on the operation schemes; finally, there are not enough theoretical tools for analyzing this type of grids. The first part of the book presents some aspects of the operation of MT-HVDC. It is important to notice that although the first power-electronics based DC-grids appears in high-power applications, the same problems appears in low-voltage applications. Therefore we require a unified framework for these grids. The second part of the book is concentrated on low-voltage applications.

Figure 1.4 Five terminal HVDC grid proposed by the CIGRE B4 working group.

1.5 Low-power applications

Some renewable energy sources and most of the energy storage devices are inherently DC. Therefore it is a natural step to integrate these components into DC-grids. These grids are called DC-microgrids for low-voltage levels and DC-distribution for medium voltage [1]. Although the difference between DC-microgrids and DC-distribution can be blurred,² we consider a DC-microgrid as low-voltage network with a centralized bus-bar as depicted in Fig. 1.5.

Figure 1.5 General configuration of a DC microgrid.

On the other hand, a DC-distribution system as a network with radial topology as shown in Fig. 1.6. Chapter 8 presents more details about the operation of these grids. Both DC-microgrids and DC-distributions consist of different distributed energy sources and storage devices together with passive and active loads. The grid can be operated whether connected to the main grid or in island. In the first case the AC–DC converter maintains a constant voltage in the master or slack node (e.g., Node 0 in the DC-distribution and the centralized bus-bar in the DC-microgrid). This operation type is denominated as master–slave since the slack node imposes the voltage, and the rest of the nodes adjust to this operative value. Island operation is more complicated and requires some control strategy to maintain a suitable voltage and stable operation. The most conventional strategy is the hierarchical primary/secondary/tertiary control presented in the next section.

Figure 1.6 Schematic representation of a DC-distribution system with high penetration of solar energy.

Finally, Chapters 10, 12, and 13 deal with applications of the DC technology in electric transportation systems. These include trains, electric vehicles, and aircraft systems. All these applications face similar challenges as DC-microgrids and include DC–DC converters, motor loads, and energy storage devices.

References

[1] Ahmed T. Elsayed, Ahmed A. Mohamed, Osama A. Mohammed, Dc microgrids and distribution systems: an overview, Electric Power Systems Research 2015;119:407–417.

[2] P. Fairley, Dc versus ac: the second war of currents has already begun [in my view], IEEE Power and Energy Magazine Nov. 2012;10(6):104.

[3] Catalin Gavriluta, Ignacio Candela, Costantino Citro, Alvaro Luna, Pedro Rodriguez, Design considerations for primary control in multiterminal vsc hvdc grids, Electric Power Systems Research 2015;122:33–41.

[4] Reyes S. Herrera, Patricio Salmerón, Instantaneous reactive power theory: a comparative evaluation of different formulations, IEEE Transactions on Power Delivery 2007;22(1):595–604.

[5] Dirk Van Hertem, Mehrdad Ghandhari, Multi-terminal vsc hvdc for the European supergrid: obstacles, Renewable and Sustainable Energy Reviews 2010;14(9):3156–3163.

[6] João Abel Peças Lopes, André Guimarães Madureira, Carlos Coelho Leal Monteiro Moreira, A view of microgrids, Wiley Interdisciplinary Reviews: Energy and Environment 2013;2(1):86–103.

[7] Ned Mohan, First Course on Power Electronics, 2009.

[8] Scott D. Sudhoff, Paul C. Krause, Oleg Wasynczuk, Analysis of Electric Machinery and Drive Systems. 2002 p. 632.

[9] Muhammad H. Rashid, Power Electronics Handbook 2007.

[10] Bob Somerville, Georges Constable, A Century of Innovation: Twenty Engineering Achievements That Transformed Our Lives. Joseph Henry Press, National Academy of Engineering; 2003.

[11] Remus Teodorescu, Marco Liserre, Pedro Rodriguez, Grid Converters for Photovoltaic and Wind Power Systems. Chichester, UK: John Wiley & Sons, Ltd; Jan. 2011.

[12] Raymundo E. Torres-Olguin, Alejandro Garces, Marta Molinas, Tore Undeland, Integration of offshore wind farm using a hybrid HVDC transmission composed by the PWM current-source converter and line-commutated converter, IEEE Transactions on Energy Conversion 2013;28(1):125–134.

---

¹  The main advantage of DC transmission is the lack of reactive power. Therefore an HVDC line is viable only in the cases in which the cost of reactive power compensation along the line is higher than the cost of the capacitors. The breaking point is usually 100 km, but the costs of converter is constantly reduced, and hence this distance.

²  Some authors differentiate another class of grids, namely, nanogrids [6]. These are microgrids at very low voltage levels. We maintain the term microgrid for networks of this type.

Part 1: High power applications

Outline

2. HVDC transmission for wind energy

3. DC faults in HVDC

4. Eigenvalue-based analysis of small-signal dynamics and stability in DC grids

5. Inertia emulation with HVDC transmission systems

6. Real-time simulation of a transient model for HVDC cables in SOC-FPGA

7. Probabilistic analysis in DC grids

2: HVDC transmission for wind energy

A slow-dynamics model

Alejandro Garcésa; Raymundo E. Torres-Olguinb    aUniversidad Tecnológica de Pereira, Department of electric power engineering, Pereira, Risaralda, Colombia

bSINTEF Energy, Department of energy systems, Trondheim, Norway

Abstract

This chapter presents a slow-dynamics model for HVDC transmission systems that integrates wind farms, especially in offshore applications. This model considers the slow dynamics of the system such as variations on the wind velocity, the dynamics of the mechanical axis of the turbines, and the HVDC cable. However, it neglects the fast dynamics of the converter. The model is nonlinear/nonautonomous, and hence it is analyzed using nonlinear theory instead of the conventional small-signal approach. A formal demonstration of stability is presented. Numerical experiments performed in Open-Modelica complement different sections of the chapter.

Keywords

wind farms; slow-dynamics model; HVDC transmission; voltage source converter; modular-multilevel converter

Chapter points

•  A slow-dynamics model for wind farms is presented.

•  A review of the main type of converters and their control.

•  Stability analysis for the slow-dynamics model.

It is not surprising that wind power is one of the fastest growing renewable technology in the world, wind is a resource available almost everywhere, its price is competitive with conventional sources, and the technology has matured in the last decades. However, the best places for wind generation are usually open spaces without skyscraper or mountains that generate turbulence. Therefore wind farms can be far away from consumption centers such as major cities, and hence HVDC transmission becomes a very attractive solution, especially for offshore wind farms [1].

Early research about offshore wind farms was developed by Nordic countries [2]. Currently, the United Kingdom, Germany, and China lead the development of large-scale commercial offshore wind farms [3]. Offshore wind farms have been also considered in the United States [4], Spain [5], Kuwait [6], Turkey [7], and Brazil [8], among other countries.

On the other hand, HVDC transmission has been considered as a solution for the integration of conventional wind farms that are distant from the main cities. For example in Colombia, more than 1 GW of wind generation is planned in the north of the country, and this generation requires to be interconnected to the center of the country, where the largest cities are located [9,10]. In China the use of HVDC transmission has been extensively used, and there are many upcoming projects [11].

The rest of the chapter is divided as follows. Section 2.1 shows basic concepts about wind energy. Next, the slow-dynamics model of the wind turbine is presented in Section 2.2. After that, the model for HVDC transmission is described. A general stability analysis is presented in Section 2.4. Finally, conclusions are given in Section 2.5. Most of the sections show numerical experiments using Open-Modelica.

2.1 Wind energy

There are several models for wind energy. However, our approach is based on the output power, which is proportional to the cube of the wind velocity:

(2.1)

where P is the power in MW, ρ is the air density (1.28 ), A is the area swept by the blades, v is the wind speed, and is a coefficient of performance, which depends on the turbine technology and its control. This coefficient is limited by the Betz law, which states that no turbine can capture more than of the kinetic energy of the wind. In practice, this limit can be even lower [12]. The coefficient of performance can be approximated to (2.2) for large wind turbines [13]:

(2.2)

where β is the pitch angle (i.e., the angle of the blade with respect to the direction of the wind), and is a function of the tip ratio λ given by

(2.3)

Note that A, ρ are parameters of the turbine, and v is a noncontrollable variable. Therefore, any power control in the wind turbine must be done through . The maximum value of is obtained for and . Consequently, the rotational speed must be modified to achieve as the wind speed changes; this implies variable speed operation.

Wind turbines come in different forms and sizes, from small-power vertical axis turbines to high-power horizontal axis turbines. The most common configuration for both onshore and offshore applications is the horizontal axis wind turbine with variable speed depicted in Fig. 2.1.

Figure 2.1 Common configuration of wind turbines: double fed induction generator (DFIG) and permanent magnet synchronous generator (PMSG).

A variable speed can be obtained in different ways, for example, using a double fed induction generator (DFIG) or a permanent magnet synchronous generator (PMSG). The first option is based on an induction machine with access to the rotor windings in order to control the rotational speed. This is done by a back-to-back converter, which is usually less than 30% the power of the machine. In this way the cost of the converter is reduced, and the rotational speed is independent of the frequency of the grid. The second option is based on a permanent magnet synchronous generator integrated to the grid by a full-size converter. The additional cost associated with this converter is compensated by the high efficiency of the machine and the possibility to eliminate the gearbox by using a machine with high number of poles.

The cubic growth of power with respect to wind speed generates very-high-power output at high speeds. However, these speeds have little probability of occurrence, so it is not economically feasible to build turbines for these high velocities. Therefore the nominal power of the turbine is usually designed for probable speeds, and a controller of the pitch angle is used to maintain the power at its nominal value, resulting in the operation curve depicted in Fig. 2.2.

Figure 2.2 Typical control strategy for a wind turbine: for v  ∈ [ v cut-in , v nom ], the turbine changes its rotational speed to achieve maximum power extraction of wind, whereas for v  ∈ [ v nom , v cut-out ], the pitch angle acts maintaining a constant power.

A wind turbine does not operate for wind speeds below a value known as . The turbine is adjusted to achieve the maximum tracking point from this value to the nominal speed. This is done by changing rotational speed by means of the power electronic converter. The pitch angle is maintained in zero during this type of operation. After the turbine achieves , the pitch angle acts to maintain the constant power ; this operation is maintained until wind speed achieves (the turbine can be damaged for values above this cut-out velocity, and hence it must be mechanically blocked).

Fig. 2.2 constitutes a