Modeling, Operation, and Analysis of DC Grids: From High Power DC Transmission to DC Microgrids 0128221011, 9780128221013

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MODELING, O P E R AT I O N , A N D A N A LY S I S O F D C GRIDS

MODELING, O P E R AT I O N , A N D A N A LY S I S O F D C GRIDS From High Power DC Transmission to DC Microgrids Edited by

ALEJANDRO GARCÉS

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

Contents

Contents List of contributors . . . . . . . . . . . . . . . . . . . . . . xi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 1.2 1.3 1.4 1.5

Alejandro Garcés The battle of the currents . . . . . . . . . . . . . . . . . 1 DC grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Power electronics . . . . . . . . . . . . . . . . . . . . . . . . 3 High-power applications . . . . . . . . . . . . . . . . . . 5 Low-power applications. . . . . . . . . . . . . . . . . . . 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Part 1 High power applications Chapter 2 HVDC transmission for wind energy . 13 2.1 2.2 2.3 2.4 2.5

Alejandro Garcés and Raymundo E. Torres-Olguin Wind energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Slow-dynamics model of the wind turbine 16 HVDC transmission for wind farms . . . . . . 18 Stability of HVDC transmission lines . . . . . . 19 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Chapter 3 DC faults in HVDC . . . . . . . . . . . . . . . . . 27 3.1 3.2 3.3 3.4 3.5

Raymundo E. Torres-Olguin Minimum requirements for the protection system of MTDC. . . . . . . . . . . . . . . . . . . . . . . . Impact of DC faults in VSC . . . . . . . . . . . . . . Analysis of the MTDC-HVDC during DC faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detection and identification strategies in MTDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clearance strategies for MTDC . . . . . . . . . .

28 29 31 48 53

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3.6 HVDC circuit breakers. . . . . . . . . . . . . . . . . . . 59 3.7 Fault current limiters. . . . . . . . . . . . . . . . . . . . 61 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 4 Eigenvalue-based analysis of small-signal dynamics and stability in DC grids . . . . . . . . . . . . . . . . . . . . . . . 69 4.1 4.2 4.3 4.4 4.5

4.6 4.7

Salvatore D’Arco, Jef Beerten, and Jon Are Suul Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Introduction to state-space modeling of electrical systems . . . . . . . . . . . . . . . . . . . . . . 72 Synthesis of system-level state-space models of HVDC grids . . . . . . . . . . . . . . . . . . 82 Examples of sub-system modeling. . . . . . . 96 Practical considerations for modular and automated generation of system-level small-signal state-space models . . . . . . . . . 110 Example of small-signal analysis . . . . . . . . 114 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Chapter 5 Inertia emulation with HVDC transmission systems. . . . . . . . . . . . . 129 5.1 5.2 5.3 5.4 5.5

Santiago Bustamante, Hugo A. Cardona, and Jorge W. Gonzalez Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . Basis for a need of virtual inertia with VSC HVDC systems . . . . . . . . . . . . . . . . . . . . VSC HVDC control approaches for inertia emulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fast frequency response service by VSC HVDC systems . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . .

129 130 132 135 142 143

Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Chapter 6 Real-time simulation of a transient model for HVDC cables in SOC-FPGA. . . . . . . . . . . . . . . . . . . . . . . 149 6.1 6.2 6.3 6.4 6.5 6.6

Santiago Sanchez-Acevedo Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Frequency domain model formulation . . 150 Cable model with difference equations . . 152 VHDL conceptual design of the HVDC cable model. . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Integration and development of the HVDC cable in VHDL . . . . . . . . . . . . . . . . . . . . . . . . . 167 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . 174 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Chapter 7 Probabilistic analysis in DC grids . . 175 7.1 7.2 7.3 7.4 7.5 7.6

Carlos D. Zuluaga R. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . DC power grid model . . . . . . . . . . . . . . . . . . Probabilistic power flow analysis in DC grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bayesian modeling of DC grids . . . . . . . . . Experimental validation . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 176 177 179 184 189 189

Part 2 Low power applications Chapter 8 Stationary-state analysis of low-voltage DC grids . . . . . . . . . . . . . 195 8.1 8.2 8.3 8.4

Oscar Danilo Montoya and Walter Gil-González Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Modeling the grid . . . . . . . . . . . . . . . . . . . . . 196 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . 210

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Chapter 9 Stability analysis and hierarchical control of DC power networks . . . . . 215

9.1 9.2 9.3 9.4 9.5 9.6 9.7

Alberto Rodríguez-Cabero, Miguel Jiménez Carrizosa, Javier Roldán-Pérez, and Milan Prodanovic Literature review and scope of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Power system and control system overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Small-signal modeling of the DC microgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Case study and prototype description . . . 232 Validation of the model predictive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Validation of the small-signal modeling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Chapter 10 Digital control strategies of DC–DC converters in automotive hybrid powertrains . . . . . . . . . . . . . . . . . . . . . 245 10.1 10.2 10.3 10.4 10.5

Carlos Restrepo and Catalina González-Castaño Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of the DC–DC power converters Digital current control strategies . . . . . . . . Simulation results . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245 253 256 260 281 283 283

Chapter 11 Adaptive control for second-order DC–DC converters: PBC approach. . 289

Contents

11.1 11.2 11.3 11.4 11.5 11.6

Walter Gil-González, Oscar Danilo Montoya, and Gerardo Espinosa-Perez Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . DC–DC converter modeling. . . . . . . . . . . . . Passivity-based control method . . . . . . . . Control design for DC–DC converters . . . Simulation results . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 292 296 298 301 306 308 308

Chapter 12 Advances in predictive control of DC microgrids . . . . . . . . . . . . . . . . . . . 311 Ariel Villalón, Marco Rivera, and Javier Muñoz 12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 12.2 Predictive control of DC microgrids . . . . . 314 12.3 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . 333 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Chapter 13 Modeling and control of DC grids within more-electric aircraft . . . . . . 337 13.1 13.2 13.3 13.4

Cheng Wang, Habibu Hussaini, Fei Gao, and Tao Yang Introduction to more-electric aircraft . . . . Modeling of aircraft EPS . . . . . . . . . . . . . . . Control development . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337 340 354 364 364

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

ix

List of contributors

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

xi

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List of contributors

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

List of contributors

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

xiii

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

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 Modeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00006-X Copyright © 2021 Elsevier Inc. All rights reserved.

1

2

Chapter 1 Introduction

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.

Chapter 1 Introduction

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 ∗ and hence, to transmit high power, it is necesby Skm = Vk Ikm sary 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 rkm |Ikm |2 . 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. 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

3

AC-side

Thyristor

IThyristor

Chapter 1 Introduction

DC-side

4

turn on

turn off (line commutation) t

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

details). These types of converters have been widely used in highvoltage 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]. A recent evolution of the voltage source converter for highvoltage 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 highpower 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]

IA

Chapter 1 Introduction

pulse width modulation

AC-side

DC-side

IGBT

t

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.

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 1 The main advantage of DC transmission is the lack of reactive power. There-

fore 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.

5

6

Chapter 1 Introduction

AC grid

AC DC

4

DC AC 5

DC AC 6

offshore wind farm

3

2 DC AC DC AC

AC grid

1

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

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 powerelectronics 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.

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 DCmicrogrids for low-voltage levels and DC-distribution for medium voltage [1]. Although the difference between DC-microgrids and

Chapter 1 Introduction

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. AC

AC DC

AC DC

v1

AC DC

DC

vk

v2

vn

... i1

i2

... ik

in v0

c0 Load ρ

Load g

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 DCdistribution 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. 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. 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.

7

Chapter 1 Introduction

≈

=

constant voltage 1pu

50m

0

Connection to the AC grid

15m

1

20m

3

18m

4

23m

2

17m

5

6

13m

8

21m

8

9

15m

7

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

References [1] Ahmed T. Elsayed, Ahmed A. Mohamed, Osama A. Mohammed, Dc microgrids and distribution systems: an overview, Electric Power Systems Research 119 (2015) 407–417. [2] P. Fairley, Dc versus ac: the second war of currents has already begun [in my view], IEEE Power and Energy Magazine 10 (6) (Nov. 2012) 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 122 (2015) 33–41. [4] Reyes S. Herrera, Patricio Salmerón, Instantaneous reactive power theory: a comparative evaluation of different formulations, IEEE Transactions on Power Delivery 22 (1) (2007) 595–604. [5] Dirk Van Hertem, Mehrdad Ghandhari, Multi-terminal vsc hvdc for the European supergrid: obstacles, Renewable and Sustainable Energy Reviews 14 (9) (2010) 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 2 (1) (2013) 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.

Chapter 1 Introduction

[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, John Wiley & Sons, Ltd, Chichester, UK, 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 28 (1) (2013) 125–134.

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2 HVDC transmission for wind energy A slow-dynamics model Alejandro Garcésa and Raymundo E. Torres-Olguinb a Universidad Tecnológica de Pereira, Department of electric power engineering, Pereira, Risaralda, Colombia. b SINTEF Energy, Department of energy systems, Trondheim, Norway

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 Modeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00007-1 Copyright © 2021 Elsevier Inc. All rights reserved.

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Chapter 2 HVDC transmission for wind energy

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: 1 P = Cp ρAv 3 , 2

(2.1)

where P is the power in MW, ρ is the air density (1.28 kg/m3 ), A is the area swept by the blades, v is the wind speed, and Cp 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 16/27 ≈ 59% 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]:     151 −18.4 , (2.2) − 0.58β − 13.2 exp Cp (λ, β) = 0.73 λi λi where β is the pitch angle (i.e., the angle of the blade with respect to the direction of the wind), and λi is a function of the tip ratio λ given by 1 0.003 1 . = − λi λ + 0.02 · β β 3 + 1

(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 Cp . The maximum value of Cp is obtained for β = 0 and λ = λoptimum . Consequently, the rotational speed must be modified to achieve λoptimum as the wind speed changes; this implies variable speed operation. Wind turbines come in different forms and sizes, from smallpower vertical axis turbines to high-power horizontal axis turbines. The most common configuration for both onshore and off-

Chapter 2 HVDC transmission for wind energy

shore applications is the horizontal axis wind turbine with variable speed depicted in Fig. 2.1.

Gearbox Generator β

DFIG

grid

∼ AC DC

DC AC

PMSG

∼

grid AC DC

DC AC

Converter

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.

15

16

Chapter 2 HVDC transmission for wind energy

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

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. A wind turbine does not operate for wind speeds below a value known as vcut-in . 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 vnom , the pitch angle acts to maintain the constant power pnom ; this operation is maintained until wind speed achieves vcut-out (the turbine can be damaged for values above this cut-out velocity, and hence it must be mechanically blocked). Fig. 2.2 constitutes a stationary state model of a wind turbine, since it relates accurately the wind speed with the output power.

2.2 Slow-dynamics model of the wind turbine The dynamic behavior of the converters in a wind turbine is fast in comparison to the dynamics of its mechanical axis. There-

Chapter 2 HVDC transmission for wind energy

Model Wind Random βmax ω0

±

kp

Model Turbine Eq. (2.1)

0

pM

±

1 ωJm

1 S

ω

pE kω3

Figure 2.3. Nonlinear slow-dynamics model for a wind turbine considering the inertia of the turbine and its control.

fore a slow-dynamics model is represented by the axis equation ωJm

dω = Pm − Pe , dt

(2.4)

where Jm is the inertia of the group turbine-generator, ω is the angular speed, Pm is the mechanical torque, and Pe is the electrical torque. It is common in power system applications to represent this equation as a function of the inertia constant H = Jm ω2 /2; however, ω may be highly variable, and hence it is not realistic to assume that H is constant. Therefore the term in the left-hand side of the equation is maintained nonlinear. The electric power is maintained proportional to the cube of the rotational speed to achieve the maximum power extraction from the wind. A simple proportional controller is used to regulate the pitch angle when the wind velocities are higher than its nominal. The model of the turbine is presented in Fig. 2.3. A module for random generation of wind velocities is used; the turbine is represented by (2.1), and the nonlinear dynamics of the axis given by (2.4). Note that this model maintains the main nonlinear performance of the turbine, but it is simple and general enough to represent any wind turbine. The model was evaluated in Open-Modelica, and the results are presented in Fig. 2.4. As a remark, the output power describes a smooth curve even for fast changes of the wind velocity. For completeness, we present the code in Modelica: model Wind Real Vwind, Vrand; equation Vwind = 8 + 6*sin(0.1*time)+Vrand;

17

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Chapter 2 HVDC transmission for wind energy

Figure 2.4. Power output of the wind turbine using the proposed slow-dynamics model.

Vrand = sin(tan(2*sin(2*time))); end Wind; model Turbine Real Cp, lambda, lambdai,beta, Pm, P_available; equation Cp = 0.73*(151/lambdai-0.58*beta-13.2)*exp(-18.4/lambdai); lambdai = 1/(1/(lambda+0.02*beta)-0.003/(beta^3+1)); lambda = (ws/Vwind)*(7.05*12); Pm = (Cp/0.51)*(Vwind/12)^3;

// lambda_opt is 7.05 at 12 m/s // Cp_opt is 0.51 at 12 m/s

P_available = (Vwind/12)^3; end Turbine; model Slow_Model_Turbine extends Wind; extends Turbine; parameter Real Jm = 2; parameter Real kpitch = 400; Real Pm, Pe, ws; initial equation ws = 0.7; equation ws*Jm*der(ws) = Pm - Pe; Pe = ws^3; if ws > 1 then beta = kpitch*(ws-1); else beta = 0; end if; end Slow_Model_Turbine;

2.3 HVDC transmission for wind farms Places with high potential of wind energy are usually far away from large cities in both offshore as onshore emplacements. Therefore HVDC is required to transmit the power generated by the wind farm to the main grid. Two key components are the power electronic converters and their control. Power electronic converters can be classified as line-commutated and force-

Chapter 2 HVDC transmission for wind energy

commutated converters. The former is based on thyristors and constitutes one of the first technologies for HVDC transmission. The later is based on Insulated Gate Bipolar Transistors (IGBT), which allow interrupting the current or the voltage using a modulation strategy. Converters in wind turbines and converters in the HVDC system are usually based on forced commutation. Among force-commutated converters, stand out the voltage-source converter (VSC), the pulse-width-modulated current-source converter (PWM-CSC), and the modular-multilevel converter (MMC). The first and the third are the most common in practical applications. On the other hand, the control strategy for HVDC transmission in wind farms is different from a conventional HVDC system. In a conventional HVDC system, one converter is in charge of controlling the DC-voltage, whereas the other converter controls the active power. In a wind farm the active power is determined by the availability of the wind, and hence it is not controlled by the converter. Therefore the control of the converter in the side of the wind farm must be modified to maintain the voltage in the AC side and supply a reference for the frequency and voltage as shown in Fig. 2.5. This aspect is presented in Section 2.4.

control objectives vDC , Q AC DC

control objectives vAC , f DC AC

Figure 2.5. Control objectives of an HVDC line for wind farms applications. In this configuration, one converter is in charge of the DC voltage, whereas the other converter controls the AC voltage and the frequency in the wind farm.

2.4 Stability of HVDC transmission lines HVDC systems exhibit a remarkable stable behavior despite being a nonlinear time-varying dynamical system with high variability of the generated power. This can be explained by the fol-

19

20

Chapter 2 HVDC transmission for wind energy

r

l

v1

v2 i12 PI

c

p(t) v1

c

offshore

Figure 2.6. Simplified model of an HVDC transmission line for offshore applications.

lowing analysis. Consider the simplified model of an HVDC line depicted in Fig. 2.6. This model considers the main characteristics of an HVDC system: the PI control for the outer loop in the converter onshore, the dynamics of the DC line and, the nonlinear behavior of the generated power offshore. The dynamical model of the system is presented by the equations dv1 p = − i12 , dt v1 dv2 c = i12 − u, dt di12 = v1 − v2 − ri12 , l dt dz = 1 − v2 , dt u = kp (1 − v2 ) + ki z.

(2.5)

c

(2.6) (2.7) (2.8) (2.9)

The system is represented in per unit, and hence v2 must remain close to 1 pu. A new set of state variables is defined as follows: x1 = cv1 , x2 = cv2 , x3 = li12 , x4 = ki z. Then the dynamical system can be represented as ⎞ ⎛ pc2 x˙1 x12 ⎜ x˙2 ⎟ ⎜ ⎜ ⎟=⎜ 0 ⎝ x˙3 ⎠ ⎜ ⎝ 1 x˙4 0 ⎛

0

−1

−kp −1 −ki

1 −r 0

⎞⎛

⎞ ⎛ 0 x1 ⎟⎜ ⎟ ⎜ x ⎟ ki ⎟ ⎜ 2 ⎟ + ⎜ 0 ⎝ ⎠ ⎝ 0 0 ⎠ x3 x4 ki /c 0 0

⎞ ⎟ ⎟ , (2.10) ⎠

Chapter 2 HVDC transmission for wind energy

where x1 > 0 and x2 > 0. The equilibrium point for this dynamical system with constant p is given by the following equations:  

1 (2.11) 1 + 1 + 4prc2 , x¯1 = 2 x¯2 = c, (2.12) lpc , (2.13) x¯3 = x¯1 kp x¯2 x¯3 − . (2.14) x¯4 = c l Therefore it is possible to define an incremental model with state variables x = x − x¯ as given by ⎞⎛ ⎛ ⎞ ⎛ ⎞ −pc2 x1 /c x˙1 0 −1 0 x ¯ ( x ¯ +x ) 1 1 1 ⎟⎜ ⎜ x˙2 ⎟ ⎜ ⎟ 0 −kp 1 ki ⎟ ⎜ ⎟=⎜ ⎜ x2 /c ⎟ . (2.15) ⎜ ⎟ ⎝ x˙3 ⎠ ⎝ ⎝ x3 / l ⎠ 1 −1 −r 0 ⎠ x4 /ki x˙4 0 −ki 0 0 This system defines a port-Hamiltonian system [14], that is, a system of the following structure: x = (J − R)∇H,

(2.16)

where J and R are square matrices given by (2.17) and (2.18), respectively, ⎛ ⎞ 0 0 −1 0 ⎜ 0 0 1 ki ⎟ ⎟, J =⎜ (2.17) ⎝ 1 −1 0 0 ⎠ 0 −ki 0 0 

 pc2 R = diag , kp , r, 0 , x¯1 (x¯1 + x1 )

(2.18)

and H is the Hamiltonian given by H=

x12 x22 x32 x42 . + + + 2c 2c 2l 2ki

(2.19)

Note that J is a skew-symmetric matrix (i.e., J = −J  ), and R is a positive semidefinite matrix. Under these conditions, the equilibrium point is Lyapunov-stable with Lyapunov function H . The point is also asymptotically stable for initial points in the set D = (−x¯1 , ∞] × R3 , as can be easily concluded from LaSalle invariance principle [15].

21

22

Chapter 2 HVDC transmission for wind energy

Figure 2.7. Simulation results for an HVDC line with constant power generated by the wind farm.

The power generated by the offshore wind farm is highly variable. Therefore it is necessary to extend this analysis for the timevarying case, in which the power is given by p = p¯ + p(t) with p(t) ≤  < p¯ (this implies that the power is never negative). Under these conditions, the matrix R is still positive semidefinite, and the system remains Lyapunov-stable. In addition, the following properties are fulfilled for all t ≥ 0: H (0) = 0,

(2.20)

α x ≤ H ≤ β x , H˙ ≤ 0, 2

with

2

1 1 1 , , , α = min 2c 2l 2ki

1 1 1 β= + 2 + 2. 2 2c 4l 4ki

(2.21) (2.22)



(2.23) (2.24)

Therefore the dynamical system is uniformly Lyapunov-stable (see Theorem 4.6 in [15].) Fig. 2.7 shows the dynamical performance of the voltages v1 , v2 and the current i12 for a profile p(t). Initially, p(t) is constant until t = t0 , where p(t) starts varying continuously. In the first case (t < t0 ) the system is asymptotically stable. After that, for t ≥ t0 , the system varies but remains in a welldefined region. Fig. 2.8 shows the output variables for variations of the power generated by the wind farm. For completeness, we present the code in Modelica: model HVDC_wind parameter Real c = 2*0.1;

// capacitor with a inertia of 0.1 s

Chapter 2 HVDC transmission for wind energy

Figure 2.8. Simulation results for an HVDC line with variable power generated by the wind farm.

parameter Real l = 0.1;

// line inductance

parameter Real r = 0.01;

// line resistance

parameter Real kp = 1; parameter Real ki = 0.2; parameter Real Vref = 1; Real x1, x2, x3, x4, p; Real v1, v2, i12, z; initial equation x1 = 1.0*c; x2 = 1.0*l; x3 = 0.0; x4 = 0.0; equation der(x1) = p*c/x1-x3/l; der(x2) = -kp*x2/c + x3/l + x4; der(x3) = x1/c - x2/c -r*x3/l; der(x4) = -ki*x2/c + ki*Vref; x1 = c*v1; x2 = c*v2; x3 = l*i12; x4 = ki*z; if time < 4 then p=1; else p=1.2; end if; end HVDC_wind;

2.5 Summary In this chapter, we have shown a simplified model for an HVDC system for applications of wind farms. This model considers the main nonlinear aspects of both the HVDC line and the wind farm. This model is simple enough to be implemented in practice. A formal demonstration of stability was presented taking into account variations of the wind velocity. A remarkable result is that the system remains stable even for large variations of the wind speed.

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Chapter 2 HVDC transmission for wind energy

All models were simple for implementation in Modelica. However, these models can be extended to analyze more complex systems.

References [1] J.L. Peters, T. Remmers, A.J. Wheeler, J. Murphy, V. Cummins, A systematic review and meta-analysis of GIS use to reveal trends in offshore wind energy research and offer insights on best practices, Renewable and Sustainable Energy Reviews 128 (2020) 109916, https:// doi.org/10.1016/j.rser.2020.109916. [2] J. Ladenburg, P. Hevia-Koch, S. Petrovi´c, L. Knapp, The offshore-onshore conundrum: preferences for wind energy considering spatial data in Denmark, Renewable and Sustainable Energy Reviews 121 (2020) 109711, https://doi.org/10.1016/j.rser.2020.109711. [3] K.Y. Oh, W. Nam, M.S. Ryu, J.Y. Kim, B.I. Epureanu, A review of foundations of offshore wind energy convertors: current status and future perspectives, Renewable and Sustainable Energy Reviews 88 (2018) 16–36, https:// doi.org/10.1016/j.rser.2018.02.005. [4] X. Costoya, M. deCastro, D. Carvalho, M. Gómez-Gesteira, On the suitability of offshore wind energy resource in the United States of America for the 21st century, Applied Energy 262 (2020) 114537, https:// doi.org/10.1016/j.apenergy.2020.114537. [5] A. Colmenar-Santos, J. Perera-Perez, D. Borge-Diez, C. de Palacio-Rodríguez, Offshore wind energy: a review of the current status, challenges and future development in Spain, Renewable and Sustainable Energy Reviews 64 (2016) 1–18, https://doi.org/10.1016/j.rser.2016.05.087. [6] W. Al-Nassar, S. Neelamani, K. Al-Salem, H. Al-Dashti, Feasibility of offshore wind energy as an alternative source for the state of Kuwait, Energy 169 (2019) 783–796, https://doi.org/10.1016/j.energy.2018.11.140. [7] C. Emeksiz, B. Demirci, The determination of offshore wind energy potential of Turkey by using novelty hybrid site selection method, Sustainable Energy Technologies and Assessments 36 (2019) 100562, https:// doi.org/10.1016/j.seta.2019.100562. [8] M.S. de Sousa Gomes, J.M.F. de Paiva, V.A. da Silva Moris, A.O. Nunes, Proposal of a methodology to use offshore wind energy on the southeast coast of Brazil, Energy 185 (2019) 327–336, https:// doi.org/10.1016/j.energy.2019.07.057. [9] J.G. Rueda-Bayona, A. Guzmán, J.J.C. Eras, R. Silva-Casarín, E. Bastidas-Arteaga, J. Horrillo-Caraballo, Renewables energies in Colombia and the opportunity for the offshore wind technology, Journal of Cleaner Production 220 (2019) 529–543, https:// doi.org/10.1016/j.jclepro.2019.02.174. [10] O. Pupo-Roncallo, J. Campillo, D. Ingham, K. Hughes, M. Pourkashanian, Large scale integration of renewable energy sources (res) in the future Colombian energy system, Energy 186 (2019) 115805, https:// doi.org/10.1016/j.energy.2019.07.135. [11] Y. Li, X. Huang, K.F. Tee, Q. Li, X.P. Wu, Comparative study of onshore and offshore wind characteristics and wind energy potentials: a case study for southeast coastal region of China, Sustainable Energy Technologies and Assessments 39 (2020) 100711, https://doi.org/10.1016/j.seta.2020.100711. [12] M.D. Lellis, R. Reginatto, R. Saraiva, A. Trofino, The Betz limit applied to airborne wind energy, Renewable Energy 127 (2018) 32–40, https:// doi.org/10.1016/j.renene.2018.04.034.

Chapter 2 HVDC transmission for wind energy

[13] J.G. Slootweg, S.W.H. de Haan, H. Polinder, W.L. Kling, General model for representing variable speed wind turbines in power system dynamics simulations, IEEE Transactions on Power Systems 18 (1) (2003) 144–151. [14] A. Van der Schaft, D. Jeltsema, Port-Hamiltonian Systems Theory, Now Publishers, 2014. [15] H. Wassim, S.C. Vijaya, Nonlinear Dynamical Systems and Control, Princeton University Press, 2011.

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3 DC faults in HVDC✩ Detection, identification, and handling Raymundo E. Torres-Olguin SINTEF Energy, Department of energy systems, Trondheim, Norway

Chapter points • Multiterminal HVDC. • Analysis of DC faults. • Detection of DC faults.

In the last decade, multiterminal HVDC (MTDC) systems, HVDC systems consisting of three or more terminals, have gained much attention. All the advancements of power electronics in the last 50 years has made possible the implementation of MTDC using mainly voltage-source converters (VSCs). A VSC-based MTDC represents a potential solution to interconnected multiple AC systems for transfer of bulk power over long distances with significant technical and economical advantages [1,2]. An MTDC system can offer several advantages; among them: • MTDC can facilitate power trading between remote market areas [3]. • MTDC can be used for exchanging ancillary services between more than three remote AC areas [3]. • This solution suits certain applications such as the large-scale integration of remote renewable sources or the interconnection of asynchronous AC regions [3]. • MTDC can increase the reliability of power transmission. If MTDC is connected in meshed configuration, then MTDC can offer alternative paths in critical situations [3]. Among the advantages mentioned, one of the most desirable features of meshed MTDC systems is the ability to increase the reliability of power transfer to abnormal events [4]. One of the most ✩ The information collected and material developed here was part of “Protection

and Fault Handling in Offshore HVDC Grids” (ProofGrids), funded by Research Council of Norway and associated industry partners. Modeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00008-3 Copyright © 2021 Elsevier Inc. All rights reserved.

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Chapter 3 DC faults in HVDC: detection, identification, and handling

challenging abnormal events is DC faults in VSC. When a DC fault occurs in an MTDC system, there is a rapid fault propagation to the rest of the system due to the low impedance. A proper overall protection system is crucial to provide alternative transmission paths in MTDC [5]. Therefore the protection system for MTDC plays a fundamental role in minimizing the impact of DC faults and guarantees reliable power transfer in MTDC systems [6]. In this chapter, we explore the minimum requirements in Section 3.1 and the DC fault characteristic in an MTDC in Section 3.2.

3.1 Minimum requirements for the protection system of MTDC VSC-based HVDCs are the trend for MTDCs because of many their advantages compared to LCC technology [7]. However, VSCs are vulnerable to DC faults. So, the protection system is one of the major issues for realization of MTDC systems. There are certain characteristics that the protection system must meet: • Fast response. DC transmission lines and cables are characterized by the small series impedances. These small series inductances are unable to limit the rise time of the fault current. So, a protection system for HVDC applications requires strict specifications in terms of opening time to prevent large overcurrents during faults. • Selectivity. The protection system must identify and isolate only the faulted DC line. Moreover, fault detection algorithms must be based on local measurements. • Sensitivity. The protection system should be able to discriminate between normal operation or external disturbances and DC faults. • Reliability. The protection system should have a backup system in the case of primary protection system fails. The overall DC fault protection for MTDC includes several steps. Fig. 3.1 shows a schematic diagram of the fault clearing process, and below each of them is described. First, the process starts when a DC fault occurs. Before that, there is a normal operation monitoring. At normal conditions the measurement equipment should have enough small sampling time to guarantee the minimum data for accurate fault detection. After the fault, the fault propagates to the sensor. It takes only a few microseconds. Then DC fault detection is performed. A fast fault detection method is needed, preferably with no communications. After DC fault is detected, the identification algorithm allows determining the faulted line, whereas the healthy ones are discriminated. Once the fault

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.1. Block diagram of the fault clearing process [8].

is detected and located, the circuit breakers can open and isolate the faulted lines. After the faulted current decays to zero, isolators are operated to isolate the faulted line. It takes two milliseconds to complete this process using the latest technology. The whole process should take three or four milliseconds; if the isolation process takes two milliseconds, the detection and location algorithm must take one or two milliseconds. Therefore fault has to be detected and isolated in the transient phase of the current (or voltage). This feature makes the problem of detecting and isolating DC faults highly challenging.

3.2 Impact of DC faults in VSC A DC fault is the most serious condition for VSC [9]. DC faults may occur in different parts of the system: inner-convert faults, DC cable faults, and junction parts. Faults at the cable occur more frequently, and they are commonly related to insulation deterioration and breakdown or due to external causes primarily from maritime activities, for example, anchoring and fishing [10]. Regardless of the number of terminals of an HVDC system, three types of DC faults may occur at the DC cables: (i) pole-to-pole (positive-tonegative pole) (ii) positive pole to ground, and (iii) negative pole to ground.

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Chapter 3 DC faults in HVDC: detection, identification, and handling

Pole-to-pole faults are very unlikely to happen, especially in submarine cables, since they are related to physical damage, whereas pole-to-ground faults are more likely since they are related with insulation deterioration of the cable. However, any kind of fault has to be handled by the protection system. The following aspects regarding DC faults in VSC are important to emphasize: • The voltage drop due to faults is spread much faster in the DC grid than AC grids. This is due to the relatively small inductance at the DC side, so a very high rate-of-rise of DC current is expected. Therefore DC protections must be faster than for AC protections. The detection should be within a few milliseconds of the fault occurrence. • Existing protection methods used for LCC-HVDC cannot necessarily be used for VSC HVDC since LCC is a current source converter with fundamentally different operating principles than VSC [4,5]. • IGBTs can break the current when a faulted condition is detected (e.g., overcurrent and undervoltage). However, the antiparallel diodes cannot break the current and will continue conducting the fault current. To understand the response of a VSC during DC faults, Fig. 3.2 shows the equivalent circuits to study the fault current path for the different periods of a short circuit on the DC transmission cable. Each period is described as follows. 1. Phase 1: Capacitor discharge phase. Immediately after the fault has occurred, the DC capacitor(s) on the converter terminals is(are) discharged. The equivalent circuit, shown in Fig. 3.2(a), can be written as a second-order differential equation, which gives an oscillatory solution if R < 2(L/C)1/2 [11]. The magnitude and steepness of the first current peak are determined by the DC voltage and the cable resistance and inductance. 2. Phase 2: Freewheel diode phase. In this phase the capacitor voltage is zero, and the cable is discharged through the antiparallel diodes in the converter bridge. If a fault occurs on the DC-side of the converter, and the IGBTs are blocked, then the current can still flow through the antiparallel diodes. This is the most challenging phase for freewheel diodes since the overcurrent can be very large because the current in the cable inductor cannot change instantaneously when the DCcapacitor voltage goes to zero. 3. Phase 3: Grid feeding phase. Finally, in the third phase the fault is fed from the AC-grid through the diodes. In this phase the current depends on the short circuit capacity of the grid, the

Chapter 3 DC faults in HVDC: detection, identification, and handling

31

Figure 3.2. Equivalent circuits for a fault on DC-side during (a) capacitor discharge phase, (b) freewheel diode phase, and (c) grid feeding phase.

filter impedance, the DC-cable impedance, and the fault resistance.

3.3 Analysis of the MTDC-HVDC during DC faults In this section, we analyze the transient characteristic during a pole-to-ground DC fault. The main objective is providing a better understanding of the transient of DC fault. A better understanding of the transient characteristic during DC faults allows designing a proper fault detection mechanism and gives the minimum requirements for DC circuit breakers. A DC grid test system was proposed by the Cigre working group B4.52, which is called the “HVDC grid feasibility study”. The system consists of: • Two large onshore AC systems, system A (A0 and A1) and system B (B0, B1, B2, and B3) • Four independent offshore AC systems, which represent either a remote renewable energy source or remote load. • Two DC nodes, B4 and B5. • Three interconnected DC systems, DCS1, DCS2, and DCS3. Fig. 3.3 shows a schematic of the test system, which includes the buses, converters, and all line lengths in km. First, regarding the AC systems, AC system A has two buses interconnected by overhead lines. System A connects converters and one slack bus at each end. It is expected that in system A, there is a power surplus, and then its main objective is to export the electric power to the rest of the system. System B has four buses. One bus, B0, represents a slack bus, whereas the rest are connected to power converters. It is pretended that system B imports the electric power from the rest of the system. There are four independent AC grids. Systems C, D, and F correspond to remote renewable sources like

32

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.3. Reference DC grid test system proposed by Cigre.

offshore wind farms, and System E corresponds to one large load, like and offshore oil and gas platform. Second regarding the DC systems, there are three main DC subsystems: DCS1, DCS2, and DCS3. DSC1 is a point-to-point monopole HVDC link. It connects the offshore wind park C1 to AC system A. DCS2 is a 4-terminal monopole HVDC link. It connects the converters F1 and E1 to system B. System B consists of overhead lines and cable in series. DCS3 is a 5-terminal bipolar system that connects systems C, D, and E to both systems A and B in a ring configuration. It contains both transmission lines and cables. Additionally, DCS3 contains DC-DC converters, and the main function is as DC transformers, which are able to control power flow as well. The DC grid test system contains both monopolar and bipolar configurations. Fig. 3.3 shows monopolar configuration in light blue (light gray in print version) and the bipolar in dark blue (gray in print version). Overhead lines are solid lines and cables dashed lines. Each power converter is a half-bridge modular multilevel converter (MMC) with the converter arm inductance of 15% and

Chapter 3 DC faults in HVDC: detection, identification, and handling

Table 3.1 Converter parameters. Parameters

Converters A1, B1, B2, A1, B2, C1, C2 B3 D1, F1

pu E2

Power rated

1200 MVA

800 MVA

400 MVA

200 MVA

1

Converter reactor

33 mH

49 mH

98 mH

196 mH

0.25

Converter resistance 0.403 

0.605 

1.21 

2.42 

0.01

Capacitance

300 µF

150 µF

75 µF

60 ms

450 µF

transformer inductance of 18%. The main parameters are presented in the following Table 3.1. In [12], all converters used average models. This kind of models suit for power flow analysis and other studies with sufficient accuracy. However, this research is focused on the behavior of the system when a DC fault occurs. So, detail electromagnetic transients are needed. The AC systems are represented by equivalent AC sources with corresponding impedance for the generators and dependent source currents for the loads or renewable energy sources. The MMC control system contains two control levels: upperlevel controllers and lower-level controllers as specified in [12]. First, upper-level controllers are in charge of the primary objectives such as regulating DC voltage or active power. They use a vector control strategy, which divided the controller into two loops, the inner loop and outer loop. The inner loop controller regulates the current at the AC side of the converter, whereas the outer controller regulates DC voltage or active power, and AC voltage or reactive power depending on the control objectives. In this case, converters A1, A1-m, B1, B2, B3, and B2-m regulate the DC voltages, A1 regulates the active power, and C1, C2, D1, E1, and F1 are controlled in island mode, that is, they regulate AC voltage magnitude and frequency. The vector control strategy uses a synchronous reference frame, also called the Park transformation, to facilitate the control design. Second, lower-level controllers are in charge of the internal objectives in the MMC such as the circulating current suppression, voltage balancing, and firing signals. A voltage droop controller has been considered. The active power between systems A and B has been distributed using a droop control.

33

34

Chapter 3 DC faults in HVDC: detection, identification, and handling

Table 3.2 Data for the transmission line and cables.  R [ km ]

L [ mH ] km

μF C [ km ]

DC OHL ± 400 kV

0.0114

0.9356

0.0123

DC OHL ± 200 kV

0.0133

0.8273

0.0139

–

3000

DC cable ± 400 kV

0.0095

2.1120

0.1906

0.048

2265

DC cable ± 200 kV

0.0095

2.1110

0.2104

0.062

1962

AC cable 145 kV

0.0843

0.2526

0.1837

0.041

715

AC cable 380 kV

0.020

0.8532

0.0135

–

3555

Line/cable

μS G [ km ] Max. current [kA]

–

3500

The test system includes AC and DC cables and overhead lines. Table 3.2 summarizes the proposed parameters. The transmission and line parameters suggest that the proposed models are Bergeron. These models represent L and C in a distributed manner. The main limitation is that this model can represent the system parameters at the chosen frequency, usually the fundamental. The main focus of this research is to study the DC grid under fault, so AC generators are simplified. So, all AC systems are emulated by an equivalent AC source voltage of 380 kV, which operates at 50 Hz. It is assumed that all AC systems are relatively strong systems with the corresponding short circuit ratio of 20. In this section, we analyze pole-to-ground faults in a bipolar three-terminal configuration. All the converters are interconnected using cables. Pole-to-ground faults are mainly investigated since they are more likely to happen in a system with cables compared with pole-to-pole faults.

3.3.1 Steady state in MTDC We performed the following simulations using a subsystem consisting of three converters. Basically, two remote generation units, converters C and E in Fig. 3.4, provide energy to system A. So, converters C and E work in island mode, that is, they control the AC voltage magnitude and frequency to transfer automatically the power produced by the generation units. In this case, it is considered that offshore wind farms are installed as generation units. They are modeled using a Thevenin equivalent, that is, dependent source currents. All results presented in this chapter can be extended easily to the whole system, but for simplicity, we show only the three-terminal case.

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.4. Proposed simplified Cigre benchmark.

We give some remarks for this configuration: (i) The cable lengths were modified. All cables have the same length for explanatory purposes; (ii) The overhead line are substituted by cables; (iii) The AC system is simplified as Thevenin equivalent. The AC overhead line are discarded; and (iv) Only one converter is controlling the DC-voltage. Fig. 3.4 shows the energization and steady-state behavior of the three-terminal system. Converter A1 controls the DC voltage. The setpoint is 800 kV pole-to-pole. Converters C1 and D1 control the active power that is injected into the system. Here, the setpoints are set at 500 MW and 1000 MW, respectively. First, all converters are disabled. At 0.2 s, the converter A is enabled, so it is able to regulate the DC voltage as shown in Fig. 3.4(b). At t = 0.4 s, both converters C1 and D1 are enabled, so they can regulate the AC voltage and frequency, that is, they work in island mode. After this, all the power generated in the generation units can be transmitted automatically. During steady-state, Fig. 3.4(a) displays the active powers. Converters C1 and D1 reach their respective references, 500 MW and 1000 MW. The power at converter A1 is about 1500 MW. Fig. 3.5(b) shows that DC voltages A1, C1, and D1 are well regulated to 800 kV pole-to-pole. Fig. 3.5(c) shows the AC voltages that are set at 380 kV in all the buses. Fig. 3.6 shows the current at different locations in the three terminal subsystem based on Fig. 3.4. Note that the current is circulating mainly in cables 1 and 2, and the current circulating for

35

36

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.5. Steady-state response for the three-terminal system (from top to bottom): (a) active powers, (b) DC voltages, and (c) rms AC voltages.

cable 3 is almost zero. This is the expected behavior, because the converters C1, and D1 interconnect the generation units through cables 1 and 2. Cable 3 represents a redundancy in the system, so in normal conditions, it is not transferring active power.

3.3.2 Fault transient Fig. 3.5 shows the transient behavior of the three-terminal system during a pole-to-ground fault. The fault occurs at 10 km from converter A over cable 1. Fig. 3.7(a) displays the active powers.

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.6. Time response of the current for the three-terminal system.

We can notice that the active power at converter A1 drops to zero since the fault is close to the converter that regulates the DC voltage. The active power at converters C1 and E1 take longer to drop since they are located far from the fault. Fig. 3.7(b) shows the DC voltages at A1, C1, and D1. The DC voltage at converter A drops faster than C and E since the fault is located closer to converter A1. The voltage drops to half the value of their nominal value, as is expected for a pole-to-ground fault in a bipolar configuration. Fig. 3.7(c) shows the AC voltages. Converters C and E work in island mode, so they regulate the AC voltage magnitude and frequency. Therefore it is expected that converters C and E are unable to regulate the AC voltage magnitude as shown in Fig. 3.7(c).

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38

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.7. Fault transient response for the three-terminal system (from top to bottom): (a) active powers, (b) DC voltages, (c) rms AC voltages.

Fig. 3.8 shows the current at different locations in subsystem 1 during a pole-to-ground fault. The fault is simulated in cable 1, and the distance is relative to the position of converter A1. Observe the following characteristics: • The maximum overcurrent occurs close to the converters; e.g., the current at A1 (C1) when the fault occurs at 1 km or at C1 (A1) when the fault occurs at 199 km, as shown in Fig. 3.8.

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.8. Fault transient response of the current for the three-terminal system.

• A huge overcurrent occurs in the adjacent relay, in this case, A1 (E1). Observe that the direction of the current in A1 (E1) is opposite to A1 (C1). • The current has forward direction in the affected cable, in this case, cable 1; see A1 (C1) and C1 (A1) in Fig. 3.8. • The current at E1 (A1) has almost the same magnitude and direction compared with C1 (A1) when the fault occurs close to convert A (at 1 km). The same happens between A1 (C1) and E1 (C1). We can asseverate that there is a problem to identify the faulted cable when the fault occurs close to one of the converters. • The currents at C1 (E1) and E1 (C1), that is, in cable 3, have a small magnitude compared with cables 1 and 2. The direction is not well defined.

39

40

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.9. Fault transient response of the DC voltages for the three-terminal system.

• The corresponding voltage at each terminal is shown in Fig. 3.9.

3.3.3 Critical interruption time In this section, we illustrate the influence of key parameters in transients when pole-to-ground DC faults occur. The selected test system is the so-called subsystem 1; see details in the previous section. The system consists of three half-bridge modular multilevel converters in a bipolar configuration, which interconnects two offshore AC systems (C1 and E1) with one onshore system

Chapter 3 DC faults in HVDC: detection, identification, and handling

(A1) using a DC grid configuration. Each MMC is modeled using a switching model and contains 98 levels each. The converters are connected through three long cables, that is, of 200 km. Cables were modeled using a frequency-dependent (phase) model available in PSCAD. At each cable terminal, there are protection relays where all measurements are taken. It is assumed that all AC systems are relatively strong systems with a corresponding short circuit ratio of 10. All AC systems are simulated by equivalent AC sources of 380 kV at 50 Hz. The VSC-based HVDCs are made using insulated-gate bipolar transistors (IGBT). These kinds of devices cannot conduct in a reverse direction, so an additional diode, which is named freewheeling diode (FWD) or antiparallel diode, is placed in parallel with the IGBT to be able to conduct current in the reverse direction. When a DC fault occurs, IGBTs are blocked, and the FWDs are suddenly exposed to high currents. The thermal stress introduced by the large overcurrent will potentially damage the device permanently due to the thermal fatigue that is accumulated. This stress can be measured in terms of the accumulated energy during a period and is generally referred to as I 2 t, which is calculated as follows:  2 I 2 t = Idiode t, (3.1) where Idiode is the current flowing in the diode. I 2 t can be used as an indicator of converter failure, that is, when I 2 t, calculated by numerical simulations, exceeds its specified limit given by the manufacturer in the data sheets, then a converter failure can be expected. The critical time for clearing the fault can be estimated using the period between the start of the fault and when the thermal limit is reached [13]. The influence of the key parameters is studied estimating I 2 t. Table shows the rating of range in different devices, which are suitable for HVDC applications [13]. Fig. 3.10 shows the estimated I 2 t when the fault occurs at 1 km from converter A1. The rms value of the line current Is is estimated using the DC current Idc , which is available from the simulations. So, in the idealized case, rms value can be calculated as follows:  2 Is = Idc . (3.2) 3 I 2 t is estimated using the sum of the trapezoidal integration of I 2 t. Fig. 3.10 indicates when the limit is exceeded. In this case the limit is set at 500 kA2 s, gives a critical time of 1.54 ms. It should be

41

42

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.10. Estimation of

 2 I t from diode currents in case of a fault located in cable 1 in subsystem 1.

noted that the critical time is not so small for the other power stations. If the limit is set at 900 kA2 s, the critical interruption time is 1.92 ms. Fig. 3.10 shows the estimation of the critical time in the other terminals; the critical interruption time is greater than 10 ms as shown in Fig. 3.10.

3.3.4 Influence of the DC capacitor on the critical interruption time Fig. 3.11(a) shows that the influence of relatively small equivalent capacitance on the critical interruption time. The equivalent capacitance is 700 µF. Two values have been tested for low filter

Chapter 3 DC faults in HVDC: detection, identification, and handling

Table 3.3 IGBT rating of different manufactures. Ic

Ipeak

I 2t

[V]

[A]

[A]

[kA2 s]

Semikron

1200

1500

10200

520

ABB

3300

1500

13500

911

Infineon

3300

1500

12000

730

Dynex

3300

1500

12000

720

Manufacturer Vce

capacitance: one-third of the equivalent capacitance, 200 µF, and one-sixth of the equivalent capacitance, 100 µF. As indicated in Fig. 3.11(a) the I 2 t decreases as the equivalent capacitance decrease. So, the critical interruption time increases with the equivalent capacitance in the MMC, i.e. from 1.52 ms to 1.72 ms with one-third of the capacitance, and from 1.52 to 2.25 ms with onesixth of the capacitance. Notice that Fig. 3.11(a) shows two critical interruption times, one is calculated using the 520 kA2 s, and the second one is calculated using the 911 kA2 s which were described in Table 3.3. Fig. 3.11(b) shows that the influence of relatively large equivalent capacitance on the critical interruption time. The equivalent capacitance is 700 µF. Two values have been tested for low filter capacitance: three times the equivalent capacitance, 2.1 mF, and six times the equivalent capacitance, 4.2 mF. As shown in Fig. 3.11(b), I 2 t increases as the equivalent capacitance increases. Therefore the critical interruption time decreases with the equivalent capacitance in the MMC. As in the previous figure, Fig. 3.11(b) shows two critical interruption times, one is calculated using 520 kA2 s, and the second one is calculated using 911 kA2 s, which were described in Table 3.3.

3.3.5 Influence of the DC smoothing inductance on the critical interruption time DC smoothing inductances serve mainly to reduce the circulating current in the MMC. However, they can limit the rate of rising of the fault current. Fig. 3.12(a) shows the effect of different smoothing inductances: 10 mH, which is the base case, one-third of the base case, which is 3.33 mH (see Fig. 3.12(a)), and one-sixth of the base case, which is 1.66 mH. As indicated in Fig. 3.12(a), the rate of rising increases as the value of the inductors decreases.

43

 2 I t.

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.11. Influence of the DC capacitor on estimation of

44

Figure 3.12. Influence of the DC reactor on estimation of

 2 I t.

Chapter 3 DC faults in HVDC: detection, identification, and handling

45

 2 I t.

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.13. Influence of the SCR on estimation of

46

Chapter 3 DC faults in HVDC: detection, identification, and handling

Therefore it is expected that the critical interruption times decreases as the inductors decrease as well as shown in Fig. 3.12(a). The minimum critical interruption time is 1.52 ms, calculated using the I 2 t of 520 kA2 s. This time is reduced to 0.7 ms when the inductor is one third of the base case and is reduced to 0.4 ms when is one sixth of the base case as indicated in Fig. 3.12(a).

3.3.6 Influence of the short circuit ratio of the AC system Fig. 3.13(a) illustrates the influence of relatively large short circuit ratio (SCR) on the critical interruption time. Fig. 3.13(a) is the base case, then SCR increases to 20 in Fig. 3.13(b), and SCR is 30 in Fig. 3.13(c). As the figure indicates, the influence in the critical interruption time is negligible when the SCR is increased. Fig. 3.13(b) shows the influence of a relatively small short circuit ratio (SCR) on the critical interruption time. Fig. 3.13(b) shows when SCR is 5 and critical time remains the same as in the base case. However, when SCR is 2.5, then the critical interruption time is decreased. All these suggest that a higher SCR has little influence in the critical interruption time; however, a low SCR can reduce the critical time substantially as is shown in Fig. 3.13(b).

3.3.7 Influence of the fault resistance on the critical interruption time Fig. 3.14 illustrates the influence of fault impedance on the critical interruption time. Fig. 3.14(a) is the base case, the fault resistance is quite small. Fig. 3.14(b) shows the effect of increasing the fault resistance up to 0.1 . There is little effect on the critical interruption time. Fig. 3.14(c) shows the effect of increasing the fault resistance up to 1 . It results in that the critical interruption time is slightly increased. Fig. 3.14(c) shows the results when the fault resistance is 10 , then is clear that the interruption time is drastically increased as shown in Fig. 3.14.

3.3.8 Remark of the section This subsection has illustrated the steady state and transient during a fault in a simple meshed three-terminal HVDC system. It has given a remark to the estimation of critical interruption time to evaluate the thermal capacity of the power electronic devices subjected to DC faults. Some measurements to increase the critical interruption time are the following: • The reduction of the equivalent DC capacitor size.

47

48

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.14. Influence of fault resistance on the estimation of resistance of 1 , (d) fault resistance of 10 .

 2 I t: (a) base case, (b) fault resistance of 0.1 , (c) fault

• The increase of the smoothing reactor in the MMC. • SCR has a minor effect on the critical interruption time. • The increase in the fault resistance leads to higher critical interruption times.

3.4 Detection and identification strategies in MTDC The main challenge for detection and identification of the fault is to produce a fast algorithm able to provide high selectivity on

Chapter 3 DC faults in HVDC: detection, identification, and handling

faults within its protection zone and discriminate faults outside its protection zone [6]. The operation speed of a few milliseconds is required to minimize the effect on the sensitive power electronic equipment of MTDC systems [6]. This is due to expected high DC fault currents because of the transmission line and cable characteristics, as was mentioned before [11].

3.4.1 Selectivity problem Two cases are shown to illustrate graphically the selectivity problem in the MTDC systems. Subsystem 1 is used as a test system. Briefly, the system consists of three half-bridge modular multilevel converters in a bipolar configuration that interconnects two offshore AC systems (C1 and E1) with one onshore system (A1) using a DC grid configuration. The converters are connected through three long cables: cable 1 with 400 km, and cables 2 and 3 with 200 km. Cables have been modeled using a frequencydependent (phase) model available in PSCAD. At each cable terminal, there are protection relays where all measurements are taken. It is assumed that all AC systems are strong systems with a corresponding short circuit ratio of 20. All AC systems are simulated by equivalent AC sources of 380 kV at 50 Hz. Fig. 3.15 shows DC currents seen by the protection relays when a pole-to-ground DC fault occurs at 2.7 s. First, the fault is located at 2 km from converter A1 (see Fig. 3.4) over cable 1, which is shown in solid line in Fig. 3.15, and second, the fault is applied at 2 km from converter A1 over cable 2, which is displayed in dashed line in Fig. 3.15. In the protection relays that are close to the fault, that is, A1 (C1) and A1 (E1), it is easy to determine that a fault has occurred using either the direction or magnitude as seen in Fig. 3.15. The problem is to determine the fault location in the protection relays far away from the fault. Fig. 3.15 shows the current seen by relays C1 (A1) and E1 (A1). Either magnitude or direction is almost the same when the fault is located either in cable 1 or cable 2. Therefore it is useless to use magnitude or direction-based location techniques, such as overcurrent protection or directional overcurrent protection.

3.4.2 Proposed detection and location methods for MTDC The traditional fault detection algorithms in AC grids cannot be applied directly in DC grids. Some detection and identification methods have been previously proposed in the literature for MTDC [14–18]. A brief description of each of them is given further.

49

50

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.15. Performance of the reference system under a fault at 2 km in cable 1 (solid line) and at 2 km in cable 2 (dashed line) near converter A1. From top to bottom: DC current at A1 (C1), DC current at C1 (A1), DC current at A1 (E1), and DC current at E1 (A1).

Chapter 3 DC faults in HVDC: detection, identification, and handling

3.4.2.1 Overcurrent protection and undervoltage DC voltage level protection Overcurrent and undervoltage protections are the most straightforward possible solutions. They are based on detecting if the current exceeds a predetermined threshold or if the voltage is below a set threshold. The method is quite effective for point-to-point connection; however, the selectivity is limited for MTDC. Overcurrent protection scheme is the simplest but the least selective solution. Directional protection can provide more selectivity since its operation depends upon the direction of the fault the current with respect to the voltage. However, some breakers near to the fault may be tripped unnecessarily as well. Nevertheless, overcurrent may be used as backup protection scheme. For DC grids, overcurrent and undervoltage protection may have some limitations: 1. Lack of selectivity since many breakers near to the fault may be tripped unnecessarily even using directional protection. 2. High-impedance fault may not detect by this scheme [19]. 3. The rate-of-change of this scheme is considered instantaneous but cannot be made sensitive enough to cover the whole line length without making it too prone to unnecessary operations [20]. 4. The reliability of this scheme is subjected to noise and incorrect sample data.

3.4.2.2 Differential current protection In this method the so-called differential current is calculated, which is simply the sum or the difference of the current at each cable end. Following the Kirchhoff law, when the link is healthy, the differential current has a very low value. When a fault occurs, the differential current will have a large value. This method can provide good reliability and protection coverage for MTDC. However, for DC grids, it may have some limitations: 1. It requires high-speed and high-quality communications. The cost of providing this kind of communication infrastructure is very high and in many cases is not justified. 2. The signal propagation delay due to the communications may influence the response time of this protection method. Propagation delay has to be considered. 3. The long line length of HVDC cables may introduce error in the measured currents due to the charging and discharging currents caused by any voltage variation.

51

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Chapter 3 DC faults in HVDC: detection, identification, and handling

3.4.2.3 Traveling waves This method is based on the traveling wave theory. This theory is based on the fact that any disturbance on a transmission line produces traveling waves along the transmission line. These traveling waves are the result of charge and discharge of the line capacitance and line inductance of the transmission line. Each wave is a composite of frequencies, from a few kilohertz to several megahertz, which has a propagation speed near the speed of the light. Traveling wave-based methods require measurements of the arrival time of fault generated surges at the terminals to determine the distance to the fault. Traveling wave can be monitored using the transient voltage and current signal on one bus. In AC transmission lines the conventional current and voltage transformers give simple and cost-effective measurements. In DC transmission lines the current and voltage transformers are behind harmonic filters, since they cannot provide the output for detecting traveling waves. Global positing systems (GPS) can provide an accurate measurement of the arrival time to determine the distance to the fault. Traveling wave-based protection is well suited for HVDC lines due to the fact that a DC line has a simple structure and the converters reflect waves instead of refracting waves [21]. A traveling wave-based fault location scheme is presented in [15]. This technique is based on the arrival times at different points within MTDC. However, this method requires fast communications. A wavelet-based method without communication is proposed in [16]. Three different selection criteria are used and demonstrated by numerical simulation. However, the selectivity is not fully demonstrated. The traveling wave-based methods have some limitations: 1. The key to traveling wave fault location is the detection of the wave head. This method fails if the wave head is not detected. Traveling wave signals become weak when the line is grounded through a large resistor or by a fault caused by a gradual change in the transition resistance. A weak signal may be not detected [22]. 2. The accuracy of the method highly depends on the parameters of the line since the fault distance is the product of the wave head arrival time and the wave speed [22]. 3. The method depends on a high sampling frequency. Since the wave travels almost to the light speed, a very high sampling frequency is needed. The fault locator requires a sampling time not less than 1 MHz, the higher the sampling frequency of the input signal, the more accurate the result [23]. 4. This method is vulnerable to interface signals [22].

Chapter 3 DC faults in HVDC: detection, identification, and handling

3.4.2.4 Based on rate of change In [18] fault location is based on the derivatives of DC currents. Both the sign and magnitude are considered. The fault can be located and detected quickly without using communications. In [24] a rate of change of DC voltage is proposed for detection of DC faults. Other variation is the change of rate in the DC reactor voltage proposed in [25], where the voltage across the DC reactor is monitored. This voltage is near zero in steady-state operation, but it increases significantly during faults. However, in general, the main disadvantage of all these methods is that the derivative terms can easily amplify the noise from the signals.

3.4.2.5 Other methods A fault location based on electromagnetic time reversal is presented in [17]. This method estimates the distance to the fault, but it requires communication to give selectivity in the case of MTDC systems. In [26] a detection method based on the high-frequency content in the current is proposed. The short-time Fourier transform is used to extract the high-frequency content. The method is able to detect DC fault; however, the selectivity was not tested completely. A distance-based detection method is proposed in [14]. However, this method is more suitable for low-power applications. Recently, in [4] a nonpilot protection scheme based on the voltage magnitude and derivative is proposed. [4], [18]. and [24] rely on the use of inductor terminations to achieve the selectivity in the system. The next section is focused on the clearance strategies.

3.5 Clearance strategies for MTDC Once the DC fault is detected, the protection system must isolate the faulted line. Current methods to interrupt DC faults can be classified as: • Protection system with AC breakers. • Protection system with DC breakers. • Protection system embedded on the power converter. In what follows, we discuss in more detail each of the methods mentioned above.

3.5.1 Protection system with AC breakers The existing DC fault protection system for VSC-based HVDC normally uses AC breakers to disconnect faulted lines [27,28]. AC circuit breakers are placed on each AC side of VSCs as shown in

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Figure 3.16. Handshaking method: (a) AC breakers are opened, (b) DC breakers are opened, (c) faulted line is isolated, and (d) DC breakers of healthy line are closed.

Fig. 3.16. During a DC fault, all circuit breakers are opened to clear the DC fault. Since AC circuit breakers are available to handle AC faults, this approach is the most simple and economical method to protect against DC faults. However, in a multiterminal configuration the entire system is disconnected each time a single fault occurs at any line. Once all AC breakers are opened, it may take up to seconds to restart the entire system [29,30]. Therefore this method has a lack of selectivity since it is not possible to isolate a single faulted line [31]. The protection system using AC breakers provide more selectivity with the handshaking method presented in [32,33]. This method uses AC circuit breakers together with fast DC switches instead of DC breakers. A handshaking method is summarized as follows: 1. Once a DC fault is detected, AC breakers are opened. At the same time, all IGBTs in the VSCs are blocked. The potential faulted DC line is identified. 2. Once the fault current decays to zero, fast DC switches of the faulted line are opened. The fault line is isolated opening its ends. 3. Once the fault is isolated, all AC breakers are reclosed. The DC capacitors are recharged through the free-wheeling diodes of the VSC.

Chapter 3 DC faults in HVDC: detection, identification, and handling

4. Finally, the fast DC switch is connected when the DC voltages on both sides reach their set values, and the IGBTs are unblocked. Since AC circuit breakers are available commercially, this approach is the most simple and economical method to protect against DC faults. However, for MTDC, the entire system is disconnected for a while each time a DC fault occurs at any line. Moreover, it may take up to several seconds to restart the entire system.

3.5.2 Protection system with DC breakers The second strategy deals with the use of DC breakers. The development of DC breakers has been a great challenge due to the demanding requirements such as (i) nonzero-crossing, (ii) the large amount of energy to dissipate, and (iii) large overvoltages after the interruption. Recently, a new high-voltage DC breaker was developed. This breaker has a fast current interruption capability with low conduction losses. In general, the use of DC breakers can provide the following advantages: • Fast interruption time. DC breakers use power electronics devices instead of mechanical devices, which makes AC breakers have the longest interruption times. • Selectivity. The use of DC breakers gives the possibility to isolate each transmission line independently. There are two main strategies for setting the DC breakers [8], (a) moderate use of DC breakers or (b) full use of DC breakers. The first option uses a combination of DC breakers with disconnectors. So once the DC fault occurs, the first action is locating the faulted line. Then the two closest DC breakers are opened. So, one station is isolated. Once the fault current decays to zero, the disconnector can be opened. Once the faulted line is isolated, the other DC breaker can be closed, and it connects the isolated station. This process is based on the AC approach for handling DC fault, so the entire system is shut down for a while but this time using the DC breakers. In [31], it was proposed to use isolators to minimize the number of DC breakers. The operation is summarized as follows: 1. Once the DC fault is detected, the first action is locating the faulted line. 2. The second action is opening the two closest DC breakers as shown in Fig. 3.17(b). One station is disconnected as can be noticed in Fig. 3.17(b). 3. Once the fault current decays to zero, the isolator can be opened as shown in Fig. 3.17(c).

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Figure 3.17. DC fault protection with DC breakers and isolators: (a) location of the fault line, (b) open the closest DC breakers to the faulted line, (c) open isolator, and (d) close the DC breaker.

4. Once the faulted line is isolated, the other DC breaker can be closed, and it connects the losing station as shown in Fig. 3.17(d). In the second option, each transmission line has two breakers, one at each end as shown in Fig. 3.18. Once a fault is detected, every single transmission line can be completely isolated. This solution is very fast and flexible without question; however, it is also expensive for the number of breakers required. In general, the use of DC breakers has the following advantages: • The fastest interruption time. The use of AC breakers is the cheapest solution, but it also provides the longest interruption times due to their mechanical restrictions. • More flexibility. Unlike the solution with AC breakers, where the whole system is interrupted when a fault occurs at the DC grid. The use of DC breakers gives the possibility to isolate each transmission line independently. So, it is very suitable for DC grids.

3.5.3 Protection system embedded on the power converter Finally, another alternative to limit the DC fault currents is using converters with DC fault ride-through capabilities. The main options available in the market are following:

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.18. DC fault protection with DC breakers.

• Full-bridge multimodulated converter (FB-MMC) • Alternate-arm converters (AAC). • Double clamped submodules (DCSM). These topologies can clear a DC fault current since they can decouple the direct diode path between AC and DC side of the converter. In case of an FB-MMC, all IGBTs are blocked during the fault; therefore the DC current, which increases rapidly, is able to flow to the capacitor of each submodule as shown in Fig. 3.19. However, FB-MMC topologies contain the double of semiconductors devices of the conventional half-bridge MMC (HB-MMC). Therefore FB-MMCs have higher power losses. As was shown above, FB-MMC is able to handle DC faults. However, FB-MMC topologies have higher power losses compared with a half-bridge MMC (HB-MMC) topology because of the additional IGBTs, which in fact are unnecessary in normal operation. Some alternative topologies with lower losses have been proposed [34,35]. For instance, in [29] a clamped circuit was proposed. MMC works the same way as an HB-MMC during normal operation using the clamped circuit on each submodule. The middle semiconductor device allows us to bypass half of the bridge on each module when it is as shown in Fig. 3.20. When a fault occurs, the device is turning off resulting in an FB-MMC that is able to handle a DC fault [34]. AAC is simply a VSC with series connected full bridge cells. ACC aims to reduce the overall power losses and to gain the ability to control DC faults [35]. HB-MMC is VSC topology with the lowest power losses (it is estimated that FB-MMC and DCSM produce 70% and 35% more

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Figure 3.19. Full-bridge MMC during DC fault.

Figure 3.20. Double clamped submodules.

losses than HB-MMC, respectively [34]). However, HB-MMC is unable to ride through a DC fault. Recently, a new modification has been proposed by [29], which is added an extra bypass switch as shown in Fig. 3.21. In the standard HB-MMC configuration a thyristor is parallel-connected with the lower diodes in each submodule as shown in Fig. 3.21(a). In normal operation, these thyristors are switched off. During a DC fault, thyristors are switched

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.21. MMC with bypass: (a) single thyristor switch, (b) double thyristor switch.

on to bypass the submodules. Fault current flows through thyristors, so they protect the diodes from overcurrents as they have higher thermal overstress capacity compared with diodes. In the modification proposed by [29], double thyristors are used in each submodule as shown in Fig. 3.21(b). Both thyristors are controlled with the same signal. In normal operation the thyristors are turned off, whereas they are turned on during a DC fault condition. If all thyristors are switch on, then a sort of AC short circuit is forced. In this condition the converter cannot act as a uncontrolled rectifier, so the DC fault current decays to zero. Once the fault is cleared, the induced AC fault current can be removed simply using the gate signal. MMC is able to restart and resume automatically [29].

3.6 HVDC circuit breakers A current limitation for the massive deployment of HVDC is a cost-effective high-voltage DC circuit breakers (HVDC-CB). Four promising technologies are the most advanced and mature: • Mechanical HVDC circuit breakers. • Solid-state HVDC circuit breaker. • Hybrid HVDC circuit breaker.

3.6.1 Mechanical HVDC circuit breakers This breaker is a combination of a mechanical switch with surge arrestors as shown in Fig. 3.22 [36]. During normal operation, the mechanical switch is closed and is able to conduct the current. During a fault, an additional LC circuit is placed in parallel to the switch. The objective is to create a current oscillation between the two paths and thus an artificial zero-crossing. The oscillation can be achieved by an active current injection from a precharged capacitor or excited by the arc. Once the current zerocrossing is achieved, the mechanical switch can be opened, then

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Figure 3.22. Mechanical HVDC circuit breakers.

Figure 3.23. Solid-state HVDC circuit breakers.

the fault current is commuted to the surge arrestor, and this can produce a countervoltage to extinguish the fault current. One of the disadvantages of this technology is the response time, which is typically in the order o several tens milliseconds [8,37].

3.6.2 Solid-state HVDC circuit breakers A solid-state HVDC CB replaces the mechanical switch and LC circuit by a series of connected semiconductor devices such as Gate-Commutated Thyristors (GCTs), Gate Turn-Off Thyristors (GTOs), or IGBTs as shown in Fig. 3.23. A surge arrestor is connected in parallel to the main switch to limit the voltage and absorb the great amount of energy during faults. The main advantage of this type is the reaction time, which can be in less than 1 millisecond [38]. However, there are substantial on state losses and cost due the semiconductor devices. This technology will progress with the development of the new generation of semiconductor devices [37,39].

3.6.3 Hybrid HVDC circuit breaker This solution consists of a mechanical breaker in parallel with a solid-state breaker as shown in Fig. 3.24 [36,40]. The mechanical breaker serves as the conduction path during normal operation, whereas the solid-state one operates during DC faults. The principle of operation is the following: On normal operation, all semiconductor devices are switched on; however, current flows

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.24. Hybrid HVDC circuit breaker.

through the fast disconnector and the auxiliary breaker since they are designed to have lower on-state voltage compared with the main switch. Once a fault is detected, the auxiliary breaker opens, and the current commutes to the main breaker; this process takes a few microseconds. Once the current commutes to the other path, the disconnector can be open since the current is zero. Then the main switch is disabled, and the fault current commutes to the arrestor, which can provide a reverse voltage to counteract the fault current. The hybrid solution is faster than the mechanical breakers and has low on state losses [8,37].

3.7 Fault current limiters Fault current limiters (FCLs) are devices that insert an additional impedance to limit the fault current to levels, which can be interrupted by the protection devices. There are different types of FCLs; some of them are in the path of the fault permanently, other alternatives are connected only during faults, and other alternatives change the impedance when the current increases through them. In the following, we present an overview of the fault current limiters.

3.7.1 Inductors Inductors are a simple and cost-effective solution to limit the fault current in any power system device. Inductors are normally used in substations or feeders in the AC power grids. However, this solution increases power losses. Inductor-based FCLs are permanently connected and need no any special equipment or control strategy for its operation. In particular, for VSC-based HVDC during a DC fault, the capacitor discharge current, cable current, and diode current can be limited in a small range by inductors as was shown in [41]. However, in [41], there is no analysis that shows the power losses. A conventional transformer may be mod-

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ified to produce a high-impedance transformer. This transformer is characterized by high leakage impedance, which can limit the fault current. The special winding design to produce high leakage impedance is reflected in the cost. The performance of the highimpedance transformer to limit the fault current is the same as the inductor, and it has the same issues with power losses.

3.7.2 Tuned LC circuit The tuned LC circuits reduce the line impedance using the impedance compensation principle, which is employed in transmission lines. The tuned LC circuit has a combination of a series inductor with parallel LC circuit. The idea behind is that the equivalent impedance of LC circuit has a capacitive impedance and is equal to the impedance of the series inductor under normal conditions. Therefore the total impedance is zero in normal conditions. When the fault occurs, the inductor of the LC circuit is saturated, and this limits the voltage drop across the capacitor, which makes the equivalent impedance inductive. LC circuits have been proposed for MMC HVDC [42]. Fig. 3.25(a) shows a simplified MMC, which has two arms, and each arm contains series modules of H-bridges; in the figure, each arm has only two modules. When a DC fault is detected, all the modules in one arm are turned on, and all the modules in the other arm are turned off. When all modules are turned on, the only impedance is the MMC inductance, that is, all capacitors are out of the line as shown in Fig. 3.25(b). When all modules are turned off, all capacitors are in to the line as shown in Fig. 3.25(b). The equivalent circuit, shown in Fig. 3.25(c), is the above described LC circuit, which is able to limit the current.

3.7.3 Polymer PTC resistor-based FCL This type of FCL is based on a polymer composite that is sensitive to the temperature. The polymer composite contains conductive particles dispersed therein, which are in contact with each other under ambient temperature. These conductive particles provide a path for the current. So, the polymer composite has a low resistance under normal conditions. When a fault occurs, there is a significant increase in the current, and thus also temperature increases. The polymer composite expanses with the increase in the temperature, and the conductive particles are disconnected causing a highly resistive path [43].

Chapter 3 DC faults in HVDC: detection, identification, and handling

Figure 3.25. 2 MMC HVDC using a FCL based on LC circuit.

3.7.4 Liquid metal FCL Fault current limiters based on liquid metal uses the principle that the liquid metal changes its states from liquid to vapor when the current increases significantly [43]. Under normal conditions, the liquid metal has a low impedance. When a fault occurs, part of the liquid metal becomes vapor due to the increase of the temperature. The change of state provokes a high-impedance path. When the fault is cleared, the high current is interrupted, and the vapor becomes liquid again.

3.7.5 Superconductive FCL An FCL based on superconductive material exhibits a superconductive state, the near-zero impedance at normal conditions and normal state (high impedance) when a fault occurs. Under normal conditions, the impedance is near to zero, but the superconductive material requires a cooling system to regulate the temperature. When a fault occurs, the fault current heats the su-

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perconductive material and reverts to its normal state, that is, high impedance. The high impedance is able to limit the fault current to levels suitable for the protection system [43]. Superconductors possess a highly nonlinear behavior. Current limiting characteristics depend on the three parameters: current density, magnetic field, and temperature. If any of these three parameters changes, a transition between the semiconducting to normal conducting condition occurs. At present, superconducting FCLs are mostly concentrated in AC systems. A variety of FCL based on superconductors are now in the prototype stage. The peak current, after the occurrence of the fault, is expected to remain the same for AC or DC systems. However, not all concepts can be used directly in DC systems since many of them are based on magnetic coupling as the shieldcore superconducting FLC and saturable-core superconducting FCL [44]. Superconducting FCL for DC system and in particular for HVDC has received limited attention [45]. Resistive superconducting appears as the only concept able to be used in HVDC applications. Resistive superconductive FCL consists of a superconducting material as the main current path, and shunt impedance is a combined inductor and resistor. The current flows through the superconducting material under normal conditions. When a fault occurs, the current increases and causes that superconductor changes from superconducting condition to conducting condition. The sudden increase in resistance produces an increase of the voltage across the superconductor, so eventually the current changes to the shunt path. The shunt path also limits the voltage increase across the superconductor during the quench [45]. The dynamics of the process after the quench is governed by the shunt elements. Moreover, the quench process involves an increase in the temperature, which must be carried away using a cooling system. Typically, there is a period of time when the fault is cleared and the cooling system carries away the heat; this period is known as the recovery cycle. The recovery cycle is a critical parameter for an FCL based on superconductors. The recovery time may be reduced by the joint action of a circuit breaker [45].

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[2] D.V. Hertem, M. Ghandhari, Multi-terminal {VSC} {HVDC} for the European supergrid: obstacles, Renewable and Sustainable Energy Reviews 14 (9) (2010) 3156–3163. [3] T.M. Haileselassie, K. Uhlen, Impact of DC line voltage drops on power flow of MTDC using droop control, IEEE Transactions on Power Systems 27 (Aug. 2012) 1441–1449. [4] W. Leterme, J. Beerten, D.V. Hertem, Nonunit protection of HVDC grids with inductive DC cable termination, IEEE Transactions on Power Delivery 31 (April 2016) 820–828. [5] J. Wang, B. Berggren, K. Linden, J. Pan, R. Nuqui, Multi-terminal DC system line protection requirement and high speed protection solutions, in: 2015 Lund International Symposium by Cigre, 2016. [6] V. Akhmatov, M. Callavik, C.M. Franck, S.E. Rye, T. Ahndorf, M.K. Bucher, H. Müller, F. Schettler, R. Wiget, Technical guidelines and prestandardization work for first HVDC grids, IEEE Transactions on Power Delivery 29 (Feb. 2014) 327–335. [7] L. Xu, B. Andersen, Grid connection of large offshore wind farms using HVDC, Wind Energy 9 (4) (2006) 371–382. [8] J. Descloux, Protection contre les courts-circuits des réseaux à courant continu de forte puissance, PhD thesis, Université de Grenoble, 2013. [9] J. Yang, Fault analysis and protection for wind power generation systems, PhD thesis, University of Glasgow, 2011. [10] J. Yang, J.E. Fletcher, J. O’Reilly, Short-circuit and ground fault analyses and location in VSC-based DC network cables, IEEE Transactions on Industrial Electronics 59 (10) (2011) 3827–3837. [11] Y. Yeng, M. Kazerani, V. Quintana, Current-source converter based STATCOM: modeling and control, IEEE Transactions on Power Delivery 20 (April 2005) 795–800. [12] T.K. Vrana, Y. Yang, D. Jovcic, S. Dennetire, J. Jardini, H. Saad, The CIGRE B4 DC grid test system, System 500 (2013) D1. [13] F. Page, G. Adam, S. Finney, D. Holliday, L. Xu, DC fault parameter sensitivity analysis, in: Developments in Power System Protection (DPSP 2014), 12th IET International Conference on, IET, 2014, pp. 1–6. [14] J. Yang, J. Fletcher, J. O’Reilly, Multiterminal DC wind farm collection grid internal fault analysis and protection design, IEEE Transactions on Power Delivery 25 (Oct. 2010) 2308–2318. [15] S. Azizi, S. Afsharnia, M. Sanaye-Pasand, Fault location on multi-terminal DC systems using synchronized current measurements, International Journal of Electrical Power & Energy Systems 63 (2014) 779–786. [16] K. De Kerf, K. Srivastava, M. Reza, D. Bekaert, S. Cole, D. Van Hertem, R. Belmans, Wavelet-based protection strategy for dc faults in multi-terminal VSC HVDC systems, IET Generation, Transmission & Distribution 5 (April 2011) 496–503. [17] R. Razzaghi, M. Paolone, F. Rachidi, J. Descloux, B. Raison, N. Retiere, Fault location in multi-terminal HVDC networks based on electromagnetic time reversal with limited time reversal window, in: Power Systems Computation Conference (PSCC), 2014, Aug. 2014, pp. 1–7. [18] J. Marvik, S. Darco, K. Sharifabadi, Protection scheme for multi-terminal radial VSC HVDC system without communication between terminals, in: Across Borders-HVDC Systems and Markets Integration, 2015. [19] J. Arrillaga, High Voltage Direct Current Transmission, vol. 29, IET, 1998. [20] C. Kim, V. Sood, G. Jang, S. Lim, S. Lee, HVDC Transmission, Wiley Online Library, 2009.

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4 Eigenvalue-based analysis of small-signal dynamics and stability in DC grids Salvatore D’Arcoa , Jef Beertenb,c , and Jon Are Suula,d a SINTEF

Energy Research, Trondheim, Norway. b Department of Electrrical Engineering (ESAT-electa), KU Leuven, Heverlee, Belgium. c EnergyVille, Genk, Belgium. d Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway

Chapter points • Introduction to state-space modeling and small-signal analysis of electrical systems. • Definition of interfaces for modular state-space modeling of HVDC systems. • Outline of workflow and required numerical procedures for modular model generation and linearization to obtain system-level small-signal state-space models of HVDC grids. • Review and application of small-signal state-space models of main HVDC grid components including long HVDC cables and modular multilevel converters. • Examples of results from eigenvalue-based analysis of a multiterminal HVDC grid.

4.1 Introduction Over the last decades, DC power systems have attracted an increasing level of interest within a wide range of applications, including low-voltage microgrids, medium-voltage industrial applications and large-scale high-voltage DC (HVDC) transmission systems [1]. The commercial application of high-power, high-voltage DC transmission systems based on voltage source converter (VSC) technology was initiated with the demonstration of two-Level VSC HVDC converters by ABB [2]. More recently, the introduction of the modular multilevel converter (MMC) by Siemens has paved the way for increased voltage and power levels by enabling reModeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00009-5 Copyright © 2021 Elsevier Inc. All rights reserved.

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duced losses and a highly scalable design [3]. Thus, MMC-based VSC HVDC transmission has become a viable solution for long distance bulk power transfer and for interconnections between different synchronous power systems, becoming often the only option for connecting large-scale offshore wind power plants. Furthermore, VSC-based technology is suitable for operation of multiterminal HVDC grids, which have been envisioned as a future overlay transmission network or “Supergrid” for increasing longdistance transmission capacity and for supporting large-scale grid integration of offshore renewable power generation [4]. The first demonstrations of full-scale multi-terminal VSC HVDC systems [5,6] and a meshed HVDC grid [7] have been developed in China during the last decade. In the same period a significant scientific effort has been dedicated towards the development of methods for simulation, control system design, modeling and stability analysis of VSC HVDC transmission [8–10]. The prospects of a growing share of VSC-based HVDC transmission, either as radial or meshed multiterminal systems, have also spurred the development of a wide range of models and tools for assessing the stability and dynamic interactions within HVDC systems [11] or with the existing AC power systems [12]. In the general power system context, stability analysis can be largely categorized into transient stability and small-signal stability [13]. Since the transient stability of power systems is mostly influenced by their nonlinear characteristics, analysis of large-signal transients in complex systems is normally conducted by time-domain simulations. Small-signal stability, on the contrary, can be analyzed by applying linearization. Thus, several methods and tools available for analyzing linear systems have been extensively applied to the study of stability and dynamic interactions in power systems. However, the traditional underlying models have typically been based on representation of machine dynamics, which are inherently slower than the fast dynamics that can appear in VSC HVDC systems [14]. Furthermore, the presence of fast dynamics combined with multiple cascaded control loops implies that VSC-based transmission systems can be sensitive to interactions between the converter control loops and the electrical power grid or between different converter units [15,16]. Such phenomena are typically associated with the small-signal characteristics and the controller parameters of the system. Within the recent development of methods for analyzing smallsignal dynamics, two mainstream approaches have been pursued, relying on either state-space or impedance-based modeling [17]. Both approaches refer to small-signal analysis around a specified steady-state operating point. The approaches based

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

on state-space models have the advantage of providing an inherently global stability assessment and allow for utilization of a large set of methods for analyzing the dynamic properties of linear systems. However, state-space-based analysis requires the knowledge of the dynamic models and parameters of each and every component in the system. Instead, the impedance-based analysis provides the advantage of a relatively simple interpretation of the system characteristics in the frequency domain, including the possibility for including nonrational functions such as timedelays in the model. Impedance-based methods are also suitable for measurement-based analysis and for the representation of black-box systems [18]. However, they generally provide only a location-specific view of stability since they rely on impedance characteristics at a specified node in the system. Thus, stability problems caused by interactions between different parts of a system might not be observable from all points of analysis where the impedance characteristics are evaluated. This chapter focuses on the state-space-based modeling and analysis of HVDC systems, which allows for a global stability analysis of complex system configurations. The fundamental concepts and the nomenclature applied for state-space modeling are introduced as a background for further discussions. Since the presented analysis relies on a time-invariant representation, the modeling of three-phase AC electrical systems in a single synchronously rotating reference frame (SSRF) by applying Park’s transformation is introduced as a premise for obtaining a nonlinear time-invariant state-space equations. Furthermore, the requirements for calculating the steady-state operating point needed for linearization and small-signal analysis, and the most common tools for eigenvalue-based analysis of small-signal dynamics are briefly reviewed. On this basis, the synthesis of systemlevel state-space models of an HVDC grid with multiple converter terminals and DC-side interconnections is discussed. The main subsystems in a multiterminal DC grid are identified, and standardized interfaces between the subsystems are defined. Subsequently, a general approach for obtaining state-space models of large-scale HVDC systems from the subsystem models is outlined together with considerations on aspects related to software implementation. To illustrate how small-signal state-space models for HVDC transmission systems can be developed, examples of state-space models for the main subsystem components, including MMCbased converter terminals and HVDC cables, are presented. Furthermore, a numerical example of small-signal state-space analysis for a test case of a 4-terminal meshed HVDC grid is in-

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troduced. The example demonstrates how the linearized statesspace model can be assembled and how the eigenvalue-based analysis of participation factors and parametric sensitivity can be utilized to gain further insights into the small-signal dynamics of a system.

4.2 Introduction to state-space modeling of electrical systems A brief introduction to state-space modeling and eigenvaluebased small-signal analysis is presented to summarize the theoretical background and to introduce the nomenclature applied in the following. The presentation assumes that a nonlinear timeinvariant state-space representation on explicit form can be obtained for the system to be studied. Modeling approaches required to a time-invariant representation of AC power systems are also briefly described before introducing how eigenvalue-based analysis can be utilized to assess the small-signal stability properties of the linearized system model.

4.2.1 Nonlinear time-invariant state-space models A continuous-time nonlinear time-invariant (NTI) dynamic system can be generally represented as a set of state-space equations on the vector form [13] x˙ = f (x, u) , y = g (x, u) ,  x = x1 x 2  y = y1 y2

...

xn

...

yk

T T

,

u=



u1

u2

...

um

T

,

, (4.2.1)

where x is the vector of state variables, and u is the vector of input signals. Thus, the derivative of the states is given as a generic nonlinear function f of the states and the inputs. Furthermore, an output vector y is given as a generic nonlinear function g of the states and the input signals. The state variables and the input signals are all functions of time, but the time notation is omitted for brevity. The set of state-variables describing the system dynamics is not unique and, consequently, a variety of sets can be used to describe the internal dynamics of a system. To convey practical interpretation of the modeled dynamics, a state variable xi is typically associated with an element linked to energy storage or a

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

memory function. For example, the currents of inductors and the voltages of capacitors are suitable candidates for state variables in an electrical system as they describe the stored energy in the magnetic field of an inductor and in the electric field of a capacitor, respectively. Similarly, all integrators in a control system can be adopted as state variables since they represent a memory function in the control. However, a model of an electrical conversion system obtained directly from a circuit diagram combined with a block diagram of the control system may include both state equations and algebraic equations. Thus, the general form of expressing the system equations according to (4.4.1) also assumes that all algebraic variables can be eliminated from the model and included into the state equations by substitution to achieve a state-space model with explicit expressions for all the states.

4.2.2 Time-invariant representation of three-phase electrical systems Since all currents and voltages in a DC power system should settle to constant values in steady-state operation, such systems are inherently time-invariant and can be straightforwardly represented in the above manner. AC systems, on the contrary, present a periodic behavior at the fundamental frequency in steady-state operation. To represent the AC system quantities in a time-invariant framework, the periodicity at the fundamental frequency has to be eliminated from the state variables. For threephase AC systems, this can be directly obtained by the transformation to a rotating reference frame. In traditional power system analysis the periodicity in AC system quantities is commonly accounted for by representation as complex phasors. Instantaneous values of variables such as currents and voltages are then represented as projections of rotating vectors, called “phasors”, on a stationary axis. These vectors rotate with the constant grid frequency and are completely defined by their magnitude and angle, which are expressed with respect to a complex plane rotating at the same angular frequency. Under unbalanced system conditions, each phase requires its own phasor and, consequently, three phasors are needed to completely describe the voltages on a bus or the currents through a line. Alternatively, a transformation to symmetrical components can be considered to describe the three-phase quantities in terms of positive, negative, and zero sequence (homopolar) components. For both cases, six dimensions are needed to fully describe the threephase quantities. These dimensions are typically the amplitude

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and phase angle of all phase quantities or of the symmetrical components. Under balanced system conditions, a single phasor suffices to describe the three-phase quantities, since the three phases are merely displaced versions of each other with predefined phase shifts of 120 degrees. As a result, only two dimensions are necessary to fully describe three-phase quantities. These dimensions can be the phasor’s amplitude and angle or, alternatively, its projection in terms of real and imaginary components in a rotating complex plane. The concept of phasor-based analysis usually entails a notion of a steady-state or quasi-steady-state assumption as, for example, in the assessment of transient or small-signal stability of a multimachine system or in power flow analysis [13]. However, it is also possible not to apply steady-state assumptions to these phasor quantities, leading to a concept called “dynamic phasors” [19]. From a mathematical point of view, phasor modeling shows clear similarities to the description of three-phase quantities in a rotating reference frame as detailed below, especially when no steady-state modeling assumptions are applied in the derivation of the resulting equations. However, attention should be paid when adapting phasor-based AC component models for use in small-signal analysis with power-electronic converters, especially when the validity over a wide frequency range is required. For instance, existing phasor-based models might come equipped with implicit assumptions of a quasi-steady-state description of electrical quantities as explained above, or imply a fixed system frequency in the analysis, for instance, by the use of reactances instead of inductances and capacitances. In modeling and control of electrical machines and power electronic converters, it is common to describe system quantities such as currents and voltages as space vectors in a synchronously rotating reference frame (SRRF). The path to obtain the variables in this form typically involves coordinate transformations by means of the Clarke and Park transformations. A three-phase power system can be described in an αβ0 framework, in which the α and β axes form an orthogonal set, and in which the zero-axis represents the component common to all three phases (i.e., the homopolar or zero-sequence component). This transformation is typically denoted as the Clarke transformation and is given, together with its reverse operation, by the

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

following transformation matrices: ⎡ ⎤ 1 xα ⎣ xβ ⎦ = 2 ⎢ ⎣ 0 3 x0 1

2 ⎡

− 12

⎤ ⎡ 1 xa ⎣ xb ⎦ = ⎢ ⎣ − 12 xc − 12

⎡

− 12

√

−

3 2 1 2

√

3 2

1 2



Tαβ0

0

√

3 2√

− 

3 2

T−1 αβ0

⎤⎡

⎤ xa ⎥⎣ ⎦ xb ⎦ , xc

⎤⎡

⎤ xα ⎥ 1 ⎦ ⎣ xβ ⎦ . x0 1

1

(4.2.2)

It is worth noting that this transformation of instantaneous values of the abc signals to the αβ0 framework does not imply any assumption on the relationship between the three-phase abc variables. However, under the assumption of a balanced system, the zero-sequence component x0 is equal to zero and can be omitted. Thus, the transformation matrix can be reduced to a 2 × 3 matrix. Under the assumptions of balanced system conditions, in which only the instantaneous αβ variables are retained, these can be transformed to a rotating dq-frame by means of the Park transformation. The 2 × 2 transformation and back-transformation matrices are respectively given by 



xd xq xα xβ





= 

  cos θ sin θ xα , xβ − sin θ cos θ





=

Tdq

cos θ sin θ

  − sin θ xd . xq cos θ 

(4.2.3)

T−1 dq

Combining both Clarke and Park transformations and retaining the zero-sequence component modeled separately results in a direct transformation from abc to dq0 variables with the corresponding reverse transformation from dq0 to abc variables, given

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

as ⎡

⎡

⎤

cos θ

xd ⎢ ⎣ xq ⎦ = 2 ⎢ ⎢− 3 ⎣ sin θ x0

1 2

    2π cos θ + cos θ − 2π 3 3     2π − sin θ + 2π − sin θ − 3 3 1 2



1 2

⎤

⎡ ⎤ ⎥ xa ⎥⎣ ⎥ xb ⎦ , ⎦ x c

Tdq0

⎡

⎤ ⎤ cos θ − sin θ 1 ⎡     xa ⎢ ⎥ xd 2π 2π − sin θ − 3 1 ⎥ ⎣ xq ⎦ . (4.2.4) ⎣ xb ⎦ = ⎢ cos θ − 3 ⎣ ⎦     xc x0 2π − sin θ + 1 cos θ + 2π 3 3 

⎡

⎤

T−1 dq0

For a balanced system with no zero-sequence component, the AC system variables are then described in terms of two orthogonal variables, namely d- and q-variables, which are constant under steady-state conditions. The definition of AC power system quantities such as voltages and currents in terms of space vectors in an SRRF by the Park and Clarke transformations yields an important advantage for the design of converter controllers. Indeed, the system to be controlled (e.g., the converter currents) is transformed from a timeperiodic into a time-invariant system under steady-state conditions. As a result, the control loops can be conveniently designed in terms of equivalent d- and q-components of the quantities (e.g., the converter currents), which are now constant in steady state. With the converter control taking the form of a DC-like control of the AC system quantities, it is commonplace to extend this dqtransformation also to the circuit equations of the external AC network under consideration. In this way the model of the external network is also transformed under steady-state conditions from time-periodic to time-invariant and becomes compatible with the time-invariant modeling approach described in the previous section.

4.2.3 Linearization Representation of a time-invariant system on the form given by (4.2.1) results in a steady-state condition characterized by an equilibrium point of the system equations where all variables settle to constant values and, consequently, all time derivatives are equal to zero. This condition can be expressed as f (x0 , u0 ) = x˙ 0 = 0

(4.2.5)

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

with x0 and u0 representing the steady-state values of the state variables and input signals. Thus, the equilibrium point around which the system has to be linearized can be found by solving (4.2.5) for x0 as a function of u0 . The system variables after linearization can then be generally expressed as x = x0 + x, u = u0 + u, y = y0 + y,

(4.2.6)

where  indicates small deviations from the equilibrium point defined by x0 and u0 . Then it is only necessary to evaluate the dynamics of the deviations from the operating point, as given by x, to study the small-signal stability of the system [13]. The linearized small-signal dynamics of the studied system on the form given by (4.2.6) can be obtained from the first-order Taylor-expansion of the nonlinear equations given by (4.2.1), resulting in   ∂f (x, u)  ∂f (x, u)  x + u, ˙x = ∂x  x=x0 ∂u  x=x0 u=u0 u=u0





A  B  ∂g (x, u)  ∂g (x, u)  y = x + u.  x=x0 ∂x ∂u  x=x0 u=u0 u=u0





(4.2.7)

D

C

As an example, the detailed expression for linearization of the state equation for one state variable is given as x˙1 =

  ∂f (x, u)  ∂f (x, u)  x + x2 + ... 1 ∂x1  x=x0 ∂x2  x=x0 u=u0 u=u0





A11

A12

  ∂f (x, u)  ∂f (x, u)  + u1 + u2 + .... ∂u1  x=x0 ∂u2  x=x0 u=u0 u=u0





B11

(4.2.8)

B12

The linearized small-signal model of the system can then be expressed as ˙x = A (x0 , u0 ) · x + B (x0 , u0 ) · u, y = C (x0 , u0 ) · x + D (x0 , u0 ) · u,

(4.2.9)

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where the matrices A, B, C, and D contain the coefficients resulting from the linearization, and A is the Jacobian matrix of the system. Assuming the general case of a nonlinear system, the elements of these matrices depend on the linearization point, as expressed by the dependency on x0 and u0 in (4.2.9). However, for any linear dynamics included in the system model, the corresponding elements of the matrices will be constant and independent from the operating point. Independently of the number of nonlinear equations included in the initial system model, the small-signal model resulting from the linearization will be a linear time-invariant (LTI) system, implying that all established tools for linear system analysis can be used to study the small-signal dynamics and associated stability properties.

4.2.4 Eigenvalue-based analysis of small-signal dynamics Under the condition that the system to be studied can be represented by a time-invariant model and linearized to reach the general formulation given by (4.2.9), the eigenvalues of the Amatrix can be utilized to evaluate the small-signal dynamics. Then the eigenvalues λ are defined as the solution to the characteristic equation according to det (A − λI) = 0.

(4.2.10)

The eigenvalues satisfy the following equation: Ai = λi i ,

(4.2.11)

where the column vector i is the right eigenvector associated with the eigenvalue λi . Furthermore, the corresponding left eigenvector is defined as a row vector by  i A = λi  i .

(4.2.12)

Any multiple of an eigenvector is also a solution to the equations above, but usually the eigenvectors are scaled to result in a vector product of 1, given as  i i = 1.

(4.2.13)

Accordingly, the right and left eigenvector matrices are defined by (4.2.14) and (4.2.15), respectively, whereas the diagonal eigenvalue matrix  is defined by (4.2.16).   (4.2.14)  = 1 2 ... n ,

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

=



1 ⎡

λ1 ⎢ 0 =⎢ ⎣ 0 0

T

2

...

n

0 λ2 0 0

... ... ... ...

⎤ 0 0 ⎥ ⎥. 0 ⎦ λn

,

(4.2.15)

(4.2.16)

From definitions (4.2.11)–(4.2.16) we can also derive the following identities: A =   = I

−→ −→

−1 A = , A = .

(4.2.17)

Considering for simplicity the autonomous system ˙x = A · x

(4.2.18)

resulting from ignoring the inputs signals and the dependency on the equilibrium point in (4.2.9), we can obtain a decoupled system representation by introducing a transformed state vector defined by x =  · z.

(4.2.19)

Substituting this definition into (4.2.18), premultiplying the resulting expression by −1 , and using the identities given by (4.2.17) result in the following definition: z˙ = A · z =  · z.

(4.2.20)

Since the eigenvalue matrix  is diagonal, (4.2.20) provides a transformed system description where the dynamics associated with the eigenvalues of the original system are decoupled. This can be confirmed by solving the differential equations from (4.2.20) to obtain the time domain response, resulting in the independent dynamics of each of the transformed states zi as zi (t) = zi (0) eλi t .

(4.2.21)

Thus in this transformed system representation, each mode or eigenvalue is associated with a decoupled time response. Considering that the eigenvalues can be complex numbers of the generic form λi = α + j ω,

(4.2.22)

the time-response of the transformed system provides an intuitive basis for assessment of the small-signal dynamics of the linearized

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system as defined by the eigenvalues of its A-matrix. Indeed, considering either a single real eigenvalue or a complex conjugate eigenvalue, the generic time response can be expressed by (4.2.23) or (4.2.24), respectively. λi = α −→ zi (t) = keαt , λi,i+1 = α ± j ω −→ zi,i+1 (t) = keαt sin (ωt + ϕ) .

(4.2.23) (4.2.24)

From these expressions, we can easily interpret the conditions for small-signal stability according to linear system theory. Indeed, a purely real negative eigenvalue leads to an exponential decay according to (4.2.23), whereas a complex conjugate eigenvalue with negative real part leads to a damped sinusoidal response as indicated by (4.2.24). Equivalently, a positive real part of the eigenvalue leads to an exponential increase, resulting in instability. Thus the stability limit of a system is defined by the zero real part of an eigenvalue. Furthermore, the asymptotic stability of the system (within the range of validity for the small-signal model) requires the real parts of all eigenvalues to be negative. The general expressions in (4.2.23) and (4.2.24) also clearly show how the real part of an eigenvalue represents the inverse of the time constant for the time response associated with an eigenvalue, whereas the imaginary part of a complex conjugate eigenvalue is associated with the oscillation frequency of the time response. By applying (4.2.19) and (4.2.17) the initial value zi (0) in (4.2.21) can be calculated from the initial value x(0) of x and  i . Then by substituting the result back into (4.2.19) the time response for state variable i can be expressed as xi (t) =

n  p=1

i,p  i x (0) ·eλi ·t ,



(4.2.25)

zi (0)=ci

which in expanded form can be written as xi (t) = i1 c1 · eλ1 ·t + i2 c2 · eλ2 ·t + ... + in cn · eλn ·t .

(4.2.26)

Thus we can see that the time response of the original state variables is a linear combination of the decoupled time responses, or the modes, of the transformed system defined by the eigenvalues of the system. This expression also shows how the elements of the eigenvector matrices of the system determine how the different modes appear in the time response of the state variables. From this basis it is clear that the eigenvalues of the linearized system on the form (4.2.9) can be utilized to assess the smallsignal stability and dynamic response of a system. Furthermore, a

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

wide range of techniques based on linear algebra and control theory can be applied to the linearized system model, for instance, to assess controllability, reachability, observability, or internal interactions within the system [13]. In the following we show the concepts of participation factors and parametric sensitivity to provide useful information about the small-signal dynamics of large systems. The participation factors of a system are defined from the elements of the left and right eigenvector matrices according to pki = ki ik .

(4.2.27)

By this definition the participation factor pki represents the participation of state k in mode i or vice versa. Although several modified definitions of participation factors in linear systems have been proposed [20], the definition in (4.2.27) has been widely used in traditional small-signal analysis of large-scale power systems [13]. Although the participation factors provide mainly indicative measures, they can reveal important information on which states interact in a specific mode. Thereby they reveal which state variables can be influenced by control to damp oscillations or to prevent stability problems associated with critical modes. The parametric sensitivity of the eigenvalues also provides important information on how the system parameters and the controller tuning of a system influences the small-signal dynamics. The sensitivity αi,k of eigenvalue λi with respect to parameter ρk can be defined as αi,k =

∂A i  i ∂ρ ∂λi k = . ∂ρk  i i

(4.2.28)

Thus the sensitivity represents the partial derivative of the eigenvalue with respect to a parameter, which is a direct measure of how much the location of the eigenvalue moves due to a small change in one specific parameter. The sensitivity is generally a complex number and represents how the time constant and oscillation frequency of a mode are influenced by the parameter. Note that the sensitivity calculated from this expression represents the derivative of the trajectory that an eigenvalue will follow when the parameter is changed. Therefore the calculated sensitivity will only be valid in a small range around the location of the eigenvalue corresponding to the initial parameter values. Thus the parametric sensitivity must be recalculated for each value of the system parameters. Although the numerical value of the sensitivity can be very useful to identify the controller parameters that can have the highest impact on a critical model, the influence of parameter

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

variations is commonly illustrated by showing the trajectory of the system eigenvalues for a range of parameter variations.

4.3 Synthesis of system-level state-space models of HVDC grids The main elements that must be modeled in a DC network are the converters for interfacing to the AC system and the passive elements of the DC network. Considering dynamics far below the switching frequency of the converters, the aim when modeling the DC network is to capture the dynamics that can interact with the control of the converters in the system. For geographically limited systems like small DC microgrids, this implies that the DC network often can be modeled with its shunt capacitance and the equivalent series resistance of the lines. However, for HVDC systems with long transmission distances, the internal resonances of the DC cables may appear at frequencies that are low enough to interact with the converter control. Considering the complexity of future multiterminal configurations, a systematic approach is needed to effectively obtain suitable small-signal state-space models. For this purpose, the following discussions present a definition of subsystems and interfaces that can be suitable for ensuring scalability of state-space modeling in the analysis of large system configurations.

4.3.1 Definition of interfaces between sub-systems A multiterminal HVDC transmission system can be composed by multiple converters and cables, which may result in a dynamic system model of relatively high order. A first approach for the modeling could be to directly express the dynamic equations of the entire system. Thus the system should be represented on state space form and then linearized. Although directly deriving and linearizing all the state equations would be a straightforward application of the theory outlined in Section 4.2, this approach presents a few major disadvantages. The first disadvantage for relatively high-order systems is associated with the complexity of handling the corresponding number of equations. Moreover, such an approach would not benefit from the structure of the system where quite distinct components such as the converter terminals and the cables can be clearly identified. The presence of these components with well-defined interfaces and possibly with repetitions or at least strong similarities would not be leveraged when modeling the system directly as a whole. From this point of view,

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

a more modular approach could benefit from the similarities of subsystems while also possibly simplifying the modeling effort. The first step toward a modular derivation of the state-space model of a DC grid is to identify subsystems can be modeled separately and to define interfaces. The decomposition of the system into subsystems should also preferably result in a logical grouping of the states. A reasonable choice could be the isolation of parts of the converter models and their control functions for being modeled separately. Thus, the inner loop current controllers and the outer control loops for reactive power and active power or DC voltage, as well as AC-side filter of the converter, could be modeled as individual subsystems. This would result in a high number of very simple systems, as, for instance, utilized in [21]. An alternative choice is treating each converter or cable like a subsystem. This corresponds to a lower number of moderately complex systems. For further discussions, in this chapter, the second option is preferred. A decomposition of a generic HVDC transmission system according to the requirements described above is indicated in Fig. 4.1. All converter terminals and cables are assumed to be represented as separate subsystem models. However, the definition of converter terminals and cables as subsystems still allows for several arbitrary choices in defining the boundaries between subsystems and the corresponding interface variables. For this purpose, note that a node can be identified at the connection between subsystems. Indeed, considering the DC-side connection between a converter terminal and a cable as a node allows for interfacing multiple cables and converters to the same point. Independently from the number of converter terminals or cables, we will assume that the DC node is characterized by an equivalent capacitance and that it can be modeled as a capacitor with multiple current inputs determined by the connected subsystems. Furthermore, the capacitance associated with the node is equal to the sum of all the equivalent capacitances of the components connected to that node. In the case of a cable modeled as a π-equivalent, the capacitance contribution will be the equivalent capacitor at the connected cable terminal. For a converter terminal, the contribution to the node capacitance will be the DC bus capacitor, if present. This leads to a natural choice of defining the subsystems of a larger system configuration to be classified as converter terminals, cables, and DC nodes. To establish generic definitions of subsystem models that can be later interconnected to a global system model, the following interfaces and naming conventions are specified for the three types of subsystems identified above:

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

Figure 4.1. General example of HVDC transmission system with converter terminals, DC nodes, and cables indicated as subsystems.

1. DC nodes: – The name of a generic DC node is defined as Nj (i.e., N 1 for node 1, etc.). – The DC nodes are modeled as equivalent capacitances and are associated with one state variable only, that is, the node voltage, which for DC node Nj is defined as vNj . – Each node is characterized by one output (i.e., the capacitor voltage) and a current input for each connected subsystem. 2. Converter terminals: – Since a converter terminal Ti would be connected to a specific DC node Nj , a generic terminal is labeled as T iNj (i.e., T 1N 1 would label converter terminal T 1 connected to DC node N 1). Note that this allows for considering multiple converter terminals connected to the same DC node. – The converter terminal will have one output (i.e., the DCside current) and one input (i.e., the DC voltage) associated with the node. The subsystem model of a converter terminal can also have additional input and output signals (i.e., reference signals and grid voltage conditions as inputs and measurements as outputs), but they will be independent of the DC node. – Since the state-space model of a converter terminal includes multiple states, the state vector of a generic terminal is referred to as xT iNj , whereas the input vector is labeled as uT iNj . 3. HVDC cables: – A cable is always connected between two DC nodes. Thus the generic cable connecting nodes Nj and N k is simply referred to as Cj k (i.e., C12 is the cable between nodes N 1 and N 2). – Cable models have two separate interfaces toward two nodes. Thus a cable model has two outputs (i.e., the currents into the two nodes) and two inputs (the two node voltages).

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

– It will be later shown how a cable can be represented on state space form, and the generic state vector of cable Cj k is defined as xCj k . Since the cable model has no other inputs than the node voltages at its ends no generic input vector is defined for the cable models. It should be noted that the presented definitions of subsystems and interfaces is intended for the analysis of multiterminal DC grids, with focus on the DC-side interconnection. However, for analysis of systems including AC-side interconnections and/or larger AC networks, AC nodes could be defined equivalently to the DC nodes defined above. Then the main difference would be the dq-frame modeling of the AC-system, which requires the definition of two state variables (i.e., the d- and q-axis voltage components) and two-dimensional input signals (i.e., the dand q-axis current components of each connected subsystem) for representing the node. Furthermore, the modeling of the connection to an AC-node should include the transformation between different local reference frame orientations. However, large-scale AC system modeling is beyond the scope of the following discussions.

4.3.2 Generic definition of subsystem models In the following, the presented definitions are first applied to clarify the generic models of the three types of identified subsystems before presenting an example of how these subsystem models can be assembled into an overall system model.

4.3.2.1 Definition of per-unit scaling and requirements for subsystem interconnection Since per-unit scaling is typically used for implementation of control systems and also provides a more intuitive representation of variables and parameters, we further assume that all subsystem models are scaled to per-unit quantities. The basis for defining the per-unit system on subsystem level is chosen as the nominal MVA rating, the nominal phase-to-ground voltage in the three-phase system, and the nominal frequency of the AC grid. Thus the relationship between the base values for the per-unit system and the rated values of a converter terminal is defined by √ 3 3Vnom,LL,RMS Inom,RMS = Vb Ib , 2 ωb = 2πfb = 2πfnom , Sb = Snom =

(4.3.1)

where the subscript b refers to a base value, and the subscript nom refers to a nominal or rated value. Then the base values for

85

86

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

the active and reactive powers will be equal to the base value Sb for the apparent power. From these identities we can derive the base values for all other variables and parameters, where the main quantities for the AC-side parameters of a converter terminal are Zb =

Vb , Ib

Lb =

Zb , ωb

Cb =

1 . ωb Zb

(4.3.2)

Since no per-unit base is used for time, this definition implies that the state equations for voltages and currents expressed in perunit quantities will contain the base value of the angular grid frequency ωb , as will be clear from the expressions presented further onward in the chapter. Alternatively, Zb can be directly used to define per-unit values for inductive and capacitive components, in which case their “per-unit” equivalents have the physical dimension of seconds (i.e., [p.u.· s]) [22]. Considering that the DC voltage of a converter terminal needs to be equal to or higher than the peak-to-peak phase voltage for a VSC to operate without third harmonic injection, the base value for the DC-side voltage is defined as twice the AC-side base voltage. Defining the same base value for power on the AC- and DCsides of a converter terminal, this implies: Pb,dc = Sb ,

Vb,dc = 2Vb ,

Ib,dc =

Sb . Vb,dc

(4.3.3)

Then the DC-side base values derived from the DC-side rating of a converter terminal can also be applied for per-unit scaling of parameters related to the DC-side, including the cables. With defined base values for the power and the AC- and DCside voltages of an HVDC system, the dynamic equation of a DC node can be directly scaled to per-unit quantities and expressed as ωb  iNj , (4.3.4) v˙Nj = cNj j

where  cNj is the equivalent per-unit capacitance of the node, and iNj represents the sum of the DC-currents flowing into the node. Note that the base value of the angular frequency ωb of the AC-system appears as a scaling factor in the equation due to the time derivative. For a two-terminal HVDC transmission system, the per-unit base values defined above can be directly applied, and per-unit representation of all quantities can be obtained by dividing any physical variable or parameter by the corresponding base value. However, for multiterminal systems having multiple converter

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

terminals and cables with different power ratings, it is natural to model the subsystems in their local per-unit systems. This requires the variables and parameters to be scaled to a global perunit system when interconnecting the subsystems to an overall system model. One approach to ensure this is using the same set of base values for the per-unit scaling of all subsystems to be modeled. This approach has the advantage of an easy interpretation of all variables in the global system context but loses the advantage of having an intuitive interpretation of local variables and parameters with respect to the rating of the subsystem. Another approach, as outlined in the following, is introducing a scaling of variables between global and local per-unit systems used for the subsystem modeling. Considering the base values associated with a certain DC node to define the global per-unit system according to (4.3.3), the scaling of the voltage input to the connected subsystems and the current inputs to the DC node Nj can be defined as V

b,dc = kv,subsystem · vNj , vNj,local = vNj Vb,local

iNj = iNj,local

Ib,local Ib,dc

= ki,subsystem · iNj,local .

(4.3.5)

The scaling of the DC node voltage inputs to the subsystems according to (4.3.5) ensure consistency of the subsystem models, whereas the scaling of the DC currents resulting from the subsystem models ensures that the DC nodes are represented in a unified global per-unit system. It should be noted that in most cases the converter terminals and cables in an HVDC system have the same voltage rating. Thus the scaling factor k v,subsystem in (4.3.5) will typically be equal to or close to 1.0, and then the scaling factor k i,subsystem will be equal to the ratio between the power rating of the subsystem and the global per-unit base for the power. As a convenient choice, the global base value for the power can be set to the nominal power of the largest converter terminal in the system. Note that this approach can be easily reverted to the case of defining all variables and parameters in a global per-unit system by setting the scaling factors k v,subsystem and k i,subsystem equal to 1.0.

4.3.2.2 Models of converter terminals According to the specified conventions, the generic nonlinear state-space model of a converter terminal can be defined as   x˙ T iNj = fT iNj xT iNj , uT iNj , kv,T iNj · vNj .

(4.3.6)

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

Thus the state equations are defined as a generic function fT iNj of the state vector xT iNj , representing the states of converter terminal T i connected to DC node Nj , the corresponding input vector uT iNj , and the voltage vNj of the DC node scaled to the local perunit system of the converter. In addition, it is necessary to define the DC-side current of the converter, which is needed as an input to the DC node model. This current is defined as a generic function gT iNj of the state vector xT iNj , the node voltage vNj , and the input vector uT iNj and is scaled back to the global per-unit system of the DC node:   iT iNj = ki,T iNj · gT iNj xT iNj , kv,T iNj · vNj , uT iNj .

(4.3.7)

4.3.2.3 Cable models As already mentioned, the cable represents a connection between two DC nodes, and the only input signals to the cable model are the voltages at these nodes. Thus, assuming that the cables can be represented on state-space form, we can express the model of a generic cable between nodes Nj and N k as   x˙ Cj k = fCj k xCj k , kv,Cj k · vNj , kv,Cj k · vN k .

(4.3.8)

The outputs of the cable model should be the current at the cable extremities, which will be input signals to the DC node models. These two currents are not necessarily equal and must be expressed separately. This leads to the following generic expressions for the currents into nodes Nj and N k from the cable Cjk connecting these two nodes:   iNj,Cj k = ki,Cj k · gNj,Cj k xCj k , kv,Cj k · vNj ,   iN k,Cj k = ki,Cj k · gN k,Cj k xCj k , kv,Cj k · vN k .

(4.3.9)

From these equations we can notice that the currents that constitute the interfaces to the nodes can depend on the internal states of the subsystem models representing the cables and on the node voltage itself. Whereas the general characteristics of detailed dynamic cable models are based on distributed and frequency-dependent parameters, we will show later how the cable dynamics can be approximated by an equivalent lumped circuit representation resulting directly in a linear state-space model. Thus the cable behavior can be generally represented in state-space form (4.3.10) with the corresponding linear functions for calculating the currents at the

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

DC node interfaces given by (4.3.11): x˙ Cj k = FCj k xCj k + LCj k vNj + MCj k vN k ,

(4.3.10)

iNj,Cj k = GNj,Cj k xCj k + HNj,Cj k vNj , iN k,Cj k = GN k,Cj k xCj k − HN k,Cj k vN k .

(4.3.11)

The dependency of the interface currents on the node voltage represents in this case a resistive element, as will be shown later. Then the scaling to the local per-unit systems according to (4.3.8) and (4.3.9) is included in the matrices FCj k , LCj k , MCj k , and the scaling back to the global per-unit system is included in the matrices GNj,Cj k .

4.3.2.4 Model of DC nodes As already mentioned, the DC nodes are modeled as capacitances, resulting in a first-order model with the node voltage as the only state. Considering the specified current interfaces with subsystems corresponding to converter terminals and cables, the state equation for the per-unit voltage at a specific node can be defined as ⎛ ⎞    ωb ⎝ iT iNj + iNj,Cij − iNj,Cj k ⎠ . (4.3.12) v˙Nj = cNj j

j

j

This equation shows that the currents resulting from all converter terminals and cables having j in the subscript should be considered for modeling the voltage dynamics of the corresponding node Nj . The equivalent node capacitance cNj expressed in the per-unit system of the node is given by the sum of the equivalent capacitance of all subsystems connected to the node:   kv,T iNj · ki,T iNj · cdc,T iNj + kv,T iNj · ki,Cij · cCij,Nj cNj = j

+



j

kv,T iNj · ki,Cj k · cCj k,Nj .

(4.3.13)

j

Note that the equivalent capacitances of the cable models usually are quite small, especially if the cables are modeled with multiple π-sections. Thus, if there is a 2L VSC terminal connected to a DC node, then the DC-side capacitance of the converter terminal will typically be dominant in the equivalent node capacitance cNj . For MMC-based converter terminals, depending on the design, there may not be any DC-side capacitor, implying that the equivalent

89

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

capacitance would result only from the cables connected to the DC node.

4.3.3 System model synthesis From the generic subsystem models defined above, the overall model of any DC grid configuration can be synthesized. However, this process requires the interconnection of the subsystems and the calculation of the steady-state operating point, as well as the linearization of the subsystem models, before the overall smallsignal model of the system can be generated.

4.3.3.1 Organization of system equations and reduction to state-space form Considering a system with any number of converter terminals, cable interconnections, and DC nodes, the state equations can be organized in any arbitrary sequence. The approach utilized in the following is listing all state-equations associated with converter terminals consecutively, followed by all states associated with the cables before listing the state equations of the DC nodes that will interconnect the other subsystem models. Furthermore, the model must be reduced to state-space form by eliminating the algebraic equations from (4.3.7) and (4.3.11). Thus, substituting (4.3.7) and (4.3.11) into (4.3.12), we can list the generic set of state equations for an HVDC grid as .. .

  x˙ T iNj = fT iNj xT iNj , uT iNj , kv,T iNj · vNj , .. . x˙ Cj k = FCj k xCj k + LCj k vNj + MCj k vN k , .. . ⎛ (4.3.14) ωb ⎝ ki,T iNj · gT iNj (xT iNj , uT iNj , kv,T iNj · vNj ) v˙Nj = cNj j   + GNj,Cij xCij − HNj,Cij vNj j

⎞   GNj,Cj k xCj k + HNj,Cj k vNj ⎠ . − j

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

These equations define the complete system model, which can be utilized for calculating the equilibrium point and for obtaining the linearized small-signal model.

4.3.3.2 Calculation of steady-state operating point As can be implied by the generic form of the presented subsystem models, the overall system model (4.3.14) contains a mix of nonlinear and linear equations. For the linear equations, the behavior is not affected by the operating point. Thus the system matrices are constant, independently of the linearization, and do not depend on the steady-state operating point. However, for the nonlinear equations, the linearized models will explicitly contain elements depending on the operating point. Thus the calculation of the operating point for the overall system is necessary to numerically evaluate the system matrices and eigenvalues. Once the system is expressed in the state-space form, imposing the condition of zero time derivatives according to (4.2.5) leads to a system of algebraic equations. The solutions of these algebraic equations for a specified set of input signals will provide the equilibrium or steady-state conditions required for linearization. Note that since parts of the algebraic system are nonlinear, multiple solutions may be possible. However, for most practical examples, only one of the solutions offers realistic and desired operating conditions. For instance, solutions corresponding to values of the voltages or currents outside the acceptable range of operation should be discarded, despite being a mathematical solution to the system equations. The general expression of the system of algebraic equations to be solved for finding the equilibrium point needed for linearization of the system from (4.3.14) can be expressed as

. 0 = ..   0 = fT iNj xT iNj,0 , uT iNj,0 , kv,T iNj · vNj,0 , . 0 = .. 0 = FCj k xCj k,0 + LCj k vNj,0 + MCj k vN k,0 ,

91

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

. 0 = .. ⎛ ωb ⎝  0= ki,T iNj · gT iNj (xT iNj,0 , uT iNj,0 , kv,T iNj · vNj,0 ) cNj j   + GNj,Cij xCij,0 − GNj,Cj k xCj k,0 j

j

⎞   HNj,Cij + HNj,Cj k vNj,0 ⎠ , − j

(4.3.15) where the subscript 0 denotes the steady-state operating point to be found. Then, these equations also imply how the general solution of the state variables x0 at the linearization point will result from the specified input vector u0 , which will define the operating point of the system. Thus a new solution to the equations should be obtained for any new combination of input signals to be evaluated. From (4.3.15) we can obtain the equilibrium point for linearization by any numerical algorithm for solving nonlinear equations. Typically, implementations based on the Newton–Raphson iteration can be used for this purpose, although some practical adaptations for ensuring a good initial estimation of the different state variables can be necessary, as further discussed in Section 4.5.

4.3.3.3 Linearization and assembly of the small-signal model The symbolic expressions for the linearized model can be directly obtained from the general nonlinear state-space model from (4.3.14). However, the numerical value of the matrix elements for the small-signal model cannot be calculated without knowledge of the equilibrium point as defined by (4.3.15). Still, for illustrating the procedures and the resulting structures of the small-signal model, it is convenient to show the process of linearization and model assembly in general symbolic form. Thus (4.3.16) shows on generic form the linearization of the defined subsystem models and how the different elements of these equations result in different submatrices to be used in the resulting

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

system-level small-signal model. .. .

 ∂fT iNj (xT iNj , uT iNj , kv,T iNj · vNj )   xT iNj =xT iNj,0 xT iNj ∂xT iNj u =u

˙xT iNj =



T iNj T iNj,0 vNj =vNj,0



AT iNj

 ∂fT iNj (xT iNj , uT iNj , kv,T iNj · vNj )      ∂ kv,T iNj · vNj

+





kv,T iNj vNj

AT iNj,v

+

xT iNj =xT iNj,0 uT iNj =uT iNj,0 vNj =vNj,0

∂fT iNj (xT iNj , uT iNj , kv,T iNj ∂uT iNj

 · vNj )  



xT iNj =xT iNj,0 uT iNj =uT iNj,0 vNj =vNj,0

uT iNj ,

BT iNj

.. . ˙xCj k = FCj k xCj k + LCj k vNj + MCj k vN k , .. .   ωb ∂gT iNj (xT iNj , uT iNj , kv,T iNj · vNj )  ki,T iNj · v˙Nj =  xT iNj =xT iNj,0 xT iNj cNj ∂xT iNj u =u j

⎡ ⎢ +⎢ ⎣

T iNj T iNj,0 vNj =vNj,0



ANj,T iNj



ωb j cNj ki,T iNj

−





ωb j cNj

·



∂gT iNj (xT iNj ,uT iNj ,kv,T iNj ·vNj )   xT iNj =xT iNj,0 ∂vNj uT iNj =uT iNj,0 vNj =vNj,0

  HNj,Cij + HNj,Cj k 

⎤ ⎥ ⎥ vNj ⎦

ANj

+

 ωb  ωb GNj,Cij xCij − GNj,Cj k xCj k cNj cNj j j





ANj,Cij

A

Nj,Cj k    ωb ∂gT iNj (xT iNj , uT iNj )  + ki,T iNj ·  cNj ∂uT iNj x

j





uT iNj .

T iNj =xT iNj,0 uT iNj =uT iNj,0

BNj,T iNj

(4.3.16) Assembling the overall small-signal model from the states and input signals of all subsystem models and the matrices resulting from the linearization in the general state-space form according to (4.2.9), the resulting structure of the system level A- and Bmatrices is indicated in (4.3.17). Thus eigenvalue-based analysis of the small-signal dynamics and stability of the overall system can

93

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

be conducted on the resulting A-matrix. ⎡

.. .

⎤

⎡

⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ˙xT iNj ⎥ ⎢ ⎢ ⎥ ⎢ .. ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ . ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ .. ⎢ ⎥ ⎢ . ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ˙xCj k ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ .. ⎢ ⎥=⎢ ⎢ ⎥ ⎢ . ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ .. ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ . ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ v˙Nj ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ v˙N k ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎣ ⎦ ⎢ ⎣ .. .





. .. 0

0 AT iNj

0

0

0 0 .. .

0 0

0 0

0 0

0

0

0

. . .. .. 0 AT iNj,v .. 0 .

0 FCj k 0

0 0 .. .

. .. 0 .. .

0 0

0 0

0 0

. .. 0

0

0

0

0

0 .. . .. . .. .

.. . .. . .. . .. .

.. . .. . .. . .. .

0 ANj,T iNj 0 0

˙x

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ·⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

.. .

⎤

⎡

. ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ xT iNj ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ 0 . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ 0 . ⎥ ⎢ ⎥ ⎢ xCj k ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ .. ⎥+⎢ 0 ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ . . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ v˙Nj ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ v˙N k ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎦ ⎣ .. .. . . 



x

0 ANj,Cj k AN k,Cj k 0



.. . .. . .. . .. .

  A x0 ,u0

0 BT iNj 0 0 0 0 0 BNj,T iNj 0 0 

  B x0 ,u0

⎤ 0 0 .. .

0 0 .. .

. ..

. ..

LCj k .. .

MCj k .. .

. .. 0 .. .

.. .

0

0

0

0

ANj

0

0

0

0

AN k

0

0

0

0 .. .

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ .. ⎥ ⎥ . ⎥ ⎥ ⎥ ⎥ 0 ⎥⎡ ⎥ ⎥ . .. 0 ⎥ ⎥⎢ ⎢ ⎥⎢ uT iNj 0 ⎥ ⎥⎢ ⎥⎣ . ⎥ .. . ⎥ .. ⎥  ⎥

⎥ u .. ⎥ ⎥ . ⎥ ⎥ .. ⎥ ⎥ . ⎥ ⎥ ⎦ .. .

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

(4.3.17)

4.3.3.4 Example of system-level small-signal state-space model To give a generic example of how the small-signal state-space model of a multiterminal DC grid can be assembled, we consider the four-terminal meshed DC grid illustrated in Fig. 4.2. By applying the generic expressions for the linearized subsystem models resulting from (4.3.16) and considering the interconnections be-

⎡

⎤

⎡

⎤ AT1N1,v 0 0 0 ⎥ ⎢ ⎥ 0 AT2N2,v 0 0 AT2N2 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 0 0 A 0 0 A T3N3 T3N3,v ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 0 0 0 A 0 0 A ⎥ ⎢ T4N4 T4N4,v ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 0 0 0 FC12 LC12 MC12 0 0 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 0 FC23 0 LC23 MC23 0 0 0 0 ⎥=⎢ ⎥ ⎥ ⎢ 0 0 FC24 0 LC24 0 MC24 ⎥ 0 0 0 ⎥ ⎢ ⎥ ⎥ ⎢ 0 0 0 0 0 0 FC34 0 0 LC34 MC34 ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ AN1,T1N1 ⎥ 0 0 0 AN1,C12 0 0 0 AN 1 0 0 0 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 0 0 0 A A 0 0 A 0 0 0 A N2,T2N2 N 2 N2,C23 N2,C24 ⎥ ⎢ ⎥ ⎦ ⎣ ⎦ 0 0 AN3,C23 0 AN3,C34 0 0 AN 3 0 0 0 AN3,T3N3 0 0 AN4,C24 AN4,C34 0 0 0 AN 4 0 0 AN4,T4N4 ⎡ ⎤ ⎡ ⎤ BT1N1 0 0 0 xT1N1 ⎢ xT2N2 ⎥ ⎢ ⎥ 0 0 0 BT1N2 ⎢ ⎥ ⎢ ⎥ ⎢ x ⎥ ⎢ ⎥ 0 0 0 BT1N3 ⎢ ⎥ T3N3 ⎥ ⎢ ⎢ ⎥ ⎢ 0 0 0 BT1N4 ⎥ ⎢ xT4N4 ⎥ ⎢ ⎥⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ xC12 ⎥ ⎢ ⎥ uT 1N 1 0 0 0 0 ⎢ ⎥ ⎢ ⎥⎢ ⎢ xC23 ⎥ ⎢ ⎥ ⎢ uT 1N 2 ⎥ 0 0 0 0 ⎥ ⎥+⎢ ⎥⎢ ·⎢ (4.3.18) ⎥. ⎢ x ⎥ ⎢ ⎥ ⎣ u 0 0 0 0 T 1N 3 ⎦ C24 ⎥ ⎢ ⎢ ⎥ ⎢ x ⎥ ⎢ ⎥ u 0 0 0 0 ⎢ ⎥ T 1N 4 C34 ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ vN1 ⎥ ⎢ BN1,T1N1 ⎥ 0 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ vN2 ⎥ ⎢ ⎥ 0 0 0 BN2,T1N2 ⎢ ⎥ ⎢ ⎥ ⎣ vN3 ⎦ ⎣ ⎦ 0 0 0 BN3,T1N3 vN4 0 0 0 BN4,T1N4 AT1N1 0 0 0 0 0 0 0

0

0 0

0 0 0

0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0

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

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

˙xT1N1 ˙xT2N2 ˙xT3N3 ˙xT4N4 ˙xC12 ˙xC23 ˙xC24 ˙xC34 v˙N1 v˙N2 v˙N3 v˙N4

95

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

Figure 4.2. Schematic of multiterminal HVDC grid used for example of small-signal state-space modeling.

tween the subsystem models according to the structure of the grid as defined by Fig. 4.2, we can obtain the linearized state-space model as given by (4.3.18). A similar exercise could be conducted with any other configuration of the DC grid. Furthermore, the approach is general in terms of the internal subsystem dynamics as long as the interfaces are compatible with the given specifications. Thus models of different converter topologies or converter terminals with different control strategies can be easily combined into the same overall system model. Similarly, the different cables can be represented by models specified from the modeling requirements of a specific case together with the corresponding cable parameters and lengths.

4.4 Examples of sub-system modeling According to the defined structure for model synthesis, the converter terminals and cable sections of the DC grid are modeled as a separate subsystem. The models of the converter terminals should include the AC–DC power conversion, as well as the ACside filters and the control loops of the converter. If the system to be modeled is used for interconnection of asynchronous AC systems, a simplified equivalent model of the AC grid can also be included in the model of each converter terminal. However, if further descriptions of the AC-side dynamics should be included or if the modeled system includes a larger synchronous AC grid with multiple converter terminals interfaced to different locations in the DC grid, then a similar modular approach should also be followed for the modeling of the AC system. Since the main focus here is to study the small-signal dynamics associated with the DC system and its control, this approach is not further pursued. Furthermore, the modeling of DC nodes has been already defined in Section 4.3.2.4. Thus only examples of state-space models for representing converter terminals and HVDC cables will be presented.

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

4.4.1 AC–DC converter terminals As mentioned in the Introduction, significant efforts have been dedicated to the modeling and analysis of converter terminals in HVDC systems during the last two decades. The initial efforts were mainly concentrated on the modeling of two-level (2L) or threelevel (3L) VSCs with their associated control systems, whereas the efforts during the last years have been mainly directed toward modeling MMC-based converter terminals. Thus the state-space representation of VSCs for HVDC applications based on average modeling of three-phase converters is well established [23,24]. As a regular three-phase VSC can be conveniently modeled by the instantaneous power balance between the AC and DC sides of its power conversion stage, the main variations on VSC modeling are associated with associated with the studied control system. However, for the MMC topology, the internal capacitance can have significant impact on the dynamic response between the AC- and DC-side terminals, and the control system implementation can have significant influence on how a state-space model should be expressed [25]. Furthermore, the models of MMC-based converter terminals can become significantly more complex compared to the models of 2L VSC-based terminal. Thus, in the following, a detailed example of how a 2L VSC can be modeled will be first presented, followed by an outline of how a simplified approach for MMC-modeling can be applied for the same purpose. Note that any model of the converter terminals, independently of converter topology and control objective, can be utilized in the presented framework as long as the interfaces are defined to be compatible with the presented approach for system-level model synthesis.

4.4.1.1 Example of AC-power controlled HVDC terminal with two-level voltage source converter As an example, we consider the modeling of a 2L VSC with AC-side power control for illustrating in detail how the subsystem state-space models can be derived and utilized in the presented modeling approach. A detailed block diagram of the studied configuration and the assumed control system is given in Fig. 4.3, adapted to the DC node interfaces specified in Section 4.3. Considering a conventional approach for dq-frame modeling of the assumed AC-side system configuration and representing all the integrators in the assumed control system as state variables, the state and input vectors for the model of the converter terminal are

97

98

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

Figure 4.3. Overview of model representing a converter terminal based on a two-Level voltage source converter with AC-side power control, adapted from [27].

defined by  xT iNj = vo,d uT iNj =



ϕq ∗ il,q

vo,q

il,d

il,d

γd

vP LL,d

vP LL,q

vˆg

∗ pac

ωg

γq

P LL T ∗ vdc .

io,d δθP LL

io,q

ϕd

vdc,f

ρ

pac,m

T

(4.4.1) This state-space model includes four states related to the representation of the phase-locked loop (PLL) that are not further described here but explained in [26,27]. Thus the subsystem model presents 17 state variables and 5 input signals. A list of all the variable names with a brief explanation is given in Table 4.1. Furthermore, the voltage VNj at the DC node where the converter terminal is connected is an additional input to the model defined in (4.3.6). It should also be mentioned that the model is presented on perunit quantities based on the local set of base values defined from the power and voltage rating of the converter terminal. The perunit quantities are denoted by lower case symbols in the equations and in Fig. 4.3, whereas the variables and parameters expressed in physical units are shown with upper case symbols. The detailed procedures of deriving the different stateequations, and for eliminating algebraic equations resulting from the block diagram structure, to obtain a nonlinear state-space

,

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

model in the general form given by (4.2.1) has been discussed and documented in several recent publications [16,26,27]. However, the resulting set of state equations for the specific case considered here is explicitly given in (4.4.2). For simplicity, the subscripts TiNi for identifying the converter terminal T i and the DC node N i that it is connected to, are omitted in the equations. The interface to the DC node, as given by the DC-side current flowing to the equivalent DC-side capacitance according to (4.3.7) is expressed by (4.4.3). Note that this current is the negative value of the DC-current Idc indicated in Fig. 4.3, which is calculated from the instantaneous power balance between the AC and DC sides of the converter model. ωb ωb il,d − io,d + ωb ωg vo,q , cf cf ωb ωb 2) v˙o,q = il,q − io,q − ωb ωg vo,d , cf cf kp,c + rf ωb il,d − ki,P LL ωb P LL il,q 3) i˙l,d = − lf   vP LL,q kAD − kF F,v + 1 il,q − − kp,P LL ωb tan−1 ωb vo,d vP LL,d lf kδ,vdc kp,c kp,pac kAD ωb ki,c ωb + φd − ωb vdc,f + γd lf lf lf ki,pac kp,c ωb kp,c kp,pac ωb + ρ− pac,m lf lf ∗ + p ∗ )ω kp,c kp,pac (kδ,vdc vdc ac b + , lf   vP LL,q il,d 4) i˙l,q = ki,P LL ωb P LL il,d + kp,P LL ωb tan−1 vP LL,d ∗ k il,q p,c ωb kp,c + rf − ωb il,q + lf lf kAD − kF F,v + 1 kAD ωb ki,c ωb − ωb vo,q + φq + γq , lf lf lf

1) v˙o,d =

5)

γ˙d = −il,d − kδ,vdc kp,pac vdc,f + ki,pac ρ − kp,pac pac,m ∗ ∗ + kp,pac (kδ,vdc vdc + pac ),

6)

∗ , γ˙q = −il,q + il,q

7)

i˙o,d = −

8)

i˙o,q

rg ωb io,d + ωb ωg io,q − lg rg ωb = −ωb ωg io,d − io,q + lg

vˆg ωb ωb cos(δθP LL ) + vo,d , lg lg vˆg ωb ωb sin(δθP LL ) + vo,q , lg lg

99

100

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

Table 4.1 List of state variables and input signals for model of power-controlled 2L VSC. Symbol Description State variables vo,d

d-axis voltage at AC-side filter capacitors

vo,q

q-axis voltage at AC-side filter capacitors

il,d

d-axis current component in the filter inductor

il,q

q-axis current component in the filter inductor

γd

Integrator state for d-axis current controller

γq

Integrator state for q-axis current controller

io,d

d-axis current component in the equivalent grid impedance

io,q

q-axis current component in the equivalent grid impedance

ϕd

Internal d-axis low-pass filter state used for active damping of LC-oscillations

ϕq

Internal q-axis low-pass filter state used for active damping of LC-oscillations

vPLL,d

Internal low-pass filtered d-axis voltage component in the PLL

vPLL,q

Internal low-pass filtered q-axis voltage component in the PLL

P LL

Integrator state for PI controller in the PLL

δθPLL

Integrator state representing relative phase angle displacement between the grid voltage and the PLL orientation

vdc,f

Low-pass filtered measurement of the DC voltage input signal vNj

ρ

Integrator state of PI controller for active power flow

pac,m

Low-pass filtered measurement of AC-side active power flow

Input signals vˆg ∗ pac iq∗ ∗ vdc

Amplitude of equivalent grid voltage AC-side active power reference Reactive current reference DC-side voltage reference

9) φ˙ d = ωAD vo,d − ωAD φd ,

10) φ˙ q = ωAD vo,q − ωAD φq ,

11) v˙P LL,d = ωLP ,P LL vo,d − vP LL,d ωLP ,P LL , 12) v˙P LL,q = ωLP ,P LL vo,q − ωLP ,P LL vP LL,q   −1 vP LL,q , 13) ˙P LL = tan vP LL,d ˙ P LL = ki,P LL ωb P LL + kp,P LL ωb tan−1 14) δθ



vP LL,q vP LL,d

 ,

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

15) v˙dc,f = ωvdc vNj − ωvdc vdc,f , ∗ ∗ + kδ,vdc vdc , 16) ρ˙ = −pac,m − kδ,vdc vdc,f + pac   17) p˙ ac,m = ωpac il,d vo,d + il,q vo,q −ωpac pac,m .



  gT iNj xT iNj , uT iNj =

(4.4.2)

pac

∗ −v kp,c kp,pac kδ,vdc (vdc kp,c 2 kAD φd dc,f ) il,d − il,d − il,d vNj vNj vNj ∗ −p kp,c kp,pac (pac ki,pac ρ + ki,c γd ac,m ) − il,d − il,d vNj vNj kp,c 2 kAD − kF F,v + il,d vo,d + i vNj vNj l,q ∗ +k kp,c il,q AD φq + ki,c γq kAD − kF F,v − il,q + il,q vo,q . vNj vNj (4.4.3)

This model is obtained under the assumptions that the converter is always operating in the linear range of modulation and that the average modeling of the converter is valid for all relevant operating conditions to be studied. We also assume that the switching frequency of the converter and the sampling frequency of the digital control system implementation are relatively high so that the continuous approximation can be directly applied. However, it could be possible to extend the model with a continuoustime rational approximation of the delays associated with the control system and the PWM switching operation of the converter. This could typically include a first-order filter approximation or a Padé approximation, which would introduce additional states and also an additional cross-coupling between the d- and q-axis dynamics. The matrices defining the small-signal model resulting from linearization of the presented equations are according to the discussions in the previous section defined by AT iNj , AT iNj,v , ANj,T iNj , and BT iNj . Since explicit representation of all the matrix elements can be easily obtained from (4.4.2) and (4.4.3), the details are not shown here. However, an overview of the nonzero elements in AT iNj and BT iNj is shown in Fig. 4.4. As can be seen from this figure, AT iNj has 49 nonzero (nz) elements, whereas BT iNj has 13 nonzero elements. Furthermore, we can find from (4.4.3) that ANj,T iNj is a row vector with 11 nonzero elements.

101

102

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

Figure 4.4. Overview of nonzero matrix elements in AT iNj and BT iNj for a generic 2L VSC HVDC converter terminal with AC-side power control.

4.4.1.2 Example of modular multilevel converter-based HVDC terminal As already mentioned, the modular multilevel converter (MMC) topology is becoming the preferred choice for VSC-based HVDC transmission systems due to its modularity, low losses, reduced filtering requirements, and scalable design, which makes it suitable for high voltage levels [8]. Therefore significant efforts have been recently dedicated to the modeling and analysis of MMC HVDC terminals. Indeed, the MMC topology exhibits internal dynamics that in some cases can influence its interactions with the AC- or DC-side networks [28–32]. The main approaches recently proposed in the literature for detailed state-space modeling of MMCs include methods based on harmonic linearization [28,29], dynamic phasor modeling [33], and modeling in multiple synchronously rotating dq reference frames [25,31,32], as well as harmonic state-space modeling [34]. However, the required detailing level of an MMC model depends on the intended application and is significantly influenced by the strategy used for calculating the modulation index signals. In case

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

the calculation of the insertion index signals includes a compensation for the continuous variations in the equivalent arm capacitor voltages, it has been shown that a simplified model of the MMC can be developed [25,30,31]. This approach for calculating the insertion index signals can be referred to as “compensated modulation” (CM). With a CM-based control system, the dynamic response at the AC- and DC-side terminals of the converter can be effectively decoupled from the dynamics of the internal circulating currents of MMC and the associated dynamics of the equivalent internal capacitor voltages. Thus, under the assumption of CM-based control, the MMC can be represented by a dynamic model that includes only the aggregated dynamics of the internal capacitive energy storage of the MMC in addition to the AC- and DC-side currents [30,31]. In the following, we will use the simplified model of an MMC with CM-based control presented in [30] as an example of how MMC HVDC terminals can be included in the presented approach for modular small-signal modeling of HVDC grids. Thus we will briefly introduce the applied modeling approach and indicate how this model can be directly adapted to the presented modular approach for small-signal analysis. An overview of the MMC configuration and the corresponding simplified control system assumed for the presented example is shown in Fig. 4.5. This figure indicates how an aggregated arm model (AAM) is assumed as a starting point for the modeling of the MMC topology [35]. In addition to the arm inductances of each phase, the figure also indicates how the assumed model includes an external filter and/or a simplified representation of a transformer modeled by its equivalent leakage inductance. If only the converter terminal itself is considered, then the model can include only the inductance and consider the grid-side voltages as inputs. However, for representing the equivalent grid, an LC circuit corresponding to a π-equivalent of the AC grid and an ideal equivalent voltage source are included in the following. The assumed control system of the MMC is based on independent control of the AC-side and circulating currents. Thus a conventional dq-frame controller is assumed for the AC-side currents, with a reference signal for the active current component provided by an outer loop controller for the power or the DC voltage. Similarly, the circulating currents are assumed to be controlled by an inner loop current controller, with a reference signal for the DCcomponents given by an outer loop energy controller, as explained in the following. For analyzing the MMC and the assumed control system, it is useful to remind how the assumption of the CM-based control

103

104

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

Figure 4.5. Overview of MMC HVDC converter terminal and a simplified representation of a CM-based control system used as an example for modeling and analysis, adapted from [27].

system simplifies the modeling of the MMC and the representation of the control system. For this purpose, the basic equations defining the relationship between the AC-side currents, the circulating currents, and the individual arm currents, as well as the corresponding voltages can be defined as [30,36,37] ivk = iuk − ilk , k, vu,l =

N 

ick =

k,SMi vu,l ,

i=1 vlk − vuk

iuk + ilk 2

for

k ∈ abc,

k, k vu,l ≈ nku,l · vu,l ,

nkl · vlk, − nku · vuk, , 2 2 v k + vuk nk · v k, + nku · vuk, vck = l ≈ l l , 2 2 vvk =

≈

(4.4.4) (4.4.5)

(4.4.6) (4.4.7)

where iv is the AC-side current, ic is the circulating current component, and vv and vc are the associated voltage components driving these currents. The subscripts u and l refer to the upper and lower arms, whereas the superscript  refers to the sum of the capacitor voltages in one arm. Furthermore, the insertion index calculation under CM-based control is given by

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

nku = nkl =

−vvk,∗ +·vck,∗ , vuk, vvk,∗ +·vck,∗ . vuk,

(4.4.8)

By substituting (4.4.8) into (4.4.6) and (4.4.7) we can see that the voltage components vv and vc , are equal to the reference values resulting from the control system. This simplification will be valid when the converter is in a normal operating condition and a signal representing the instantaneous value of the sum capacitor voltage of each arm is available in the control system, and as long as the impact of delays can be considered negligible. Under these conditions, the AC-side and circulating currents can be controlled independently, and the control of these current components can be considered independent from the capacitor voltage dynamics. Thus the only dynamics that needs to be modeled for representing the MMC as observed from its AC- and DC-side terminals will be the currents at the terminals and the equivalent energy dynamics that influences the power transfer between the AC and DC sides of the converter. Applying a complex space vector representation (x = xd + j xq ), the dynamics of the AC-side currents in the synchronous dq reference frame can be expressed by (4.4.9). Furthermore, the DC-side current, which will equal to the zero-sequence components of the circulating currents according to (4.4.4), is given by (4.4.10) [30].   ra /2 + rf ωb ωb ωb div = vv − vo − + j · ωg ωb iv , dt la /2 + lf la /2 + lf la /2 + lf (4.4.9) ωb ra ωb dic,z ωb = vdc − vc,z − ic,z . (4.4.10) dt la la la From these equations we can see that the expressions for the ACside currents will have the same form, although based on different parameters, as for the 2L VSC. Assuming that the AC grid can be represented as indicated in Fig. 4.5, the general modeling approach and the control structure for the AC side of the HVDC converter terminal can be the same as for the 2L VSC. Thus also the state equations for this part of the model will be the same. Since the DC-side current of the model will be given directly by the zero sequence circulating current ic,z according to (4.4.10), this state variable will also provide the direct interface to the DC node. Thus g TiNj according to (4.3.7) will be equal to −ic,z . In addition to the AC-side current, represented by the same model as for the 2L VSC, and the DC-side current given by (4.4.10), the simplified model of an MMC with the CM-based control must

105

106

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

include a representation of the equivalent aggregated energy balance of the internal capacitance. The corresponding energy dynamics can be defined from the power balance between the ACand DC-terminals and the total equivalent capacitive energy storage within the MMC, as given by [30]: ⎤ ⎡   dw ⎥ ωb ⎢  . (4.4.11) = ⎣− vv,d · il,d + vv,q · il,q + 4vc,z · ic,z ⎦ dt



 8ceq pac

pdc

Note that the scaling in this case originates from the equivalent capacitance ceq of the MMC being represented in a per-unit system based on the AC-side quantities. From these considerations it is clear that the simplified modeling of the MMC topology implies two additional state variables compared to the model of a 2L VSC. These two states are necessary for representing the zero-sequence circulating current according to (4.4.10) and the equivalent energy storage dynamics according to (4.4.11). Thus two corresponding control loops should also be included in the modeling, as indicated in Fig. 4.5, which shows how the controller for the total stored energy in the MMC provides the current reference for the inner loop control of the zerosequence circulating current. With this modeling approach, any additional control loop implemented for regulating other components of the circulating current and the stored energy in the MMC does not have to be represented in the model. Thus the currents and voltages commonly associated with the control of second harmonic circulating current components are indicated in gray in Fig. 4.5. Assuming that PI-controllers are used for the control of the total equivalent energy and for the zero-sequence current controller, two additional state equations, in this case labeled as κ and ξz must be included in the state-space model. Thus the states and input vectors for an MMC with the same basic control system as presented for the 2L VSC can be listed as xT iNj =



vo,d

vo,q iv,d iv,d γd γq io,d io,q ϕd ϕq vP LL,d vP LL,q  δθP LL vdc,f ρ pac,m ic,z T w,z κ ξz , T  ∗ ∗ ∗ ∗ vˆg ωg pac vdc w . uT iNj = il,q ,z (4.4.12) With four additional states compared to the 2L VSC model in (4.4.2) and the reference for the total energy as an additional input signal, the resulting MMC model has 21 states and 6 inputs.

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

Since the model complies with the defined interfaces to the DC node, it can be used for linearization and synthesis of a systemlevel model in the same way as the presented model for the 2L VSC. Furthermore, note that both the presented model for the 2L VSC and this briefly introduced MMC model are based on ACside active power control with a droop-gain depending on the DC voltage. However, the models can also be configured for other control objectives, for instance, by having DC voltage control as the primary objective, possibly with an AC-power-dependent droop function, as documented in [27]. Furthermore, the MMC model can be adapted for different configurations of the equivalent energy control, as discussed in [38].

4.4.2 Modeling of long cables for analysis of HVDC grids Cables are an essential component in HVDC grids since they physically interconnect the terminals and transfer the power over the long distances required. State-of-the-art wide-band models like the universal line model (ULM) [39] ensure a high level of accuracy and an efficient computational implementation for timedomain simulation using electromagnetic transient (EMT)-type software, but their formulation is inherently unsuitable for statespace modeling. Thus a modeling approach specifically developed for obtaining state-space representation of the dynamic characteristics of HVDC cables within a predefined frequency range have been recently developed [16,40,41]. This model, denoted as a frequency-dependent π (FD-π) model, reproduces the behavior of a DC cable by assuming a lumped parameters circuit representation composed by cascaded π sections with multiple longitudinal RL-branches in parallel within each section. An example of an FD-π model of a cable in generic form, adapted to the DC node interfaces specified in the previous section, is represented in Fig. 4.6. Thus the model representing the cable can be specified with n sections, where each section is consisting of m parallel RL branches. The parameters of the parallel RL branches can be calculated from the frequency-dependent series impedance of the cable by vector fitting, as further discussed in [27,41]. Furthermore, the values at each section for the capacitance cg and the equivalent resistance rg are resulting from the cable capacitance and conductance per length, the total length of the cable, and the selected number of sections. For consistency with the modeling of the converter terminals, the equivalent cable model in Fig. 4.6 is illustrated with per-unit quantities.

107

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

Figure 4.6. Frequency-dependent π -model of HVDC cable, adapted from [27,40,41].

In the generic structure of Fig. 4.6 the equivalent capacitances at the cable ends will be included in the equivalent capacitance of the DC nodes if the cable is connected directly to the nodes. Therefore, they are indicated in gray in the figure. However, if the cable is interconnected to the nodes with inductive fault current limiters, then the lumped inductor model should be included in the subsystem model representing the cable and will directly define the current at the interface to the DC nodes [16]. A state-space representation of the cable can be directly obtained from the lumped circuit representation in Fig. 4.6. The state variables for the resulting model are the voltages at each of the internal nodes and the currents in the inductors of each section. Thus the state vector can be specified as T  . xCj k = i1,1 · · · i1,m v1 i2,1 · · · i2,m v2 · · · vn−1 in,1 · · · in,m (4.4.13) For simplicity, the reference to the specific cable section is omitted in the local state variables of the cable as a subsystem. The corresponding state equations for a cable with n sections and m parallel branches in each section can be obtained by applying the Kirchhoff laws on the nodes and on each parallel RL branch, and can be expressed on compact form as [27] ! m "  vs dvs ωb   is,t − is+1,t − ∀s : 1 ≤ s ≤ n, (4.4.14) = dt cg rg t=1

 dis,t ωb  vs−1 − vs − rt is,t ∀s : 1 ≤ s < n, ∀t : 1 ≤ t ≤ m, = dt lt (4.4.15)  di1,t ωb  = vNj − v1 − rt i1,t ∀t : 1 ≤ t ≤ m, (4.4.16) dt lt  ωb  din−1,t = vn−1 − vN k − rt in−1,t ∀t : 1 ≤ t ≤ m. (4.4.17) dt lt

⎡

− r1lωb

1

.. ωb cg

⎤

− ωl b

1

.

.. .

− rml ωb − ωl b m m b · · · ωcgb − rω − ωcgb · · · − ωcgb g cg ωb − r1lωb l 1

1

.. . ωb lm

.. ωb cg

.

− ωl b 1

.. .

− rml ωb − ωl b m m b − ωb · · · − ωb · · · ωcgb − rω cg cg g cg ..

.

ωb cg

..

.

..

.

b · · · ωcgb − rω − ωcgb · · · − ωcgb g cg ωb − r1lωb l 1

.. . ωb lm

1

..

. − rml ωb m

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.4.18)

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

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ FCj k = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

109

110

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

Since the state-state model that can be obtained from Fig. 4.6 is linear, the state equations can be expressed directly in matrix form according to the defined structure and the specified interfaces introduced in Section 4.3. Thus the following expressions define the subsystem models of cables in the DC grid when assuming that the scaling constants for the local per-unit system equal to 1.0. ⎤T

⎡ ⎢ LCxy = ⎣ ⎡



ωb l1

···

⎥ 0 ··· 0 ⎦ , 

ωb lm

m non-zero elements

⎤T

⎥ · · · − ωlmb ⎦ ,

⎢ MCxy = ⎣ 0 · · · 0 − ωl1b



(4.4.19)

m non-zero elements

⎤

⎡ ⎢ GNj,Cj k = ⎣ 1 · · · 1 0

 ⎡

⎥ ··· 0 ⎦,

m non-zero elements

⎢ GN k,Cj k = ⎣ 0 · · · 0 1 

⎤

⎥ ··· 1 ⎦,

(4.4.20)

m non-zero elements

HNj,Cj k =

1 , 2rg

HN k,Cj k =

1 . 2rg

(4.4.21)

Then these expressions can be directly applied for the calculation of the steady-state operating point of the nonlinear parts of the system and for assembling the overall system-level smallsignal model.

4.5 Practical considerations for modular and automated generation of system-level small-signal state-space models The modeling approach presented in the previous sections is characterized by a modular construction and standardized interfaces between the components. This ensures a high degree of scalability and facilitates the software implementation of the process required for synthesizing a system-level small-signal model. This section highlights general implementation aspects that should be considered without referring to any specific software. The exact

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

workflow depends on the software environment and the level of functions for symbolic manipulation and interconnection of systems expressed on state-space form.

4.5.1 Synthesis of state-space matrices for the system A first obvious possibility for generating system-level statespace models according to the structure presented in Section 4.3.3 is creating a repository of subsystem models and assembling these manually according to the studied system configuration. This will imply that the system-level model should be expressed as a set of nonlinear (and linear) state equations in symbolic form as presented in Section 4.3.3 and used as input to a software for symbolic manipulation (e.g., Mathematica, Maple, Matlab Symbolic Toolbox, etc.). In this case the small-signal model can be derived by symbolic linearization of the entire set of equations. The process would in general be relatively less scalable than other more numerical oriented alternatives and would be mainly recommended for systems of limited complexity. Indeed, note that even multiterminal configurations with three converter terminals can easily require the number of states to exceed 100, depending on the requirements for the cable modeling. Moreover, whereas the availability of the system matrices in symbolic form can be an advantage, the individual expression of some of the matrix elements can be complicated and of limited practical use. In any case the eigenvalues need to be calculated numerically. As another approach, subsystem models can be directly represented as simulation blocks in software for graphical representation of system dynamics, like Simulink or Modelica. In this case the assembly of the subsystems and the linearization can be handled by the software directly. The interconnection of the models can typically be performed manually or be scripted if a higher level of automation is desired. A third possibility is to explicitly code the procedure of assembling matrices obtained from subsystem models into the statespace matrices for the overall system. Implementation of this approach requires that the matrices representing the linearized models of each subsystem should be coded in symbolic form and then converted into numerical matrices when assembled into the system-level model. The process of assembling the components of the studied system from defined subsystems can be managed by predefined commands available in some software environments (e.g., “Connect” in Matlab) or coded according to the expressions presented in Section 4.3.3. In this case the code should be de-

111

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

signed to derive from the overall topology and the interconnection of the subsystems how the matrices should be combined. Then the matrices should be calculated and converted into numerical form for the linearization point to be evaluated.

4.5.2 Calculation of the steady-state operating point All approaches described above depend on the values of the state variables at the studied equilibrium for calculating the matrix elements resulting from linearization. In principle, this implies the solution of a set of nonlinear and linear equations given by (4.3.15), which could be obtained by any algorithm capable of providing a numerical solution to the set of equations defining the steady-state operating point. However, the solution to these equations should determine the load-flow and the operating point of all variables influenced by nonlinearities. Thus, depending on the system configuration, ensuring the convergence to the desired operating point can be a challenge. The numerical solution for obtaining the steady-state solution required for linearization can be especially sensitive to the initial estimation of the state variables. In case the process of obtaining the model is based on interconnecting blocks in numerical simulation software, a simple solution for obtaining the equilibrium point needed for linearization is simulating the time response of the system until it reaches steady-state operation and then using the results for calculating the matrices of the small-signal model. Although this approach is relatively straightforward to execute, it can be very time consuming, especially for large systems containing one or more modes with slow dynamics. Furthermore, this approach will not be applicable if the system is unstable and will consequently not be useful for analysis of how a system experiencing small-signal instability can be stabilized. To ensure a fast and accurate solution that can also be applicable for finding equilibrium points that experience smallsignal instability, a direct numerical solution to (4.3.15) should be obtained. The Newton–Raphson iterations or similar numerical techniques are suitable for this purpose. It is also worth noting that the Newton–Raphson method requires the Jacobian matrix, which in this case is the matrix A for the overall system model. Thus the main intermediate step for the numerical solution will be available from the linearization of the subsystems and the assembly of the system-level A-matrix. Furthermore, practical considerations could be included to avoid initial estimates of the state variables that will cause the Newton–Raphson iterations to converge to an unintended operating point. Although the initial

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

condition for voltages in the system can be reasonably assumed to be close to 1.0 pu for DC-side voltages and 1.0 pu in amplitude for the AC-side variables, specifying the status of integrators in the control system or the currents in the equivalent inductances of the assumed cable model can require further considerations. One approach that can be utilized to improve the initialization of the variables is using the linear equations that are closely related to the input signals and substituting the initial values that can be obtained from simplified calculations on the subsystem models into the state vector used as the initial guess for the numerical solution.

4.5.3 Applied procedure for generating system-level state-space models in the presented framework for modular subsystem modeling Here, we apply a custom implementation in Matlab for obtaining the presented results. This implementation is based on the general principles and considerations presented in the previous sections, and the resulting procedure can be summarized by the following workflow: 1. Define the topological configuration of the grid. The configuration of the MT HVDC system is defined by specification of the DC nodes, the converter stations connected to each node, and the cable interconnections between nodes. This step defines the topology of the MT HDVC grid, which is converted into a matrix variable. For practical reasons, each DC node, converter and cable are marked with a tag identifier. The base for the per-unit system is specified by indicating which ratings to consider as a reference for each subsystem, whereas the base values corresponding to the converter terminal with the highest power rating are selected as the global per-unit base for the DC nodes. 2. Input the submodules parameters and the values of the input variables. The parameters for each submodule (converter or cable) are specified. This should include the value of circuit parameters (inductances, capacitances, resistances, and corresponding model configuration for cables) as well as controller parameters and input signals at the studied operating point. 3. Generate system-level model: The structure of the systemlevel matrices is algorithmically generated as a function of the linearization point according to the basic principles specified in Section 4.3. Thus the matrices can be generated from symbolically predetermined matrices for representing the subsys-

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tem models and the subsystem parameters specified in the previous stage. The matrix A is then first utilized for the calculation of the steady-state conditions in the next step and for calculating the eigenvalues once the steady-state operating point is determined. 4. Calculate steady-state conditions for the system. The steadystate solution for the system is calculated by the Newton– Raphson algorithm, by applying the system-level A-matrix and an initial estimation of the state variables. The initial conditions for the Newton-Raphson iterations are generated by a script that calculates the initial value for each state by starting from initial values for a subset of states that could be easy to guess (i.e., voltages close to 1.0 and state variables controlled by closed-loop controllers close to their reference values). 5. Generate the system-level small-signal model. When the steady-state operating point is calculated, the numerical values for the matrices of each submodule and for the overall system can be calculated. 6. Reiteration for new operating conditions. Step 4 and 5 in this procedure can be repeated for updating the model with changes in the parameters and/or the operating conditions. For example, this is necessary for parametric sweeps. Then for small changes in operating conditions, the initial estimation of the state variables in the Newton–Raphson iteration can be based on the solution resulting from the previous calculation of the steady-state conditions. When the system-level small-signal model is generated, it can be directly applied for assessing the small-signal stability and the corresponding dynamics by calculating the eigenvalues of the Amatrix. Then the model can also be directly applied for participation factor analysis. However, parametric sensitivity analysis requires further processing of the model, since such studies will require either the partial derivative of the A-matrix with respect to the parameter to be studied or a programmed reiteration of the procedure for generating the small-signal model and calculating the eigenvalues resulting from a certain set of parameters and input signals. In the following, this second approach, based on point 6 in the above list, is preferred for illustrative purposes.

4.6 Example of small-signal analysis In this section, we illustrate the application of the eigenvaluebased approach for the analysis of an electrical dynamic system by considering a numerical example based on the multiterminal HVDC system presented in Section 4.3.3.4. The aim is highlighting

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

how the modular approach can be utilized to generate the systemlevel model for performing state-space-based small-signal analysis of a relatively complex configuration and how additional insights into the system dynamics can be gained by applying tools for linear system analysis such as the participation factors. An additional objective is demonstrating that the eigenvalue analysis can be implemented in a modular approach so that it can be easily applicable to larger and more complex systems configurations. Even if not directly indicated by the example, the modularity leads to a high degree of scalability and could be extended on the AC-side to enable analysis of configurations including larger AC systems.

4.6.1 Case description The presented example is based on analysis of the four-terminal HVDC configuration from Fig. 4.2, which is further specified with information about the power ratings, control objectives, and cable lengths in Fig. 4.7. The AC-side voltage rating for all converter terminals is 380 kV RMS line-to-line voltage, which, together with the power ratings, defines the basis for the per-unit scaling in the subsystems. The system is composed by three MMC converters and one 2L VSC terminal interconnected by four HVDC cables. In the investigated case, a single converter is operated with a closed loop DC voltage control, whereas the remaining three are controlling the power flow. However, the 2L VSC terminal with AC power control also contributes to the DC voltage regulation by a nonzero droop gain responding to variations in the DC voltage. For simplicity, the model assumes a monopolar configuration with ideal grounding. The main parameters of the cables and the converter terminals are listed in Table 4.2, whereas the input signals defining the operating conditions of the system are listed in Table 4.3.

4.6.2 Linearized state-space model The small-signal model of the system in Fig. 4.7 is obtained by applying the general principles of the modular approach described in Section 4.3 and models based on the examples presented in Section 4.4 by applying the procedure detailed in Section 4.5.3. The nonlinear state-space model of the 2L VSC terminal with power control, T2N2 was given in Section 4.4.1.1, whereas the models for the MMC-based converter terminals T1N1, T3N3, and T4N4 are based on the modeling approach from Section 4.4.1.2 by adaptation of the models documented in [27]. The cable models are obtained with parameters from [40,42] and configured with

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Table 4.2 List of system parameters for the studied example. Subsystem and main function

Parameter

Value

Cables: C12, C23, C34, C24

Number of parallel branches, m

3

Parameters and model configuration from [40,42] Converter terminals: T1N1, T2N2, T3N3, T4N4 Electrical system models

Number of cascaded pi sections, n

4

Filter inductance conv side, lf

0.16428 pu

Filter resistance conv side, rf

0.00285 pu

Filter capacitance, cf

0.0887 pu

Equivalent grid-side inductance, lg

0.1121 pu

Equivalent grid-side resistance, rg Arm resistance, ra

0.0001 pu 0.005 pu

Arm inductance, la

0.08 pu

Converter terminals: T1N1, T3N3, T4N4 Additional electrical system parameters for MMCs Converter terminals: T1N1, T2N2, T3N3, T4N4 Common control parameters

PI controller proportional gain of PLL kp,PLL

0.0844

PI controller integral gain of PLL ki,PLL

4.691

PLL low pass filter time constant, Tf,PLL

0.002 s

Current controller proportional gain, kp,c

0.26

Current controller integral gain, ki,c

2.14

Active damping gain kAD

0.2

Active damping filter time constant, Tf,AD

0.05 s

Converter terminals: T1N1, T3N3, T4N4

Energy controller proportional gain, kpw Energy controller integral gain, kiw

20

PI controller for total MMC energy Converter terminals: T1N1, T4N4 PI controller for active power

Power controller proportional gain, kp,pac

1

Power controller integral gain, ki,pac

10

DC voltage droop gain, kδ,vdc

0

DC voltage droop gain, kδ,vdc

−10

Converter terminal T2N2 – power control Converter terminal T3N3 PI controller for dc voltage

10

DC voltage controller proportional gain, kp,vdc 20 DC voltage controller integral gain, ki,vdc

20

Active power droop gain, kδ ,pac

0

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

117

Figure 4.7. Overview of studied system configuration with specified power ratings and control objectives.

Table 4.3 List of reference signals and set points defining the steady-state operating conditions. Subsystem Converter terminals: T1N1, T2N2, T3N3, T4N4 Converter terminal: T2N2

Converter terminals: T1N1, T3N3, T4N4

Parameter

Value

∗ Reactive current reference, il,q,f 0

0 pu

∗ DC voltage reference, vdc,0

1.0 pu

∗ Active power reference, pac,0 ∗ Active power reference, pac,0 ∗ DC voltage reference, vdc,0 ∗ energy reference, w,z,0

n = 4 π-sections and m = 3 parallel branches per section, as specified in Table 4.2, resulting in 15 states for each cable. Thus, considering the model definitions from Section 4.4, the final systemlevel state-space model will include 21 states for each of the three MMC-based converter terminals, 17 states for the 2L VSC terminal, four cables with 15 states and 4 states representing DC nodes, resulting in a model with in total 144 states. The system also has six inputs for each MMC terminal and five input signals for the 2L VSC terminal, resulting in totally 23 input signals to the model. The dynamics of the resulting linearized model is defined by the A- and B-matrices, and an overview of the nonzero terms of these matrices is shown in Fig. 4.8. The figure highlights a relevant aspect of the system structure that is relatively common for power systems. Indeed, the matrix A is relatively sparse with only a minority of the elements having nonzero values. In this specific example, only 502 elements of a 144 × 144 matrix have nonzero values, cor-

−0.5 pu 0.2 pu 1.2 pu 2.25 pu

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

Figure 4.8. Overview of nonzero elements in A- and B-matrices for the overall system model.

responding to a sparsity of about 2%. Furthermore, the figure reflects the structure of the general A-matrix from Section 4.3.3, with the blocks associated with the subsystem A-matrices along the diagonal and the elements defining the interconnection between the subsystems in the rows and columns associated with the DC node voltages.

4.6.3 Small-signal stability analysis The most direct application of eigenvalue-based analysis for the study of small-signal stability is assessing the location of the eigenvalues in the complex plane. An overview of the eigenvalues in the studied system is shown in Fig. 4.9(a). The location of the eigenvalues is according to recurring patterns for HVDC transmission systems, as, for instance, shown for smaller systems in [16,27], with the eigenvalues clustering together in defined regions of the plane. Several modes are located quite on the left on the complex plane and associated with very small time constants. These poles can be shown to originate from well-damped RL-responses within the applied cable model and will not have any noticeable influence on the overall system dynamics. A more enlarged view of

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

the eigenvalues with real part above −600 is shown in Fig. 4.9(b). The clustering of several almost overlapping poles resulting from similar dynamics in the different converter terminals and cables can be noticed in the figure, especially, for the poles with relatively high oscillation frequency close to the imaginary axis. These eigenvalues are normally associated with LC-oscillations in the output filters and/or the equivalent AC-side grid models of the converter terminals. Clusters of eigenvalues with relatively high oscillation frequency and high damping can also be noticed in the left halfs of the figures, and these modes are mainly related to the cables. A set of poles very close to the origin is normally associated with the outer loop controllers like the active and reactive power controllers, the DC voltage controller, and the energy control of the MMC, as well as the interaction between the control system and the AC grid via that PLL. An enlarged view of the figure highlighting the eigenvalues close to the origin is shown in Fig. 4.9(c).

4.6.4 Analysis of participation factors and system interaction To provide further insight into the dynamics of the system and how different eigenvalues can be associated with different parts of the system, we conduct a participation factor analysis on a selected set of modes. An explicit identification of the modes selected for further analysis is shown by colored markers in Fig. 4.10, whereas the main results from the participation factor analysis of these eigenvalues are listed in Table 4.4. In this table the marking in Fig. 4.10 and the corresponding numerical value of the eigenvalue is given in the two first column, whereas the third column shows which subsystem the participating states are associated with. Finally, the fourth and fifth columns show the name and the corresponding participation of each identified state in the evaluated mode. The participation is shown as the absolute value of the participation factors associated with the specified state, and only states with participation above 1% in the studied mode are explicitly listed. Note that although the participation factor matrix is scaled so that the sum of elements associated with each state is equal to 1.0, the participation factors are complex numbers, implying that the sum of the absolute values can be larger than 1.0. Also note that the participation factors are generally an indicative measure, implying that mainly the relative values should be considered, whereas the specific numerical values are of limited importance. The first mode investigated in Table 4.4 and marked with a yellow circle (light gray in print version) in Fig. 4.10 is a real mode

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Figure 4.9. Eigenvalues of the studied system at the specified operating point.

representing one of the slowest time-responses in the system. Indeed, this mode corresponds to a first-order response with a time constant of 2 s and has a dominant participation from the integral state of the energy controller, κ , of the MMC-based converter terminal T4N4. The mode also has smaller participation from the equivalent states in the other MMC-based terminals. Thus it represents a small degree of interaction between different terminals in the studied system.

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

Figure 4.10. Eigenvalues selected for participation factor analysis.

The low-frequency mode, marked with blue stars (black in print version) in Fig. 4.10, with eigenvalues at −5.45±0.019 i is mainly associated with the integral states of the current controllers in the converter terminal T2N2. Similarly, the high-frequency at −17.26±3617i, marked with black squares has participations from the voltages and currents at the AC side of terminal T1N1. Thus both these modes represent local dynamics associated with individual converter terminals. Although not clearly visible from Fig. 4.10, there are also several eigenvalues with almost the same numerical values as identified for these two modes, which are associated with the other terminals. The eigenvalues at −24.74±67.32 i, marked with cyan colored diamonds in Fig. 4.10, demonstrate a more complex mechanism of interaction within the studied system. Although this mode has the highest participation from the state ρ in terminal T2N2, which is the integrator state of the PI-controller for the active power flow, it also has significant participations from the voltages in all the DC nodes, the filtering of the DC voltage in T2N2 and T3N3, the ACside currents in T3N3, and the energy w,z in the MMC-based converter terminals. Furthermore, the mode has small participations

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Table 4.4 Participation factor analysis of selected modes. Markers

Eigenvalue

Subsystem State

Participation factor

Yellow circle

−0.50

Blue stars

−5.45±0.019 i

T1N1 T3N3 T1N4 T2N2

Black squares

−17.26±3617 i

T1N1

Cyan diamonds −24.74±67.32 i

N1 T1N1 N2 T2N2

0.188 0.013 0.826 0.510 0.517 0.252 0.250 0.020 0.020 0.229 0.228 0.0851 0.0280 0.0717 0.0567 0.1293 0.3688 0.1173 0.0580 0.0721 0.0144 0.1719 0.0169 0.0629 0.0179 Max 0.0324 Max 0.0150 Max 0.0171 Max 0.0201 0.426 0.484 0.062 0.065 0.026 0.022 0.027 0.164 max 0.0884 max

N3 T3N3

N4 T4N4 C12 C23 C24 C34 Green triangles −352.50±39.552 i T2N2

Red crosses

−475.1±3034 i

C34 C12

κ κ κ γd γq vo,d vo,q il,d il,q io,d io,q vN 1 w,z vN 2 il,d vdc,f ρ pac,m vN 3 il,d io,d vdc,f w,z vN 4 w,z 3 states 3 states 3 states 3 states il,d il,q io,d io,q vq,PLL δθPLL ρ 11 states 11 states

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

from all the cables in the system. Thus, since this mode has significant contributions from several subsystems, it can be clearly identified to represent an interaction in the system, as could be formally analyzed by the aggregation of participation factors with respect to the different subsystems as proposed in [16]. The last two modes studied in Table 4.4 correspond to relatively fast dynamics with short settling times. Whereas the mode marked with green triangles in Fig. 4.10 is a highly damped local mode associated with T2N2, the mode marked with red crosses is mainly associated with the internal dynamics of the HVDC cables. These modes have limited influence on the dynamics and operation of the overall system and are therefore not discussed in further detail.

4.6.5 Analysis of parametric sensitivity The participation factor analysis presented above can provide qualitative understanding of the interactions in a system and can be utilized to identify the variables that are involved in modes causing poorly damped oscillations or small-signal instability. However, the participation factor analysis does not give a clear link to the system parameters and how they will influence the modes. Instead, parametric sensitivity of the modes can be utilized to assess how system parameters could be modified to directly influence the small-signal dynamics of the system. Indeed, the eigenvalue parametric sensitivity given by (4.2.28) can highlight the effects of a specific parameter on the position of an eigenvalue. The result from calculations based on (4.2.28) is a complex number that expresses the tangent to the trajectory of a pole location with respect to a parameter in the complex plane. However, the sensitivity typically has a nonlinear dependency on the parameter itself, implying that a calculated sensitivity is only valid in a small range around the operating point and the parameter values used for calculation. Thus, for illustrative purposes, the parametric sensitivity can be more easily assessed by repeatedly calculating the eigenvalues while varying a parameter within a certain range. This approach can provide a simple interpretation of the dependence of the system dynamics on a certain parameter. However, if the parameter under study influences the steady-state operation of the system, the equilibrium point should be recalculated for each numerical value of the parameter. An example of a parametric sensitivity analysis of the mode at −24.74±67.32 i, marked with cyan diamonds in Fig. 4.10, is shown in Fig. 4.11(a). In this figure the traces of the eigenvalue for a change of the integral gain constant for the DC voltage controller, ki,vdc , in terminal T3N3 is shown within the range from

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1 to 100. This parameter is selected since the DC voltage of all converter terminals and the filtered DC voltage measurements of converter terminals responding to the DC voltage have relatively high participation in the mode. The direction of the change of the eigenvalues is indicated with a color gradient in the figure. The poles corresponding to the lowest value of the parameters are marked in blue (dark gray in print version), whereas red (gray in print version) is associated with the higher values. The eigenvalues are marked with black triangles if the real part increases above 0, causing small-signal instability in the system. Thus we can clearly see from the figure that increasing the value of ki,vdc will bring the system closer to the stability limit and can even cause instability for very high values. This can be confirmed from the time domain responses of the system as presented in Fig. 4.10, which clearly shows how the response of the DC voltage in the system to a small step change in the reference value becomes more oscillatory (i.e., approaches the red curve (gray in print version)) and finally reaches instability (illustrated with the black curve) when the value of ki,vdc is increased. Such analysis can be used for identifying suitable tuning of controller parameters in the system, especially, for influencing modes caused by interaction between several parts of the system and therefore influenced by many parameters. An example of how the parametric sensitivity can be utilized to improve the small-signal dynamics and stability margins of other types of systems is presented in [43]. Similar approaches, potentially combined with participation factor analysis to identify the main functions and subsystems associated with a critical oscillation mode, could also be further developed for application to multiterminal HVDC systems.

4.7 Conclusion In the chapter, we presented a modular approach to statespace modeling and eigenvalue-based analysis of small-signal dynamics in DC grids. Although the general principles of statespace modeling, linearization, and eigenvalue-based stability assessment are well established, the presentation in this context is specifically adapted for analysis of multiterminal HVDC transmission systems. Thus the chapter includes a brief introduction of the theoretical basis and applied methods. This background information is followed by a description of a modular approach for the derivation of the state-space models specifically tailored to HVDC systems. For this purpose, converter terminals, HVDC cables, and DC nodes are defined as the typical groups of subsystems, and the interfaces between these subsystems are defined to aid the

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125

Figure 4.11. Example of parametric sensitivity analysis and corresponding response in the time domain, with k i,vdc changing from 1 to 100 (marked with color transition from blue (dark gray in print version) toward red (gray in print version)).

modular generation of models for representing multiterminal systems. The modeling approaches that can ensure time-invariant state-space representation of the considered subsystems are also reviewed by introducing examples of how applicable models for 2L or MMC-based HVDC converter terminals and HVDC cables can be obtained. Practical considerations are also presented regarding implementation of the presented modular approach for state-space modeling based on the proposed definitions of subsystems and interfaces. The chapter is concluded by reporting numerical examples from analysis of a multiterminal HVDC configuration. These results illustrate how the eigenvalues can be interpreted and how tools as the participation factor analysis and the parametric sensitivity can offer valuable insight into the dynamics, internal interactions, and parameter tuning of such systems.

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[4] D. Van Hertem, M. Ghandhari, Multi-terminal VSC HVDC for the European supergrid: obstacles, Renew. Sustain. Energy Rev. 14 (9) (Dec. 2010) 3156–3163. [5] G. Bathurst, P. Bordignan, Delivery of the Nan’ao multi-terminal VSC-HVDC system, in: IET ACDC 2015, IET, Birmingham, UK, 10–12, Feb. 2015, 6 pages. [6] NR Electric, World’s first 5-terminal VSC-HVDC links, [Online]. Available http:// www.nrec.com/en/web/upload/2019/05/13/15577104617777z7mzh.pdf. (Accessed 12 March 2021). [7] X. Guo, Y. Zhou, N. Mei, B. Zhao, Construction and characteristic analysis of Zhangbei flexible DC grid, Power Syst. Technol. 42 (11) (Nov. 2018). [8] K. Sharifabadi, L. Harnefors, H.-P. Nee, S. Norrga, R. Teodorescu, Design, Control, and Application of Modular Multilevel Converters for HVDC Transmission Systems, IEEE-Wiley Press, Oct. 2016. [9] D. Van Hertem, O. Gomis-Bellmunt, J. Liang (Eds.), HVDC Grids: for Offshore and Supergrid of the Future, IEEE-Wiley Press, Feb. 2016. [10] D. Jovcic, K. Ahmed, High Voltage Direct Current Transmission: Converters, Systems and DC Grids, IEEE-Wiley Press, Jul. 2015. [11] J. Beerten, S. Cole, R. Belmans, Modeling of multi-terminal VSC HVDC systems with distributed DC voltage control, IEEE Trans. Power Syst. 29 (1) (Jan. 2014) 34–42. [12] N.R. Chaudhuri, R. Majumder, B. Chauduri, J. Pan, Stability analysis of VSC MTDC grids connected to multimachine AC systems, IEEE Trans. Power Deliv. 26 (4) (Oct. 2011) 2774–2784. [13] P. Kundur, Power System Stability and Control, McGraw-Hill Inc, New York, 1993. [14] J. Beerten, O. Gomis-Bellmunt, X. Guillaud, J. Rimez, A. van der Meer, D. Van Hertem, Modeling and control of HVDC grids: a key challenge for the future power system, in: Power Systems Computation Conference (PSCC), Wroclaw, Poland, 2014. [15] N.R. Chaudhuri, B. Chaudhuri, R. Majumder, A. Yazdani, Multi-Terminal Direct-Current Grids: Modeling, Analysis, and Control, IEEE-Wiley Press, Sep. 2014. [16] J. Beerten, S. D’Arco, J.A. Suul, Identification and small-signal analysis of interaction modes in VSC MTDC systems, IEEE Trans. Power Deliv. 31 (2) (Apr. 2016) 888–897. [17] M. Amin, M. Molinas, Small-signal stability assessment of power electronics based power systems: a discussion of impedance- and eigenvalue-based methods, IEEE Trans. Ind. Appl. 53 (5) (Sept.-Oct. 2017) 5014–5030. [18] A. Bayo Salas, Control Interactions in Power Systems with Multiple VSC HVDC Converters, PhD. Thesis, KU Leuven, Aug. 2018. [19] T.H. Demiray, Simulation of power system dynamics using dynamic phasor models, PhD Thesis, Swiss Federal Institute of Technology Zürich (ETH), 2008. [20] W.A. Hashlamoun, M.A. Hassouneh, E.H. Abed, New results on modal participation factors: revealing a previously unknown dichotomy, IEEE Trans. Autom. Control 54 (7) (July 2009) 1439–1449. [21] Y. Wang, X. Wang, Z. Chen, F. Blaabjerg, Small-signal stability analysis of inverter-fed power systems using component connection method, IEEE Trans. Smart Grid 9 (September 2018) 5301–5310. [22] J. Beerten, Modeling and Control of DC Grids, PhD Thesis, KU Leuven, May 2013.

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

[23] G.O. Kalcon, G.P. Adam, O. Anaya-Lara, S. Lo, K. Uhlen, Small-signal stability analysis of multi-terminal VSC-based DC transmission systems, IEEE Trans. Power Syst. 27 (4) (November 2012) 1818–1830. [24] A. Yazdani, R. Iravani, Dynamic model and control of the NPC-based back-to-back HVDC system, IEEE Trans. Power Deliv. 21 (1) (January 2006) 414–424. [25] G. Bergna, J.A. Suul, S. D’Arco, State-space modeling of modular multilevel converters for constant variables in steady-state, in: Proceedings of the 17th IEEE Workshop on Control and Modeling for Power Electronics, COMPEL 2016, Trondheim, Norway, 27–30 June, 2016, 9 pp. [26] S. D’Arco, J.A. Suul, M. Molinas, Implementation and analysis of a control scheme for damping of oscillations in VSC-based HVDC grids, in: Proceedings of the 16th International Power Electronics and Motion Control Conference and Exposition, PEMC 2014, Antalya, Turkey, 21–24 September 2014, 2014, pp. 586–593. [27] S. D’Arco, J.A. Suul, J. Beerten, Configuration and model order selection of frequency-dependent π -models for representing DC-cables in small-signal eigenvalue analysis of HVDC transmission systems, IEEE J. Emerg. Sel. Top. Power Electron. (January 2020), 18 pp. [28] A. Jamshidifar, D. Jovcic, Small-signal dynamic DQ model of modular multilevel converter for system studies, IEEE Trans. Power Deliv. 31 (1) (February 2016) 191–199. [29] T. Li, A.M. Gole, C. Zhao, Harmonic instability in MMC-HVDC converters resulting from internal dynamics, IEEE Trans. Power Deliv. 31 (4) (August. 2016) 1738–1747. [30] G. Bergna Diaz, J.A. Suul, S. D’Arco, Small-signal state-space modeling of modular multilevel converters for system stability analysis, in: Proceedings of the 2015 IEEE Energy Conversion Congress and Exposition, ECCE 2015, Montreal, Quebec, Canada, 20–24 September 2015, 2015, pp. 5822–5829. [31] G. Bergna-Diaz, J.A. Suul, S. D’Arco, Energy-based state-space representation of modular multilevel converters with a constant equilibrium point in steady-state operation, IEEE Trans. Power Electron. 33 (6) (June 2018) 4832–4851. [32] G. Bergna-Diaz, J. Freytes, X. Guillaud, S. D’Arco, J.A. Suul, Generalized voltage-based state-space modelling of modular multilevel converters with constant equilibrium in steady-state, IEEE J. Emerg. Sel. Top. Power Electron. 6 (2) (June 2018) 707–725. [33] Ö.C. Sakinci, J. Beerten, Generalized dynamic phasor modeling of the MMC for small-signal stability analysis, IEEE Trans. Power Deliv. 34 (3) (Jun. 2019) 991–1000. [34] J. Lyu, X. Zhang, X. Cai, M. Molinas, Harmonic state-space based small-signal impedance modeling of a modular multilevel converter with consideration of internal harmonic dynamics, IEEE Trans. Power Electron. 34 (3) (March 2019) 2134–2148. [35] H. Saad, X. Guillaud, J. Mahseredjian, S. Dennetière, S. Nguefeu, MMC capacitor voltage decoupling and balancing controls, IEEE Trans. Power Deliv. 30 (2) (April 2015) 704–712. [36] A. Antonopoulos, L. Ängquist, H.-P. Nee, On dynamics and voltage control of the modular multilevel converter, in: Proceedings of the 13th European Conference on Power Electronics and Applications, EPE 2009, Sep. 2009, pp. 1–10. [37] L. Harnefors, A. Antonopoulos, S. Norrga, L. Ängquist, H.-P. Nee, Dynamic analysis of modular multilevel converters, IEEE Trans. Ind. Electron. 60 (7) (July 2013) 2526–2537.

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5 Inertia emulation with HVDC transmission systems Santiago Bustamantea , Hugo A. Cardonab , and Jorge W. Gonzalezc Universidad Pontificia Bolivariana, Electrical Engineering Department, Medellín, Colombia

Chapter points • Current models and strategies for inertia emulation using HVDC VSC systems are detailed. • A comparison is made for some methods to increase inertia in networks with HVDC VSC systems. • Comments on potential requirements for inertia emulation implementation are presented.

5.1 Introduction Traditionally, in terms of frequency, electrical power systems have related their behavior to synchronous generators and machines. Based on electromechanical interaction, the dynamic response of the systems has been traditionally solved, widely studied, and detailed in the literature. Due to a great flexibility and controllability, power electronics via HVDC interfaces are increasingly used in electrical systems, bringing great changes in dynamics by displacing and decoupling the inertia of rotating machines. a Obtained his BSc (Eng) in 2017 and MSc in 2021 and is PhD Student at Univer-

sidad Pontificia Bolivariana. He is currently a researcher in the Energy Transmission and Distribution group at the same university. b Electrical Eng, MSc and PhD Student. At present, researcher and Professor at UPB Colombia in Electrical Eng. Faculty. He has worked for multiple industrial projects and for National System Operator, XM, as a guest researcher. c Obtained his BSc (Eng) in 1992 and MSc and PhD. degrees in 2003 and 2006, respectively. He is Titular professor at Universidad Pontificia Bolivariana, Colombia, in the department of Electric and Electronics Engineering. He worked for HMV Consulting, Siemens PTD in Germany, and Empresas Publicas de Medellin, EPM, Utility. In Germany, he was a guest researcher at Power Systems Institute Erlangen University, and at Werner von Siemens Laboratory in Kempten University. Modeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00010-1 Copyright © 2021 Elsevier Inc. All rights reserved.

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With loss of inertia, the stability of the system is diminished, preventing the kinetic energy of the rotating masses from supplying the instantaneous imbalances between generation and demand, reducing the critical times for fault clearance, increasing the magnitudes of the frequency deviations, and decreasing its damping. In this chapter, we show how VSC HVDC systems can be used to achieve virtual inertial delivery and compensate transient imbalances between generation and load, helping to maintain frequency stability in large-power systems by using various strategies. This work is organized as follows: in Section 5.2, we present basic concepts on inertia and frequency stability and show how it is affected when inertia decreases; in Section 5.3, we expose different approaches and control strategies for inertia emulation; and, finally, in Section 5.4, we comparatively present three methods to increase power systems inertia using VSC HVDC links, which are extraction of energy from wind turbines, from VSC HVDC link capacitors, and power exchange between asynchronous areas using MTDC links.

5.2 Basis for a need of virtual inertia with VSC HVDC systems Traditionally, the inertia of electrical power systems directly depends on rotating machines (such as synchronous generators and motors) and on the electromagnetic coupling between their rotating masses and electrical network [1]. With the inclusion of power electronics interfaces, a decoupling occurs between the synchronous generators and the network, which prevents energy from reducing the transients present in the system [2]. Due to this, it is necessary to define methodologies that allow the network to maintain inertia levels H such that the magnitude of the frequency deviation is minimized. In this way the dynamics of the system exposes a slower and damped behavior, increasing the critical times for clearing faults and allowing gaps in carrying out switching actions to counteract imbalances in active power as a result of faults or events such as loss of transmission lines, generation, loads, and so on [3]. In Fig. 5.1, it is appreciable how the frequency is altered from the changes in the levels of inertia in the system. When an imbalance occurs between the generation and the load resulting in frequency excursions, this imbalance is compensated in the first

Chapter 5 Inertia emulation with HVDC transmission systems

131

Figure 5.1. Simulation of the impact of inertia H on the system frequency for the 2022 scenario in Northern Ireland. Taken and adapted from [4,5].

instance by the inertia of the Hs system represented in Eq. (5.1), where the inertia Hg of each generator is presented in Eq. (5.2) [4]. Hs =

 Hg,i Sg,i i

Hg,i =

Ss

,

Ekin,i J (2πfn )2 = , Sg,i 2Sg,i

(5.1) (5.2)

where Ekin,i is the kinetic energy of the generator, J is the moment of inertia, Ss is the total power of the system, fn is the nominal frequency of the system, and Sg,i is the power of each generator. Next, it is the system control task to respond to the power imbalance. Fig. 5.2 shows the control stages that the European Network of Transmission System Operators for Electricity (ENTSO-E) has established [7]. RoCoF is the immediate rate-of-change-of-frequency after a system unbalance because of a disturbance. Loss of generation,

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Figure 5.2. Control categories. Adapted from [6].

sudden loss of load, and islanding are examples of disturbances. The initial RoCoF assuming null load damping [8] is expressed as follows: RoCoF =

df P = f0 , dt 2H

(5.3)

where P is the size of the contingency (MW lost), H is the system inertia (MW-seconds), f0 is the frequency at the time of disturbance (Hz), and df/dt is the rate of change frequency (Hz/s). Eq. (5.3) states no control action other than the physical inertia. A slow time response scenario for RoCoF, from seconds to minutes, is related to the inertia provided by synchronous generators with their governors to decrease this RoCoF. Grid inertia control improvement in this physical time frame can be naturally accomplished by synchronous condensers. On the other side, a fast response scenario to control RoCoF can be a quick injection of energy, provided by either increasing generation or decreasing load. Inverter-Based Power Sources (IBPS), like wind, solar, batteries, supercapacitors, and flywheels, can provide fast frequency response. But in this task, there are a variety of schemes in which delays or latencies are a challenge. It has to be taken into account that the conventional physical inertia of synchronous generators is instantaneous, so delays of control systems should be fast enough to preserve stability.

5.3 VSC HVDC control approaches for inertia emulation Several authors have proposed different strategies to face the problem of inertia reduction. In [9–11] the use of battery storage systems (BESS) is proposed for the emulation or subsequent delivery of inertia. Flywheels can also contribute to solve the problem

Chapter 5 Inertia emulation with HVDC transmission systems

Table 5.1 Control strategies and resources for virtual inertia with VSC HVDC systems. Control strategy

Resource

MTDC + wind generation Two areas AC Wind generation Derivative control Capacitor + Wind Generation Inertia emulation control Capacitor Wind generation MTDC Capacitor + Wind Generation Virtual synchronous generator MTDC MTDC + wind generation Two areas AC HVDC VSC + AC network PLL-based vector control Weak AC network Proportional control

[4,12]. On the other hand, some countries such as Ireland proposed limiting the percentage of instantaneous generation penetration connected to electronic power interfaces as a way of preserving inertia and stability of the system [13]. VSC HVDC systems (see Fig. 5.3) base their operation on the Isolated Gate Bipolar Transistor (IGBT), which enables Pulse Width Modulation (PWM) or multilevel converters (Modular Multilevel Converters, MMC) to control active and reactive power, in addition to managing voltage and frequency [14]. This feature also decouples the two ends, avoiding the exchange of transients between interconnected systems and the propagation of faults [2]. Basic control strategies can be reviewed in the following categories: proportional control, derivative control, inertia emulation control, virtual synchronous generator, and Phase Locked Loop (PLL)-based vector control. In each control a resource or strategy to provide an output for modulating power or leading the system to recover frequency balance can also be defined. A resource can be the power from the wind generation, energy injections from different terminals of a Multiterminal DC system (MTDC), external actions from AC areas, energy from capacitors, or energy storage assets. In addition, photovoltaic systems can produce actions to suddenly modify power injections. Table 5.1 reviews such approaches reported in the literature, also including tags of the application.

References [15–18] [19–21] [22] [23,24] [25,26] [27] [28–30] [2,31,32] [33,34] [35] [36] [37–39] [40,41]

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Figure 5.3. Point-to-point link HVDC VSC.

In order for an electrical system to maintain the frequency within acceptable limits after a major disturbance, HVDC system can be used to provide frequency regulation services from VSC controls. An alternative is delivering power based on the rate of change of frequency (RoCoF) [42], also allowing primary energy reserves to be shared between the connected ends [43]. Some regulatory entities, such as the case of ENTSO-E [44], have included within their network codes requirements for HVDC VSC connections frequency support services. Regarding this, several authors have proposed different strategies seeking to guarantee stability in the system, mainly focused on proportional controls (droop control), which are included in a complementary way to VSC control and react to frequency deviations [15–17]. In [2] a variation of this method is proposed to allow synthetic inertia emulation using the energy stored in the capacitors of the HVDC-VSC link. However, the aggregation of primary reserves between AC areas should also be performed to compliment the ancillary solution. Not only the transient response as an autoregulation action using capacitor energy may solve all situations [45]. Reference [45] proposes a control scheme based on Receding Horizon Control (RCH) to support the frequency through MTDC links between AC areas using local measurements. Some authors have focused their efforts on ensuring that the inertia provided by the HVDC-VSC links is the product of the inertia of the wind turbines added to the capability of the link as shown in [18,46,47]. Although there are several publications in this regard, most of them propose to take the energy of the turbine rotor as changes in the DC level are perceived, which in turn is modified as the frequency of the AC network varies. It is possible to affirm that this methodology is really related to a virtual coupling between wind power plants and the grid. The great advantage of this control strategy is that it does not require communication between the ends of the HVDC to provide the inertial response and can be used in MTDC systems [23]. In contrast, it has the disadvantages that when faced with a variation in frequency on the AC grid side, the Maximum Power Point Tracking (MPPT) of the wind farm must be modified, which can affect the economics of the in-

Chapter 5 Inertia emulation with HVDC transmission systems

stallation. Extracting a lot of kinetic energy from wind turbines can cause them to stagnate; from this point it is often difficult to restore the rotor speed [16]. Another alternative to procure the synthetic inertia levels with HVDC VSC is using the derivative of the frequency with respect to time as shown in [22], taking out the energy from the wind farms and delivering it during variations in the DC voltage level. For control types that use DC voltage variation, a very large inertia value needs an equally large voltage variation range [48]. This relationship can be obtained from the following equation [2]: Hvsc =

N CVN2 V 2 2Svsc [ VN + 1] 2f fn

−1

,

(5.4)

where N is the number of capacitors connected in the DC link, C is the capacitance of one capacitor, VN is the nominal voltage or DC set point, V is the variation of the DC voltage, f is the frequency variation, and fn is the nominal frequency of the AC system. In the work carried out in [2] a control system for HVDC VSC is proposed to emulate the behavior of a synchronous generator. The energy to achieve this behavior comes from the link capacitor (featured in Fig. 5.3), which can eventually increase with the installation of more capacitors. This last point is important because a large number of BESS battery storage systems will be installed in the world and eventually compliment the energy requested to the capacitor energy. This strategy can also be used in MTDC multiterminal systems due to its characteristics.

5.4 Fast frequency response service by VSC HVDC systems HVDC systems can be used for fast frequency response services. A fast ramp up or ramp down (runback) of power through the HVDC can help to arrest frequency declines at events during the critical period when generator governors start to respond but have not yet reached their full output. The maximum power transfer of an HVDC system depends on the capabilities of its components. Nevertheless, converter valves have very low thermal inertia. HVDC systems are fast enough to respond. Longer times from 100 ms to 300 ms are found in the literature for special situations [8]. Special attention should be devoted to modular multilevel converters VSC HVDC (MMC VSC HVDC) and time to recover

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from faults, which may be related to the longest delay times of this technology. For example, 500 ms was obtained in Caprivi VSC HVDC project [49]. Some other examples of real cases for HVDC projects where power is modulated responding to grid frequency changes are: New Zealand, Pacific HVDC Intertie, Sardinia-Corsica-Italy HVDC, Intermountain HVDC, Gotland HVDC, Square Butte HVDC [8]. Then, in terms of frequency unbalances, for example, 5-second overload capability could be beneficial for a fast frequency response service for HVDC systems [8]. But latencies in some schemes where telecommunications are included should be specially accounted for. In Sections 5.4.1–5.4.3, we detail two control strategies that emulate inertia and use different principles. A third approach that involves a control strategy that is conceived to have the ability to be adapted to the system state is also described. The inertia emulation of the first two approaches is based on energy management from offshore wind power plants and from capacitors of the DC side. The third strategy, to make a contribution, had a starting point in which it analyzed, or even judged, different approaches including inertia emulation using the DC side voltage, and then incepted this feature into a distributed scheme, which makes it possible for different converters to share a power action for frequency control.

5.4.1 Inertia emulation with offshore wind power plants In this case the analysis is based on a point-to-point HVDC VSC system, which on one side has an onshore AC grid and on the opposite side has the wind power plant. In this control strategy the first aim is extracting the kinetic energy in the rotor of the wind turbine [47]. Initially, the model considers a communication scheme between the wind plant and the VSC connected to the AC network. This VSC measures the frequency and issues a control order to the wind plant to modify the power. Because communication introduces delays in the system response time, the dynamics of the system will be determined by the dynamic behavior of the DC voltage in the link, the response of the wind turbines, and the delays imposed by the communication system. Wind plants are sets of turbines, and it is not feasible to establish direct communication between the VSC at the end of the network with each turbine; therefore communication is performed between the VSC at both ends and after the VSC on the side of the wind plant with each

Chapter 5 Inertia emulation with HVDC transmission systems

Figure 5.4. Control model for inertia emulation from the energy stored in the capacitance of the HVDC link. Adapted from [2].

wind generator. Each link in communication would increase delays, further limiting the responsiveness of this control model for the first few moments after the failure. A variation of this method consists on coupling both ends of the HVDC VSC virtually or by means of controls, linking their response with the DC voltage present on the link. In this way, when at the VSC end of the onshore network frequency deviations take place, the link voltage will be affected, and this, in turn, will activate the VSC controls of the wind plant to extract the energy from the turbines. The following equation represents this behavior:   iwpp − iAC dVDC = , (5.5) dt C where iwpp and iAC are the currents that deliver the wind power plant and the current that delivers the VSC to the AC grid, respectively.

5.4.2 Inertia emulation using the capacitor of the HVDC VSC link A control structure for including inertia emulation, which was adapted from the one used in [2] but modified to make it possible to involve primary reserve services is presented in Fig. 5.4. Once a first response is obtained to serve as inertial action by using DC capacitor energy, a primary delayed control action switches the PI control output (inertia service) to PAC order (PAC*) as of primary control ancillary function.

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The basic method of extracting the energy stored in the link capacitor responds to changes in frequency in the AC network, reducing the RoCoF and increasing the time frame for the operation of the primary controls. The time constant τ of a capacitor can be determined by the equation [2] CN V 2

DC WE 2 τ= = , Svsc Svsc

(5.6)

where WE is the energy stored in the capacitor, Svsc is the nominal apparent power that the VSC can deliver, C is the capacitance of a single capacitor of the link, and N is the number of capacitors in the HVDC VSC. The energy stored in the DC capacitors is somewhat similar to the kinetic energy stored in the rotors of synchronous generators. In order that the power stored or released in the capacitor be equivalent to the kinetic power of the absorbed or delivered machine with the frequency changes, we present the following equation [2]: N CVDC dV DC = Pin − Pout = P2 , SV CV SC dt

(5.7)

where Pin is the power going from the VSC to the capacitors, Pout is the power going out from the capacitors to the VSC, and P2 is the dynamics of the energy stored or released by the capacitors. Note from Eq. (5.7) that when the DC voltage is altered, the capacitors react by storing or delivering energy. If the previous relationship is compared with the equation that describes the dynamic behavior of the synchronous generator for frequency variations, then it is integrated and also the values of the total capacitance of the link are considered, so the nominal DC voltage and the power of the converter (to find the integration constant) will be obtained from the equation [2] 2 N CVDC 2HV SC f = + f0 2SV SC



2 N CVDC0 2HV SC f − f0 2SV SC

 .

(5.8)

To rewrite this as a function of the inertia delivered by the VSC HVSC and relate the change in the DC voltage level with the variation of the AC frequency, we can obtain Eq. (5.4). Finally, with the intention that the voltage on the DC side varies with the frequency on the AC side, the equation must be trans-

Chapter 5 Inertia emulation with HVDC transmission systems

139

Figure 5.5. DC voltage and frequency behavior with the Control model for inertia emulation from the energy stored in the capacitance of the HVDC link. Adapted from [2].

formed into the equation [2]  4SV SC HV SC 4SV SC HV SC ∗ 2 . + VDC0 VDC = f− N Cf0 NC

(5.9)

The control system of Fig. 5.4 shows the inclusion of the pursued inertia the fed by energy in the VSC capacitors according to Eq. (5.9). Fig. 5.5 shows the behavior of DC voltage and frequency at different levels of kinetic inertia HVSC with a load variation of 5%. Observe that as the amount of inertia HVSC delivered by the HVDC VSC increases, the system has a better response and a smaller frequency nadir. On the other hand, DC voltages are also affected as the inertia is delivered, since it is necessary to decrease it to push to the AC network the energy stored in the capacitor. Therefore in the scenario in which the proposed control is not considered, the frequency reaches a higher nadir, and the DC voltage remains constant over time.

5.4.3 Frequency support through MTDC based in (RCH) This method addresses a strategy to support frequency stability between asynchronous AC areas connected by means of an HVDC VSC link with multiple terminals (MTDC) using P-V droop control as expressed in Eq. (5.10). Voltage droop control is analogous to

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Figure 5.6. Control model for Frequency support through MTDC based in (RCH). Adapted from [45].

the primary/governor frequency-power control shared between synchronous generators. In this case the converters of an MTDC grid share power imbalances. Some of the converters follow a P-V characteristic according to a power order P ∗ , a voltage order V ∗ , and a droop KP V . This relationship to define the expected power output P of a VSC related to the DC voltage V in the HVDC VSC [45] is exposed by the equation P = P ∗ − KP V (V − V ∗ ).

(5.10)

See Fig. 5.6, where the frequency is conceived for emergency situation. The system will not react to small changes in frequency, avoiding unnecessary operations and reacting only to large changes in frequency seen by the VSC, which would modify the power depending on the variation, taking advantage of the reserves of other asynchronous areas connected to the MTDC. In order for the proposed method to have a fast and reliable response, local measurements are used to eliminate the need for communication between VSCs , which can introduce delays or failures [45]. Because it is necessary to maintain the integrity of the MTDC network when power exchanges occur, the control must be able to maintain the DC voltage at acceptable levels. The proposed control is based on a predictive control model that aims to minimize frequency deviations considering restrictions such as the voltage stability of the MTDC network. Since the proposed model is focused on operation exclusively under emergency situations, its main characteristic is that it has a dead band (frequency deviation) in which the control remains inactive. When the magnitude of the frequency deviation exceeds the limits, lower f(Min−On) or higher f(Max−On) of the band, the control is activated until reducing the frequency value to a nar-

Chapter 5 Inertia emulation with HVDC transmission systems

141

Figure 5.7. Emergency frequency control activation scheme. Adapted from [45].

Figure 5.8. DC voltage and frequency behavior with the Control model for Frequency support through MTDC based in (RCH). Adapted from [45].

rower range from f(Min−Off) to f(Max−Off) , which can be seen in Fig. 5.7. The main objective of the proposed control is adjusting the steady-state power P flowing through the VSC so that it is proportional to the frequency deviation in the AC network (at time ti ). The model uses three measurements, namely the power flow through the VSC, the voltage on the DC side, and the frequency on the AC side. From this a reference is defined, which aims to bring the power of the VSC from the measured value to a value that drives to zero the following equation in a finite number of steps [45]:  (5.11) lim P (t) − P (ti) − Kf (f (t) − fN ) t→∞

From these reference values an optimization process is carried out, which seeks to minimize the deviations from these values. Furthermore, as restrictions, the DC voltage and power do not exceed the minimum or maximum values, and we consider a model of prediction for these two variables, which start from the measured values.

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Table 5.2 State of the art virtual inertia with VSC HVDC systems. Control strategy

Description

Advantage

Disadvantages

Inertia emulation with offshore wind power plants [18].

Links the wind plant frequency with the DC voltage and the network frequency using “Control Droop”.

No need of communication between the ends, and the response is very similar to systems that use communications.

Extraction of kinetic energy from the turbine rotor can cause stagnation making speed recovery difficult [16].

Inertia emulation using the capacitor of the HVDC VSC link [2].

The DC voltage level is used to match the power output from the link capacitor, responding to frequency variations in the AC network.

It can emulate from the energy stored in the capacitor the natural behavior of synchronous generators when faced with frequency deviations in the system.

It does not allow the exchange of inertial energy between AC areas [45]. This method can be extrapolated to large BESS energy storage systems.

Frequency support through MTDC based (RCH) [45].

The frequency between different AC areas is supported by means of an MTDC link. The control strategy focuses on local controls based on predictive control that links the frequency with the DC voltage level.

It does not require communication between the ends of the MTDC, it allows the exchange of energy between different AC areas.

The method was tested for two AC areas connected by the MTDC. Additional validations including more areas is expected [50].

In Fig. 5.8, we compare actions using this method. The objective was to compare actions from different converters (East and West) in the VSC MTDC for a modified Nordic Network. Shared powers and effects on DC voltage can be seen, with comparable actions like in the second strategy outlined in Fig. 5.5. In Table 5.2, we present comparisons of the three approaches.

5.5 Summary Although the control methods for inertia emulation with HVDC VSC that include communication systems within their topology

Chapter 5 Inertia emulation with HVDC transmission systems

can help to reduce the frequency nadir, they have limitations due to the delays introduced by the communication systems that limit the response in the first cycles after a contingency. The integration of suitable controls for HVDC VSC can contribute significantly to increase inertia levels and contribute to the stability of the system, allowing greater controllability and flexibility, increasing the penetration margin of renewable energies with power electronics interface. It is important to note that in recent times the possibility of implementing IBPS technology has been studied for the implementation of Grid Forming (GFM), which has the ability to increase the strength of the system and perform black starts. This could also be a solution to the loss of inertia problems due to the high penetration of power electronics [51]. Nevertheless, a consensus of what could be regulated or applied in either country must be stated clearly. From another side, the solutions for synthetic inertia could be in the form of a tailored designed SPS (Special Protection System) [8,49].

Acknowledgment Convocatory of Universidad Pontificia Bolivariana, UPB Innova, Convocatoria 20 de 2018. Project “Estrategia de transformación del sector energético Colombiano en el horizonte de 2030” financiado en la convocatoria 778 de MinCiencias Ecosistema Científico. Contrato FP44842210-2018. Special thanks to Eros Escobar, Idi Isaac, Gabriel Lopez, Oscar Vasco, and Anderson Quintero.

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6 Real-time simulation of a transient model for HVDC cables in SOC-FPGA✩ A practical example Santiago Sanchez-Acevedoa SINTEF Energi AS, Department Energy Systems, Trondheim, Norway

Chapter points • The main objective is to help the engineer to develop a real-time-simulation module using VHDL for SOC-FPGA devices and to validate the electromagnetic transient dynamics. • There are many power system engineering applications that require the use of hardware in the loop validations. In this chapter, we aim to support the development of an HVDC cable component for real-time simulation and hardware in the loop implementation.

6.1 Introduction The aim of this chapter is to provide an understanding of the steps required to implement a frequency dependent model of an HVDC cable for real-time simulation and hardware in the loop applications. The reader may need other sources to complement the work described in this chapter. First, we introduce a frequency model of HVDC cables. This model offers a point of view of the full cable dynamics. The second part shows the discrete form of the cable model and the final circuit diagram required for implementation in a programming-simulation tool such as the modeling through VHDL or C. The third part describes the required tools in VHDL to obtain a model of the HVDC cable and a basic concept to convert the physical values to fixed point arithmetic. The ✩ RT simulation of HVDC cable with SoC-FPGA. a Santiago Sanchez-Acevedo is a research scientist at SINTEF Energi AS in Norway.

His research interest is stability of power systems, real-time and power hardware in the loop validation, digital substations and communications in power systems. Modeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00011-3 Copyright © 2021 Elsevier Inc. All rights reserved.

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third part describes two main steps: the first is an overview of the blocks, and the second presents an example of the VHDL description. The fourth section of this chapter shows the integration of the model to develop the full HVDC cable model.

6.1.1 What is a SoC-FPGA? “SoC FPGA devices integrate both processor and FPGA architectures into a single device” [2]. This architecture provides a communication between the processor and FPGA, with a large number of peripherals and on-chip memory. In the processor, it is possible to develop software with C code, and in the FPGA, we can develop core modules with VHDL or verilog. These characteristics are very useful for the power-system community to develop fast simulations or real-time simulations and integrate the simulations with some hardware to validate the performance of multiple devices.

6.2 Frequency domain model formulation In this section, we describe the basic concepts for modeling transient dynamics of cables. For more details on the model of electromagnetic transients of cables, the reader can use [3]; the book explains the calculation of the parameters necessary for the dynamics. We start with the formulation of the phase-domain transmission line (i.e., the same for cable modeling) presented in [1,5]. The voltage and current relations at the ends of the cable shown in Fig. 6.1 can be written as − Ik (ω) + Yc (ω)Vk (ω) = H (ω)(Yc (ω)Vm (ω) + Im (ω)),

(6.1)

− Im (ω) + Yc (ω)Vm (ω) = H (ω)(Yc (ω)Vk (ω) + Ik (ω)),

(6.2)

with the following parameters: the frequency ω, the subindices k, m represent the k and m extremes, Vx (ω) and Ix (ω) with x ∈ k, m are the voltage and current at the cable ends, Yc (ω) is the characteristic admittance described in (6.4), and the propagation function H (ω) is presented in (6.5).  (6.3) (ω) = Z(ω)Y (ω), Yc (ω) = Z −1 (ω)(ω), H (ω) = e

−(ω)l

,

(6.4) (6.5)

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

Figure 6.1. Cable terminals.

where (ω) is the propagation coefficient, l is the length of the cable, Z(ω) is the series impedance, and Y (ω) is the shunt admittance. We need to obtain an approximation for the frequency responses of Yc (ω) and H (ω) to model the cable in the time domain. This approximation is performed with the help of the rational approximations using the vector fitting technique [4]. Hence by a vector fitting technique Yc (ω) and H (ω) can be written in the Laplace domain by sums of rational functions as p 

ci + d, i = 1, . . . , p. s + ai i=1   n  ci −sτ H (s) ≈ e + d , i = 1, . . . , n, s + ai Yc (s) ≈

(6.6)

(6.7)

i=1

where, a, c, d are used as the fitting parameters, p is the number of poles used for fitting of Yc (s), and n the number of poles used for fitting of H (s). The parameter τ is the time delay introduced by the propagation function. It was described in [1] that these approximations are the main functions to model the cable dynamics. Our aim is to get a discrete model to implement in the FPGA with VHDL language. Hence, following a similar procedure as that in [1], we first describe the state space model in the Laplace domain and calculate the discrete form of this model. This procedure is well known for implementation of the cable and line models in electromagnetic transient programs. We discuss this approximation in the next section. We can split (6.5) into two functions as shown in (6.8); the first part defines the transfer function H0 (s), and the second is the delay e−sτ . H (s) = H0 (s)e−sτ .

(6.8)

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Figure 6.2. Circuit and blocks representation for the sampled current at node k.

Before obtaining the discrete model, we represent the rational function with state space system that takes into account the imaginary residues and poles with (6.9)–(6.10) and x = (xR , x1 , x2 ) . The state variables are xR ∈ Rre×1 , where re is the number of all the real poles, and x1 , x2 ∈ Rcc×1 , cc is the number of pairs of complex conjugate poles [1]. ⎡

⎤ ⎡ ⎤⎡ ⎤ 0 0 AR x˙R xR s ⎣ x˙1 ⎦ = ⎣ 0 Acreal Acimag ⎦ ⎣ x1 ⎦ x2 x˙2 0 −Acimag Acreal ⎡ ⎤ 1 + ⎣ 2 ⎦ u, 0

 y = CR Ccreal Ccimag x,

(6.9) (6.10)

where AR , Acreal , Acimag are the state matrices for the real coefficients and complex conjugated coefficients, respectively. Finally, the matrices CR , Ccreal , Ccimag are the output matrices for the real coefficients and complex conjugated coefficients.

6.3 Cable model with difference equations The model for the two Norton circuits is presented in Fig. 6.2. In [1], there was described the trapezoidal rule for numerical integration of model (6.9)–(6.10). The discrete form of the system is presented by

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

t t −1 t (I + A) A)xn−1 + Bun−1 2 2 2

t + Bun , 2 yn = Cxn + Dun , xn = (I −

(6.11) (6.12)

where I is the identity matrix, the subscripts n and n − 1 define the present and one-delay samples, xn , xn−1 are the states, y is the output for each system, and u is the input. Using the same notation as in [1], we apply the following state transformation to represent the system: xn = xn − (I −

t −1  A) Bun . 2 2

(6.13)

The final system is described by  xn = αxn−1 + (α + I )γ un−1 ,

(6.14)

yn = Cxn

(6.15)

+ (Cγ + D)un ,

t −1 where the matrices are α = (I − t 2 A) (I + 2 ) and γ = (I − t t −1 2 A) 2 B. Hence Yc and H are discrete forms, and we can model the nodal Eqs. (6.1)–(6.2). Let us invoke Kirchhoff’s current law for the discrete models. Eq. (6.1) in the discrete time domain can be written as

ik,n − yok,n = −ih,n ,

(6.16)

where ik,n is the current at terminal k at the discrete instant n, yok,n is the discrete output current at node k (i.e., the output of the system Yc (ω)Vk (ω) and defines a current), and ih,n is the resulting discrete output current from the system H (ω)(Yc (ω)Vm (ω) + Im (ω)). The discrete current yok,n is defined xk,n = αk xk,n−1 + (αk + I )γk vk,n−1 , yok,n = Ck xk,n + (Ck γk + Dk )vk,n ,

(6.17) (6.18)

where the subscripts for the matrices indicate the system type, xk,n is the discrete state variable of the system, the input is the voltage at the node at the time instants n and n − 1. Moreover, the product (Ck γk + Dk )vk,n = gk vk,n = igk,n is a current measured directly from the system under simulation, and gk is an admittance. Therefore (6.18) is calculated by yok,n = Ck xk,n + gk vk,n = Ck xk,n + igk,n .

(6.19)

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The calculation of current ih,n considers the delay δ (i.e., the discrete form of the delay τ ) and is defined by the system xh,n = αh xh,n−1 + (αh + I )γh it,(n−1−δ) ,

(6.20)

ih,n = Ch xh,n + (Ch γh + Dh )it,(n−δ) ,

(6.21)

where it,n is a temporary current variable, which represents the output from the system (Yc (ω)Vm (ω) + Im (ω)): xt,n = αt xt,(n−1) + (αt + I )γt vm,(n−1) ,

(6.22)

it,n = Ct xt,n + (Ct γt + Dt )vm,n + im,n .

(6.23)

Finally, (6.16) can be modeled with the sampled variables: ik,n = igk,n + Ck xk,n − ih,n = igk,n + is,n ,

(6.24)

where is,n is the current based on past samples of the system and described by is,n = Ck xk,n − ih,n .

(6.25)

6.4 VHDL conceptual design of the HVDC cable model In this section, we describe the VHDL package development that calculates the currents described in the previous section.

6.4.1 Floating to fixed point conversion and arithmetic In this subsection, we aim to present a basic application of fixed point arithmetic. The values obtained during the discretization of the cable model are floating point values, and it is necessary to convert them to fixed point arithmetic to use them in the FPGA. A representation with hexadecimal numbers will be used in VHDL to set the initialization and configuration of the parameters used in the cable model. The conversion requires the word length N used for fixed point arithmetic, the number of fractional bits n used in the representation. The conversion equation is X = x × Q, where Q = 2n , X is the number in fixed point representation, and x is the number in floating point representation. Hence X represents a floating number x, for example, if a word has N = 32 bits

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

and we use the fractional part of n = 16 bits, then we have that X = x × Q = x × 216 . This conversion can be implemented in a simple script in Python. We present a Python script example for this type of conversion, also for negative values −x. Example 6.1. Conversion of a floating point number to hexadecimal. # f l o a t i n g p oi n t number x =2.5 #word d e s c r i p t i o n (N−1 downto 0 ) N= 3 2 ; # wordleght n= 1 6 ; #n : number o f f r a c t i o n a l b i t s m=N−n−1; #m: number o f i n t e g e r b i t s Q=2∗∗n ; #n : number o f f r a c t i o n a l b i t s # c o n v e r s i o n t o f l o a t i n g p oi n t x=X∗2∗∗(−n ) # f i x e d p o i n t c o n v e r s i o n X=x∗2∗∗n X= i n t (Q∗x ) ; # c o n v e r t s −x t o two ’ s complement Xhex_neg=hex ( 2 ∗ ∗ (N)−abs ( X ) ) # n e g a t i v e X hexadecimal

It is well known that a hexadecimal number can be easily converted to a binary representation. Additionally, the application with VHDL is more simple passing from hexadecimal to binary numbers and vice versa. Finally, we can use some basic functions with binary representation of the numbers. Basically, the application requires multiplication, division, addition, and subtraction of binary numbers.

6.4.2 Blocks architecture of the HVDC cable with VHDL The idea is to build two circuits of the cable and to split the tasks with multiple blocks. The blocks will be used to interconnect the complete system. It is a good practice when designing with VHDL to create the library where multiple circuits can be hosted and we can reuse them for multiple purposes. It was shown in the previous section that one side of the cable model has three state space modules, one block with delays and two adders used in the construction of the system. The following subsection explains the design and development of the VHDL library used for this purpose.

6.4.3 Description of the blocks used in the HVDC cable It is necessary to develop the one side of the cable and use the symmetry of the design to build the second side.

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We start by describing the main package developed with VHDL. This package has the main components used in the development of the side of the cable modules. First, we list the components used in the package: • Delay: This component produces a delay of the signal when used in feedback mode. The application will be presented below. • Delay_1: This component produces a delayed signal when used in feed-forward mode. • product_afloop: This component multiplies two vectors using a recursive description with VHDL. • product_vecscalar: Component with the product of a vector and an scalar number. • product_vecmatr: Component with the product of a vector with a matrix. • delay_1sc: This component produces one delay of a scalar number. • sum: Component with the addition of two arrays. • ss_siso: This component defines the state space system with a single input and single output behavior. Let us present the constants of the package: • N : integer with the length of the word used for computations. • nn: integer that defines the number floating point bits. • M: size of the arrays. We have created two variables for this package: • vars: one array of variable size with signed elements of length N (e.g., the length is from N − 1 down to 0, with N defined above). It is important to highlight that the number of elements of the array was not defined yet. • Matriz: is a matrix of variable size of signed elements of size N . In the next subsections, we will show the code for each of the components listed.

6.4.3.1 Delay feedback: delay component This component is used to obtain a delay of a signed signal used in the state space representation and is applied to a feedback signal. We present a code: −−Signed d e l a y when f e e dback library ieee ; use i e e e . numeric_std . a l l ; use i e e e . s t d _ l o g i c _ 1 1 6 4 . a l l ; use work . my_pkg_matrix . a l l ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− e n t i t y delay i s generic (M: i n t e g e r : = 1 ;

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

N: i n t e g e r : = 1 6 ) ; port ( u : in v a r s ( 0 to M) : = ( others => ( others => ’ 0 ’ ) ) ; −−input c l k : in s t d _ l o g i c ; x_k1 : out v a r s ( 0 to M) : = ( others => ( others => ’ 0 ’ ) ) −−delayed ); end delay ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− a r c h i t e c t u r e bhv of delay i s begin process ( c l k ) begin i f c l k ’ event and c l k = ’ 1 ’ then x_k1 ( others => ’ 0 ’ ) ) ; 12 −−input 13 c l k : in s t d _ l o g i c ; 14 x_k1 : out v a r s ( 0 to M) : = ( others => 15 ( others => ’ 0 ’ ) ) −−delayed 16 );

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

17 end delay_1 ; 18 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 19 a r c h i t e c t u r e bhv of delay_1 i s 20 s i g n a l c_temp : v a r s ( 0 to M) : = ( others = >( others => ’ 0 ’ ) ) ; 21 begin 22 process ( c l k ) 23 begin 24 i f c l k ’ event and c l k = ’ 1 ’ then 25 c_temp ’ 0 ’ ) ) ; 16 s i g n a l uy : v a r s ( 0 to My ) ; 17 s i g n a l yoy : v a r s ( 0 to My) : = ( others => ( others => ’ 0 ’ ) ) ; 18 begin 19 Unit : delay_1 generic map(M=>M) 20 port map( u=>u1 , c l k => clk1 , x_k1=>yo ) ; 21 Unity : delay_1 generic map(M=>My) 22 port map( u=>uy , c l k => clk1 , x_k1=>yoy ) ; 23 process begin 24 clk1 < = ’ 1 ’ ; 25 wait f o r 50 ns ; 26 clk1 < = ’ 0 ’ ; 27 wait f o r 50 ns ; 28 end process ; 29 process begin 30 u1 ( others = > ’ 0 ’ ) ) ; 20 begin 21 ok : f o r i in 0 to M generate 22 temp ( i ) ’ 0 ’ ) ; 31 OKacum : f o r j in 0 to M loop 32 ytemp : = ytemp+temp ( j ) ; 33 −−r e s u l t s sum from 0 t o M 34 end loop ; 35 y( others = > ’ 0 ’ ) ) ; 12 b : in signed (N−1 downto 0 ) ; 13 y : out v a r s ( 0 to M) : = ( others => ( others = > ’ 0 ’ ) ) ; 14 c l k : in s t d _ l o g i c 15 ) ; 16 end p r o d u c t o _ v e c s c a l a r ; 17 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 18 a r c h i t e c t u r e bhr of p r o d u c t o _ v e c s c a l a r i s 19 s i g n a l temp : v a r s ( 0 to M) : = ( others => ( others = > ’ 0 ’ ) ) ; 20 21 begin 22 ok : f o r i in 0 to M generate 23 temp ( i ) 12 ( others = > ’ 0 ’ ) ) ) ; 13 b : in v a r s ( 0 to M) : = ( others = >( others = > ’ 0 ’ ) ) ; 14 y : out v a r s ( 0 to M) : = ( others = >( others = > ’ 0 ’ ) ) ; 15 c l k : in s t d _ l o g i c 16 ) ; 17 end producto_vecmatr ; 18 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 19 a r c h i t e c t u r e bhr of producto_vecmatr i s 20 s i g n a l temp : Matriz ( 0 to M, 0 to M) : = ( others = >( others => 21 ( others = > ’ 0 ’ ) ) ) ; 22 s i g n a l temp1 : v a r s ( 0 to M) : = ( others = >( others = > ’ 0 ’ ) ) ; 23 begin 24 vectmanip : f o r j in 0 to M generate 25 ok : f o r i in 0 to M generate 26 temp ( j , i ) ( 34 others = > ’ 0 ’ ) ) ; 35 begin 36 i f ( c l k ’ event and c l k = ’ 1 ’ ) then 37 ytemp : = ( others = >( others = > ’ 0 ’ ) ) ; 38 okaa : f o r j in 0 to M loop 39 OKacum : f o r i in 0 to M loop 40 ytemp ( j ) : = ytemp ( j )+temp ( j , i ) ; 41 −−i n d i v i d u a l product r e s u l t s 42 sum from 0 to M 43 temp1 ( j ) ’ 0 ’ ) ) ; c l k : in s t d _ l o g i c ; y : out v a r s ( 0 to M) : = ( others = >( others = > ’ 0 ’ ) ) ); end suma ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− a r c h i t e c t u r e bhv of suma i s s i g n a l temp : v a r s ( 0 to M) : = ( others = >( others = > ’ 0 ’ ) ) ; begin ok : f o r i in 0 to M generate temp ( i ) ’ 0 ’ ) ) ; s i g n a l A_t : Matriz ( 0 to M, 0 to M) : = ( others = >( others => ( others = > ’ 0 ’ ) ) ) ;

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

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

−−temporal matrix s i g n a l B_t , C_t : v a r s ( 0 to M) : = ( others = >( others = > ’ 0 ’ ) ) ; −−temporal matrix −−I n i t i a l i z a t i o n s i g n a l tempu , tu : signed (N−1 downto 0 ) : = ( others = > ’ 0 ’ ) ; s i g n a l y t : signed (N−1 downto 0 ) : = ( others = > ’ 0 ’ ) ; s i g n a l xk_1 : v a r s ( 0 to M) : = ( others = >( others = > ’ 0 ’ ) ) ; −−x ( k−1) s i g n a l x_k : v a r s ( 0 to M) : = ( others = >( others = > ’ 0 ’ ) ) ; begin B_t x_k , c l k => c l k , x_k1=>xk_1 ) ; −−d e l a y f o r s t a t e ( feedback ) U2 : producto_vecmatr generic map(M=>M,N=>N, nn=> nl ) port map ( a=>A , b=>xk_1 , y=>temp , c l k => c l k ) ; U3 : d e l a y _ 1 s c generic map(N=>N) port map ( u=>au , c l k => c l k , x_k1=>tempu ) ; −−s c a l a r d e l a y f o r t he input ( feed −forward ) U5 : p r o d u c t o _ v e c s c a l a r generic map(M=>M,N=>N, nn=> nl ) port map ( a=>B_t , b=>tu , y=>temp1 , c l k => c l k ) ; U6 : suma generic map(M=>M,N=>N) port map ( a=>temp4 , b=>temp2 , c l k => c l k , y=>x_k ) ; U7 : producto_afloop generic map(M=>M,N=>N, nn=> nl ) port map ( a=>C_t , b=>temp3 , y=> yt , c l k => c l k ) ; temp2A , B=>B , C=>C , c l k => c l k ,ym=> t y ) ; 26 y ’ 0 ’ ) ; −−input 19 s i g n a l t y : signed (N−1 downto 0 ) : = ( others = > ’ 0 ’ ) ; 20 −−output 21

169

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

22 −−Parameters f o r c a b l e H _ d i s c r e t e 23 constant A : Matriz ( 0 to M, 0 to M) : = ( ( a r r a y 0 ) , . . . , 24 ( arrayM ) ) ; 25 26 constant B : v a r s ( 0 to M) : = ( element_B0 , . . . , element_BM ) ; 27 28 constant C : v a r s ( 0 to M) : = ( element_C0 , . . . , element_CM ) ; 29 30 constant D: signed (N−1 downto 0 ) : = element_D0 ; 31 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 32 begin 33 prodesc : p r o d u c t o _ s c l r s c l r generic map(N=>N, nn=>nn ) 34 port map ( a=>u_d , b=>D, y=>tu ) ; −−Du 35 u n i t : s s _ s i s o generic map(M=>M,N=>N, nl =>nn ) 36 port map( au=>u , A=>A , B=>B , C=>C , c l k => c l k ,ym=> t y ) ; 37 −−s s _ s i s o 38 y ’ 0 ’ ) ; downto 0 ) : = ( others = > ’ 0 ’ ) ; downto 0 ) : = ( others = > ’ 0 ’ ) ; downto 0 ) : = ( others = > ’ 0 ’ ) ; downto 0 ) : = ( others = > ’ 0 ’ ) ; downto 0 ) : = ( others = > ’ 0 ’ ) ; downto 0 ) : = ( others = > ’ 0 ’ ) ; downto 0 ) : = ( others = > ’ 0 ’ ) ;

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− begin Yc1 : s y s _ y generic map(M=>M,N=>N, nn=>nn ) port map( u=> v_loc , c l k => c l k , y=> t y 1 ) ; Yc2 : s y s _ y generic map(M=>M,N=>N, nn=>nn ) port map( u=>v_os , c l k => c l k , y=> t y 2 ) ; ty3 N) port map( u=>ty4 , c l k => c l k , x_k1=>tu1 , x_k2=>tu2 ) ; H: sys_h generic map(M=>Mh,N=>N, nn=>nn ) port map( u=>tu2 , u_d=>tu1 , c l k => c l k , y=> t y 5 ) ; ty ’ 0 ’ ) ; −−ism −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− begin Kside : h a l f _ c l b port map( v _ l o c =>v_k , v_os=>v_m , i _ g o s =>i_gm , i _ o s =>i_m , c l k => c l k , i _ s l o c => t y 1 ) ; −−k s i d e model i _ s k v_m , v_os=>v_k , i _ g o s => i_gk , i _ o s => i_k , c l k => c l k , i _ s l o c => t y 2 ) ; −−m s i d e model i_sm 0, around x0 into a linear function. This nonlinear function presented by (8.8) in matrix form can be rewritten as follows: diag−1 d (vd )Ad [pgd − pd ] = [Gdg vg + Gdd vd ].

(8.12)

Now the main idea of the hyperbolic linearization is transforming the left-hand side of (8.12) into a linear expression; for doing so, let us apply Taylor’s series expansion for this matricial expression by neglecting the higher-order terms, which produces the following approximation presented in [11]: −1 0 −2 0 diag−1 d (vd ) ≈ 2 diagd (vd ) − diagd (vd ) diagd (vd ).

(8.13)

In addition, we assume that the net injected current provided by all the distributed generators in (8.12) can be approximated, as recommended in [13], as follows: −1 0 diag−1 d (vd )Ad pgd ≈ diagd (vd )Ad pgd .

(8.14)

Note that the approximation reported in (8.14) is possible since the effect of the variables pgd is higher in comparison to the impact of the voltage profiles [13]. Now if we use expressions (8.13) and (8.14) to substitute the left-hand side component in (8.12), then we get the following linear approximation:

−1 0 −2 0 0 (v )A p − 2 diag (v ) − diag (v ) diag (v ) A d pd diag−1 d gd d d d d d d d d = Gdg vg + Gdd vd .

(8.15)

Expression (8.15) yields a new convex version of Model 8.1: Model 8.3. Hyperbolic approximation minimize ploss = v T GL v, subject to Ag pg = diagg (vg )[Ggg vg + Ggd vd ],

−2 0 0 2 diag−1 (v ) − diag (v ) diag (v ) Ad pd d d d d d d 0 = −Gdg vg − Gdd vd + diag−1 d (vd )Ad pgd ,

pgmin

≤ pg ≤ pgmax ,

vmin ≤ v ≤ vmax , − Ri max ≤ Av ≤ Ri max .

(8.16)

203

204

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

8.2.2.3 Second Taylor-based method: product linearization This method also works with Taylor’s series expansion method for multivariate functions [14]. In general terms, we suppose that a scalar function f (x, y) is continuous and differentiable around the operation point (x0 , y0 ) and thus can be approximated via a hyperplane as follows: f (x, y) = xy, ← f (x, y) = y0 x + x0 y − x0 y0 + H.O.T.

(8.17)

As reported in the previous Taylor-based methods, the effect of the high-order terms is negligible for electrical networks due to undernormal operating conditions; this point is near to the exact solution [9]. On the other hand, if expression (8.17) is generalized for matrices and vectors as the case of the right-hand side component of (8.8), then it can be rewritten as follows: diagd (vd )[Gdg vg + Gdd vd ] ≈ diagd (vd )Gdg vg + diagd (vd0 )Gdd vd + diagd (vd )Gdd vd0 − diagd (vd0 )Gdd vd0 , (8.18) Now, considering expression (8.18) on the nonconvex original Model 8.1, we get the third convex version of the OPF problem: Model 8.4. Hyperbolic approximation minimize ploss = v T GL v, subject to Ag pg = diagg (vg )[Ggg vg + Ggd vd ], Ad [pgd − pd ] = diagd (vd )Gdg vg + diagd (vd0 )Gdd vd + diagd (vd )Gdd vd0 − diagd (vd0 )Gdd vd0 , pgmin

(8.19)

≤ pg ≤ pgmax ,

vmin ≤ v ≤ vmax , − Ri max ≤ Av ≤ Ri max .

8.2.3 Convex reformulations Convex optimization is a field of mathematical optimization that focuses on studying problems with convex functions and convex constraints. In the last years the convex optimization has taken great interest due to its theoretical and practical support, which can be applied to many engineering problems (finance, statistics, control systems, communication, and power systems, among others) [15]. This has been supported by recent advancements in computing and optimization algorithms.

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

8.2.3.1 Semidefinite programming model The Semidefinite programming (SDP) method is a subfield of convex optimization that focuses on working with the space of matrices. This method transforms the vector space Rn to a matricial space Rn×n , which presents better geometric features [16]. An SDP model is represented as follows: minimize z = Tr(CX) subject to Ai X = Bi , i = 1, 2, ..., n,

(8.20)

X  0, where X ∈ Rn×n represents decision variables, C ∈ Rn×n is the cost function (power losses for the OPF problem), Ai ∈ Rn×n is a matrix, d Bi ∈ Rn is a vector, which shapes the m-linear equations, Tr corresponds to the matrix trace, and  denotes positive semidefiniteness. We define a new variable X = vv  to transform the OPF problem described in Model 8.1 into the canonical formulation of the SDP. This new variable is a matrix, which has to be positive semidefinite of rank = 1. These constraints can be represented as X  0,

(8.21)

rank(X) = 1.

(8.22)

Observe that the matrix X may have a rank greater than 1 and meet the constraint (8.22) maintaining Model 8.1 nonconvex. Therefore it is necessary to relax this constrain to reach a convex optimization model with semidefinite constraints. Model 8.5 (SDP Approximation). minimize ploss = T r(GL X), subject to

   pg − pd = diag GL + GN X ,

(8.23) (8.24)

pgmin ≤ pg ≤ pgmax , 2 2 JN vmin ≤ X ≤ JN vmax ,    max −F ≤ diag GL + GN X ≤ F max ,

(8.25)

X  0.

(8.28)

(8.26) (8.27)

where JN ∈ Rn×n is an all-ones matrix, and F max is a vector that contains the whole maximum flow of lines.

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

Model 8.5 can be solved successfully with different toolboxes for convex optimization, for example, CVX [17]. Now it is necessary to recover v from the matrix X to achieve it; we employ the decomposition method based on eigenvalues and eigenvectors [18] as follows: X=

n 

λk Wk Wk ,

(8.29)

k=1

where λk represents eigenvalues associated with the matrix X, and Wk is its corresponding eigenvector. In most practical cases, the eigenvalues are all close to zero except one. Hence rank(X) ≈ 1, and X can be represented as X ≈ λn Wn Wn ,

(8.30)

where λn denotes the maximum eigenvalue. Finally, the voltages of the system can be recovered as v≈



λn Wn .

(8.31)

It is important to mention that the accuracy of Model 8.5 depends on how close are the other values to zero [16].

8.2.3.2 Second-order cone programming model Second-order cone programming (SOCP) model is also a subfield of convex optimization. The SOCP model has recently taken great importance for OPF in DC grids since it is very efficient. In addition, it can solve this problem reliably and efficiently by guaranteeing a unique solution, and, in some cases, it can be a global optimum [19]. The SOCP model minimizes a linear or quadratic function over a convex region. This region consists of the intersection of two spaces, which are affine linear and second-order cones [20]. The canonical SOCP formulation can have the following representation: min c x subject to Ak x + bk ≤ αk x + βk ,

(8.32)

Fl x = gl , where k and l denote the numbers of conical and affine constraints, respectively. The Euclidean norm is represented by · , Ak ∈ Rn×n and Fl ∈ Rp×n are real matrices, c, bk , αk ∈ Rn and gl ∈ Rp

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

are vectors, βk ∈ R is a scalar, and x ∈ Rn is the optimization variable. Two new variables X = vv  must be defined to transform the OPF problem presented in Model 8.1 into the SOCP model as follows: Vi = vi2 , i ∈ N , Wij = vi vj , i ∼ j, and a matrix

 Rij =

Vi Wj i

Wij Vj

(8.33) (8.34)

 .

(8.35)

The matrix Rij has the same properties as the matrix X in the SDP model, which has to be positive semidefinite of rank = 1. These constraints can be represented as Rij  0, rank(Rij ) = 1.

(8.36) (8.37)

As for the SDP model, constraint (8.37) is relaxed to obtain a convex model with linear and second-order cone constraints. Model 8.6. SOCP Approximation minimize ploss =

n n     Vi − Wij gij ,

(8.38)

i=1 j =1

subject to p i g − pi d =

  Vi − Wij gij , i ∈ N ,

(8.39)

j :j ∼i

pgmin ≤ pg ≤ pgmax , 2 vmin

2 ≤ Vi ≤ vmax ,

i ∈ N,

Wij ≥ 0, i −→ j, Wij = Wj i , i −→ j, Rij  0, i −→ j,   − Fij ≤ Vi − Wij gij ≤ Fij , i ∈ N ,

(8.40) (8.41) (8.42) (8.43) (8.44) (8.45)

where pig and pid are generated and demand power at node ij , respectively, and Fij is the power flow through of transmission lines ij . Model 8.6 can also be solved with CVX. In this model the original variables are recovered as follows:  vi = Vi , i ∈ N . (8.46)

207

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

Figure 8.3. 35-node test feeder for LVDC applications.

8.3 Results To verify the ability of the optimal power flow methods based on Taylor’s’ methods and the convex reformulation via semidefinite and second-order cone programming, in this section, we present a DC test feeder composed by 35 nodes, 34 lines, two ideal nodes (i.e., slack nodes), and three distributed generators. The electrical connection between nodes for this test feeder is presented in Fig. 8.3, and the information related to branches, loads, and distributed generators is presented in Tables 8.1 and 8.2, respectively. Note that the voltage and power bases for this low-voltage direct current (LVDC) network are 380 V and 10 kW, respectively. Note that in Table 8.1 the columns Pj correspond to the constant power consumption at node j . For this test system, considering that the power outputs in all the DGs are null, the total power loss in all the branches of the net-

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

Table 8.1 Information of the 35-node test feeder. Node i Node j Rij [p.u.] Pj [p.u.] Node i Node j Rij [p.u.] Pj [p.u.] 1 2 1 4 5 5 7 7 9 9 9 4 13 14 14 14 17

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.00260 0.00245 0.00148 0.00256 0.00365 0.00450 0.00865 0.00255 0.00444 0.00500 0.00458 0.00654 0.00885 0.00200 0.00338 0.00265 0.00245

0.850 0.450 0.000 0.000 0.900 0.000 1.250 0.000 0.750 0.450 0.500 0.000 0.000 0.750 0.300 0.000 0.355

17 13 20 21 22 22 20 25 26 27 28 28 25 25 32 33 33

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

0.00358 0.00355 0.00389 0.00397 0.00456 0.00150 0.00420 0.00356 0.00550 0.00364 0.00470 0.00125 0.00321 0.00165 0.00146 0.00186 0.00456

Table 8.2 Information about distributed generators and slack nodes. Location [node] Type Pgmin [p.u.] Pgmin [p.u.] Voltage [p.u] 1 7 17 24 32

Slack DG DG Slack DG

– 0.000 0.000 – 0.000

– 3.50 3.00 – 5.00

work is 6.5717 kW, and the power outputs in the ideal generators (slack nodes 1 and 24) are 87.760 kW and 5.7702 kW, respectively. Table 8.3 presents the results of each one of the proposed optimal power flow approaches when applied to the test feeder illustrated in Fig. 8.3. Observe that all the proposed methods reach the same optimal solution via sequential quadratic programming and convex reformulations. The main difference between these

1.000 – – 1.000 –

0.654 0.000 0.125 0.800 1.050 0.000 0.000 0.610 0.440 0.000 0.360 0.540 0.700 0.000 0.680 0.865 0.450

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

Figure 8.4. Voltage profile at load nodes before and after dispatching the distributed generators.

methods is only the number of iterations that each one of them takes. In this sense, both Taylor-based methods take only 4 iterations, followed by the Newton–Raphson with 5 iterations, and finally by the second-order cone programming and the semidefinite one with 43 and 49 iterations, respectively. This behavior regarding the number of iterations can be explained by the fact that the semidefinite and second-order cone programming models generate n2 variables [7], that is, 1225 variables associated with voltages, whereas the first three-methods work only with 35 variables related to the voltages. Regarding power losses reduction, we can mention that after locating distributed generators, this reduces from 6.5717 kW to 1.0235 kW, which implies a reduction of about 84.4256%, confirming the results reported in the recent literature [7,13]. An additional important fact regarding voltage profiles is that the worst voltage before the location of distributed generators is 0.9199 at node 29, whereas after locating all the distributed generators, the minimum voltage is 0.9767 at the same node. Note that all the voltage profiles in both cases are depicted in Fig. 8.4. According to the voltage profiles reported in Fig. 8.4, we can observe that there are multiple nodes lower than 0.95 p.u., that is, nodes from 14 to 19 and from 25 to 35, when there is no presence of DGs, notwithstanding that all the DGs are optimally dispatched.

8.4 Conclusions In this chapter, we reviewed the main methodologies for optimal power flow via numerical methods that guarantee convergence based on sequential quadratic formulations and convex ap-

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

Table 8.3 Numerical comparisons between OPF method. Approach Pow. Loss [kW] Min. voltage [p.u] Gen. Power [kW] Iter.

Model 8.2

Model 8.3

Model 8.4

Model 8.5

Model 8.6

1.0235

1.0235

1.0235

1.0235

1.0235

{29, 0.9767}

{1, 19.2493} {7, 34.7434} {17, 20.5035} {24, 19.3697} {32, 45.4477}

5

{29, 0.9767}

{1, 19.2493} {7, 34.7434} {17, 20.5035} {24, 19.3697} {32, 45.4477}

4

{29, 0.9767}

{1, 19.2493} {7, 34.7434} {17, 20.5035} {24, 19.3697} {32, 45.4477}

4

{29, 0.9767}

{1, 19.2493} {7, 34.7434} {17, 20.5035} {24, 19.3697} {32, 45.4477}

49

{29, 0.9767}

{1, 19.2493} {7, 34.7434} {17, 20.5035} {24, 19.3697} {32, 45.4477}

43

proximations. Five numerical approaches were studied, namely: (i) Netwon–Raphson based method, (ii) the first Taylor-based method (product approximation), (iii) the second Taylor-based method (hyperbolic approx.), (iv) the semidefinite programming model, and v) the second-order cone programming model. Numerical results on a 35-node test feeder in low-voltage DC applications show that each of them converges to the same solution, which reduces the total power losses about 85% when DGs are optimally dispatched in comparison to the based case. Additionally, the main difference observed between those OPF approaches

211

212

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

is regarding the number of iterations that each takes, since the first three methods converge at maximum 5 iterations, whereas the SDP and SOCP approaches take more than 40 iterations. This difference was explained by the fact that convex models require at least 1225 variables regarding voltages, whereas the Taylor-based and Newton–Raphson methods only require 35 iterations.

References [1] S. Parhizi, H. Lotfi, A. Khodaei, S. Bahramirad, State of the art in research on microgrids: a review, IEEE Access 3 (2015) 890–925. [2] J. Li, F. Liu, Z. Wang, S.H. Low, S. Mei, Optimal power flow in stand-alone DC microgrids, IEEE Transactions on Power Systems 33 (5) (2018) 5496–5506. [3] O.D. Montoya, On linear analysis of the power flow equations for DC and AC grids with CPLs, IEEE Transactions on Circuits and Systems II 66 (12) (2019) 2032–2036. [4] C. Jin, P. Wang, J. Xiao, Y. Tang, F.H. Choo, Implementation of hierarchical control in DC microgrids, IEEE Transactions on Industrial Electronics 61 (8) (2013) 4032–4042. [5] K. Rouzbehi, J.I. Candela, G.B. Gharehpetian, L. Harnefors, A. Luna, P. Rodriguez, Multiterminal DC grids: operating analogies to AC power systems, Renewable & Sustainable Energy Reviews 70 (2017) 886–895. [6] B. Stott, J. Jardim, O. Alsaç, DC power flow revisited, IEEE Transactions on Power Systems 24 (3) (2009) 1290–1300. [7] O.D. Montoya, W. Gil-González, A. Garces, Sequential quadratic programming models for solving the OPF problem in DC grids, Electric Power Systems Research 169 (2019) 18–23, https:// doi.org/10.1016/j.epsr.2018.12.008. [8] A. Garcés, O.D. Montoya, A potential function for the power flow in DC microgrids: an analysis of the uniqueness and existence of the solution and convergence of the algorithms, Journal of Control, Automation and Electrical Systems 30 (5) (2019) 794–801, https://doi.org/10.1007/s40313-019-00489-4. [9] A. Garcés, On the convergence of Newton’s method in power flow studies for DC microgrids, IEEE Transactions on Power Systems 33 (5) (2018) 5770–5777, https://doi.org/10.1109/TPWRS.2018.2820430. [10] A. Garces, A linear three-phase load flow for power distribution systems, IEEE Transactions on Power Systems 31 (1) (2016) 827–828, https:// doi.org/10.1109/TPWRS.2015.2394296. [11] O.D. Montoya, L. Grisales-Noreña, D. González-Montoya, C. Ramos-Paja, A. Garces, Linear power flow formulation for low-voltage DC power grids, Electric Power Systems Research 163 (2018) 375–381, https:// doi.org/10.1016/j.epsr.2018.07.003. [12] O.D. Montoya, V.M. Garrido, W. Gil-González, L.F. Grisales-Noreña, Power flow analysis in DC grids: two alternative numerical methods, IEEE Transactions on Circuits and Systems 66 (11) (2019) 1865–1869, https:// doi.org/10.1109/TCSII.2019.2891640. [13] O.D. Montoya, W. Gil-González, A. Garces, Optimal power flow on DC microgrids: a quadratic convex approximation, IEEE Transactions on Circuits and Systems II 66 (6) (2019) 1018–1022, https:// doi.org/10.1109/TCSII.2018.2871432.

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

[14] O.D. Montoya, A convex OPF approximation for selecting the best candidate nodes for optimal location of power sources on DC resistive networks, Engineering Science and Technology, an International Journal (2019). [15] Z.Q. Luo, W.K. Ma, A.M.C. So, Y. Ye, S. Zhang, Semidefinite relaxation of quadratic optimization problems, IEEE Signal Processing Magazine 27 (3) (2010) 20–34. [16] A. Garcés, Convex optimization for the optimal power flow on DC distribution systems, in: Handbook of Optimization in Electric Power Distribution Systems, Springer, 2020, pp. 121–137. [17] M. Grant, S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1, http://cvxr.com/cvx, 2014. [18] W. Gil-González, O.D. Montoya, E. Holguín, A. Garces, L.F. Grisales-Noreña, Economic dispatch of energy storage systems in dc microgrids employing a semidefinite programming model, Journal of Energy Storage 21 (2019) 1–8. [19] H. Hindi, A tutorial on convex optimization, in: Proceedings of the 2004 American Control Conference, vol. 4, IEEE, 2014, pp. 3252–3265. [20] F. Alizadeh, D. Goldfarb, Second-order cone programming, Mathematical Programming 95 (1) (2003) 3–51.

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9 Stability analysis and hierarchical control of DC power networks Alberto Rodríguez-Caberoa , Miguel Jiménez Carrizosab , Javier Roldán-Péreza , and Milan Prodanovica a IMDEA

Energy, Electrical Systems Unit, Móstoles, Madrid, Spain. Politecnica de Madrid, CEI & ETSIME, Madrid, Spain

b Universidad

Chapter points • A DC power network for railway applications is modelled and controlled. • A hierarchical controller is applied. • A model predictive controller (MPC) is used to minimise the operation costs. • A small-signal analysis is used to verify the stability limits. • All the developments are validated in a laboratory.

9.1 Literature review and scope of the chapter 9.1.1 Introduction The massive integration of renewable energy sources has modified the conventional structure of electrical energy systems, where large generation plants were operated far away from energy consumers. This operation paradigm simplifies the control system but requires huge investments in infrastructure and leads to additional losses due to the energy transmission. Nowadays, governments are promoting distributed energy generation at lowvoltage levels, where energy resources placed close to consumers area are used (i.e., distributed generation). In these new schemes, DC and AC supply systems can coexist, and this increases the control complexity. Therefore new control challenges have arisen [1]. In the context of railway applications the conventional design Modeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00014-9 Copyright © 2021 Elsevier Inc. All rights reserved.

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based on an AC-traction substation is also changing. New solutions based on high-voltage DC (HVDC) feeders are suggested [2]. Railway networks are a good example of a hybrid grid where AC distribution lines, storage systems, AC traction electrical machines, and even renewable energy sources can be interconnected via a high-voltage DC link [3]. For this type of applications and with the purpose of taking into account the different dynamics of each involved actor, several controls with different time scales and application areas comprise the whole system control in a hierarchical way. From local controllers, which deal with the frequency, rotor angle, power, or voltage stability issues in AC nodes and voltage or power in DC nodes (in the short term), until supervisor controllers, which deal with energy balancing vis-a-vis the existing resources in the grid [4]. Several advantages can be achieved if a supervisor controller is implemented in a grid with intermittent production: minimized cost, power smoothing, reduction in peak power, better reliability, or greenhouse gas emission reduction, to name a few. To reduce the energy storage requirements and to maximize the revenues, solutions based on optimal control philosophies have notably proliferated, and many of these techniques have been applied to multiterminal DC systems [5]. For example, in [6] a multiobjective optimization with a Pareto-optimal solution is proposed. In [7] a load frequency controller is discussed having changes in communication topology via a multiagent system for smart grids. In [8] a robust energy management formulation based on Lagrange multipliers is presented, which minimizes the microgrid net cost and the utility of dispatchable loads. Also, it takes into account the worst-case transaction cost stemming from the uncertainty in renewable energy sources. However, none of these works provides experimental results, and the results are based on simulations. However, other authors handle this problem by means of a model predictive control (MPC) approach [9]. In [10] a distributed MPC philosophy for controlling large-scale systems is discussed, as well as in [11], where a supervisor controller based on MPC is used to minimize transmission losses in a multiterminal HVDC grid with high penetration of renewable energy resources in the presence of storage elements. Nevertheless, they do not provide experimental results neither. In [12] a control for multiterminal DC networks, which includes local, primary, and secondary controllers, is analyzed in a real environment. However, it does not operate with real storage devices. On the other hand, for railway applications, the MPC is also a popular alternative. For example, in [13] a model-based predictive direct power control is suggested

Chapter 9 Stability analysis and hierarchical control of DC power networks

to calculate the voltage vector that minimizes the error between the reactive and active powers and theirs references. However, in this paper the prediction horizon uses only the available data of the next step. On the other hand, in [14] a robust MPC for train regulation in underground railway transportation is analyzed, with the purpose of designing a state feedback control law at each decision step to optimize a metro system cost function subject to safety constraints on the control input. Nevertheless, this method presents calculation problems when the number of variables is too large. Finally, in [15] a railway energy management system is presented, where significant cost and energy consumption reductions are achieved. However, it lacks a complete control scheme that manages all the electrical and electronics devices involved. Regarding low control levels, one of the main hurdles is that electronics interfaces used for energy conversion in DC microgrids have fast internal controllers (e.g., current and voltage) that can interact between them [16]. This problem has already been analyzed and solved in the literature by using different approaches. Du et al. [17] modeled different AC and DC elements and studied how they interact when they operate in multiterminal AC and DC networks. For this analysis, they developed a smallsignal model of the power system, including AC and DC dynamics. In this analysis, they reported that, under certain conditions, the interaction between AC and DC networks may become strong, and multiterminal medium voltage DC (MVDC) systems need to be carefully examined to avoid undesired effects. Several works have also studied stability of multiterminal DC grids that are used for the integration of offshore wind farms [18]. In particular, the values of the AC and DC droops have been identified as a source of instability in this type of networks. However, the parameters of the internal controllers can also have a relevant effect on the system stability, especially if the system dynamics are fast [19]. In these cases, it is important to perform a small-signal analysis that includes the low-level control layers so that possible interactions and instabilities could be detected. This task is commonly addressed by using the system eigenvalues or participation factors, although other alternatives, such as singular value decomposition (SVD), have also been reported in the literature [20]. Constant power loads (CPLs) are commonly used in DC microgrids. This type of load has received much attention in recent years since it is an important source of instabilities. For example, in [21] a DC controller that stabilizes a DC microgrid is proposed, and the system is analyzed by using small-signal tools to detect possible instabilities. From the literature review it is clear that if this type of load is used, then stability should be carefully studied.

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9.1.2 Contents of the chapter This chapter presents a full control scheme (primary, secondary, and supervisory) and shows experimental results of the MPC, as well as of the rest of control layers. The primary controllers consist of current and voltage regulators that are adequately tuned to provide fast transient responses. This control layer also includes droop controllers to share the load when sudden variations of the load take place. Then a secondary controller is used to restore the value of the DC-link voltage. Finally, an MPC is applied in the tertiary level to optimize the performance of the microgrid. The proposed MPC takes into account the actual power flow and the electricity prices along a prediction horizon to calculate the optimal references. This chapter also includes a detailed small-signal modeling of the DC microgrid under study. This model takes into account all the internal elements of the electronic interfaces (such as filters and current controllers) and AC– DC interconnections. Then all the system equations are linearized and merged to form a state-space representation. The system stability is analyzed by using the system eigenvalues and SVD. Then this model is applied to calculate the stability limits of the microgrid. The accuracy of the proposed modeling method is verified by using a dedicated laboratory test network composed of four ACDC converters and a battery system forming a DC power network. Also, the theoretical conclusions obtained from the eigenvalue analysis were validated experimentally. Some parts of this chapter were previously used for a conference [3].

9.2 Power system and control system overview 9.2.1 Microgrid description In this chapter, a battery system is installed in a DC railway substation. The battery helps maintaining the voltage level of the DC link and plays a key role in the economic operation of the system in real-time. In this respect the main objective is to flatten the power demand consumption to avoid the peaks in the price of electricity. Fig. 9.1 shows a simplified diagram of the DC microgrid studied in this work. The system consists of four voltage source converters (VSCs). All they are connected to the same DC busbar and form a DC microgrid. VSC1 is connected to the main AC grid by using an LCL filter, and it can absorb/inject active (P ) and reactive (Q) power from the AC grid. VSC2 is connected to a railway system by using an LC filter. This converter sets the volt-

Chapter 9 Stability analysis and hierarchical control of DC power networks

Figure 9.1. Simplified electrical diagram of the DC microgrid studied in this work.

age and frequency levels of the railway system. VSC3 is connected to an energy storage system by using an LCL filter and controls the active power P that is absorbed/injected into the battery. VSC4 is connected to an islanded (auxiliary) network by using an LC filter. This converter arbitrarily sets the network voltage and frequency. The connection of the railway system to the DC microgrid provides some degrees of freedom, which can be used to reduce the operational cost of the traction system. Also, the battery can be used to manage the energy recovered from regenerative brake systems. This energy is stored when trains are breaking and is realized to reduce costs when electricity prices are high. However, as locomotive feeders are commonly operated at 25 kV, AC–DC converters with conversion ratios of 1/20 would be needed to connect the railway system to the DC microgrid. Other alternatives such as operating the DC microgrid at higher voltages can also be used.

9.2.2 Microgrid control system structure A hierarchical control structure has been used to operate the DC microgrid. This controller is divided into three control levels, as it is typically used in the literature: local controllers for each VSC, which ensures an adequate control of currents and voltages, a primary controller, which ensures an adequate sharing of activepower transients among the VSCs, a secondary controller, which restores the DC voltage to its reference value after an active power transient, and a tertiary controller, which optimizes the operation of the microgrid and reduces the operation costs. Fig. 9.2 shows a detailed control diagram of the four VSCs, where each one is controlled according to the application requirements. The details of the control system will be explained in the following sections.

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Figure 9.2. Control structure of VSC1 and VSC3, including the local controllers (green (dark gray in print version)), the droop-based primary controller (blue (mid gray in print version)), and the secondary controller (orange (light gray in print version)). VSC1 and VSC3 controllers are implemented in the same control board. Therefore the DC voltage error is integrated only once for the controller implementation.

9.2.3 Local and primary controllers The role of local and primary controllers is controlling the output variables (currents and voltages), the DC-link voltage, and the power flows. Therefore any variation in the power demanded by the application will be immediately compensated by using the energy available on the DC side. Fig. 9.3 shows a detailed control diagram of all the VSCs, which includes the local and primary controllers. The control system of VSC1 consists of a current controller implemented in a synchronous reference frame (SRF) that is synchronized with the daxis component of the PCC voltage (vpcc1 ). The control system of VSC3 consists of a current controller applied to the DC current. The current references of VSC1 and VSC3 are generated by using two independent DC-voltage controllers. The control systems of VSC2 and VSC4 consist of two independent voltage controllers for the filter capacitor voltage (vcf2 and vcf4 ). The control system of both VSCs are implemented in an internally generated SRF. PI controllers were used to control the currents and voltages of all VSCs: KII , s KV CV (s) = KPV + I , s

CI (s) = KPI +

(9.1) (9.2)

Chapter 9 Stability analysis and hierarchical control of DC power networks

221

Figure 9.3. Electrical and control system diagram of a railway substation. It consists of four VSCs connected via a DC link. VSC1 and VSC3 are connected to the grid and the battery system via LCL filters, respectively, and they operate as current-controlled sources. VSC2 and VSC4 are connected to the railway system and the auxiliary network via LC filters, and they operate as voltage-controlled sources.

Cdc (s) = KPdc +

KIdc , s

(9.3)

where KPI , KPV , and KPdc and KII , KIV , and KIdc , are the proportional and integral gains of the current and voltage controllers, respectively. This type controller achieves zero steady-state error when neither negative sequence nor harmonic component is present in the error signal. Otherwise, dedicated controllers such as repetitive or resonant controllers can be used [22,23]. To share the control of the DC voltage, VSC1 and VSC3 include a droop-based primary controller, where the droop constants are Kd1 and Kd3 , respectively. The objective of the primary controller is to share the power instantaneously. Meanwhile, the DC-voltage level should be kept within reasonable limits. The DC-voltage references of both VSCs are modified by using two independent droop controllers. This can be written as follows: ∗2 ∗2 vdc1 = vdc − Kd1 (p1∗ − p1 ),

(9.4)

∗2 Vdc2

(9.5)

∗2 = vdc

− Kd3 (p3∗

− p3 ),

∗2 and v ∗2 are the DC-voltage references that are used where vdc1 dc2 in the DC-voltage controllers, whereas p1 and p3 are the instantaneous active powers delivered by VSC1 and VSC3, respectively; ∗2 is the DC-voltage reference, and p ∗ and p ∗ are VSC1 and VSC3 vdc 1 3 active-power references, respectively.

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Chapter 9 Stability analysis and hierarchical control of DC power networks

The local controllers were designed to achieve the desired control specifications (settling time and stability margins). The droop controllers were designed to achieve an adequate distribution of active power between the VSCs during transients and disturbances. Therefore their time response was chosen to fulfill the requirement tset < 50 ms.

9.2.4 Secondary controller The objective of the secondary controller is to level the primary control reserves, which are used to balance the active power in the short term (during transients) by means of restoring the DCvoltage level to its nominal value after any disturbance or modification of the reference values. It consists of an integral controller that modifies the active power references of VSC1 and VSC3 (ps1 and ps3 , respectively) until the active power mismatch and the DC-voltage deviation are eliminated. The active power references p1∗ and p3∗ are generated by using two independent controllers. First, the integral of the DC error is defined as  ∗2 2 ev = vdc − vdc dt. (9.6) Then the references for VSC1 and VSC3 are calculated as follows: p1∗ = p1set − KI 1 ev ,

(9.7)

p3∗

(9.8)

= p3set

− KI 3 ev ,

where KI 1 and KI 3 are the integral gains of the secondary controller of VSC1 and VSC3, respectively, whereas p1set and p3set are the active-power references provided by the supervisory controller. Note that both secondary controllers use the same feedback variable (vdc ) and will be implemented on the same control platform in the experimental setup. The secondary controllers were designed to restore the DC voltage to its nominal value in less than one second. Therefore tset < 1 s.

9.2.5 Supervisor model predictive controller A tertiary controller has been implemented in the system and its main goal is to reduce the electricity bill. This controller provides power references for all dispatchable converters to optimize a cost function that will be defined later. Therefore an optimal controller based on an MPC was implemented. This controller receives information (predictions) of the power consumed by the

Chapter 9 Stability analysis and hierarchical control of DC power networks

loads (train and resistive load), as well as the prices of the electricity market. Within the scope of this work, it will be assumed these values are known in advance. Then an optimization with a prediction horizon Np will be carried out. The solution of this problem guarantees the minimum cost and energy balance in the DC microgrid. A battery system coupled via a DC–DC converter has been included in the problem to help in balancing the energy in the microgrid and minimizing the cost of energy supplied by the AC grid. The value of the prediction horizon Np has an important impact on the solution of the optimization problem. Clearly, if more information is available in advance (load, railway, and electricity price), then the power scheduling will provide a more suitable (economic) solution. In many power systems applications, it is common to have weather and load forecasts one day in advance. Also, for some railway applications (such as long-distance trains), the braking and acceleration instants are calculated several hours in advance. In this context, the use of an MPC for this specific application is well justified, even more if the dynamic model of the battery is taken into account in the optimization problem. There are several works published in the literature that already used MPC for railway applications [13,14,24,25]. However, to the best of our knowledge, none of these works has applied an MPC combined with an energy storage element to reduce the electrical bill. In this chapter, we present an explicit linear MPC. This controller makes it possible to optimize a cost function subjected to constraints. These constraints will be used to model restrictions such as the maximum power absorbed/supplied from the main AC grid, the battery dynamics, the battery charging/discharging performance, and the maximum power absorbed/supplied by the energy storage element at each instant. Besides, we consider that for a given period, the net energy supplied/absorbed from the battery can be controlled. This assumption is realistic since a fully controllable DC–DC converter is used to interface the battery system. The objective function can be written as follows: ⎛ ⎞ Np  J = min ⎝ p(k) · PAC (k)⎠ k=1

⎛ ⎞ Np  = max ⎝ p(k) · ( Pbat (k) + Ptrain (k) + Pload (k) + δ )⎠ ,

(9.9)

k=1

subject to Emin ≤ Ebat (k) ≤ Emax ,

(9.10)

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Pbat,min ≤ Pbat (k) ≤ Pbat,max , α 

(9.11)

Ebat (k) = ,

(9.12)

k=1

where PAC (k), Pbat (k), Ptrain (k), and Pload (k) represent the power of the AC grid, the battery, the train, and the load, respectively, at the instant k (note that Pbat (k) = p3set (k)). The criterion is to use a positive sign when power enters in the DC link, and a negative sign in any other case. The energy price at the instant k is p(k), E(k) represents the energy stored in the battery at the instant k, and δ is linked to the disturbances that cannot be measured (the system losses are included here). On the other hand, Pbat,min , Pbat,max , Emin , and Emax are the maximum powers absorbed/supplied from the battery and the minimum/maximum energies stored on it, respectively. Finally,  is a parameter that models the net of energy supplied/absorbed by the battery for a given period α ≤ Np . The battery dynamics can be modeled by using the following equation: E(k + 1) = E(k) + μ · Pbat (k) · T ,

∀k,

(9.13)

where μ is the efficiency of the battery during the charge and discharge process, whereas T is the period considered for the controller implementation. Here the same efficiency is considered for the charge and discharge process, although different values can be easily taken into account.

9.3 Small-signal modeling of the DC microgrid In this section, each element of the DC microgrid is modeled to perform the small-signal stability analysis. Each set of equations is then linearized and written in a state-space representation. Finally, all the state-state representations are merged by means of the DC-link equation, which is modeled by the energy balance equation. Fig. 9.4 shows a graphical representation of the smallsignal model, where all the blocks are aggregated by using the DC-link voltage equation.

9.3.1 Model of the grid-connected VSC VSC1 is interfaced to the grid by using an LCL to filter out ripples and high-frequency noise. The small-signal state-space

Chapter 9 Stability analysis and hierarchical control of DC power networks

225

Figure 9.4. Block diagram showing the connections between the VSCs and the DC-network small-signal models. The aggregated model of the DC microgrid includes the output filters and the control system models of each VSC and the DC-network model.

model of the LCL filter of VSC1 in dq can be written as [26] d dq dq dq dq xf 1 = Af 1 xf 1 + BfS 1 vs1 + BfV1 vpcc1 + Bfω ωs1 , dt

(9.14) dq

where the symbol  stands for the incremental operator, xf 1 = dq

dq

dq

[iI 1 vcf 1 io1 ]T (T means transposed), vcf 1 is the AC capacitor voltage, iI 1 is the converter-side inductor current, vs1 is the converter output voltage, and ωs1 is the grid frequency. The state-matrices Af 1 , BfS 1 , BfV1 , and Bfω1 include the coefficients of the filter, and their values can be found in [26]. VSC1 includes a current controller to follow current references. A possible state-space representation of a PI current controller like (9.1) (one for each axis) is d dq dq dq∗ dq I ϕ1 = AIC1 ϕ1 + BC1 (io1 − io1 ), dt dq dq dq∗ dq I I ϕ1 + DC1 (io1 − io1 ), vs1 = CC1

(9.15) (9.16)

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Chapter 9 Stability analysis and hierarchical control of DC power networks

where ϕ1 is an auxiliary state variable for the integral action, dq∗ io1 is the reference current reference, AIC = 02×2 , BCI = KII I 2×2 , CCI = I 2×2 , and DCI = KPI I 2×2 . The reference current is generated by using the nonlinear instantaneous active and reactive power equations [27]. This calculation method provides good results if the grid is not too weak. Otherwise, other alternatives should be considered. These equations can be linearized around an operating point, yielding io1 = MV 1 P Q∗1 − MV I 1 vpcc , dq∗

dq

(9.17)

,

(9.18)

with  MV 1 =

d Vpcc1

q

Vpcc1

q

−1

d Vpcc1 −Vpcc1  q d Io1 Io1 , MV I 1 = MV 1 · q d −Io1 Io1

(9.19)

where P Q∗1 = [p1∗ q1∗ ]T , and Io1 and Vpcc1 are the grid-side current and the PCC voltage operating points, respectively. Uppercase symbols stand for “operating point”. The DC-controller includes a droop term so that all converters can operate in parallel configuration. A possible representation of a DC-voltage controller like (9.3) with droop control is dq

dq

d sp dc ∗2 2 γ1 = Adc C γ1 + BC (vdc − vdc − Kd1 (p1 − p1 )), dt sp ∗2 2 − vdc − Kd1 (p1 − p1 )), p1∗ = CCdc γ1 + DCdc (vdc

(9.20) (9.21)

dc dc dc dc dc where Adc C = 0, BC = KI , CC = 1, DC = KP , kd1 is the droop sp gain, γ is an auxiliary state variable for the integral action, p1 is the active power set point of VSC1, and p1 is the instantaneous active power delivered to the grid. A synchronous reference frame phase-locked loop (SRF-PLL) was used in [26]. This algorithm aligns the dq reference frame of the converter with the fundamental component of the grid voltage space-vector. A possible small-signal state-space representation of the SRF-PLL is 



 d x x 0 pcc DQ = Apll +Bpll vpcc + ωs , (9.22) θpll −1 dt θpll     xpll

ω Bpll

Chapter 9 Stability analysis and hierarchical control of DC power networks



  θ

  θ

 Cpll Dpll θpll x 0 DQ = + v + ωs , pcc ω ω ωpll θpll −1 Cpll Dpll    

(9.23)

pcc

Cpll

Dpll

where x is an auxiliary variable used to model the internal state of the SRF-PLL. More advanced PLLs can be taken into consideration, but their representation requires additional modeling efforts. pcc ω , C , and D pcc can be The state-space matrices Apll , Bpll , Bpll pll pll found in [26]. The small-signal models of the current controller, the DCvoltage controller, and the SRF-PLL can be merged together to form the state-space representation of the converter. Therefore d pcc sp sp DQ x1 = A1 x1 + B1 vpcc1 + B1P p1 + B1Q q1 dt dq ∗2 2 +B1V ∗ vdc + B1V vdc + B1I io1 + B1ω ws , dq vs1

pcc DQ = C1 x1 + D1 vpcc1

sp + D1P p1 dq ∗2 2 +D1V ∗ vdc + D1V vdc + D1I io1

(9.24)

sp Q + D1 q1

+ D1ω ws ,

(9.25)

dq

where x1 = [ϕ1 , γ1 , xpll ]T .

9.3.2 Battery-system VSC The battery converter includes an LCL filter that operates in DC. This filter can be represented as follows: ⎤ ⎡ −RI 3 /LI 3 iI 3 d ⎣ vcf 3 ⎦ = ⎣ −1/Cf 3 dt io3 0   ⎡

xf 3

−1/LI 3 0 1/Lo3



⎤ 0 1/Cf 3 ⎦ xf 3 −Ro3 /Lo3 

Af 3

⎤ ⎡ ⎡ ⎤ 1/LI 3 0 + ⎣ 0 ⎦ vs3 + ⎣ 0 ⎦ vbat3 , −1/Lo3 0     BfS 3

Bfbat 3

where iI 3 is the converter-side inductor current, vcf 3 is the filter capacitor voltage, vbat3 is the battery voltage, and vs3 is the converter output voltage. A PI current controller like (9.1) is used to control the output current of the LCL filter. This filter can be represented as a state-

227

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space model by using the following set of equations: d ∗ ϕ3 = AC3 ϕ3 + BC3 (io3 − io3 ), dt ∗ vs3 = CC3 ϕ3 + DC3 (io3 − io3 ),

(9.26) (9.27)

∗ is the reference curwhere ϕ3 is an auxiliary state variable, io3 I rent reference, AC = 0, BC = KI , CC = 1, and DC = KPI . The names of the filter parameters are shown in Fig. 9.3. The DC-voltage controller of the VSC3 can be represented by sp sp using (9.20) and (9.21) and replacing p1 by p3 , p1 by p3 , and Kd1 by Kd3 . The reference current allows the converter to follow the activepower reference. The active-power flow in the battery can be represented as p3∗ = vbat · i3∗ . This equation is nonlinear and has to be linearized before it is included in the full state-state representation. This operation can be performed as follows: ∗ io3 =

1 Vbat

p3∗ −

Io3 vbat3 , Vbat

(9.28)

where Vbat3 and Io3 are the operating points of the battery voltage and current, respectively. The small-signal models of the current and DC-voltage controllers can be merged, yielding d sp ∗2 x3 = A3 x2 + B3bat vbat3 + B3P p3 + B3V ∗ vdc dt 2 + B3V vdc + B3I io3 , (9.29) ∗2 vs3 = C3 x2 + D3bat vbat3 + D3P p3 + D3V ∗ vdc sp

2 + D3V vdc + D3I io3 ,

(9.30)

9.3.3 Railway and auxiliary-network VSCs The railway and auxiliary network converters are connected by using LC filters. A possible state-space representation of this filter in a dq-reference frame is [28] ⎤ ⎤ ⎡ RI 2 ⎡ −1 d − ω 0 s2 i LI 2 LI 2 ⎥ ⎢ Iq2 ⎥ ⎢ −1 ⎥ ⎢−ωs2 − RI 2 ⎥ 0 i d ⎢ ⎢ ⎥ dq L L I 2 I 2 I 2 ⎥ ⎢ (9.31) ⎥ xf 2 d ⎥=⎢ 1 ⎢ ⎥ dt ⎢ v 0 0 ω s2 ⎣ cf 2 ⎦ ⎣ Ccf 2 ⎦ q 1 vcf 2 0 −ωs2 0 Ccf 2  

  dq

xf 2

Af 2

Chapter 9 Stability analysis and hierarchical control of DC power networks

⎡ ⎤ ⎤ q 0 0 II 2 ⎢ 0 ⎢ ⎥ ⎢ 0 ⎥ ⎥ dq ⎢ IIq2 ⎥ 1 ⎥ ⎥ dq ⎢ ⎢ 0 ⎢ ⎢ ⎥ ⎥ LI 2 ⎥ vs2 + ⎢ −1 +⎢ 0 ⎥ il2 + ⎢V q ⎥ ωs2 , ⎣ 0 0 ⎦ ⎣ Ccf 2 ⎣ cf 2 ⎦ ⎦ −1 d 0 0 0 vcf Ccf 2 2

   

  ⎡

1 LI 2

0

BfS 2

⎤

⎡

Bfl 2

Bfω2

dq

where il2 is the current consumed by the load, and ωs2 is the frequency of the railway system (imposed by the converter). A PI controller like (9.2) was used to control the output voltage of VSC2. The voltage and frequency references can be set arbitrarily since the AC system is isolated. This controller can be represented in a state-space fashion as follows: d dq dq dq∗ dq ψ2 = AC2 ψ2 + BC2 (vcf − vcf 2 ), dt dq dq∗ dq vs2 = CC2 ψ2 + DC2 (vcf 2 − vcf 2 ).

(9.32) (9.33)

The filter used here commonly leads to undesired oscillations due to the medium-frequency resonance. Several strategies can be applied to damp this oscillation. In this work a lead-lag network was used to damp the resonance without additional passive elements [29,30]: Cad (s) = Kad ·

1 + Ds , 1 + fm Ds

(9.34)

where Kad is a proportional gain, D is a differential gain, and fm is the gain of a filter used to limit the effect of high-frequency noise. This lead-lag network was designed to maximize the damping of the resonant poles. A possible state-space representation of the filter network such as (9.34) (one for each axis) can be written as d dq dq dq γ2 = Aad γ2 + Bad icf 2 , dt dq dq dq vs2 = Cad γ2 + Dad icf 2 ,

(9.35) (9.36)

with 1 2×2 , Bad = I 2×2 , I fm D   1 Kad Kad 2×2 I 2×2 , Dad = 1− = I , fm D fm fm

Aad =

(9.37)

Cad

(9.38) dq

where I is the identity matrix of appropriate size, and γ2 auxiliary variable used to model the additional filter state.

is an

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The voltage controller and the active damping strategy can be merged in a single state-space representation, yielding d dq dq∗ dq x2 = A2 x2 + B2V vcf 2 + B2V ∗ vcf 2 + B2I icf 2 , dt dq dq dq∗ dq vs2 = C2 x2 + D2V vcf 2 + D2V ∗ vcf 2 + D2I icf 2 .

(9.39) (9.40)

The VSC4 control system can be modeled by using the same prodq dq dq∗ dq∗ dq cedure and replacing x2 by x4 , vcf 2 by vcf 4 , vcf 2 by vcf 4 , and icf 2 by dq

icf 4 .

9.3.4 DC-capacitor modeling If the VSCs are located in the same place, then the DC cables used to connect them to the DC link are small, and their inductances and resistances can be neglected. Therefore Cdc = Cdc1 + Cdc2 + Cdc3 + Cdc4 ,

(9.41)

where Cdc is the equivalent DC capacitor of the DC power network. The equation modeling the energy stored in the equivalent DC capacitor can be written as Cdc d 2 v = −(pdc1 + pdc2 + pdc3 + pdc4 ), 2 dt dc

(9.42)

where pdc1 , pdc2 , pdc3 , and pdc4 are the instantaneous active powers taken from the equivalent DC capacitor by VSC1, VSC2, VSC3, and VSC4, respectively. Typically, conduction losses of VSCs are taken into account in the parameter RI . In this case the DC and DC-side powers are almost the same, so pdc ≈ ps . Therefore the AC-side powers can be written in terms of dq currents and voltages as [27] q q

d d ps1 = vs1 iI 1 + vs1 is1 ,

(9.43)

q q + vs2 is2 ,

(9.44)

d d iI 2 ps2 = vs2

ps3 = vs iI 3 , d d ps4 = vs4 iI 4

(9.45) q q + vs4 is4 .

(9.46)

These power expressions can be linearized, yielding dq

dq

dq

dq

ps1 = Vs1 iI 1 + Is1 vs1 ,

(9.47)

dq dq ps2 = Vs2 iI 2

(9.48)

dq dq + Is2 vs2 ,

Chapter 9 Stability analysis and hierarchical control of DC power networks

ps3 = Vs3 iI 3 + II 3 vs3 ,

(9.49)

dq dq ps4 = Vs4 iI 4

(9.50)

dq dq + Is4 vs4 .

By linearizing (9.42) and merging the result with (9.47) and (9.50) we obtain the linearized DC-capacitor equation Cdc d dq dq dq dq dq dq dq dq 2 vdc = −(Vs1 iI 1 + Is1 vs1 + Vs2 iI 2 + Is2 vs2 2 dt dq dq dq dq + Vs3 iI 3 + II 3 vs3 + Vs4 iI 4 + Is4 vs4 ). (9.51) This equation is the main link between all the converters of the DC microgrid. This type of link between the models differs from the classical one made for AC systems, where each element is coupled by using the matrices of the electrical grid [28]. The complete expression of the aggregated model will be explored in the next section.

9.3.5 Aggregated model of the DC microgrid The filter models of all the VSCs and the linearized DC-capacitor model can be merged in a single state-space model, yielding d S d ω xM = AM xM + BM vs + BM d + BM ωg , dt T  2 xM = xf 1 xf 2 xf 3 xf 4 vdc ,  T dq dq dq d = vpcc1 , vbat il2 il4 T  vs = v dq v dq vs3 v dq . s1

s2

s4

(9.52) (9.53) (9.54) (9.55)

Merging (9.52), (9.24), (9.29), and (9.39) and substituting (9.25), (9.30), and (9.40) into (9.52), we obtain the aggregated small-signal model of the DC microgrid: d d xDC = ADC xDC + BDC r ∗ + BDC d, dt T  xDC = xM x1 x2 x3 x4 ,  T dq dq dq d = vpcc1 , vbat il2 il4 T  ∗2 v dq∗ v dq∗ r ∗ = p1sp p3sp vdc . cf 2 cf 4

(9.56) (9.57) (9.58) (9.59)

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9.4 Case study and prototype description The experiments have been carried out in the Smart Energy Integration Lab (SEIL) facilities [31,32]. The case study consists of a railway system connected to a DC microgrid. The DC microgrid interconnects an islanded network (that operates in AC) and the main AC grid. A battery converter coupled via a DC–DC converter is included to balance the power flow in the DC link (see Fig. 9.1). The primary and secondary controllers were implemented on Nexcom NISE 6140 industrial computers, which were also used to control the power converters. A LabView platform was used as a gateway to manage the data exchanged between the converters and the tertiary controller. The latter was implemented in Matlab and was executed in an external PC. More details of the laboratory equipment and the infrastructure can be found in [31]. The nominal voltage of the system was set to 400 V, and the nominal frequency was set to 50 Hz. VSC2 and VSC4 were rated a power of 15 kVA. Both converters have the same LCL filter, and the values were LI = 2.3 mH, Cf = 8.8 µF, and Lo = 0.93 mH. On the other hand, the rated power of VSC1 and VSC3 was 75 kVA. Both converters had the same LCL filter, where the values were LI = 500 µH, Cf = 100 µF, and Lo = 250 µH. Since of all they were connected to the same DC link without using DC lines, they formed an equivalent DC capacitor of Cdc = 12 mF. The DC link operates at 680 V. The control systems were implemented on two industrial PCs. These PCs generated the firing signals for all the VSCs. VSC1 and VSC3 were controlled with one industrial PC, whereas VSC2 and VSC4 were controlled with another one. For VSC1 and VSC3, the switching and sampling frequencies were equal to 8 kHz, whereas for VSC2 and VSC4, they were 10 kHz.

9.5 Validation of the model predictive controller The experiment recreates two days of the microgrid operation. Real data were used to emulate the load consumption and the islanded AC network [33]. It is assumed that the railway substation allows energy recovery and that the train has regenerative brakes. Also, real electricity prices for a typical industrial customer in the main AC connection were been used. The price was regulated according to the time of year and the daily hour (see Fig. 9.5) [34]. The battery was included to reduce the demand at peak hours. Also, it was used to support the immediate energy balancing by

Chapter 9 Stability analysis and hierarchical control of DC power networks

Figure 9.5. Hourly price of the electricity used in the experiments.

Table 9.1 Control parameters. Parameter Pmax,ch Pmax,dis μc h μd is Emax Emin 

Description

Value

Maximum charging power 15 kW Maximum discharging power 15 kW Charging efficiency 95% Discharging efficiency 90% Maximum energy 45 kWh Minimum energy 5 kWh Supplied-absorbed net energy 0 kWh

providing or recuperating energy from the railway system. The scenarios were used to test the whole system in a single test, with all the control levels working at the same time. All the system changes and transients were recorded and stored to analyze the performance in a later stage. As explained in the previous sections, the main goal of the supervisory control is minimizing the cost of energy absorbed from the main AC grid. To analyze the impact of the prediction horizon in the performance of the controller, three different values were tested, namely, NP = 24, 6, and 3 hours. The same scenario was used for the three cases and included the data of two days. To speed up the tests, in the real-time implementation, each one lasted for three minutes. Therefore four seconds correspond to one hour. Also, to make a fair comparison, the same parameters were used in all the experiments, including the initial state of charge of the battery system (see Table 9.1).

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Figure 9.6. Experimental results of the DC-microgrid for Np = 24.

9.5.1 Local, primary, and secondary controllers Fig. 9.6(a) shows an overview of the experimental results obtained for a period of two days. First, it can be observed that the active power injected in the ac grid (blue line (dark gray in print version)) follows the reference given by the supervisor model predictive controller (red line (gray in print version)). Fig. 9.6(b) shows a zoom over the transient response. We can see that the converter follows the reference value. However, there is a small

Chapter 9 Stability analysis and hierarchical control of DC power networks

Figure 9.7. Energy stored in battery system for Np = 24.

mismatch caused by the primary controller (droop). We can see that the active power demanded by the islanded network and the railway system increased, and therefore the supervisor predictive controller provided updated active power references for VSC1 and VSC3 (p1set and p2set ) according to the new operating conditions. The local and primary controller tracked the new power references, and the power mismatch was shared between VSC1 and VSC3. Therefore the DC voltage deviated from the reference value. After that, the secondary controller modified the active power references to compensate the active power mismatch and to reduce the DC-voltage deviation. In steady state the instantaneous active power delivered by VSC1 and VSC3 (in blue (dark gray in print version)) presented a small deviation from the references provided by the supervisor predictive controller (in red (gray in print version)). This is consistent with the standard operation of the secondary controller, which introduces small deviations in the power references (p1set and p3set ) to eliminate steady-state DC-voltage deviations.

9.5.2 Prediction horizon set to Np = 24 hours Fig. 9.6(a) shows the performance of the supervisory control when the prediction horizon is 24 hours. We can see that the demand of the islanded network is mainly supplied by the main AC grid. Also, the battery supplies active power when the train is accelerating (increase speed). Thanks to the model predictive controller, the system is able to anticipate the whole consumption (by means of the forecasting). Therefore the battery is discharged when the electricity price is high (see Fig. 9.5) and is charged when the price is low. Also, we can see that all the constraints are fulfilled. This information is summarized in Table 9.1. Fig. 9.7 shows the energy stored in the battery when the prediction horizon is 24 hours. For simplicity, the time scale of the

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Chapter 9 Stability analysis and hierarchical control of DC power networks

Figure 9.8. Experimental results showing transient responses of the DC-microgrid for Np = 6.

experimental results has been modified to present it in hours. We cane observe that the battery energy was always within its operational limits and that it helped the system by supporting the DC voltage when the energy demand increased.

9.5.3 Prediction horizon set to Np = 6 hours Fig. 9.8 shows the experimental results of the microgrid operation for Np = 6 hours. We can see that the active power of the AC grid and the power supplied/absorbed by battery are different compared to the previous case (Np = 24 hours). In this case, it can be confirmed that the battery supplied the energy required to maintain the DC voltage. However, the volatility of electricity prices plays a more important role in the charge and discharge process of the battery. This happens because there was no enough information available in advance, and the model predictive controller can only find local minima. Therefore this controller leads to suboptimal performance. Fig. 9.9 shows the energy stored in the battery for Np = 6 hours. We can observe that the variables are always within the preestablished limits. However, the battery suffers more charging and discharging cycles compared to the previous case (Np = 24 hours). This fact may reduce the lifetime of the battery, although this was not one of the optimization objectives. However, this type of result can be expected since the variations in the controller set-points will be less abrupt if there is more information available in advance.

Chapter 9 Stability analysis and hierarchical control of DC power networks

Figure 9.9. Energy stored in battery for Np = 6.

9.5.4 Prediction horizon set to Np = 3 hours Fig. 9.10 shows the experimental results for Np = 3 hours. In this case the active power injected to the AC grid and the active power supplied/absorbed by battery are different compared to the previous cases. Clearly, this case is the most sensitive to variations in the price of energy since the predictions are only available three hours in advance. This can be verified in Fig. 9.11, which shows the maximum number of charging/discharging cycles of the battery. Table 9.2 shows the cost of energy for the main AC grid. In the first case (without an optimized planning), only the power consumed by the train and the load are considered. In the other cases (with optimization) the battery was also considered. We can observe that the cost of energy is reduced as the prediction horizon increases. This fact is not surprising, since having a better fore-

Figure 9.10. Experimental results of the DC microgrid for Np = 3.

237

238

Chapter 9 Stability analysis and hierarchical control of DC power networks

Figure 9.11. Energy stored in battery for Np = 3.

Table 9.2 Summary of energy cost for the different cases. Case

Cost

Without optimization With optimization, Np = 3 hours With optimization, Np = 6 hours With optimization, Np = 24 hours

3.71€ 3.51€ 2.99€ 2.33€

casting system allows the DC microgrid to be operated in a more economic way. In addition, we can observe that without optimization the costs are approximately 60% higher compared to the case of Np = 24 hours. In addition, with Np = 3 hours, the costs are 50% higher with respect to the case of Np = 24 hours. Finally, with Np = 6 hours, the costs are 30% higher.

9.6 Validation of the small-signal modeling approach 9.6.1 Stability analysis of the DC microgrid The system eigenvalues were used to evaluate the dynamic properties of the DC microgrid. Also, the control system parameters were modified to obtain the trace of the system eigenvalues. By using this methodology it is possible to select the controller parameters that lead to better stability margins. Special attention was paid to the selection of the droop coefficients of VSC1 and VSC3 controllers, since they have an important effect on the DC voltage regulation and on the stability properties.

Chapter 9 Stability analysis and hierarchical control of DC power networks

Figure 9.12. Trajectories of the system eigenvalues when (a) Kd1 varies between 0.01 and 1 and (b) Kd1 varies between 0.01 and 1.

Figs. 9.12(a, b) show the trajectories of the system eigenvalues when Kd1 and Kd3 vary between 0.01 and 1. We can observe that the system eigenvalues follow very similar trajectories for the variation of both parameters. This result is consistent since in this case study, VSC1 and VSC3 have the same rated power, filter parameters, and control gains. Therefore both VSCs have similar dynamic properties and exhibit the same sensitivity to variations on the droop gains (Kd1 and Kd3 ). In both cases, when Kd1 and Kd3 increase, some of the low-frequency eigenvalues move away from the unstable region, increasing their frequency. However, some medium frequency eigenvalues move into the unstable region. For Kd1 ≥ 0.7 and Kd3 ≥ 0.7, the medium frequency eigenvalues cross the imaginary axis, and the system becomes unstable. Fig. 9.13 shows the singular value decomposition (SVD) for the DC microgrid model. In this analysis the value of Kd1 was varied between 0.01 and 0.7. We can see that the system is less affected by lowfrequency disturbances (less than 1 Hz). On the other hand, the system is more sensitive to disturbances in the range of 10 to 100 Hz. We can observe that a low value of Kd1 reduces this sensitivity. However, this selection also affects the regulation accuracy of the DC voltage. Finally, we can observe that the high-frequency resonance is also affected by this value. These results highlight the effect of the droop gains over the whole frequency range of the DC microgrid.

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Chapter 9 Stability analysis and hierarchical control of DC power networks

Figure 9.13. Frequency response of the maximum singular value when Kdq varies between 0.01 and 0.7.

9.6.2 Experimental results Figs. 9.14(a, b) show the transient response of the experimental setup when Kd1 and Kd3 change from 0.5 to 0.8, respectively.

Figure 9.14. Transient response of the DC microgrid when (a) Kd1 is modified from 0.5 to 0.8 and (b) Kd3 is modified from 0.5 to 0.8.

Chapter 9 Stability analysis and hierarchical control of DC power networks

We can see that the transient response oscillates when the droop values increases. The values that lead to instability were selected from previous stability studies. Therefore this result is in line with the results obtained from the theoretical analysis (eigenvalue analysis) presented in Section 9.6.1.

9.7 Conclusion In this chapter, we presented a hierarchical control structure for DC microgrids in railway applications. This scheme includes a battery system installed to deal with the instantaneous energy balancing and to minimize the cost of energy taking into account forecasts. The implementation was carried out by using a two-day scenario scaled down in time, and executed over 3-minute period. The primary and secondary control layers guaranteed the stability during disturbances such as variations of the load. A droop controller was applied to stabilize the DC voltage and to allow a proportional sharing of the load demand. A stability analysis of the DC microgrid was carried out to find the maximum and minimum values of the controller parameters. Theoretical and experimental results were used to validate the main contributions of this work. For the tertiary controller, the results showed that the stability of each node is preserved under any power configuration. The results confirmed a significant reduction of the cost of energy when the predictive control approach was used. Three different scenarios were tested, and the experimental results were provided. Economic benefits of more than 60% were achieved, and it was demonstrated that the improvements were closely related to the available system information and the demand forecast of the railway and auxiliary loads. For the small-signal stability modeling approach, it was shown that the proposed method provides an accurate description of the system dynamics. Also, it was verified that, for large values of the droop coefficients, the system becomes unstable. The SVD analysis confirmed that the droop coefficients do have an important impact not only at low-frequency range but also at medium frequency. In future works, new challenges for the tertiary control will be addressed as, for example, multiobjective cost functions, which include demand response or battery lifetime consideration. Also, advanced demand flexibility schemes will be included in the optimization problem.

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10 Digital control strategies of DC–DC converters in automotive hybrid powertrains Carlos Restrepoa and Catalina González-Castañob a Universidad

de Talca, Department of Electromechanics and Energy Conversion, Curicó, Chile. b Universitat Rovira i Virgili, Departament d’Enginyeria Electrònica, Elèctrica i Automàtica, Tarragona, Spain

Chapter points • Buck and boost DC–DC power converters analysis. • Software-in-the-loop digital current controllers testing. • PSIM simplified C block. • Average current control based on passivity. • Discrete-time sliding-mode current control strategy. • Digital proportional-integral current control. • Predictive digital current programmed control.

10.1 Introduction The global warming and climate change are the result of human activities, mainly due to emissions of carbon dioxide (CO2 ) from fossil fuel combustion in several sectors, which include electricity generation, transportation, agriculture, and commercial and residential activities [1]. However, the reduction of the road transport carbon emissions is considered currently one of the greatest challenges of our society, and the situation can worsen in the future if immediate and decisive actions are not taken. The transportation sector is responsible for 24% of global greenhouse gas (GHG) emissions and in a large extent also blameable for the air quality degradation, which threatens population’s health and the oil dependence [2]. Moreover, the population is growing, especially in developing countries, and therefore the required energy needs will increase, and so will the oil prices. Compounding the oil situation, large reserves are in politically unstable countries, and Modeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00015-0 Copyright © 2021 Elsevier Inc. All rights reserved.

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

Figure 10.1. EVs classification.

this can worsen the situation even more. With this background, it is understandable that the great powers make large investments to search for transport alternatives to internal combustion engine vehicles (ICEVs) [3]. Against this background, different efforts are being undertaken to reduce the GHG emissions and to improve the performance of the transportation sector by means of the development of new fuels and the electrification of transport [4]. The latter is a promising approach with potential benefits in improving the energy security by the diversification of the energy sources, fostering economic growth by creating new advanced industries focused on the electric vehicles (EVs), and most importantly, its technology is environment friendly, which allows the integration of renewable energy into the power system [4,5]. Nowadays commercial EVs can be classified into four types based on their propulsion systems: hybrid electric vehicles (HEVs), plug-in hybrid electric vehicles (PHEVs), battery electric vehicles (BEVs), and fuel cell electric vehicles (FCEVs), as shown in Fig. 10.1 [4]. Furthermore, there are great similarities between electric vehicle types HEVs, PHEVs, and BEVs, as shown in Fig. 10.2. The

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

Figure 10.2. EVs powertrain configurations.

HEVs and PHEVs are driven through a combination of an ICE and an electric motor (EM), but the small HEV battery can only be charged by the ICE or the regenerative braking. There are different powertrain configurations (series, parallel, and series-parallel) developed for HEVs and PHEVs to achieve different objectives such as improving fuel economy, increasing power, and minimizing cost [4]. All power train configurations of EVs shown in Fig. 10.2 have a system in common that is formed by the battery, the power converter, and the electric motor. Each one of these components has been the subject of an extensive research and high-level development in recent years with the aim of improving the performance of the automotive traction systems. In this system the power converter represents an important share of the total cost of the EVs and has a high potential for cost reduction [6]. However, this important component of EVs is less covered in the literature on applications of electric vehicles [6]. Two basic configurations are used to implement the power converter in Fig. 10.2. The first one corresponds to a power inverter directly connected to the battery system, and the second configuration uses a bidirectional DC–DC converter between the battery pack and traction inverter. In the case of the first configuration the inverter does not have the same performance at all modulation indexes (MIs), and it is more efficient and produces better waveforms at higher MI values. Therefore, for values below the base speed, the inverter MI is less than 1 and the phase voltage of the motor is increased proportionally to the speed, as shown in Fig. 10.3 (see DC–AC block), and the inverter is operating in low-efficiency zones. Once the base speed

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is reached, the phase voltage remains constant at the rated phase voltage by means of MI = 1 with an efficient operation of the inverter [7]. With the goal of extending this efficient inverter operation, a bidirectional DC–DC converter, typically a boost converter, is used to control the voltage at the input of the inverter according to the motor speed and to optimize the efficiency of the inverter (MI = 1) in a wider range of operating speeds, as can be seen in Fig. 10.3 (see DC–DC + DC–AC block).

Figure 10.3. Torque and power requirements for EV drive systems.

Another classification of EVs is based on their energy sources, which correspond to fossil fuel, battery, and hydrogen, as shown in Fig. 10.1. Hydrogen is a flexible energy carrier, which can be produced from any primary energy source, and it is particularly well suited for using proton exchange membrane fuel cell (PEMFC), which efficiently uses hydrogen to generate electricity

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

[8]. A PEMFC is an electrochemical device that directly converts the chemical energy of a fuel into electricity and can operate continuously as long as it is provided with sufficient amounts of reactant gases. PEMFC has many advantages such as high efficiency, fast startup, low operating temperatures, nonpolluting emissions, and high energy density, which makes them suitable for mobile applications [9–11]. However, the relatively short lifespans of fuel cells, in general, are a significant barrier to their commercialization in stationary and mobile applications [12–15]. Therefore the focus of fuel cell mobile applications is the reduction of costs and increase of its lifetime [8]. The PEMFC stack is a complex system that requires an auxiliary power-conditioning system to ensure reliable, efficient, and safe operation under different operating conditions [16]. Therefore the fuel cell current and voltage characteristics are limited by the mechanical devices used to maintain the airflow within the cathode, either by using a compressor or a blower, hydrogen flow within the anode through an adjustable valve command, temperature control using a cooling fan, and the humidity of the air in the cell using a humidity exchanger, as illustrated in Fig. 10.4(a). However, from all the above-mentioned processes, the air flow associated with a compressor is the slowest auxiliary system, which limits the PEMFC dynamics [17,18]. Such an abrupt increase in the current load produces in the fuel cell a high voltage drop during a short time, which is known as oxygen starvation phenomenon [19–22]. This phenomenon, the oxygen starvation, occurs when the amount of oxygen supplied to the fuel cell is insufficient for reacting in accordance with the demand of the stack current [23]. This operational condition is evidently harmful for the FC because it accelerates the catalyst losses and the carbonsupport corrosion [21,24]. Therefore an FC is considered to be a slow dynamic-response device with respect to the transient load requirements [23]. Different approaches to prevent the oxygen starvation phenomenon have been proposed in the literature; however, only one approach has demonstrated efficacy for avoiding this phenomenon under any current transient. The solution is hybridizing the fuel cell operation with other auxiliary storage devices as batteries or supercapacitors. In the case of automotive hybrid powertrains the fuel cell system can store large amounts of energy due its high energy density, which provides the base power for constant speed driving for hours while the capacitors or batteries can pump into, or extract from, high power in microseconds or seconds, which provides additional peak power during acceleration, high load operation and recovers braking energy by regeneration,

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Figure 10.4. (a) PEM fuel cell system with main control subsystems and (b) Ragone plot.

Figure 10.5. Hydrogen fuel cell vehicle.

as shown in the Ragone plot of Fig. 10.4(b). In this way the oxygen starvation can be avoided, and the system can operate with higher efficiency [25–27]. Therefore commercial vehicles should hybridize the fuel cell with an auxiliary energy storage device as shown in Fig. 10.5. In recent years, many configurations of hybrid systems related to fuel cells and auxiliary storage devices have been proposed for mobile applications in the literature [28]. Fig. 10.6 depicts six dif-

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

251

Figure 10.6. Powertrain topologies of fuel cell hybrid electric vehicles: (T 1) floating DC bus, (T 2) controlled FC with floating battery or supercapacitor, (T 3) floating FC with controlled battery or supercapacitor, (T 4) controlled DC bus with controlled FC, battery, or supercapacitor, (T 5) controlled FC with controlled battery and floating supercapacitor, (T 6) controlled FC with controlled battery and supercapacitor.

ferent powertrain topologies (T1 to T6 ) of hybrid electric vehicles proposed in the literature. In addition, a general description of different hybrid electric topologies presented in the literature is listed in Table 10.1. As can be concluded from the table, despite the fact that the buck and boost converters are very simple topologies, they are widely used in many applications and especially in the design of powertrain topologies for fuel cell hybrid electric vehicles. Therefore in Section 10.2, we will focus on the modeling of these topologies. Furthermore, the use of classic power converter topologies will allow a better understanding of the proposed control techniques and an easy their comparison. Due to our discussion, in the next sections, we will focus on the topologies of boost and buck converters. However, in many of the powertrain topologies shown in Fig. 10.6, at all times a DC–DC converter (in blue color (gray in print version)) regulates the DC bus to ensure an efficient inverter operation (in red color (dark gray in print version)) under a wider range of speeds. There are two common closed-loop control methods for PWM DC–DC converters. The first one is designing a voltage loop, namely the

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Table 10.1 Summary of FCHEV topologies. ASD Topol. B SC Converter topologies

Ref.

T2

Buck

[29]

Boost

[30]

Three-port Bidirectional DC–DC converter

[31]

Two buck converters

[32]

T3 T4

Unidirectional boost converter and bidirectional buck–boost [33] converter (boost type) Unidirectional boost converter and bidirectional buck–boost [34] converter (boost type) Unidirectional boost converter and bidirectional buck–boost [35] converter (boost type) Interleaved boost converter and interleaved buck–boost con- [36] verter (boost type) Interleaved boost converter and interleaved buck–boost con- [37] verter (boost type) Unidirectional boost converter and bidirectional buck–boost [38] converter (boost type) Unidirectional boost converter and bidirectional buck–boost [39] converter (boost type) Unidirectional boost converter and bidirectional buck–boost [40] converter (boost type) Current–fed full bridge converter and bidirectional buck– [41] boost converter (boost type)

T5

Unidirectional buck converter and bidirectional buck–boost [42] converter (boost type) Two unidirectional boost converters and a unidirectional buck [43] converter

T6

Unidirectional boost converter and two bidirectional buck– [44] boost converters (boost type) Unidirectional boost converter, a bidirectional buck–boost [45] converter (boost type), and a bidirectional buck–boost converter (buck type) Unidirectional boost converter and two bidirectional buck– [46] boost converters (boost type) three bidirectional buck–boost converters (boost type)

[47]

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

voltage-mode control, and the second one is designing a currentmode control. This strategy consists of two nested control loops with an inductor current (inner loop) controller and a capacitor voltage (outer loop) controller. It is well known that a controlled current mode has several advantages over a voltage mode switching power supply, such as: • Higher reliability with fast cycle-by-cycle current sensing for output short circuit and overload protection. A voltage-mode control reacts slower to an overcurrent condition, which can result in a failure in some applications. This is due to the fact that the capacitor voltage has a slow dynamics compared to the inductor current. This feature justifies the selection of the current as the inner variable. • Easy and accurate current sharing. This sharing allows a parallel connection of several converters, which provides redundancy in high-reliability systems. Sharing the current equally minimizes the dynamic response required from each converter, which results in less disruption to the output and allows us to optimize the converters lifetime by means of hot spots mitigation. From the previous discussion, in Section 10.3 of this chapter, we focus on the design of four different digital current controllers to be implemented in the buck and boost converters previously modeled in Section 10.3. We provide and compare simulation results between different current mode controllers for each converter (buck or boost) in Section 10.4. These current control techniques of classic DC–DC converters can be part of a much more complex system, for example, of an electric vehicle powertrain. It is important to highlight that for each of the proposed controls, we will give the most detailed information that allows its softwarein-the-loop testing in order that all results could be easily reproduced. Finally, Section 10.5 provides a chapter summary.

10.2 Analysis of the DC–DC power converters In this section, we give the dynamic models of classic DC–DC power converters. Many studies in the literature present modeling techniques in DC–DC power converters, such as discrete mapping method [48] and the small-signal method, which describe the system around a given operating point [49], and the Euler–Lagrange modeling technique [50]. The most common DC–DC converters model is taking into account the steady-state pulse-width modulation (PWM) switched behavior [51]. In this section, we also present the second-order buck and boost converters. Analysis will be done by generating a model to obtain the proposed control

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techniques, and these can be extended to applications of EVs. The methodology to obtain the models is applying the fundamental Kirchhoff laws and defining the differential equations of the schematic circuit; then the model is interpreted as an average model, where the switch position function is the binary set {0, 1} [52]. In the presentation of each converter, we use an average model and obtain expressions for current and voltage ripples.

10.2.1 Buck converter model

Figure 10.7. Schematic of buck converter.

One of the studied systems in this section is the buck converter. The schematic circuit diagram of the buck converter is shown in Fig. 10.7. The buck converter is a type of attenuation circuit. It behaves as a step-down converter because the input voltage Vg is greater than the output voltage vo [53]. The converter is composed by an inductor L, a capacitor C at the output, and a resistive load RL connected to the output. Applying the Kirchoff current and voltage laws to the schematic circuit shown in Fig. 10.7, we get the average model of the DC–DC buck converter: diL (t) Vg u − vo = , dt L dvo (t) −vo iL = + , dt RL C C

(10.1) (10.2)

where iL is the inductor current, vo is the output voltage, and u corresponds to the input variable control taking values in the set {0, 1}. The equilibrium point associated with the input is u¯ =

v¯o . v¯g

(10.3)

For the design of the buck converter, the current ripple in the inductor L and the output voltage ripple in the capacitor C are used as the design criterion. The inductance and capacitance values can be calculated following the expressions of the current and

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

voltage ripples defining the values desired for iL and Vo . The expression for the output voltage and inductor current ripple are iL =

Vg − Vo DT , L

(10.4)

IL T . 8C

(10.5)

Vo =

The switching period is T , and D is the duty cycle for nominal values of input and output voltages.

10.2.2 Boost converter model

Figure 10.8. Schematic of boost converter.

The circuit diagram of the boost converter, also known as the stepup converter, is shown in Fig. 10.8. In this converter the output voltage vo is greater than the input voltage Vg [53]. The DC– DC boost converter is composed of an inductor L at the input, a capacitor C at the output, and a resistive load RL in the output. To obtain the dynamics of the boost converter, we apply the Kirchoff current and voltages laws to the schematic circuit shown in Fig. 10.8. Then we obtain the following system of differential equations: diL (t) Vg − (1 − u)vo = , dt L −vo (1 − u)iL dvo (t) = + , dt RL C C

(10.6) (10.7)

where iL is the inductor current, vo is the output voltage, and u ∈ {0, 1} is the control variable. The equilibrium point associated with the input for the boost converter is u¯ = 1 −

v¯g . v¯o

(10.8)

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To design a boost convert for continuous-current operation, it is useful to obtain the inductance value in terms of the desired current ripple: L=

Vg DT , iL

(10.9)

and the capacitance can be expressed in terms of the output voltage ripple: C=

DT . RL (Vo /Vo )

(10.10)

10.3 Digital current control strategies Once we obtained the average model of the converters, we introduce the main designs for linear and nonlinear methods in digital current control of power electronics converters. The techniques described in this section are: the average current control based on passivity, sliding-mode current control, proportionalintegral current control, and predictive digital current programmed control. The passivity-based control was introduced in [50] for mechanical, electrical, and electromechanical applications. The aim of this control is to allow the passivity of the closed-loop system using the Hamiltonian structure. The sliding-mode control is a discontinuous feedback control technique. This strategy is used for the regulation of switched controlled systems [52] and has been applied in the power electronics field in [54]. The proportional-integral (PI) current digital control represents the discrete-time analog of the PI regulator, which applies the backward Euler method to get the equations of a discrete-time PI in additive form and is useful for digital controls [51]. The average current programmed predictive control is introduced in [55] for three basic nonisolated DC–DC converter topologies.

10.3.1 Average current control based on passivity In this section, we present the design of a control based on this nonlinear technique for the systems under study. We apply the current control based on passivity using the exact regulation error passive output feedback in power electronics devices; this technique was introduced in [52]. The system can be written in the following Hamiltonian form: De˙ = J

∂H (e) ∂H (e) −R + beu , ∂e ∂e

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

ey = b T

∂H (e) , ∂e

where D is a positive definite diagonal matrix, J is a skewsymmetric matrix, and R is a symmetric positive semidefinite, matrix. The matrix J represents the linear terms of the variable control u, R represents the dissipative terms in the system, and the expression beu is the energy term to control the system. The exact error regulation passive output of the system is denoted by ey [56]. Defining the error as e = x − x, ¯ where x¯ is the desired equilibrium of the average states, the state vector for the systems under study is x = [iL , vo ]T , and the error of the input is eu = u − u, ¯ with the corresponding equilibrium input u. ¯ As H = 0.5x T x, the definition of the error is x − x¯ =

¯ ∂H (e) ∂H (x) ∂H (x) − = . ∂x ∂x ∂e

(10.11)

The matrices representing the model of the form (10.11) for system (10.1)–(10.2) are     L 0 0 −1 D= , J= , 0 C 1 0 ⎛ ⎞   0 0 Vg ⎝ ⎠ , (10.12) R= , b= 1 0 0 RL ey =



Vg e1



(10.13)

.

The incremental control input is eu = −γ ey , and therefore u = −γ ey + u, ¯ where γ is a positive scalar quantity. Consequently, the variable control u for the buck converter is   v¯o . (10.14) u = −γ Vg e1 + v¯g The matrices that represent the dynamics of the exact regulation error for system (10.6)–(10.7) are     L 0 0 −(1 − u) D= , J= , 0 C 1−u −1 ⎛ ⎞   0 0 vo R=⎝ , (10.15) b= 1 ⎠, −iL 0 RL ey =



v¯o e1 − i¯L e2

 .

(10.16)

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The control law for the boost converter is   v¯ u = −γ [v¯o e1 − i¯L e2 ] + 1 − g v¯o

(10.17)

with e1 = iL − i¯L and e2 = vo − v¯o .

10.3.2 Discrete-time sliding-mode current control In this section, we present the design of a discrete-time slidingmode current control (DSMCC) for the DC/DC converters studied with fixed frequency. This control has been implemented for switching systems in [57–59]. In this strategy the variable control u[n] is computed in the nth time sample period to ensure the control surface (10.18) to be reached in the next sampling period (fsamp = fs ): s[n] = iLref [n − 1] − iL [n].

(10.18)

By using (10.1) and (10.6) the inductor current waveform slopes of converters are listed in Table 10.2. The Euler approximation leads to the following discrete-time inductor current expression, taking into account the averaged model of the converter inductor current L [n] : slope didtL ≈ iL [n+1]−i T iL [n + 1] = iL [n] + T (m1 + m2 )d[n] − m2 T ,

(10.19)

where T is the switching or sampling period. Hence the resulting expression of the duty cycle is d[n] =

m2 1 e[n] + , (m1 + m2 )T m1 + m2

(10.20)

where e[n] = iLref [n] − iL [n] in (10.20) with iLref [n] = iL [n + 1]. Using the expressions for m1 and −m2 for the output current slopes from Table 10.2 in (10.20) and with the control variable u[n] = d[n], the control law for the buck converter is given by u[n] =

L vo [n] e[n] + , vg [n]T vg [n]

(10.21)

and the control law for the boost converter is u[n] =

vg [n] L e[n] + 1 − . vo [n]T vo [n]

(10.22)

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

Table 10.2 Slopes of the inductor current waveforms. Converter m1 Buck Boost

vg − vo L vg L

−m2 vo L vg − vo L

10.3.3 Digital proportional-integral current control The following PI controller can guarantee the stability and convergence to the desired operating point in the Laplace domain: Gipi (s) = Kpi +

Kii , s

(10.23)

where the gains Kip and Kii are determined to regulate the inductor current of the converter toward the desired average current value taken as a constant reference signal. The transfer function from the input variable control to the current inductor in the Laplace domain is obtained for the systems from the differential Eqs. (10.1) and (10.6). Then the transfer function for the buck converter is Vg /L , (10.24) Giu (s) = s and for the boost converter, it is Giu (s) =

vo /L . s

(10.25)

Following the forward Euler method, we represent the PI controller in the z domain as follows: Gipi (z) = Kpi +

Kii T . z−1

(10.26)

We use the forward Euler method to find the recurrence equation for the discrete-time integral PI control [51]: up [n] = Kp e[n], ui [n] = Ki T e[n] + ui [n − 1], u[n] = up [n] + ui [n],

(10.27)

where e[n] is the error from the current reference e[n] = iLref [n] − iL [n]. The coefficients Kp and Ki can be obtained taking into ac-

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count the transfer function of systems (10.24) and (10.25). For the buck converter, the gains are L , Vg T L , Ki = Kii Vg T

Kp = Kpi

(10.28) (10.29)

and for the boost converter, the gains are defined as L , v¯o T L . Ki = Kii v¯o T

Kp = Kpi

(10.30) (10.31)

The value of the output voltage is the desired average value v¯o . Since Kii = Kpi /Ti , Ti is the integral time constant. The location of the PI zero should be lower than the switching frequency fs (1/(2πTi ) < fs ).

10.3.4 Predictive digital current programmed control The predictive digital current programmed control (PACC) is presented in [55,60]. The law control for this technique can be presented taking into account the slopes for inductor current of the converters seen in Table 10.2: u[n + 1] = −u[n] +

1 m2 e[n] + 2 . (m1 + m2 )T m1 + m2

(10.32)

By the expressions for m1 and m2 in Table 10.2 for buck converter, the law is u[n + 1] = −u[n] +

L vo [n] e[n] + 2 , vg [n]T vg [n]

and the control law for the boost converter is   vg [n] L u[n + 1] = −u[n] + e[n] + 2 1 − . vo [n]T vo [n]

(10.33)

(10.34)

10.4 Simulation results The values of capacitors and inductors of the converters have been selected following the design parameters in Tables 10.3 and 10.4 for buck and boost converters, respectively. The values are

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

Table 10.3 Design buck criteria. Parameter

Value

Input voltage Vg Output voltage Vo Switching frequency fs Inductor current ripple iL Output voltage ripple vo

24 V 12 V 100 kHz 1.2 A 0.2

Table 10.4 Design boost criteria. Parameter

Value

Input voltage Vg Output voltage Vo Switching frequency fs Inductor current ripple iL Output voltage ripple vo

12 V 24 V 100 kHz 2.5 A 0.02

calculated from e Eqs. (10.4) and (10.5) for the buck converter, and from (10.9) and (10.10) for the boost converter. Simulations were carried out using PSIM. Fig. 10.9 is the diagram implemented in PSIM for the buck converter, and Fig. 10.10 for the boost converter. The simulation control of the clock block is configured with the parameter values shown in Fig. 10.11.

Figure 10.9. Schematic of buck converter using PSIM.

The switch signal is generated using a ramp signal with a peak-topeak voltage equal to 1, frequency 100 kHz, and duty cycle 0.5, as shown in Fig. 10.12.

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Figure 10.10. Schematic of boost converter using PSIM.

Figure 10.11. Parameters of the simulation control in PSIM.

Figure 10.12. Schematic of switch signal generation using PSIM.

The C simplified block shown in Fig. 10.13 is used to activate the switch of the converters and has a start signal input, which is a step voltage source to activate the pulses of the switch with the duty cycle desired. The step voltage source has a Vstep=1 and Tstep=1 ms. The reference signal (ref ) is generated using a squarewave voltage source with the following parameters: Vpeak a peak of 3, frequency 580, duty cycle 0.5, DC offset 3, T star 0, and phase delay −1. The signal samples are the output voltage vo , the input

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

Figure 10.13. Simplified C block in PSIM.

voltage vg , and the inductor current iL . The C code for the C simplified block (Fig. 10.13) is implemented to observe the current inductor waveform of Figs. 10.14(a) and 10.14(b) in open loop for boost and buck converters and is shown in Box 10.1. Box 10.1 Open-loop code // signal declaration static long n; // counter period static float Ts=10e-6; //switching period static float u=1, d=1; // variable control and duty cycle static float iL=0; // inductor current sample static float vg=1; // input voltage sample static float vo=1; // output voltage static float iLref=0, voref=1, vgref=1; // reference signals static short DIS=1; // disable driver variable static short Puls=0; // indicator static short start=0; // start signal // Sample variables iL=x1; vo=x2; vg=x3; iLref=x4; start=x5; if(start && Puls==0) { DIS=0; Puls=1; } if (t>=n*Ts+300e-9) { if( DIS==0) { d=0.5; // Duty cycle } } // Output signals y1=d; y2=DIS;

The values selected for the boost converter are L = 25 µH, C = 31.2 µF, and RL = 8 . Fig. 10.14(a) depicts the waveform for in-

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Figure 10.14. Waveform in open loop: (a) boost mode; (b) buck mode.

ductor current in boost mode with duty cycle D = 0.5, where the current ripples and the switching frequency are according to Table 10.4. The results in open loop have an inductor current of IL = 6 A. The load used in Fig. 10.10 for this simulation is a resistor of 8 , changing the output voltage source for the resistor. The element values for the buck converter are L = 50 µH, C = 6.3 µF, and RL = 2 . Fig. 10.14(b) shows the waveform for the buck mode with duty cycle D = 0.5, where the current ripples and the switching frequency have values according to Table 10.3. The results in open loop have an inductor current of IL = 6 A. The load used in Fig. 10.9 for this simulation is a resistor of 2 , changing the output voltage source for the resistor. The simulations for double loop are implemented using a proportional-integrator voltage control external loop, where the controller transfer function can be expressed in the z domain using the forward Euler method as follows: Gvpi (z) = Kpv +

Kiv , Tsamp

(10.35)

where Tsamp = 1/fsamp . The forward Euler method is used to find the recurrence equation for the discrete-time integral PI control iLp [n] = Kpv ev [n], iLi [n] = Kiv Tsamp ev [n] + iLi [n − 1], iLref [n] = iLp [n] + iLi [n],

(10.36)

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

where Kpv = C2πfc , Kpv Kiv = . Ti

(10.37) (10.38)

The value of the crossover frequency (CF) for the voltage loop fc should be lower than the CF for the current loop. Hence fc = 5000 Hz was selected for the voltage feedback loop. The location of the PI zero should be lower than fc (1/(2πTi ) < fc ), whereby Ti = 3.18 × 10−4 s was selected.

10.4.1 Average current control based on passivity simulation results The technique based on passivity for the buck converter of Fig. 10.7 and the boost converter of Fig. 10.8 was carried out implementing the law (10.14) for the buck converter and the law (10.17) for the boost converter with parameter γ = 0.005. The C codes to obtain the results for buck and boost converters are shown in Box 10.2 and Box 10.3, respectively. Box 10.2 Passivity buck code // signal declaration static long n; // counter period static float Ts=10e-6; //switching period static float u=1, d=1; // variable control and duty cycle static float iL=0; // inductor current sample static float vg=1; // input voltage sample static float vo=1; // output voltage static float iLref=0, voref=12, vgref=24; // reference signals static float de=5e-3, e1=0, e2=0; // parameter control, //current error, voltage error static short DIS=1; // disable driver variable static short Puls=0; // indicator static short start=0; // start signal

start=x5; // External variable start if(start && Puls==0) { DIS=0; Puls=1; } if (t>=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; iLref=x4; n=n+1;

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if( DIS==0) { // Control ON e1=iL-iLref; // Control law for buck u=-de*(vgref*e1)+(voref/vgref); d=u; // duty cycle } } // Output signals y1=d; y2=DIS;

Box 10.3 Passivity boost code // signal declaration static long n; // counter period static float Ts=10e-6; //switching period static float u=1, d=1; // variable control and duty cycle static float iL=0; // inductor current sample static float vg=1; // input voltage sample static float vo=1; // output voltage static float iLref=0, voref=24, vgref=12; // reference signals static float de=5e-3, e1=0, e2=0; //parameter control, // current error, voltage error static short DIS=1; // disable driver variable static short Puls=0; // indicator static short start=0; // start signal start=x5; // External variable start if(start && Puls==0) { DIS=0; Puls=1; } if (t>=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; iLref=x4; n=n+1; if( DIS==0) { // Control ON e1=iL-iLref; e2=vo-voref; // Control law for boost u=-de*(voref*e1-iLref*e2)+(1-vgref/voref); d=u; // duty cycle } } // Output signals y1=d; y2=DIS;

Figs. 10.15 and 10.16 display the simulated waveforms for the averaged current control based on passivity for the buck and boost

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

Figure 10.15. Transient responses of the buck converter under passivity current control from 6 A to 3 A and from 6 A to 3 A.

Figure 10.16. Transient responses of the boost converter under passivity current control from 6 A to 3 A and from 6 A to 3 A.

converters, respectively, when the inductor current has ± 3 A step change.

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10.4.2 Discrete-time sliding-mode current control simulation results The simulations under DSMCC were carried out with the schematic shown in Fig. 10.13. For the buck converter, the C code of the simplified C block using the law (10.21) is shown in Box 10.4, and for the boost converter, the C code of the simplified C block using the law (10.22) is shown in Box 10.5. Box 10.4 DSMCC buck code // signal declaration static long n; // counter period static float Ts=10e-6; //switching period static float u=1, d=1; // variable control and duty cycle static float iL=0; // inductor current sample static float vg=1; // input voltage sample static float vo=1; // output voltage static float iLref=0, voref=12, vgref=24; // reference signals static float L=50e-6; // inductor value static float e1=0, e2=0; //current error, voltage error static short DIS=1; // disable driver variable static short Puls=0; // indicator static short start=0; // start signal start=x5; // External variable start if(start && Puls==0) { DIS=0; Puls=1; } if (t>=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; iLref=x4; n=n+1; if( DIS==0) { // Control ON e1=iLref-iL; // Control law for buck u=e1*(L/Ts)/vg+vo/vg; d=u; // duty cycle } } // Output signals y1=d; y2=DIS;

Box 10.5 DSMCC boost code // signal declaration static long n; // counter period static float Ts=10e-6; //switching period

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

static static static static static static static static static static

float float float float float float float short short short

u=1, d=1; // variable control and duty cycle iL=0; // inductor current sample vg=1; // input voltage sample vo=1; // output voltage iLref=0, voref=24, vgref=12; // reference signals L=25e-6; // inductor value e1=0, e2=0; //current error, voltage error DIS=1; // disable driver variable Puls=0; // indicator start=0; // start signal

start=x5; // External variable start

if(start && Puls==0) { DIS=0; Puls=1; } if (t>=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; iLref=x4; n=n+1; if( DIS==0) { // Control ON e1=iLref-iL; // Control law for boost u=e1*(L/Ts)/vo+ (1-vg/vo); d=u; // duty cycle } } // Output signals y1=d; y2=DIS;

Figs. 10.17 and 10.18 illustrate the simulated results for DSMCC for the buck and boost converters, respectively, when the inductor current has ± 3 A step change.

10.4.2.1 Double-loop DSMCC results The simulation using a proportional-integrator voltage control to obtain a reference current and allow us to change the output voltage is implemented with a step voltage source (Vstep=1, Tstep=30 ms) as reference input (x4) at the simplified C block in Fig. 10.13; this step source allows to activate the change voltage reference. The parameters of the simulation control of Fig. 10.11 are modified to observe the output voltage variation; since the external loop is slower than the current loop, the new parameters are: time step 62.5 n, total time 40 m, and print time 20 m. The gains Kpv and Kiv from Eqs. (10.37) and (10.38) are calculated taking into account the output capacitor value C for each converter.

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Figure 10.17. Transient responses of the buck converter under sliding current control from 6 A to 3 A and from 6 A to 3 A.

Figure 10.18. Transient responses of the boost converter under sliding current control from 6 A to 3 A and from 6 A to 3 A.

The output voltage source of Figs. 10.9 and 10.10 are changed by a resistor of 4 . The C code of the simplified C block for the buck

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

converter is shown in Box 10.6, and for the boost converter, it is shown in Box 10.7. Box 10.6 Double-loop DSMCC buck code // signal declaration static long n; // counter period static float Ts=10e-6; //switching period static float u=1, d=1; // variable control and duty cycle static float Kpv=0.2, Kiv=628, iLp=0, iLi=0; // gains of external loop static float iL=0; // inductor current sample static float vg=1; // input voltage sample static float vo=1; // output voltage static float iLref=0, voref=5, vgref=24; // reference signals static float L=50e-6; // inductor value static float e1=0, e2=0; //current error, voltage error static short DIS=1; // disable driver variable static short Puls=0; // indicator static short start=0; // start signal static short ref=0; // active reference change start=x5; // External variable start ref=x4; // Active reference change

if(start && Puls==0) { DIS=0; Puls=1; voref=5; } if (ref==1) { voref=12; } if (t>=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; n=n+1; // External loop e2=voref-vo; iLp=Kpv*e2; iLi=iLi+Kiv*Ts*e2; if (iLi>12) iLi=12; if (iLi=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; n=n+1; // External loop e2=voref-vo; iLp=Kpv*e2; iLi=iLi+Kiv*Ts*e2; if (iLi>12) iLi=12; if (iLi=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; iLref=x4; n=n+1; if( DIS==0) { // Control ON e1=iLref-iL; // Control law for buck up=Kpi*L/(Ts*vgref)*e1; ui=ui+KiiT*L/(Ts*vgref)*e1; u=up+ui; d=u; // duty cycle } } // Output signals y1=d; y2=DIS;

Box 10.9 PICC boost code // signal declaration static long n; // counter period static float Ts=10e-6; //switching period static float u=1, ui=1, up=1, d=1; // variable controls and duty cycle

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

static static static static static static static static static static

float float float float float float float short short short

Kpi=0.35, KiiT=0.08; //gains of control iL=0; // inductor current sample vg=1; // input voltage sample vo=1; // output voltage iLref=0, voref=24, vgref=12; // reference signals L=25e-6; // inductor value e1=0, e2=0; //current error, voltage error DIS=1; // disable driver variable Puls=0; // indicator start=0; // start signal

start=x5; // External variable start

if(start && Puls==0) { DIS=0; Puls=1; } if (t>=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; iLref=x4; n=n+1; if( DIS==0) { // Control ON e1=iLref-iL; // Control law for boost up=Kpi*L/(Ts*voref)*e1; ui=ui+KiiT*L/(Ts*voref)*e1; u=up+ui; d=u; // duty cycle } } // Output signals y1=d; y2=DIS;

Figs. 10.21 and 10.22 show the simulated results for PICC for a buck and boost converters, respectively, when the inductor current has ± 3 A step change.

10.4.3.1 Double-loop PICC results The simulation using a proportional-integrator voltage control to obtain a reference current for the PICC control is implemented with RL = 4 . The reference signal (ref ) is generated using a step voltage source (Vstep=1, Tstep=30 ms) to activate an output voltage change. The parameters of the simulation in Fig. 10.11 are modified as in Section 10.4.2.1. The gains of the external loop are calculated from Eqs. (10.37) and (10.38). The C codes for buck and boost converters are shown in Box 10.10 and Box 10.11, respectively.

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Figure 10.21. Transient responses of the buck converter under PI current control from 6 A to 3 A and from 6 A to 3 A.

Figure 10.22. Transient responses of the boost converter under PI current control from 6 A to 3 A and from 6 A to 3 A.

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

Box 10.10 Double-loop PICC buck code // signal declaration static long n; // counter period static float Ts=10e-6; //switching period static float u=1, up=0, ui=0, d=1; // variable control and duty cycle static float Kpv=0.2, Kiv=628, iLp=0, iLi=0; // gains of external loop static float Kpi=0.35, KiiT=0.08; // gains of control static float iL=0; // inductor current sample static float vg=1; // input voltage sample static float vo=1; // output voltage static float iLref=0, voref=5, vgref=24; // reference signals static float L=50e-6; // inductor value static float e1=0, e2=0; //current error, voltage error static short DIS=1; // disable driver variable static short Puls=0; // indicator static short ref=0; // active reference change static short start=0; // start signal start=x5; // External variable start ref=x4; // Active reference change

if(start && Puls==0) { DIS=0; Puls=1; voref=5; } if (ref==1) { voref=12; } if (t>=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; n=n+1; // External loop e2=voref-vo; iLp=Kpv*e2; iLi=iLi+Kiv*Ts*e2; if (iLi>12) iLi=12; if (iLi=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; n=n+1; // External loop e2=voref-vo; iLp=Kpv*e2; iLi=iLi+Kiv*Ts*e2; if (iLi>12) iLi=12; if (iLi=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; iLref=x4; n=n+1; if( DIS==0) { // Control ON e1=iLref-iL; // Control law for buck udc=2*(vo/vg); u=udc+usm-u; usm=e1*(L/Ts)/vg; d=u; // duty cycle } } // Output signals y1=d; y2=DIS;

Box 10.13 PACC boost code // signal declaration

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

static static static static static static static static static static static static

long n; // counter period float Ts=10e-6; //switching period float u=1, usm=0, udc=0, d=1; // variable control and duty cycle float iL=0; // inductor current sample float vg=1; // input voltage sample float vo=1; // output voltage float iLref=0, voref=24, vgref=12; // reference signals float L=25e-6; // inductor value float e1=0, e2=0; //current error, voltage error short DIS=1; // disable driver variable short Puls=0; // indicator short start=0; // start signal

start=x5; // External variable start

if(start && Puls==0) { DIS=0; Puls=1; } if (t>=n*Ts+300e-9) { // Sample variables iL=x1; vo=x2; vg=x3; iLref=x4; n=n+1; if( DIS==0) { // Control ON e1=iLref-iL; // Control law for boost udc=2*(1-vg/vo); u=(udc+usm-u); usm= e1*(L/Ts)/vo; d=u; // duty cycle } } // Output signals y1=d; y2=DIS;

Figs. 10.25 and 10.26 show the simulated performance for PACC for buck and boost converters, respectively, when the inductor current has ± 3 A step change.

10.5 Summary • We analyzed different hybrid electric typologies presented in the literature. Certainly, the buck and boost converters are very simple topologies, and they are widely used in many applications of EVs and especially in the design of powertrain topologies for fuel cell hybrid electric vehicles. • The common closed-loop methods for PWM DC–DC converters are the voltage-mode control and the current-mode con-

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Figure 10.25. Transient responses of the buck converter under PACC current control from 6 A to 3 A and from 6 A to 3 A.

Figure 10.26. Transient responses of the boost converter under PACC current control from 6 A to 3 A and from 6 A to 3 A.

trol. This strategy consists in two nested control loops with an inductor current (inner-loop) controller and a capacitor volt-

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

age (outer-loop) controller. The presented controlled current mode has many advantages over a voltage mode for switching power supply. • In the average small-signal model approach, the steady-state PWM switched behavior is taken into account. The model results allowed us to obtain linear and nonlinear strategies in digital current control of PWM DC–DC converters. • The digital current control strategies described have proved to be a powerful tool for inductor current regulation, and when an external voltage loop id added, represent a suitable alternative to extend its application. Current control strategies can be straightforwardly implemented by simulation programs written in C code, such as PLECS, PSIM, MATLAB , and so on.

Acknowledgments This work was supported by the Chilean Government under Project CONICYT/FONDECYT 1191680 and SERC Chile (CONICYT/FONDAP/15110019).

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11 Adaptive control for second-order DC–DC converters: PBC approach Walter Gil-Gonzáleza , Oscar Danilo Montoyab,c , and Gerardo Espinosa-Perezd a Facultad

de Ingeniería, Institución Universitaria Pascual Bravo, Medellín, Colombia. b Universidad Tecnológica de Bolívar, Laboratorio Inteligente de Energía, Cartagena, Colombia. c Universidad Distrital Francisco José de Caldas, Facultad de Ingeniería, Bogotá, D.C, Colombia. d Universidad Nacional Autónoma de México, Facultad de Ingeniería, CDMX, Mexico

Chapter points • Second-order DC–DC converters. • DC–DC converters modeling. • Port-controlled Hamiltonian. • Proportional–integral passivity-based control. • General representation for second-order DC–DC converters.

11.1 Introduction Electrical networks are currently being transformed by the large-scale integration of renewable energy sources, energy storage systems, and controllable loads [1]. These technologies must be integrated into the network via power electronic converters that interface them and allow achieving the control objective such as (i) active and reactive power tracking, (ii) voltage and frequency regulation, (iii) minimization of power oscillations, or (iv) stability enhancement, which are applicable to alternating current (AC) networks. Nevertheless, a classical paradigm regarding direct current (DC) networks has attracted attention in the last two decades [2], as numerous new-generation and renewable technologies operate in DC, for example, batteries [3], supercapacitors [4], superconductors [5], and photovoltaic sources [6]. Therefore electrical networks are considered as smart entities capable of manModeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00016-2 Copyright © 2021 Elsevier Inc. All rights reserved.

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aging their energy interchange to maximize the comfort of the users while optimally reducing greenhouse emissions and operating costs. A smart grid can be composed of AC or DC technologies or of a combination of both. Nevertheless, recent advances in DC analysis have demonstrated that these grids are more efficient than their AC counterparts, as they operate only with active power, and concepts such as reactive power or frequency are irrelevant. Thus DC networks have lower power losses and better voltage profiles than AC ones. As interfaces to DC networks, new-generation technologies and energy storage devices composed of power electronic converters are required [7]. They allow controlling the voltage and power outputs and regulating the voltage across all the nodes of the network [8]. Regarding control theory, power electronic converters form the first layer, which manages all the state variables into the grid [9], that is, they execute primary control and secondary control stages [8]. In addition, they are responsible for regulating the power interchange between the sources and the loads for optimizing the grid operation, that is, tertiary control applications [10]. In other words, power electronic converters are crucial components of the new distribution paradigm. Different converter technologies may be required in the configuration of a DC electrical network with multiple distributed energy resources and energy storage devices. Their selection depends on the requirements of the devices. For example, buck or boost converters can be used in the integration of photovoltaic sources, as the energy flow is always in one direction, that is, from the renewable source to the grid [11]. In contrast, buck-boost converters are used in the integration of interconnecting batteries and supercapacitors, as energy needs to be supplied to the energy storage device from the DC grid during some periods, whereas energy must be returned to the network during other periods, which indicates bidirectional power flow. When power electronic converters are employed for interfacing distributed energy resources in DC networks, controlling them for guaranteeing stable operation in closed-loop operation by minimizing the steady-state errors becomes an important problem. Multiple control methodologies have been proposed in specialized literature to solve these problems. We summarize some of them. The authors of [12] proposed a sliding-mode control for maximum power tracking in grid-connected photovoltaic sources using a boost converter configuration. The main advantage of using the sliding control theory is that the stability of the closed-loop system is guaranteed, and the control output is directly a switched

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

signal, which indicates that a pulse-width modulation stage is not required in this implementation. In [13] a robust passivitybased control design for regulating the output voltage considering linear loads was presented by integrating wind turbines with permanent magnet synchronous machines connected with diode rectifiers. The passivity-based design allows guaranteeing closedloop stability in the Lyapunov sense by using a Hamiltonian representation of the system to determine the control law. In [14] a proportional–integral (PI) passivity-based controller (PBC) was developed for a buck converter including linear loads. Its main advantage is parametric independence regarding LC filter values. Similar approaches based on passivity-based control were reported in [15] and [16] considering resistive and constant-power loads, respectively. The authors of [17] presented a stability analysis for noninverting buck-boost converters via classical PI control. In this approach the authors ensured a correct closed-loop operation for linear and nonlinear loads. Additional approaches for controlling DC–DC converters include the extended-feedback PI control [18], classical feedback control [19], fuzzy logic control [20], artificial neural networks [21], and fractional proportional– integral–derivative (PID) design [22]. The brief review of the state-of-the-art approaches presented above confirms that there are multiple methodologies for addressing the problem of control in DC–DC converters. Furthermore, this is a topic under current development. This scenario, together with the increasing importance of DC networks for high- to lowvoltage applications, motivates this book chapter. In this sense the main contributions are summarized as follows: • The formulation of the dynamic models for four different second-order DC–DC converters (boost, buck, buck-boost, and noninverting buck-boost configurations) with a unique general port-controlled Hamiltonian (PCH) structure. • The application of the classical passivity-based control with a PI structure that allows achieving a generalized control law that stabilizes the voltage output in all the converters at their references as reported by Cisneros et al. [27]. • The implementation of a load estimator (for linear resistive loads) with a PI control structure that allows reducing the number of measurements during the system implementation. Resistive loads are selected, as they allow formulating the general control problem via PI-PBC due to the bilinear structure of the resulting dynamic models. Regarding the scope of this contribution, the main idea of this chapter is providing a generalized control law for second-order DC–DC converters by using the PI-PBC control reported in [29].

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Therefore we do not propose a new version of PBC control but apply the PI-PBC design, which guarantees the stability of the general PCH models for the second-order DC–DC converters in closed-loop operation. Notably, the proposed control design has a simple structure, as it includes only a PI action. This indicates that its tuning facilitates the main advantage that the asymptotic stability is guaranteed in closed-loop operation [23]. In addition, the review of the state-ofthe-art approaches did not reveal a similar approach that generalizes the four models of the second-order converters via a PCH representation. This approach, in addition to the proposed bilinear passivity-based control design, is the main contribution of this study to the current literature regarding the control of power electronic converters. The remainder of this chapter is organized as follows. In Section 11.2, we present the dynamic models for each of the four second-order converter structures. In Section 11.3, we show the general control design via passivity-based control for bilinear PCH models by demonstrating via Lyapunov theory that a simple proportional control applied over the passive output of the system allows guaranteeing the asymptotic stability. In Section 11.4, we develop a compact formulation of the four dynamic models of the converter by introducing some constants. In addition, we present a general control law in conjunction with the steady-state operative values and design the linear load estimator via conductance representation. In Section 11.5, we report the numerical validation of the general control design by regulating the voltage provided to a linear load with each converter structure. This section confirms that under well-defined load conditions, proportional PBC allows tracking the desired voltage reference with a minimum steadystate error and good response to step variations. Finally, in Section 11.6, we present the main conclusions derived from this study and discuss future works.

11.2 DC–DC converter modeling In this section we derive dynamic models of DC–DC converters. We only focus on the most widely used topologies in electrical networks for this type of converters, which are the buck, boost, buck-boost, and noninverting buck-boost converters [24]. This group of converters is classified as second-order converters, as they have only two dynamics corresponding to an inductor and a capacitor [9]. Fig. 11.1 illustrates all the second-order converters considering the case of a constant resistive load. This load is represented by its conductance value GL . The DC–DC converter mod-

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

els will be developed by applying the fundamental Kirchhoff laws to the DC–DC converter schemes. This allows obtaining differential equations of the schematic circuits represented with their average models [24]. We can also observe in Fig. 11.1 that each converter is made up of an inductor L, capacitor C, and constant resistance load connected in parallel to the capacitor. To achieve a dynamic model of the DC–DC converters, we make the following assumptions: Assumption 11.1. The power losses in the forced commuted switch and the diode are neglected. Assumption 11.2. The input voltage E, state variables i (current through the inductor), and voltage across the capacitor vc are measurable. Assumption 11.3. The parameters of the converters, that is, the inductance L and capacitance C, are positive definite. Assumption 11.4. The value of the constant resistor load, modmax eled as GL , is bounded and positive, that is, Gmin L ≤ GL ≤ GL , max where 0 < Gmin L ≤ GL . Assumption 11.5. The switching frequency is sufficiently high to employ the average modeling theory. Therefore the continuous dynamic model of the DC–DC converters is obtained. This indicates that the control input d (duty cycle) will be continued and contained in the closed interval from 0 to 1.

11.2.1 Buck converter The buck converter depicted in Fig. 11.1(a) is also known as a step-down converter, as the output voltage is stepped down from its input voltage, whereas the output current is stepped up. The main application of this type of converter is the regulation of its output voltage under load variations. Applying Kirchhoff’s laws to the buck converter (see Fig. 11.1(a)), its average dynamic model is obtained as di = −vc + dE, dt dvc C = i − GL vc , dt L

(11.1) (11.2)

where i is the inductor current, vc is the capacitor voltage (output voltage), and d ∈ [0, 1] is the control variable (duty cycle).

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Figure 11.1 Second-order DC–DC converters. (a) Buck converter, (b) boost converter, (c) buck-boost converter, and (d) noninverting buck-boost converter.

11.2.2 Boost converter The boost converter shown in Fig. 11.1(b) is also known as a step-up converter, as the output voltage is stepped-up from its input voltage. Applying Kirchhoff’s laws to this converter, its average

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

dynamic model is obtained as di = −(1 − d)vc + E, dt dvc C = (1 − d)i − GL vc , dt L

(11.3) (11.4)

where the variables and parameters were previously defined.

11.2.3 Buck-boost converter The buck-boost converter (see Fig. 11.1(c)) allows the magnitude of the output voltage to be higher or lower than the magnitude of the input voltage. However, the output voltage has the opposite polarity to the input voltage. The buck-boost converter has a similar circuit topology to the boost and buck converters. With the same variables and parameters defined before, Kirchhoff’s laws are applied to the buck-boost converter to obtain its average dynamic model as follows: di = (1 − d)vc + dE, dt dvc = −(1 − d)i − GL vc . C dt L

(11.5) (11.6)

11.2.4 Noninverting buck-boost converter The noninverting buck-boost converter shown in Fig. 11.1(d) is composed of a buck converter combined with a boost converter. The noninverting buck-boost converter employs an inductor that operates in both modes of operation (buck and boost modes), using switches instead of diodes [24]. In contrast to the buck-boost converter described in Section 11.2.3, the output voltage of the noninverting buck-boost converter maintains the same polarity as the input voltage, and its magnitude can be lower or higher than that of the input voltage. Following a similar procedure as in the previous cases, the average dynamic model of the noninverting buck-boost converter is obtained as di = −(1 − d)vc + dE, dt dvc = (1 − d)i − GL vc . C dt L

(11.7) (11.8)

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11.3 Passivity-based control method Passivity-based control deals with nonlinear control applications considering energy storage functions. This control technique utilizes the PCH structure of some dynamic systems (power converters, electrical machines, or mechanical systems) to propose a closed-loop controller that preserves the PCH structure guaranteeing asymptotic stability in most cases [25]. Two PBC approaches are common in the literature, interconnection and damping assignment (IDA-PBC) [26], and PI-PBC [27]. The IDAPBC typically operates with nonlinear systems with affine structures (e.g., hydro and thermal power plants), that is, when the input variables are not contained in the interconnection matrix [28]. In contrast, the PI-PBC is ideal for nonaffine structures, especially with bilinear structures, such as those integrated by electronic power converters (e.g., renewable energy generations, energy storage devices, and DC–DC applications) [27]. The PBC method allows designing a control law using a feedback loop that stabilizes a nonlinear system given by x˙ = f (x) + g(x)u + ξ,

(11.9)

where x and ξ ∈ Rn represent the state vector and external input, respectively, g(x) ∈ Rn×m is the input matrix, and u ∈ Rm is the input control. We assume that the nonlinear system (11.9) can be rewritten as a bilinear PCH structure, defined as follows. Definition 11.1 (Bilinear system). A bilinear PCH structure for dynamic systems corresponds to a nonaffine mathematical form with the product of the control inputs and state variables, which has the following structure: Qx˙ = (J0 + J1 u − R) x + bu + ξ,

(11.10)

where Q = Q  0 ∈ Rn×n contains the parameters of the com ∈ Rn×n is the interponents with storing capabilities, J0,1 = −J0,1 connection matrix, R = R   0 ∈ Rn×n is the damping matrix, and b ∈ Rn×m is the input matrix. Note that if the control inputs u and external inputs ξ are bounded, then the bilinear PCH system (11.10) has an assignable equilibrium defined as follows. Definition 11.2 (Assignable equilibrium point). An assignable equilibrium point in (11.10) is uniquely established by the (con-

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

stant) vectors x  such that   0 = J0 + J1 u − R x  + bu + ξ, 0 = (J0 − R) x  + g(x  )u + ξ

(11.11)

for some constant u . In other words, the assignable equilibrium point is given by     x  ∈ Rn | g(x  )⊥ f (x  ) + ξ = 0 ,

(11.12)

where g(x  )⊥ is a full-rank left annihilator of g(x  ), that is, g(x  )⊥ g(x  ) = 0. The corresponding equilibrium control input u for the bilinear PCH system (11.10) is univocally determined by  −1   u = − g(x  ) g(x  ) g(x  ) f (x  ) + ξ .

(11.13)

−1  g(x  ) must be Remark 11.1. To apply (11.13), g(x  ) g(x  )  column full-rank for all x . An incremental passivity model can be obtained by combining the PCH system (11.10) and its assignable equilibrium (11.11) as follows. Definition 11.3 (Incremental passivity model). The incremental passivity model for the PCH system (11.10) can be defined as        Qx˙˜ = J0 + J1 u˜ + u x˜ + x  + b u˜ + u + ξ,

(11.14)

where (˜·) = (·) − (·) are the incremental variables Now combining (11.14) and (11.11), we obtain   ˜ Qx˙˜ = (J0 + J1 u − R) x˜ + g x  u.

(11.15)

The output function y˜ of the incremental dynamic model is y˜ = g(x  ) x. ˜

(11.16)

The incremental dynamic model (11.15) is passive from the input control u˜ to the output function y, ˜ with the following storage function S : Rn −→ R: 1 ˜ S(x) ˜ = x˜  Qx, 2

(11.17)

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

where its time derivative along the trajectories of (11.15) is ˙ x) S( ˜ = x˜  Qx˙˜     = x˜  (J0 + J1 u − R) x˜ + g x  u˜   = x˜  (J0 + J1 u − R) x˜ + x˜  g x  u˜   = −x˜  R x˜ + x˜  g x  u˜   ˜ ≤ x˜  g x  u.

(11.18)

Note that the storage function S satisfies the inequality ˙ x) S( ˜ ≤ u˜  y, ˜ ∀ t > 0,

(11.19)

with any input control u. ˜ Therefore the incremental dynamic model (11.15) is passive.

11.3.1 PI-PBC design The incremental dynamic model given by (11.14) reaches an admissible equilibrium point with the following input control [29]: u˜ = −Kp y˜ − Ki z, z˙ = y,

(11.20) (11.21)

where Kp > 0, Ki > 0, and z ∈ Rm is an auxiliary variable vector. The stability properties of x ∗ can be guaranteed considering the following Lyapunov function candidate: 1 1 W (x, ˜ z) = x˜  Qx˜ + z Ki z, 2 2

(11.22)

whose time derivative along the trajectories of (11.15) is W˙ (x, ˜ z) = x˜  Qx˙˜ + z Ki z˙ ˜ = −x˜  R x˜ + y˜  u˜ − y˜  (Kp y + u) 

(11.23)



= −x˜ R x˜ − y˜ Kp y˜ ≤ 0, showing the equilibrium point x˜ = 0 of the closed-loop (11.15)– (11.20). Moreover, the study [29] proved that x˜ = 0 is asymptotically stable.

11.4 Control design for DC–DC converters The aim of the control for the second-order DC–DC converters is maintaining a constant output voltage under load variations.

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

Table 11.1 General bilinear representation of the second-order DC–DC converters. Converter

α1

α2

α3

α4

Buck

1

0

1

0

Boost

1

1

0

1

−1

−1

1

0

1

1

1

0

Buck-boost Noninverting buck-boost

The control design is based on the PI-PBC method described in Section 11.3. The dynamic models for the second-order DC–DC converters should be represented as a bilinear system, as shown in (11.10), to implement this method. Accordingly, let us define the following matrices:  Q=  R=

L 0 0 C 0 0

0 GL



 ,

J0 =



 ,

J1 =

0 −1 1 0 0 1 −1 0



 ,

b=



 ,

ξ=

E 0 E 0

, (11.24)

,

and thus any second-order DC–DC converter modeled in Section 11.2 can be described by the following bilinear representation: Qx˙ = (α1 J0 + α2 J1 u − R)x + α3 bu + α4 ξ = (α1 J0 − R)x + g(x)u + α4 ξ

(11.25)

with g(x) = (α2 J1 x + α3 b),

(11.26)

where αk ∈ k = {1, 2, 3, 4} are constants presented in Table 11.1. Remark 11.2. The control input for each second-order converter is given by u = u˜ + u = −Kp y˜ − Ki z + u

(11.27)

when the PI controller is applied. Hence it is necessary to determine u and y˜ obtained by using (11.13) and (11.16), respectively.

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

Applying (11.13) and (11.16), the control input u and output function y˜ for the general bilinear representation of the secondorder DC–DC converters are given by u = −

(Eα3 + α2 vc )(Eα4 − α1 vc ) − α2 i  (α1 i  − GL vc ) , (Eα3 + α2 vc )2 + α22 (i  )2

y˜ = (Eα3 + α2 vc )(i − i  ) − α2 i  (vc − vc ).

(11.28) (11.29)

To implement the control law (11.27), it is necessary to know the noncontrolled variable i  , which can be computed if the left annihilator of g(x  ) is defined as g(x  )⊥ =



α2 i 

α2 vc + α3 E



(11.30)

by employing (11.12). Under these conditions, the noncontrolled variable is i =

GL vc (Eα3 + α2 vc ) . α1 (Eα3 + α2 vc ) + α2 (Eα4 − α1 vc )

(11.31)

Remark 11.3. The output voltage reference vc must be nonzero to ensure that no singularities are present for any combination of αk . Remark 11.4. Note that the control input presented in (11.27) depends on knowing the conductance value GL , which is typically not possible in real applications. Hence it is necessary to apply an estimation method, as will be presented in the following section.

11.4.1 Adaptive control using I&I conductance estimator To solve the problem that GL is not known in practice, we implement the immersion and invariance (I&I) technique developed in [30], which allows estimating the conductance value of the equivalent load. The parameter error is defined as ˆ L − GL , ˜L =G G

(11.32)

ˆ L are the estimation error and the estimation of ˜ L and G where G ˆ L is defined as GL , respectively; G ˆ L = β + Cγ α(vc ), G where γ > 0, and β and α(vc ) need to be designed.

(11.33)

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

˜ L with respect to time yields Differentiating G ˙ˆ = β˙ + Cγ α  (v )v˙ ˙˜ = G G L L c c

α1 GL α2  ˙ = β + Cγ α (vc ) x1 − v c − x1 d C C C  = β˙ + γ α (vc ) (α1 x1 − GL vc − α2 x1 d) .

(11.34)

Now GL in (11.34) is substituted by using (11.32) as follows:   ˙˜ = β˙ + γ α  (v ) α x − (G ˆL −G ˜ L )vc − α2 x1 d . G L c 1 1 This results in the definition   ˆ L vc − α2 x1 d , β˙ = −γ α  (vc ) α1 x1 − G

(11.35)

(11.36)

which by substituting (11.36) into (11.35) yields ˙˜ = γ α  (v )G G L c ˜ L vc

(11.37)

The design of the estimator is completed when an adequate function α(vc ) is selected to guarantee the stability of the dynam˜ L . Therefore α(vc ) is chosen as ics G 1 α(vc ) = − vc2 , 2

(11.38)

˙˜ = −γ v 2 G ˜ G L c L,

(11.39)

as this structure yields

˜ L (0), the exponential convergence is guaranteeing that, for all G achieved as ˜ L (0)e−γ vc2 t . ˜ L (t) = G G

(11.40)

11.5 Simulation results In this section, we present a numerical implementation of the proposed general control design for the four second-order DC–DC converter models. The main idea of these simulations is to regulate the output voltage for a linear load by considering variations in its total current consumption.

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

Table 11.2 Parameters of the second-order converters. E [V] L [µH] C [µF] GL1 [] GL2 [] vc [V]

Converter Buck Boost Buck-boost Noninverting buck-boost

24 12 15 24

50 25 200 30

6.36 31 20 50

1/2 1/8 1/8 1/8

1 1/4 1/4 1/4

12 24 −20 20

11.5.1 Test system The second-order converters have been numerically validated via MATLAB /Simulink software using a switching model with an operation frequency of 100 kHz. The parameters of the secondorder converters are listed in Table 11.2. The test system for each second-order converter is illustrated in Fig. 11.1.

11.5.2 Numerical validation We consider the following validation case to demonstrate the effectiveness and robustness of the proposed adaptive PI-PBC controller to regulate the output voltage of the second-order converter: • Initially, the system starts with GL1 connected (see Table 11.2). Assume that the capacitor voltage is the load in the reference value. • Subsequently, the second load GL2 is connected in parallel to GL1 at 10 ms. • Finally, the first load GL1 is disconnected at 15 ms. The settling time, steady-state error, and the integral of timeweighted absolute error (ITAE) are used to quantify the performance of the proposed adaptive controller. The ITAE is computed as follows:  tsim   (11.41) t vc − vc  dt, I T AE = 0

where tsim is the simulation time range.

11.5.2.1 Buck converter Fig. 11.2 displays the dynamic responses of the buck converter. Figs. 11.2(a) and 11.2(b) illustrate the output voltage and inductor current of the converter, respectively, whereas Figs. 11.2(c) and (d) show the duty cycle and estimated conductance value,

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

303

Figure 11.2 Dynamic responses of the buck converter. (a) Output voltage (vc ), (b) inductor current (i), (c) duty cycle (d), and (d) conductance value (GL ).

respectively. The controller and estimator gains were selected as Kp = 0.0020, Ki = 0.0001, and α = 500. In Fig. 11.2(a), we can observe that the output voltage of the buck converter is regulated with a steady-state error of approximately 1.32%. When the load changes, the controller has a settling time of approximately 0.5 ms with an ITAE of 0.16. We can observe in Fig. 11.2(b) that the inductor current presents a behavior similar to the load estimation (see Fig. 11.2(d)). This is because there is a direct relationship between the current

304

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

and conductance when a constant voltage is maintained. In Fig. 11.2(c), we can observe that the duty cycle remains mostly constant, and it only shows small changes in response to load variations to maintain the output voltage at its reference value. Fig. 11.2(d) shows that the adaptive control estimates the load adequately with a settling time of approximately 0.55 ms when load changes occur.

11.5.2.2 Boost converter Fig. 11.3 depicts the dynamic responses of the boost converter. Figs. 11.3(a) and 11.3(b) display the output voltage and inductor current of the converter, respectively, whereas Fig. 11.3(c) and 11.3(d) present the duty cycle and estimated conductance value, respectively. The controller and estimator gains were selected as Kp = 0.0099, Ki = 0.0009, and α = 20. In Fig. 11.3(a), we observe that the output voltage of the boost converter has been adequately regulated to 24 V. The proposed controller has a steady-state error of approximately 0.16%. The settling time is approximately 1.0 ms under load variations with an ITAE of 0.02. The duty cycle shows a behavior similar to that for the buck converter (see Fig. 11.3(b)). It remains constant and only presents small changes when load variations occur. In Fig. 11.3(d), we observe that the settling time for the estimation of the load is approximately 0.78 ms.

11.5.2.3 Buck-boost converter The dynamic responses of the buck-boost converter are displayed in Fig. 11.4. The output voltage and inductor current of the converter are presented in Figs. 11.4(a) and 11.4(b), respectively. The duty cycle and estimated conductance value are illustrated in Figs. 11.4(c) and 11.4(d), respectively. The controller and estimator gains were chosen as Kp = 0.0031, Ki = 0.0003, and α = 200. We can observe in Fig. 11.4(a) that the proposed adaptive controller regulates the output voltage of the buck-boost converter with a steady-state error of 0.20% under load variations. The settling time and ITAE are 1.12 ms and 0.02, respectively. The conductance load continues to be accurately estimated with a settling time of approximately 1 ms (see Fig. 11.4(d)).

11.5.2.4 Noninverting buck-boost converter The dynamic responses of the noninverting buck-boost converter are illustrated in Fig. 11.5. The dynamic responses of the output voltage and inductor current of the converter are shown in

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

305

Figure 11.3 Dynamic responses of the boost converter. (a) Output voltage (vc ), (b) inductor current (i), (c) duty cycle (d), and (d) conductance value (GL ).

Figs. 11.5(a) and 11.5(b), respectively. Figs. 11.5(c) and 11.5(d) depict the dynamic responses of the duty cycle and estimated conductance value, respectively. The controller and estimator gains were chosen as Kp = 0.015, Ki = 0.005, and α = 25. Fig. 11.5(a) shows that the output voltage of the noninverting buck-boost converter remains at its reference (vc = 20 V), even when the conductance values change, with a steady-state error of approximately 0.69%. The settling time and ITAE of the output voltage are 1.13 ms and 0.08, respectively.

306

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

Figure 11.4 Dynamic responses of the buck-boost converter. (a) Output voltage (vc ), (b) inductor current (i), (c) duty cycle (d), and (d) conductance value (GL ).

Fig. 11.5(d) shows that the adaptive control estimates the load adequately with a settling time of approximately 0.80 ms when load changes occur.

11.6 Conclusions In this chapter, we proposed a general formulation for secondorder DC–DC converters by using a compact PCH representation.

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

307

Figure 11.5 Dynamic responses of the noninverting buck-boost converter. (a) Output voltage (vc ), (b) inductor current (i), (c) duty cycle (d), and (d) conductance value (GL ).

The proposed formulation allows achieving a general and simple control law using passivity-based control theory with PI actions. The main advantage of this control design is that it guarantees the asymptotic stability in closed-loop operation in the Lyapunov sense. Numerical simulations confirmed that it is possible to regulate the output voltage in all the converter structures with negligible steady-state errors with the proposed general control for the incremental model and the desired control input for the admissible trajectory.

308

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

The immersion and invariance technique was implemented to estimate the linear consumption behavior of the load. This technique allows reducing the number of measurements in the system. In addition, this estimation does not compromise the stability behavior of the system, as it is exponentially stable. This indicates that the magnitude of the load in real applications is not required to be known to implement the proposed controller for regulating the output voltage profile. As a future work, a control design based on the general PCH formulation can be developed for second-order DC–DC converters with constant-power loads. This could be designed via an IDAPBC formulation, as it can deal with general nonlinear PCH structures.

Acknowledgments The work of Gerardo Espinosa-Perez was supported by DGAPAUNAM (IN118019).

References [1] P.D. Lund, J. Byrne, R. Haas, D. Flynn, Advances in Energy Systems: The Large-Scale Renewable Energy Integration Challenge, John Wiley and Sons, 2019. [2] B.T. Patterson, DC, come home: DC microgrids and the birth of the “enernet”, IEEE Power and Energy Magazine 10 (6) (2012) 60–69. [3] M.F. Zia, E. Elbouchikhi, M. Benbouzid, Optimal operational planning of scalable DC microgrid with demand response, islanding, and battery degradation cost considerations, Applied Energy 237 (2019) 695–707. [4] M. Ayad, M. Becherif, A. Henni, A. Aboubou, M. Wack, S. Laghrouche, Passivity-based control applied to DC hybrid power source using fuel cell and supercapacitors, Energy Conversion and Management 51 (7) (2010) 1468–1475. [5] B. Yang, T. Zhu, X. Zhang, J. Wang, H. Shu, S. Li, T. He, L. Yang, T. Yu, Design and implementation of battery/SMES hybrid energy storage systems used in electric vehicles: a nonlinear robust fractional-order control approach, Energy 191 (2020) 116510. [6] W. Gil-González, A. Garces, O.D. Montoya, Current PI control for PV systems in DC microgrids: a PBC design, in: 2019 IEEE Workshop on Power Electronics and Power Quality Applications (PEPQA), IEEE, 2019, pp. 1–5. [7] D. Jovcic, L. Zhang, LCL DC/DC converter for Dc grids, IEEE Transactions on Power Delivery 28 (4) (2013) 2071–2079. [8] C. Jin, P. Wang, J. Xiao, Y. Tang, F.H. Choo, Implementation of hierarchical control in DC microgrids, IEEE Transactions on Industrial Electronics 61 (8) (2013) 4032–4042. [9] T. Dragiˇcevi´c, X. Lu, J.C. Vasquez, J.M. Guerrero, DC microgrids Part I: a review of control strategies and stabilization techniques, IEEE Transactions on Power Electronics 31 (7) (2015) 4876–4891.

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

[10] A. Garcés, Convex optimization for the optimal power flow on DC distribution systems, in: Handbook of Optimization in Electric Power Distribution Systems, Springer, 2020, pp. 121–137. [11] O.D. Montoya, A. Garcés, I. Ortega, G.R. Espinosa, Passivity-based control for battery charging/discharging applications by using a buck-boost DC-DC converter, in: 2018 IEEE Green Technologies Conference (GreenTech), 2018, pp. 89–94. [12] P. Valencia, C. Ramos-Paja, Sliding-mode controller for maximum power point tracking in grid-connected photovoltaic systems, Energies 8 (11) (2015) 12363–12387, https://doi.org/10.3390/en81112318. [13] S.K. Kim, Passivity-based robust output voltage tracking control of DC/DC boost converter for wind power systems, Energies 11 (6) (2018) 1469, https:// doi.org/10.3390/en11061469. [14] W.J. Gil-González, O.D. Montoya, A. Garces, F.M. Serra, G. Magaldi, Output voltage regulation for DC DC buck converters: a passivity-based PI design, in: 2019 IEEE 10th Latin American Symposium on Circuits Systems (LASCAS), 2019, pp. 189–192. [15] O.D. Montoya, W. Gil-González, F.M. Serra, G. Magaldi, PBC approach applied on a DC–DC step-down converter for providing service to CPLs, in: 2019 IEEE 4th Colombian Conference on Automatic Control (CCAC), 2019, pp. 1–6. [16] O.D. Montoya, J.L. Villa, W. Gil-González, PBC design for voltage regulation in buck converters with parametric uncertainties, in: 2019 IEEE 4th Colombian Conference on Automatic Control (CCAC), 2019, pp. 1–6. [17] Z. Chen, J. Hu, W. Gao, Closed-loop analysis and control of a non-inverting buck-boost converter, International Journal of Control 83 (11) (2010) 2294–2307, https://doi.org/10.1080/00207179.2010.520030. [18] H. Sira-Ramirez, Design of P-I controllers for DC-to-DC power supplies via extended linearization, International Journal of Control 51 (3) (1990) 601–620, https://doi.org/10.1080/00207179008934087. [19] K. Sundareswaran, V. Devi, S.K. Nadeem, V.T. Sreedevi, S. Palani, Buck-boost converter feedback controller design via evolutionary search, International Journal of Electronics 97 (11) (2010) 1317–1327, https:// doi.org/10.1080/00207217.2010.488904. [20] M.E. S¸ ahin, H˙I Okumu¸s, Comparison of different controllers and stability analysis for photovoltaic powered buck-boost DC–DC converter, Electric Power Components and Systems 46 (2) (2018) 149–161, https:// doi.org/10.1080/15325008.2018.1436617. [21] F. Kurokawa, K. Ueno, H. Maruta, H. Osuga, A new control method for DC–DC converter by neural network predictor with repetitive training, in: 2011 10th International Conference on Machine Learning and Applications and Workshops, IEEE, 2011. [22] A.G. Soriano-Sánchez, M.A. Rodríguez-Licea, F.J. Pérez-Pinal, J.A. Vázquez-López, Fractional-order approximation and synthesis of a PID controller for a buck converter, Energies 13 (3) (2020) 629, https:// doi.org/10.3390/en13030629. [23] O.D. Montoya, W. Gil-González, A. Garces, Distributed energy resources integration in single-phase microgrids: an application of IDA-PBC and PI-PBC approaches, International Journal of Electrical Power & Energy Systems 112 (2019) 221–231, https://doi.org/10.1016/j.ijepes.2019.04.046. [24] H.J. Sira-Ramirez, R. Silva-Ortigoza, Control Design Techniques in Power Electronics Devices, Springer Science and Business Media, 2006.

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[25] R. Ortega, J.A.L. Perez, P.J. Nicklasson, H.J. Sira-Ramirez, Passivity-Based Control of Euler–Lagrange Systems: Mechanical, Electrical and Electromechanical Applications, Springer Science and Business Media, 2013. [26] A.J. Van der Schaft, A. Van Der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control, vol. 3, Springer, 2017. [27] R. Cisneros, M. Pirro, G. Bergna, R. Ortega, G. Ippoliti, M. Molinas, Global tracking passivity-based PI control of bilinear systems: application to the interleaved boost and modular multilevel converters, Control Engineering Practice 43 (2015) 109–119. [28] J. Linares-Flores, J.L. Barahona-Avalos, C.A. Bautista-Espinosa, Passivity-based controller and online algebraic estimation of the load parameter of the DC-to-DC power converter Cuk type, IEEE Latin America Transactions 9 (1) (2011) 784–791, https:// doi.org/10.1109/TLA.2011.5876420. [29] M. Hernandez-Gomez, R. Ortega, F. Lamnabhi-Lagarrigue, G. Escobar, Adaptive PI stabilization of switched power converters, IEEE Transactions on Control Systems Technology 18 (3) (2010) 688–698. [30] A. Astolfi, D. Karagiannis, R. Ortega, Nonlinear and Adaptive Control with Applications, Springer Science and Business Media, 2007.

12 Advances in predictive control of DC microgrids Ariel Villalóna , Marco Riverab , and Javier Muñozb a Engineering Systems PhD. Program, Faculty of Engineering, Universidad de Talca, Campus Curicó, Chile. b Department of Electrical Engineering, Faculty of Engineering, Universidad de Talca, Campus Curicó, Chile

Chapter points • In Section 12.1, the introduction to DC microgrid concept, their control aspects, their components and predictive control schemes applied to them are given. Hierarchical control concept is briefly explained. • In Section 12.2, different predictive control techniques applied to DC microgrids are surveyed. In subsection 12.2.1 the primary control techniques are explained: finite control set model predictive control, modulated model predictive control, decentralized model predictive control, and the hybrid finite control set model predictive control/deadbeat predictive control scheme. • Also in subsection 12.2.2 the secondary control techniques are explained where model predictive-based self-adaptive inertia control and centralized model predictive control techniques are explained. • The concluding remarks of this chapter are given in Section 12.3.

12.1 Introduction Generally, electrical power systems have been developed considering transmission from the generation source to the demand source or loads; this configuration is seen as an unilinear power flow, from source to load. Nowadays, with the advancements on renewable energy technologies (RET), the electrical power systems are changing to include more distributed generation sources, with the result of having the electrical interconnected system with both-way power flows, especially at the medium- and low-voltage distribution lines of the electrical power system. Modeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00017-4 Copyright © 2021 Elsevier Inc. All rights reserved.

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Chapter 12 Advances in predictive control of DC microgrids

Figure 12.1. Hierarchical control levels in microgrids.

This development has given place to the opportunity to unite distributed generation (DG) technologies, controllable loads, and energy storage systems (ESS) into intelligent and smaller electrical power systems called microgrids (MGs). These systems are more flexible operationally and can operate autonomously or connected to the main grid, depending on the type of voltage in the point of common coupling (PCC). When there is DC voltage in the PCC, it can be referred to have a DC MG. This type of electrical systems has been recognized as more attractive for numerous reasons due to its higher efficiency, more natural interface to many types of RET, especially to the very-spread solar photovoltaic (PV) technology, and to ESS, and a better compliance with consumer electronics. In terms of control needs, only voltage regulation is needed as there are no issues with reactive power flow, synchronization, power quality, and frequency regulation. The advantage of this control is that it results in a notably less complex control than required for AC microgrids [1]. Control of a microgrid can be established at three hierarchical levels: primary, secondary, and tertiary (Fig. 12.1). The control levels differ in their speed of response and the time frame in which they operate, as well as in infrastructure requirements, such as communication requirements [2]. Additionally, the control schemes can be established as centralized or decentralized.

Chapter 12 Advances in predictive control of DC microgrids

Figure 12.2. Typical structure of a DC microgrid.

The primary (local) level is related to individual components and local controls (distributed generators, energy storage systems, loads, and power electronics interfaces, meeting voltage and frequency references, islanding detection, power-sharing, and power generation control). The secondary level is in charge of load and renewable energy sources, load shedding/management, unit commitment/dispatch, secondary voltage/frequency control, secondary active/reactive power control, security monitoring, and black start. Finally, the upper or tertiary level is related to the market participation of microgrids, managing decisions for importing or exporting energy where a distribution network operator and a market operator can be found [2–6]. It is very common to have in microgrids a number of parallel converters that have to work together and in harmony. As for AC microgrids, it is the same case for DC microgrids; nevertheless, there is no need to control the system considering frequency synchronization, reactive power flows, and harmonics. The only regulation needed is controlling the DC link voltage within proper boundaries along the operation of the DC system. As for AC microgrids, DC microgrids may have several components (Fig. 12.2) and all control objectives can be realized through power converters. Local control functions very usually cover the following: current, voltage, and droop control for each DG; sourcedependent functions, for example, MPPT for photovoltaic modules and wind turbines, or state-of-charge (SOC) estimation for energy storage systems (ESS); decentralized coordination functions such as local adaptive calculation of virtual resistances, distributed DC bus signaling or power line signaling [7]. When considering the global microgrid level, digital communication-based coordinated control strategies can be implemented

313

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Chapter 12 Advances in predictive control of DC microgrids

Figure 12.3. Typical model predictive control scheme of a DC microgrid.

to establish advanced energy management functions. It can be implemented using centralized or distributed approach via central controller or sparse communication network, respectively. When a distributed control is considered, variables are only exchanged among the local controllers [8]. In this chapter, we explain and revise predictive control techniques applied to DC microgrids, emphasizing their control and application aspects for power converters that operate under DC microgrid systems.

12.2 Predictive control of DC microgrids 12.2.1 Primary control of DC microgrids When microgrids consider embedded generators, for instance, solar photovoltaic or wind turbine, the primary control of those

Chapter 12 Advances in predictive control of DC microgrids

distributed generators is exerted by power converters, which for DC microgrids can have only a DC/DC or AC/DC stage for solar and wind technologies, respectively [9]. The power converters play the role of making the DC microgrid efficient and reliable, and it is mandatory for a control strategy to not only ensure proper local operation but also to enable coordinated interconnection between different modules. Thus a flexible local current and voltage control should be employed, and accurate power sharing among parallel connected converters should be achieved [7]. As all power electronics systems are mostly nonlinear, the model predictive control (MPC) method accounts for nonlinear dynamics, uncertainties, and constraints of the system [10].

12.2.1.1 Finite control set model predictive control In DC microgrids, most of the end-users correspond to electronic loads that employ point-of-load (POL) converters for power conditioning and voltage regulation. These loads are able to actively regulate the power that is extracted even under varying voltage at the microgrid side, making them commonly known as constant power loads (CPLs) [11]. When a CPL is connected to a DC microgrid, its incremental input impedance is negative, tending to destabilize the system [12]. This has been known for long time and has been commonly analyzed under impedance-based approach, which has three variations to stabilize the operation of the DC microgrid: passive or active damping of Zout and active damping of Zin [12]. Damping of Zout requires either to install additional passive components in the system increasing the size of the system and causing losses [13,14] or availability of actively controlled power electronic converters at the source side [15–19], whereas numerous applications of DC microgrids tend to use passive front-end converters, being too costly to be added only for stabilization purposes in such electrical systems [12,20]. Finite control set model predictive control (FCS-MPC) is used as an inner controller of a POL converter as seen in Fig. 12.4. This control strategy directly manipulates the converter switches. The model of the converter is used to predict its future actuation for all possible voltage vectors, where the one that minimizes a given cost function (CF) is then applied to the power converter at every sampling time [21]. As, in general, MPC-based strategies rely mainly on the CF, MPC strategies allow high flexibility and simple controllability for nonlinear systems, allowing also seamless integration of multiple control objectives and constraints [22]. As there are several drawbacks of the state-of-the-art stabilization approaches, the control of the point-of-load power converter allows stable control of the DC link with negligible effect on the

315

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Chapter 12 Advances in predictive control of DC microgrids

Figure 12.4. Constant power load (CPL) considered in [12]: two-level three-phase VSC fed from a DC link bus and supplying an AC load.

load regulation performance while simultaneously operating the converter within the safe limits [12]. To develop the FCS-MPC controller, a converter model and the design of the CF are essential. The voltage source converter (VSC) is modeled in a stationary αβ reference frame using the Clarke transformation T as follows [12]:  x¯ = xα + j xβ = T xa where

⎡

2 1⎢ T= ⎣ 0 3 1

xb

 xc ,

⎤ −1 −1 √ √ ⎥ 3 − 3 ⎦. 1 1

(12.1)

(12.2)

For establishing the FCS-MPC controller, the topology of the two-level VSC is considered to emulate the CPL and is shown in Fig. 12.4. The gating signals Sa , Sb , and Sc determine the voltage vector of the power converter. Considering Eq. (12.2) for transforming the voltages for all eight possible switch configurations, the corresponding voltage input vectors v¯i can be obtained in the complex αβ frame, as shown in Table 12.1. All the details of the equations that describe the dynamics of the inductor current i¯f , capacitor voltage v¯f , and its state-space form are as follows [12]:





 v¯i d i¯f i¯f =A +B , dt v¯f v¯f i¯o

(12.3)

Chapter 12 Advances in predictive control of DC microgrids

317

Table 12.1 Switch configurations and complex voltage vectors used in two-level three-phase VSI. Sa Sb Sc

where

0

0

v¯0 = 0

1

0

0

v¯1 = 23 vdc

1

1

0

v¯2 = 13 vdc + j 33 vdc

0

1

0

0

1

1

v¯3 = − 13 vdc + j 33 vdc v¯4 = − 23 vdc

0

0

1

v¯5 = − 13 vdc − j 33 vdc

1 1

0 1

1 1

v¯6 = 13 vdc − j 33 vdc v¯7 = 0

⎤ ⎡ R − Lff − L1f ⎦ A=⎣ 1 0 Cf

and B=

√

√

√

√

(12.4)

⎤

⎡

1 ⎣ Lf

Voltage vector v¯i

0

0

0 − C1f

⎦.

(12.5)

Additionally, for the case examined in [12], the dynamics on the DC side are also present, as can be seen in Fig. 12.4. For describing the inductance Ldc in series with the resistance Rdc connected between the stiff voltage source vs and the DC filter capacitance Cdc , we use only a differential equation describing vdc and the current idc , which flows through the inductor and is treated as an external disturbance, for modeling the DC link dynamics: Cdc

dvdc = idc − ipol . dt

(12.6)

The idc is considered as a disturbance due to the rather high value of DC inductor and because of the continuous nature of the DC voltage source and the DC link voltage, thus making the DC inductor current negligible during the sampling step of 25 µs, and then it is assumed constant between the two intersampling periods. In (12.6), ipol is the current flowing into the UPS inverter and

318

Chapter 12 Advances in predictive control of DC microgrids

can be synthesized from the filter currents and the gating signals as follows [12]: ipol = Sa if a + Sb if b + Sc if c .

(12.7)

Eq. (12.7) can also be represented with complex variables as follows:

  −1 if α  ipol = Sa Sb Sc T . (12.8) ifβ Then the discrete-time model is as follows:

v¯i (k) i¯f (k) i¯f (k + 1) = Ad + Bd i¯o (k) v¯f (k + 1) v¯f (k)

(12.9)

with Ad = eATs and

 Bd =

Ts

eAτ Bdτ,

(12.10)

(12.11)

0

where Ts is the sampling time. A special consideration for the DC microgrid is estimating how much the DC link capacitor is charged/discharged during each sample period:   ipol,i + ipol,p 1 idc − Ts , vdc (k + 1) = vdc (k) + (12.12) Cdc 2 where ipol,i and ipol,f are the initial and final currents going into the UPS inverter during the next sampling time, respectively. FCSMPC scheme uses Eqs. (12.9) and (12.12) to predict i¯f , v¯f , and vdc at the end of the next sampling time. Then these predictions are fed to a CF that determines the optimal actuation, providing robustness to parameter variations [12]. In [12] the stabilization strategy for the DC microgrid is carried out by adding the term gdc that penalizes actuations leading to deviations of the DC link voltage from its steady value to the proposed CF. In the CF Eq. (12.14) the added stabilization term is multiplied by a weighting factor λdc . The stabilization term is as follows: ∗ gdc = (vdc − vdc )2

(12.13)

∗ as the reference DC link voltage. with vdc Then the complete form of the CF used results in

gprop = gcon + λder gder + hlim + λsw sw 2 + λdc gdc .

(12.14)

Chapter 12 Advances in predictive control of DC microgrids

Additional terms included in the CF from Eq. (12.14) include the following ones: ∗ − vfβ )2 , gcon = (vf∗ α − vf α )2 + (vfβ

(12.15)

which is the conventional CF used for AC voltage regulation on the LC filter [23]. Additionally, as shown in [24], the steady-state performance is improved by including a tracking term for the derivative of the voltage reference: 2  2  ∗ gder = Cf ωref vfβ − if α + ioα + Cf ωref vf∗ α + ifβ − ioβ (12.16) with ωref = 2πfref as the angular frequency of the load reference voltage. Again, the term gder is included in the general CF gprop and balanced using a weighting factor λder . Additionally, other factors are included in the CF. To impose a constraint for the current, hlim (i) is defined as  0 if |i¯f | ≤ imax , hlim (i) = (12.17) ∞ if |i¯f | ≥ imax . On the other side, a term for penalizing the switching effort, sw 2 (i), is defined and can be controlled by the associated weighting factor λu . It is defined as follows:  sw(i) = |u(i) − u(i − 1)|. (12.18)

12.2.1.2 Modulated model predictive control As it was mentioned in the previous section for the FCS-MPC applied to DC microgrids, the majority of end-users are electronics loads that behave as constant power loads (CPL) that may cause instability of the system when tightly controlled [11,12]. In the work developed by [25], a modulated model predictive model (M2 PC) is proposed including a stabilization method, which, similarly to the previously mentioned FCS-MPC strategy proposed by [12], develops a CF including the stabilization term of the DC link. For establishing the M2 PC strategy, the differential equations that describe the dynamics of the inductor current iL and the capacitor voltage vC are turned into discrete time using the zeroorder hold method to obtain matrices A and B for the AC and DC sides, respectively [25]. Then the discrete model on the AC side is as follows:





 i L (k + 1) i L (k) v i (k) =A +B , (12.19) v C (k + 1) v C (k) i o (k)

319

320

Chapter 12 Advances in predictive control of DC microgrids

Figure 12.5. A passive front-end rectifier with an LC filter supplying a two-level VSC feeding a stand-alone AC load proposed in [25].

where A= and

B=



a11 a12 = eA1 Ts a21 a22

  Ts b11 b12 = eA1 τ B 1 dτ. b21 b22 0

(12.20)

(12.21)

Then, the resulting matrices are A1 and

0 − L1 1 0 C





0 B1 = . 0 − C1 1 L

(12.22)

(12.23)

For the DC side, the discrete model is expressed as follows [25]: vdc (k + 1) = 2vdc (k) cos ωTs − vdc (k − 1)   (12.24) 3 sin ωTs iLα (k)viα (k) + iLβ (k)viβ (k) + , 2idc  dc where ω = 2i Cdc . For this type of predictive control schemes, the stabilization proposed by the authors in [25] is exerted by developing a cost

Chapter 12 Advances in predictive control of DC microgrids

function, which includes three control objectives: the AC side capacitor voltage, the inductor current, and the DC side voltage. The discrete reference of the voltage and inductor current is expressed as follows [25]:  v ∗C (k) = Vref cos(ωref kTs ) + j Vref sin(ωref kTs ), (12.25) i ∗L (t) = j Cωref v ∗C (k) + i o (k), where ωref and Vref are the reference angular frequency and capacitor voltage amplitude, respectively. Then, the established cost function is as follows [25]: g = gv + λi gi + λdc gdc ,

(12.26)

where ∗ ∗ gv = (vCα (k + 1) − vCα (k + 1))2 + (vCβ (k + 1) − vCβ (k + 1))2 , ∗ (k + 1) − vdc (k + 1))2 , gdc = (vdc ∗ ∗ gi = (iLα (k + 1) − iLα (k + 1))2 + (iLβ (k + 1) − iLβ (k + 1))2 .

To find the optimal voltage vector, the second-order partial derivative of the cost function from Eq. (12.26) has to be obtained [25]: ∂ 2 (gv + λi gi + λdc gdc ) ∂ 2 (gv + λi gi + λdc gdc ) = ∂viα (k)2 ∂viβ (k)2   9(iL (k) sin ωTs )2 2 2 , = 2 λi b11 + b21 + λdc 2 4idc (12.27) Then, the second-order partial derivative of the cost function from Eq. (12.27) has to be compared to zero as it is larger than zero [25]: ∂(gv + λi gi + λdc gdc ) = 0, ∂viα (k)

∂(gv + λi gi + λdc gdc ) = 0. ∂viβ (k)

(12.28)

Then, the optimal voltage vector is obtained [25]: v ∗ovv (k) = w1 i ∗L (k + 1) + w2 v ∗C (k + 1) + w3 v ∗dc (k + 1) (12.29) + w4 i o (k) + w5 i L (k) + w6 v C (k), where w1 =

2 4λi b11 idc 2 (λ b2 + b2 ) + 9λ (i (k) sin ωT )2 4idc i 11 dc L s 21

,

321

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Chapter 12 Advances in predictive control of DC microgrids

w2 =

w3 =

2 b 4idc 21 2 λ b2 + 4i 2 b2 + 9λ (i (k) sin ωT )2 4idc i 11 dc L s dc 21

,

3iL (k)idc sin ωTs  2 , 2 λ b2 + 2i 2 b2 + 9λ i 2idc (k) sin ωT i 11 dc L s dc 21

  2 b b +λ b b −4idc 22 21 i 12 11 w4 =    2 , 2 b2 + λ b2 + 9λ i 4idc (k) sin ωT i dc L s 21 11   2 a + 4i 2 λ a b + 6i λ n − 4idc 21 21 dc dc dc i 11 11 w5 =    2 , 2 b2 + λ b2 + 9λ i 4idc (k) sin ωT i dc L s 21 11   2 a b +λ a b −4idc 22 21 i 12 11 w6 =    2 , 2 b2 + λ b2 + 9λ i 4idc (k) sin ωT i 11 dc L s 21   n = 2vdc (k) cos ωTs − vdc (k − 1) sin ωTs . Then, to implement the modulated model predictive control, the optimal voltage vector is introduced to the SVM module as can be seen in Fig. 12.5 [25].

12.2.1.3 Decentralized model predictive control Decentralized model predictive controller (DMPC) can ensure power sharing and regulate DC bus voltage in DC microgrids with constant power loads. In [10], the DMPC is proposed to replace the primary control layer of DC microgrids, that is, inner loops and droop control, using a single optimal controller and an automatic model of the DC microgrid, which is derived to formulate a cost function for the DMPC. Then, the discrete-time model can be obtained to predict the behavior of the system. For obtaining a model of the DC microgrid, boost converters are assumed as DC–DC converters, and its state-space model with any type load such as Z and considering inductor current iL , capacitor voltage vC , and input voltage vin can be found in detail in [10]; the authors considering the inductor current classify the DC– DC boost converters into two models, averaged and automatic models [26,27].

Chapter 12 Advances in predictive control of DC microgrids

The averaged model of the boost converter in discrete time has three different dynamic modes regarding to the switch position: when it is ON, energy is stored in the inductor; when the switch is OFF, the inductor is connected to the output, and energy is released through it to the load; and thirdly, when the switch remains OFF and the inductor current iL (t) = 0 with both switches S and D OFF, then the boost converter operates in discontinuous conduction mode (DCM). Then the averaged model of the boost converter is as follows [10]: ⎧ ⎪ ⎨(I + A1i Ts )x(k) + BTs vsi (k), S = 1, x(k + 1) = (I + A2i Ts )x(k) + BTs vsi (k), S = 0 & iLi (k) > 0, ⎪ ⎩ (I + A3i Ts )x(k), S = 0 & iLi (k) = 0, (12.30) where

A3i =

0 0 0 − P2

.

(12.31)

Coi voi

Then, the automatic model of a boost converter is developed considering, for this case, the inductor current situation in each sampling time in addition to the switching states. As mentioned earlier, there are three operational modes. The discrete-time version of the automatic model is as follows: ⎧ ⎪ ⎨G1i x(k) + Fi vsi (k), S = 1, x(k + 1) = G2i x(k) + Fi vsi (k), S = 0 & iLi (k + 1) > 0, ⎪ ⎩ G3i x(k), S = 0 & iLi (k + 1) = 0, y(k) = Hi x(k),

(12.32)

where G1i = I + A1i Ts , G2i = I + A2i Ts , and G3i = I + A3i Ts with the identity matrix I , Fi = Bi Ts , Hi = Ci of dimension two, and the considered sampling time Ts . The automatic model can, once in discrete-time, operate in four different modes: 1. When S = 1, iLi (k) > 0, and iLi (k + 1) > 0, that is, the inductor current is positive, and the switch is ON for the whole sampling interval. 2. When S = 0, iLi (k) > 0, and iLi (k + 1) > 0, that is, the inductor current is positive, and the switch is OFF for the whole sampling interval. 3. When S = 0, iLi (k) > 0, and iLi (k + 1) = 0, that is, the inductor current reaches 0 during the sampling interval, whereas the switch is OFF.

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4. When S = 0 and iLi (k) = iLi (k + 1) = 0, that is, the inductor current is 0, and the switch is OFF for the whole sampling interval. Subsequently, the work developed in [10] takes the discrete automatic model, with all the switching modes of the converter considered, thus giving the possibility of using a simple hybrid MPC instead of a nonlinear model. As in the automatic model, from Eq. (12.32) vsi (k) is considered as input signal to have an incremental state-space model, and the model should be changed adding an integrator to suit the controller design purposes. Then the model input has to be changed to the control increment

vsi (k), renaming it, in this way, as augmented model used in the design of the predictive control. The decentralized model predictive control is established to eliminate the undesired steady-state deviations that are associated with the use of droop-based control strategies shown in Fig. 12.3, regarding power quality issues in the microgrid system and with the shared power among the DGs [10]. The decentralized MPC is established for each power converter of the DC microgrid for ensuring proper current sharing and control of the DC bus voltage. The controller is implemented locally without any digital communication used. This control scheme provides optimal switching states for the DC–DC converter by solving an optimization problem using some real measurements as the capacitor voltage and inductor current. The proposed decentralized MPC algorithm is presented as follows. • Step 1. Prediction: in this step the model of the system is built. ∗ , u , v , and N are measured to be The variables iLi , voi , vdc 0 si used in the prediction process. Once checking the status of switching states and the inductor current, the perfect automatic mode is selected, and the variable predictions are done consequently. • Step 2. Cost function formulation: the objective function J (k) is formulated to penalize deviations from the desired trajectory. The cost function is formulated based on the error of the p p predicted value (iLi and voi ) of the inductive current and output voltage (state variables) with the reference values over the prediction horizon and optimal state switching u(k) of the converter simultaneously. Then the cost function of the MPC for each DG in the DC microgrid is as follows: 1 J (k) = N



N  p   p  (iLi (k + 1) + iLir ef  + voi (k + 1) − voir ef ) 1

   + λ u(k) ,

(12.33)

Chapter 12 Advances in predictive control of DC microgrids

where, N is the prediction horizon, voir ef is the reference for the ∗ , and the inivoltage for each DGi which is equal, finally, to vdc tial value depends on the initial value of the output current in each unit, and iLir ef is achieved using the power balance equation Pin = Pout and the desired current, which can be calculated as follows: Pin = vsi iLi ,

Pout = Pload

→

iLid es =

Pload . vsi

(12.34)

In the calculation of the reference of the inductor current, a proportional term is multiplied by the voltage error, considering the small-ripple approximation for regulating the output voltage for each DG unit. • Step 3. Optimization problem: at each sampling time, the optimization problem is stated as follows: U ∗ (k) = arg minJ (k)

(12.35)

subject to the automatic discrete-time model for the DC-DC boost converter with constant power load. Minimizing the cost  function results ina sequence in the form U ∗ , where U ∗ (k) = u∗ (k), u∗ (k + 1), . . . . There exist 2N switching sequences, and only their first elements u∗ (k) are applied in each sampling time.

12.2.1.4 Hybrid finite control set model predictive control/deadbeat predictive control Nowadays, air travel has become very used around the world. Looking to diminish the CO2 emissions by aircrafts that this technology is increasingly moving toward more-electric aircraft (MEA). Thanks to the development of power electronics, electric machines and advanced control technologies, several functions used to be conventionally driven by hydraulic, pneumatic, and mechanical power are being replaced by electrical subsystems in the MEA [28]. In this way, on-board DC power systems in the aircrafts are becoming more usual. These DC systems are considered as on-board DC microgrids. In [29], a current source converter (CSC), shown in Fig. 12.6, is considered for interfacing aircraft generators with an on-board DC microgrid. For establishing the control strategy, a hybrid predictive control is proposed for the CSC with output LC filter. The hybrid strategy is composed by a deadbeat predictive control with larger sampling time applied to the output circuit, generating reference source currents. Subsequently, a finite control set model predictive control (FCSMPC) with smaller sampling time is applied to the input circuit to achieve sinusoidal source currents.

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Figure 12.6. Scheme of the current source converter (CSC) system that interfaces an aircraft DC microgrid [29].

The CSC system shown in Fig. 12.6 is connected as the power source. This source may be variable generators or auxiliary power units. There is an LC filter composed by Lf i and Cf i , which attenuates high-frequency harmonics in input currents of the CSC. This CSC is composed by six switches with reverse blocking capability. In case bidirectional power flow is required, each switch can be composed of two transistors in common-source connection. If only unidirectional power flow is needed, then one transistor and one diode in series can properly do the job. At the output of the CSC an LC filter composed by Lf o and Cf o is installed to provide smooth DC load voltage [29]. The input circuit of the CSC is expressed in the state-space form as follows [29]:





 d is v is = Ai + Bi s , (12.36) v v ii dt i i where v s and i s are the vectors for source voltages and currents, respectively, and v i and i i are the vectors for the input voltages and currents, respectively. Then, the matrices Ai and B i are as follows [29]: ⎡ ⎤ R − Lff ii − L1f i ⎦ Ai = ⎣ (12.37) 1 0 Cf i and

⎡ Bi =

1 ⎣ Lf i

0

⎤ 0 − C1f i

⎦

(12.38)

Chapter 12 Advances in predictive control of DC microgrids

with parasitic resistance Rf i of the filter inductance Lf i . Next, this model of the input circuit from Eq. (12.36) is transformed into discrete-time one to establish an FCS-MPC, as previously commented, with a smaller sampling time, as shown in the following equation:



   i s (k + 1) i s (k) v s (k) = i + i , (12.39) v i (k + 1) v i (k) i i (k) where y(k) refers to the value of variable y at the beginning of the kth sampling period, that is, Tsi for the FCS-MPC for this input circuit of the aircraft DC microgrid. The matrices i and  i from Eq. (12.39) are described as [29] 

i11 i12 i = eAi ·Tsi = (12.40) i21 i22 and  i = A−1 i (i − I )B i =



  i11  i12 .  i21  i22

(12.41)

Then, the FCS-MPC consists of finding the optimal switching state by minimizing a defined CF from Eq. (12.43). The switching state determined in the kth sampling period which minimizes this CF is applied to the model of the input circuit of the CSC from Eqs. (12.39), (12.40), and (12.41). This switching state is applied in the (k + 1)th sampling period, and practically, a sampling period delay is thus obtained. FCS-MPC performance is sensitive to the control delay, making necessary delay compensation to be implemented. As when the kth sampling period just begins, the microcontroller measures source voltage v s (k), source current i s (k), and input voltage v i (k). Additionally, the input current i i (k) is obtained from the valid switching states of the CSC using the switching state S(k) determined in the previous sampling time. Bearing in mind the latter, we see that FCS-MPC is able to predict the source current i s (k + 1) and the input voltage v i (k + 1) based on Eq. (12.39) [29]. FCS-MPC scheme uses a prediction model of the system for predicting the switching state S(k + 1) for the next sampling period. The prediction model for the source current vector and input voltage vector is obtained from Eq. (12.39) as follows [29]:



   i s (k + 2) i (k + 1) v (k + 1) = i s + i s . (12.42) v i (k + 2) v i (k + 1) i i (k + 1) To determine the optimal actuation, a cost function (CF) is defined to find the optimal switching state that has to be applied

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to the power converter. This CF includes prediction errors for all the control objectives of the strategy. To achieve sinusoidal source currents, prediction errors of source currents are included in the cost function [29]: g = ||i ∗s (k + 2) − i s (k + 2)||2 .

(12.43)

The CF has no weighting factor adopted, reducing thus the empirical adjustment and, subsequently, the control complexity [29]. Next, the FCS-MPC scheme is established using a model of the system for predicting the switching state that minimizes the CF to be applied in the (k + 2)th sampling time in this case. This model predictive strategy is applied to the input circuit of the aircraft DC microgrid. For the output circuit, a deadbeat predictive control is developed [29]. The continuous model of the load voltage is considered but turned into discrete-time model for predicting vL using the forward Euler method [29]: vL (k + 1) = vL (n) +

Tso (io (n) − iL (n)), Cf o

(12.44)

where x(n) denotes the value of variable x at the beginning of the nth sampling period. For Eq. (12.44), the sampling time Tso , which is smaller than the sampling time Tsi , is used for the input circuit FCS-MPC, which can be found in Eq. (12.40). For the load voltage model from Eq. (12.44), the only controllable variable is the output current io . Hence, if the load voltage vL reaches the desired value at the beginning of the (n + 1)th sampling period, subsequently, the reference value of io can be obtained from this equation [29], as follows:   Cf o ∗ ∗ vL (n + 1) − vL (n) + iL (n), io (n) = (12.45) Tso where the reference load voltage vL∗ (n + 1) is a constant DC value. The discrete prediction model used for the output current io using the deadbeat predictive control strategy can be obtained from its continuous model by applying the forward Euler method [29]:    Rf o Tso Tso  io (n) + vo (n) − vL (n) . (12.46) io (n + 1) = 1 − Lf o Lf o Again, for the prediction model of the output current, the sampling time used is Tso . For this model, the only controllable variable is the output voltage vo . Hence the reference output voltage

Chapter 12 Advances in predictive control of DC microgrids

can be obtained based on Eq. (12.46) as follows [29]: !   Lf o ∗ Rf o Tso ∗ vo (n) = io (n) + vL (n). i (n + 1) − 1 − Tso o Lf o

(12.47)

For obtaining the reference for the output voltage vo∗ (n), the reference output current io∗ (n + 1) has to be first obtained at the beginning of the (n + 1)th sampling period. Considering Eq. (12.45), only io∗ (n) is generated in the nth sampling period, whereas io∗ (n + 1) is not available. In the work developed in [29] the following approximation is applied: io∗ (n + 1) ≈ io∗ (n).

(12.48)

This approximation is explained by the fact that the output current corresponds to a DC current and can be considered as constant within one sampling period if the output filter inductance is big enough. Then, substituting (12.48) into Eq. (12.47), we can rewrite the reference output voltage vo∗ (n) as follows [29]: !   Lf o ∗ Rf o Tso ∗ io (n) + vL (n). (12.49) vo (n) ≈ i (n) − 1 − Tso o Lf o Finally, the adoption of this hybrid control approach allowed avoiding the use of proportional-integral (PI) controllers and, as a matter of constant development issue related to MPC-based techniques, the inclusion of weighting factors for controlling the input and output circuits of the on-board DC microgrid. Thus, according to [29], a lower control complexity can be achieved, which created a capability to work under high sampling frequency (up to 150 kHz), achieving a superior performance for steady-state and transient conditions even under very high and variable source frequency (360–800 Hz).

12.2.2 Secondary control of DC microgrids The secondary control refers mainly to the management of the microgrid components [6]. Regarding this management and the conditions that these systems have to deal with, it is possible to ask about operating a microgrid with high presence of renewable energy-based distributed generators, variable loads, and distributed energy storage systems, especially about how it can be operated in a coordinated manner as an efficient and reliable whole. The answer to this question may lie in the use of new approaches of predictive control applied to microgrids and the

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inclusion of uncertainties that are a part of the microgrid nature since renewable energy sources are intermittent and load demands vary throughout the day and the seasons of the year. They can be predicted for proper control of the microgrids [9].

12.2.2.1 Model predictive-based self-adaptive inertia control In DC microgrids with power microsources as wind turbines, energy storage systems, and connected to the main AC grid are interfaced with voltage source converters (VSCs) that include selfadaptive inertia control of the DC microgrid [30]. Thus, in the VSCs a control scheme to add inertia to the DC microgrid is included to avoid undesired transition of converter operation mode or load shedding during disturbances as sudden voltage sags in the system. The power converters are controlled with an MPC scheme that has as a voltage reference, the value coming from the self-adaptive inertia control developed. The MPC scheme can provide rapid regulation of the power converter in front of system disturbances to control the proper power flow in the system, allowing the proper energy management of the wind energy-based DC microgrid system. In this case, DC voltage is the indicator of the system stability. This secondary level MPC strategy is applied for cooperating to have a rapid inertia adjustment strategy for the DC microgrid, to avoid control hysteresis and adjustment error. To avoid the control delay that may come using traditional vector control and the direct power control method based on deviation regulation, the MPC strategy for each VSC (in this case, the grid-side converter) uses the objective functions from Eq. (12.50) and the system prediction model with the sampled system state variables at instant k to obtain the optimal control variable of each VSC at instant k + 1 that fulfills the power fluctuation reduction and ensures system stability [30]  g1 = |P ∗ − P k+1 |, (12.50) g2 = |Q∗ − Qk+1 |. In this way, proper operation of the DC microgrid system is ensured, favoring reduction of power fluctuation and thus ensuring the DC microgrid system stability [30].

12.2.2.2 Centralized model predictive control DC microgrids that are fed with substantial intermittent renewable energy sources may face the power imbalance issue and the subsequent DC bus voltage fluctuations, leading to unacceptable instability in the system for fulfilling grid codes standards.

Chapter 12 Advances in predictive control of DC microgrids

For these cases, DC electric springs connected with series of noncritical loads is a scheme for stabilizing voltages in the DC buses to manage noncritical loads the way they are managed in smart grids like smart loads. Noncritical loads can be understood as appliances that can support a wide range of voltage and power fluctuations [31]. A multiple DC electric spring generates reference voltages to control the DC voltage in the buses of DC microgrids with renewable energy distributed generators. This is done using a high-level centralized model predictive controller with nonadaptive weighting factors and adaptive weighting factors to extend the existing functions of the DC electric spring. This becomes very important if there is not enough energy storage capacity for offsetting voltage fluctuations in the buses of the DC microgrid [31]. In the work [31], in the DC microgrid the critical loads can tolerate about 5% DC offsets, thus setting the bus voltage offsets. With this, the power flow is regulated by the control of the bus voltage. Then, the centralized MPC is established using the following objective function: min

J =α

m  (Vnom − Vbusi )2 i=1

+ (1 − α)

m−1  i=1

⎡ ⎣

m  j =i+1

⎤ (Vbusi − Vbusj )2 ⎦, Cij (12.51) Rij

subj ect to : 0 ≤ α ≤ 1,  (1 − η)Vnom ≤ Vbusi ≤ (1 + η)Vnom ,

where Vnom is the nominal bus voltage of the DC microgrid, η is  the voltage tolerance of the critical loads by percentage, Vbusi is the bus voltage with smart loads, and α is the weighting factor; α = 1 means that only the regulation of the bus voltage is concerned, and α = 0 means that only the power loss on the distribution lines is concerned.

12.3 Conclusion In this chapter, we revisited the predictive control advances in the last years applied to DC microgrids. As it was introduced, the control of microgrids has hierarchical levels, which consider different scopes and thus different control techniques. Predictive control-based techniques applied to power converters are well de-

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veloped in the academia and with some development in the industry. Additionally, when the power converters are interfacing distributed generators within DC microgrid systems, the application of these predictive control-based techniques is quite less developed as, for instance, the classical control methods. Considering the latter, the most developed hierarchical DC microgrid control level is the primary, where the scopes of the control consider the power converter itself operating under given power and voltage references to keep the DC microgrid system stable. In this way, finite control set model predictive control techniques are developed using cost functions to be minimized, which include terms to stabilize the DC microgrid system. Also, modulated model predictive control technique is used to stabilize using additional terms in the cost function. This allows having three control objectives for the DC microgrid system: the AC side capacitor voltage, the inductor current, and the DC side voltage of the system. Continuing with the primary level controllers, the use of decentralized model predictive control allows elimination of the undesired steady-state deviations associated with the use of droopbased control strategies, regarding power quality issues and power sharing among the distributed generators embedded in the DC microgrid system. Furthermore, the adoption of hybrid finite control set model predictive control/deadbeat predictive control allows avoiding the use of linear controllers and the inclusion of weighting factors for controlling the input and output circuits of the on-board DC microgrid, giving to the system the capability to achieve a superior performance for steady-state and transient conditions. For secondary level control, the capability of power converters for dealing with wind variability as disturbances to the DC microgrid aims to control the stability in the whole system. As microgrids have a small inertial reserve if compared with utility grids, the role that the voltage source converters play in the system stability is essential. To improve the inertial response, each converter of the microgrid may be controlled using MPC. The use of model predictive-based self-adaptive inertia control to cooperate to have a rapid inertia adjustment strategy to avoid control hysteresis and adjustment error allows having no control delays that classical control strategies may present. Additionally, centralized model predictive control regulates the voltage in the buses of the DC microgrid system by regulating the power flow among the renewable energy-based distributed generators embedded in the DC microgrid system. The centralized model predictive control establishes an objective function, which has to be minimized to allow no further than 5% DC offset in each bus.

Chapter 12 Advances in predictive control of DC microgrids

Acknowledgment The authors thank the support of the CONICYT PFCHA/DOCTORADO BECAS CHILE/2019–21190255 and CONICYT/FONDECYT Programme through the Regular 1191028 and the Regular 1160690 Projects and FONDAP SERC Chile 15110019.

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13 Modeling and control of DC grids within more-electric aircraft Cheng Wanga , Habibu Hussainia , Fei Gaob , and Tao Yanga a Power Electronics, Machines and Control Group, Faculty of Engineering, The University of Nottingham, Nottingham, United Kingdom. b Department of Electrical Engineering, Key Laboratory of Power Transmission and Conversion, Shanghai Jiao Tong University, Shanghai, China

Chapter points • This chapter presents a description of the modeling and control of the DC grids within the more electric aircraft (MEA). • The introduction in section 13.1 presents the concept of the MEA and the preference for the HVDC architecture. • A discussion of the various levels involved in the modeling of an electrical power systems and the detailed modeling of the components in the MEA is presented in section 13.2. • The different control strategies required for the components onboard the MEA are discussed in section 13.3. • The summary in section 13.4 provides an overview of the modeling and control of the DC grids within the MEA.

13.1 Introduction to more-electric aircraft Increasingly moving toward More-Electric Aircraft (MEA) is one of the few existing solutions available for the development of more efficient and environmentally friendly aircrafts. Driven by the development of power electronics, electric machines, and advanced control technologies, many functions that were conventionally driven by hydraulic, pneumatic, and mechanical power are being replaced by electrical subsystems in the MEA [1–4]. Such changes are shown in Figs. 13.1(a) and 13.1(b). Compared with conventional aircraft, the MEA offers significant cost benefits due to fewer parts, integration of key subsystems, and multiuse of components [5]. It also reduces the overall cost of operation and ownership because its more-electric architecture helps reModeling, Operation, and Analysis of DC Grids. https://doi.org/10.1016/B978-0-12-822101-3.00018-6 Copyright © 2021 Elsevier Inc. All rights reserved.

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Figure 13.1. Change in nonpropulsive power in conventional aircraft and more-electric aircraft. (a) Conventional aircraft. (b) More-electric aircraft.

duce fuel consumption per passenger per mile and increase overall aircraft performance and energy usage.

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

Figure 13.2. Typical DC grid in Electric Aircraft.

For an increased amount of power requirement and optimal management of onboard electrical power, the concept of MEA poses increased challenges for the aircraft electrical system. Therefore significant attention has been focused on the electric power system (EPS) architecture design and control aspects. Many architectures have recently been proposed with performance comparison studies. A potential solution would prefer a high-voltage DC (HVDC) network architecture. The benefits of an HVDC distribution system can be summarized as follows: • Lower losses in the power transmission cables. This is due to the fact that only two conductors (positive and negative) are required in DC distribution, whereas three conductors (three phases) are required in AC distributions [6]. • The reduction of the skin effect in DC results in reduced power loss and dielectric losses in the power cables. • Less corona effects with DC compared to AC conductors. • No need for any reactive power compensation equipment. Thus the capacity of wires and devices increase, and the cable size can be reduced because there is no need to distribute/process the reactive power [6]. • Convenience of paralleling power supplies and integrating energy storage systems (ESS). Due to these benefits, the DC power system has attracted more attention in recent years. An example architecture of an electric grid used for the initial study is shown in Fig. 13.2. This example EPS architecture consists of several components: 1. Generation system: The electrical power is mainly provided by two or more permanent magnet synchronous generators (PMSGs), which extract power from engine shafts. Depending

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on the flight scenario, the system can be operated with only one active source or with multiple sources feeding the same DC bus. 2. Energy storage system (ESS): Energy storage system is also integrated for emergency status operation. To achieve a flexible power flow in the system, a bidirectional DC/DC converter is implemented. Therefore the discharge and charge modes of the battery can be controlled. 3. Load: The onboard loads are represented by a combination of conventional resistive loads and by constant power load (CPL), typically driven by tightly controlled power electronic devices. Here a permanent magnet synchronous machine (PMSM) drive is used to represent electromechanical actuation system, which is essentially one type of CPLs.

13.2 Modeling of aircraft EPS Application of the MEA concept will see significant penetration of Power Electronic Converters (PEC) into aircraft EPS. These PECs and their control systems will lead to significant challenges for EPSs designers. To ensure system stability, availability, and power quality issues, the modeling and simulation of the EPS are required.

13.2.1 Modeling paradigm 13.2.1.1 Multilevel modeling paradigm Modeling the electrical power system element has been studied for decades. The model required for EPS studies is always dependent on its application. Fig. 13.3 categorizes the EPS model into four levels: architecture level, functional level, behavioral level, and component level [6]. The complexity of the model increases from the top architecture-level models to the bottom component-level models. (a) Component level The bottom component layer aims to model the behavior of components within a subsystem, especially critical components. Component models cover high frequencies, electromagnetic field and electromagnetic compatibility (EMC) behavior, and perhaps thermal and mechanical stressing. The modeling bandwidth of component models can be up to MHz if required. (b) Behavioral level The behavioral layer model uses lumped-parameter subsystem models, and the modeling frequencies can be up to hundreds of

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

Figure 13.3. The multilevel modeling paradigm [7].

kHz. Models in this level cover the converter switching behavior and the impact of harmonics. The nonlinearities and the dynamics of the subsystem up to the switching frequency are preserved in the behavioral model. (c) Functional level The next level is generally known as the functional level where system components are modeled to handle the main system dynamics up to 150 Hz, and the error should be less than 5% in respect of the behavior model accuracy [7]. The functional-level model is targeted at the study of overall power system performance, stability, transient response to loading and start-up, and aims to model the power system either in its entirety or in sections sufficiently large to obtain a holistic generator-to-load dynamic overview. Since the model complexity of the functional model is reduced, the computation time of the functional model is aimed at approaching the real simulated time. The model developed in this chapter will be targeted at this level, and a model library suitable for the simulation study of the future MEA power system will be established. (d) Architecture level The top architectural layer computes steady-state power flow and is used for weight, cost, and cabling studies [8]. The model in this level also allows event modeling such as bus configuration and step of loading, and reliability, stability, and availability studies. The architectural models are simpler and are representative only of steady-state power consumptions.

13.2.1.2 Studies of functional models Modeling and simulation of EPS are essential steps that enable the design and verification of numerous electrical energy systems including the modern electric grid and its components,

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distributed energy resources, and electrical systems of ships, aircraft, vehicles, and industrial automation. With the increasing use of power electronic devices, it is impractical to simulate and study such a complex system with detailed component-level device models or behavioral device models. As these models include high-bandwidth components, from kHz to MHz, very small simulation steps and a huge amount of computer memory are required. This leads to significant simulation time and makes the large-scale EPS simulation at these two lower levels impractical. In addition, these two lower-level models are discontinuous and therefore are difficult to use for extracting the small-signal characteristics of various modules for system-level analysis [9]. These challenges have led to the development of average modeling techniques; these can be categorized at the functional level. The average-value modeling (AVM) method removes the effect of the switching behavior of the power electronic device using the dynamic average values of the variables. Since the switching frequency of the electronic power converters is much higher than the system dynamics, the system-level study can be conducted with the dynamic average value defined over the length of a switching interval, instead of looking at the instantaneous values of currents and voltages that contain ripples due to switching behavior of PECs. The dynamic average value of a time-domain variable x(t) is defined as  1 t x¯ = x(t)dt, (13.1) Ts t−Ts where Ts is the switching period. For the DC/DC converters, x(t) may represent the input or output voltage v(t) and current i(t). The AVM of PWM DC/DC converters can be given in either an analytical or equivalent circuit form using definition (13.1). Theoretically, these two forms of models are equivalent for any given converter topology. As shown in Fig. 13.4, the left side shows a switched cell commonly used in a DC/DC converter, and the right side shows its equivalent circuit. In the average model the switch pair, an IGBT and a diode, are replaced with dependent sources, which are functions of the duty cycle and the averaged values of the cell terminal variables [10,11]. In the analytical AVM for the PWM DC/DC converter, the state equations for each topology within a switching interval Ts are firstly obtained. Using (13.1), the final average model is then derived from the weighted sum of the state-space equations for different subintervals. This concept of averaging has also been extended to modeling AC–DC and DC/AC converters. However, instead of directly aver-

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

Figure 13.4. Switched-inductor cell and its averaged circuit model.

aging the AC variables using (13.1), the AC side variables are transformed into a synchronous rotating reference frame, referred to as the dq frame [12]. In the dq frame, the three-phase AC variables are transformed to be composed of a DC (constant) term and highfrequency ripples with the same switching interval. Because the variables have the DC component that is constant in the steadystate, these variables can be used for averaging using (13.1). Compared with original signals, the DC-like averaged signals allow the variable-step solver to choose larger simulation steps under the same tolerance condition. Instead of deriving the AVM model by averaging variables after the dq transformation, the calculation of AVM models of AC/DC or DC/AC converters can use the converse process, that is, preprocessing the AC variables and then using the dq transformation. This method neglects all the switching higher harmonics and only considers the fundamental component of the AC variables and the switching functions. The variables on the AC and DC sides are related through the fundamental components of the switching functions. The application of the dq transformation on these fundamental components gives the AVM model of the AC/DC or DC/AC converters. This method gives the same result as that obtained from the first one and is more applicable in the modeling of an EPS. In this chapter, it will be referred to as the DQ0 modeling technique. The DQ0 modeling technique is based on the fact that the DC component in the d and q axes is only derived from the fundamental components of the AC variables. The DC component of the DC-side variables is also only dependent on the fundamental components of the AC-side variables and the switching functions. The DQ0 model has been successfully used in the modeling of the More Open Electrical Technologies (MOET) aircraft EPS [13–15]. There the DQ0 model has been proved to be an effective way to study a large-scale EPS with high accuracy and computation efficiency, especially under balanced conditions where the DQ0 model simulates more than 1000 times faster than the corresponding behavioral model. It is, so far, one of the most efficient models in EPS studies. In this chapter, we use the DQ0

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Figure 13.5. PMSG developed within AEGART project.

model for comparison studies. However, the efficiency of the DQ0 model dramatically decreases when the system is under unbalanced conditions. This is due to the negative sequence present in the system. In the DQ0 model, this negative sequence becomes the second harmonic in the d- and q-axis variables. Since the dq variables are no longer constant in the steady state, the simulation speed becomes much slower and is comparable to simulating with nontransformed AC variables.

13.2.2 Modeling of power generation system This section focuses on the modeling of components in the MEA EPS. As illustrated in the previous section, functional models are used because of their effectiveness.

13.2.2.1 Permanent magnet synchronous generators The 3-phase permanent magnet synchronous generator (PMSG) is one of the most widely used generators in DC grids in recent decades. Fig. 13.5 shows a PMSG developed by the University of Nottingham within the CleanSky AEGART project [16]. This project aimed to develop an electrical starter/generator system for next-generation business jet applications. To avoid the complexity associated with the calculation of 3phases, it is widely adopted that PMSG can be modeled with a synchronously rotating reference frame (dq frame). The dynamic

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

Figure 13.6. Model of PMSG.

equations for PMSG in the dq frame are as expressed in (13.2). 

vd = Rid + Ld didtd − ωe Lq iq , vq = Riq + Lq

diq dt

+ ωe Ld id + ωe ϕm ,

(13.2)

where vd , vq , id , iq , Ld , Lq , R, ϕm , ωe represent the d-axis stator voltage, q-axis stator voltage, d-axis current, q-axis current, d-axis inductance, q-axis inductance, stator resistance, flux linkage of the permanent magnet, and electrical rotor angular velocity, respectively. The model diagram based on (13.2) is shown in Fig. 13.6. In MEA EPS, surface-mounted PMSG is always used because of its mechanical benefit for high-speed operation. Then, the d- and q-axis inductances are considered to be identical (Ld = Lq = Ls ) in the case of the surface-mounted PMSG [16]. Therefore Eq. (13.2) can be rewritten as the expression in (13.3). 

vd = Rid + Ls didtd − ωe Ls iq , vq = Riq + Ls

diq dt

+ ωe Ls id + ωe ϕm .

(13.3)

The maximum allowable phase currents are determined by designed rated parameters of the converter and machine. The maximum voltage is dependent on the available DC-link voltage and modulation method. The voltage and current limitations can be written as in (13.4) by neglecting the stator resistance and tran-

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sient terms: ⎧  ⎪ ⎨ ω L i 2 + (L i + ϕ )2 ≤ v max , e s q s d m c  ⎪ 2 max 2 ⎩ id + iq ≤ ic ,

(13.4)

where vcmax and icmax are the maximal phase voltage amplitudes at the fundamental frequency and maximal phase current, respectively.

13.2.2.2 AC/DC power converters Due to increased onboard electrical power, high power quality is one of the requirements for the AC/DC power conversion in MEA. As a potential substitute for traditional transformer rectifiers, the active-front-end converters (AFE) can get better electrical behavior and keep the same performances in robustness and safety requirements. For aircraft applications, the topology of the AFE converter can potentially be two- and three-level converters (especially, the neutral point clamped topology) as shown in Fig. 13.7. A two-level converter was considered for its known simplicity and intrinsic reliability due to the minimal number of devices. A three-level converter was also considered because it brings advantages of lower electromagnetic interference emissions, better power quality, and the ability to handle higher fundamentals compared with two-level converters [20]. These are useful features when considering the high-speed application (e.g., aircraft electric starter/generator driven by an aircraft engine shaft). In addition, the voltage across the switches is only half the DC bus voltage. This feature effectively doubles the power rating of the converter for a given power semiconductor device. Fig. 13.8 shows an active front-end based on a three-level neutral point clamped (NPC) converter developed by the University of Nottingham within the CleanSky AEGART project [16]. Such AC/DC converters are also essential elements for electric actuation devices, namely, electro-mechanical actuators (EMAs), which will potentially replace the conventional hydraulic powered actuators to meet the target of removing hydraulic systems. The control scheme utilized in this chapter will be fieldoriented control (FOC), in which the stator currents of a threephase AC electric motor are identified as two orthogonal components that can be visualized with a vector. The rotating reference (or dq) frame, which is part of FOC, will be detailed. Fig. 13.9 shows the phasor diagram of the three-phase stationary and twophase rotating dq frames; v a, b, c represent the three-phase frames

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

Figure 13.7. Active front-end rectifier (AFE). (a) Two-level three-phase AFE. (b) Three-level three-phase AFE.

Figure 13.8. A three-level converter developed at the University of Nottingham within Clean Sky AEGART project.

of phase voltages, v d, q represent the dq frames, and θ is the angle between both frames. The frame transformation from three-phase to dq can be interpreted by a set of mathematical equations. If the d-axis is aligned with phase a (vd = va ) as a reference point for both frames (θ = 0), then 

vd vq



⎡

⎤ va = kdq ⎣ vb ⎦ , vc

(13.5)

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Figure 13.9. Three-phase and dq phasor diagram.

⎡

kdq

   ⎤  2π 2π cos θ + cos cos θ − (θ ) 2⎢ ⎥ 3 3    ⎦.  = ⎣ 2π 3 − sin (θ ) − sin θ − 2π − sin θ + 3 3

(13.6)

Now the converter is also modeled in the dq frame. The converter AC voltage vc can be expressed as vc =



vd2 + vq2 .

(13.7)

As shown in Fig. 13.10, the d-axis and q-axis components of the AC-side terminal voltage can be written as 1 vd = md vdci , 2 1 vq = mq vdci , 2

(13.8) (13.9)

where md , mq are the modulation indices in the corresponding directions. The AC-side terminal real power of the two-level AFE can be formulated as P=

3 vd id + vq iq . 2

(13.10)

Neglecting the power losses of the converter, as shown in Fig. 13.10, the power balance between the DC and AC sides yields the expression Vdc idc =

3 vd id + vq iq , 2

(13.11)

where idc is the output DC current flowing after the capacitor.

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

Figure 13.10. Model of AC/DC converter.

13.2.3 Energy storage system The energy storage system (ESS) provides power for starting PMSG and also absorbs the excess power generated by the PMSG. During an emergency (loss of normal electrical power supply from the PMSG(s)), the ESS can provide power to the loads for a limited time through a bidirectional DC/DC converter. Furthermore, the ESS can help meet the increasing onboard power demand without installing additional generators. This will be of great advantage in terms of the aircraft fuel economy [17]. Several technologies are currently used for energy storage. These include mechanical, hydraulic, electro-chemical, and pneumatic technologies [18,19]. The ultracapacitors, lead-acid batteries, flow batteries, and lithium-ion batteries are the common energy storage system solutions among the electro-chemical technologies [18,19]. Batteries have the advantage of higher charge efficiency, ease of installation, cost-effectiveness, and responsiveness when compared with other storage systems. Specifically, the lithium-ion batteries show about 99% charge efficiency and 86–99% energy efficiency, depending on the charge and discharge C-rate (a measure of the rate at which the battery is discharged in relation to its maximum charge) [20]. On the other hand, other energy storage systems, for example, fuel cells, have only about 66% efficiency [21]. Also, batteries can deliver power immediately, unlike fuel cells, which need a few minutes before generating power. Batteries find application in the establishment of standalone renewable power systems utilized for remote energy generation. The energy storage device (ESD) or system (ESS) helps to store and give out energy based on the generator and load profiles [22,23]. Fig. 13.11 shows a typical ESS. Here a bidirectional DC/DC converter (buck-boost type) is chosen due to its simple structure with higher reliability compared to other converters.

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Figure 13.11. Energy storage system.

Figure 13.12. Battery model overview.

13.2.3.1 Battery In general, the battery bank module comprises of battery cells connected in series and parallel to achieve the desired voltage and power level. As shown in Fig. 13.11, a simple model of a constant voltage source Vbat in series with a resistor Rbat is used to represent the battery. The battery voltage is always determined by the state of charge (SOC) shown in Fig. 13.12. There have been several computation methods for both SOC and battery voltage. The methods used vary and are dependent on the category of the battery. Hence we will not discussed them in detail.

13.2.3.2 Bidirectional DC/DC converter A bidirectional DC/DC converter allows both charge and discharge modes of the battery bank by modifying the switching duty cycle of power electronics devices. Here we consider an average model per switching period. The topology of the bidirectional buck-boost converter is shown in Fig. 13.11. When the converter operates in boost mode (discharging mode), the magnitude of the converter output current at the DC link idc is positive. In contrast, it is negative when it

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

operates in buck mode (charging). It is assumed that the converter inductor always operates in the continuous conduction mode. Then the model can be depicted as expressed in (13.12), (13.13), and (13.14). dV1 −V1 + Vbat iL Rbat = − , dt C1 Rbat C1

(13.12)

diL −V1 + (1 − d) Vdc = , dt Lbat

(13.13)

diL −V1 + (1 − d) Vdc = , dt Lbat

(13.14)

where Vbat and Rbat are the internal voltage and resistance of the battery bank, C1 and C2 are the capacitance at the battery bank side and DC link side, V1 is the voltage on the capacitor C1 , Lbat is the bidirectional buck-boost inductor, and iL is the current through it.

13.2.4 DC link modeling The dynamics on the DC-link can be expressed as dVdc idc − io = , dt Cb

(13.15)

where idc is the current generated from sources, io is the output current for loads, and Cb is the capacitance of DC-link. If the impedance of the DC cable between the converter and the main DC bus is ignored, then Vdc is equal to the main bus voltage Vb . The nominal voltage of the main bus is always defined based on a certain standard. For instance, 270 V is the nominal voltage, and the range between 250 V and 280 V is acceptable for the more electric aircraft (MEA) as defined in MIL-STD-704F.

13.2.5 Load modeling Onboard conventional aircraft, electric power mainly supplies energy for avionics, lights, entertainment, and galleys. With the development of power electronics, MEAs introduce new kinds of electric load, such as air conditioning and pressurisation, flight controls, fuel pumps, and wing ice protection.

13.2.5.1 Environmental control system Environmental control system (ECS) is one of the largest power consumption and nonpropulsive systems for civil aircraft. It pro-

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Figure 13.13. ECS in conventional aircraft and MEA. (a) Conventional aircraft. (b) MEA.

vides air supply, thermal control, and cabin pressurization for the crew and passengers. As shown in Fig. 13.13(a), ECS in conventional aircraft is designed to rely on the air bled from the engine to achieve the required temperature and pressure of the cabin air. However, the air bled from the engine will reduce the operating efficiency of the engine. For that matter, large amounts of energy will be wasted after the power optimization process of ECS. ECS under the MEA environment is shown in Fig. 13.13(b). Compared with Fig. 13.13(a), the ECS converts the power extraction source from the bleed air to the electric power. Such electric power is generated by electrical motors (mostly PMSMs) with adjustable speeds. Therefore it can realize a variable temperature and pressure control of the cabin air [24,25].

13.2.5.2 Flight controls Flight controls of aircraft are performed with a variety of control surfaces or mechanisms. Generally, modern aircraft actuation systems including primary control surfaces, secondary control surfaces, and landing gear system are powered by a combination of a hydraulic, pneumatic, and mechanical systems. With the development of power electronics, electro-mechanical actuator (EMA) has the potential to replace the hydraulic actuators, which brings the benefits of weight reduction, improved maintainability, and the potential advantage of more flexible flight control by introducing distributed actuation system architecture [26]. Fig. 13.14 shows a conceptual diagram of EMA. In such a system, each turn of the motor moves the actuator a fixed amount. A direct connection between the motor and actuator arm is

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

Figure 13.14. Electro-mechanical actuator.

achieved. Potentially, the EMA will be the most compact solution for flight controls.

13.2.5.3 Fuel pumps In the MEA, fuel pumps are driven by electric motors, instead of the accessory gearbox as in current aircraft. As electrical loads, fuel pumps pressurise and feed the fuel to the engine. Meanwhile, some fuel pumps also transfer fuel between the tanks and the collector tanks to modify the aircraft center of gravity position and reduce wing bending and structural fatigue within the wing. In B787 the engine-driven pumps (EDP) are replaced by electrical pumps giving a total load of 400 kVA.

13.2.5.4 Wing ice protection One of the new electric loads onboard MEA is the wing ice protection system (WIPS), which employs electrically produced heating using resistive thermal mats. In larger aircrafts, the WIPS can typically demand about 125 kW per wing [27,28].

13.2.5.5 General load model As illustrated in the previous sections, many kinds of loads are controlled by power electronics, and such loads include ECSs and EMAs. Using EMA as an example, the output power of one EMA can be expressed as PEMA = TEMA ωEMA = const,

(13.16)

where PEMA , TEMA , and ωEMA represent the output power, output torque, and rotor speed of EMA. In the EMA control system, TEMA and ωEMA are always constant, and thus EMA gives a CPL performance in the EPS. Therefore, in general, the load power in the electric aircraft DC grid is made up of a combination of constant impedance load

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Figure 13.15. Load model (constant impedance load + constant power load).

(wing ice protection load) and constant power load (CPL). The combination can be expressed as PL = Pres + Pcpl =

vb2 + Pcpl , Rres

(13.17)

where Pcpl and Pres are the total power of the CPL and resistive load, and Rres is the resistance of the resistive load. Then the voltage–current relationship can be written as io =

Pcpl vb + . Rres vb

(13.18)

In simulation studies, CPL can be simply represented as a controllable current source. The model diagram of the load combination is shown in Fig. 13.15.

13.3 Control development Effective control strategies should be developed to achieve stable and efficient operation of the DC grid in MEA. As shown in Fig. 13.2, the DC grid in MEA consists of a number of parallel converters that work together. A local controller (LC) is needed for each converter to stably generate the required voltage or current for the system. Meanwhile, a high-level control method is required to coordinate the controller among sources.

13.3.1 Single PMSG control In this section, we detail the control design for a single generation system based on models illustrated in Section 13.2.2. Based on the aforementioned discussion, the detailed control scheme for generator mode is as shown in Fig. 13.16. The flux-weakening

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

Figure 13.16. Control block diagram of PMSM.

(FW) controller and DC controller generate the dq-current references, which are fed to the current controller as shown in Fig. 13.16.

13.3.1.1 Current control loop The first stage is designing the current controllers of the starter generator (S/G) power system. It controls dq-current (id and iq ) following their reference (id∗ and iq∗ ). Fig. 13.16 shows the dqcurrent loops and their respective control plant. The feedforward terms within the current control plant are used for the compensation of the coupling effect in the dq frame. Then it can be reduced to a first-order transfer function as expressed in (13.19) and (13.20) for the d-axis and q-axis current, respectively: id 1 , = vd∗ Ls s + R

(13.19)

iq 1 . = vq∗ Ls s + R

(13.20)

Applying a PI controller with feedforward elements, a closedloop transfer function can be derived as kpc s + kic id

= , id∗ Ls s 2 + R + kpc s + kic

(13.21)

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kpc s + kic iq

= , ∗ 2 iq Ls s + R + kpc s + kic

(13.22)

where kpc and kic are the proportional and integral gains of the current regulator. The transfer function is similar to an ideal second-order response, which can be written as

s2

ωn2 , + 2ξ ωn s + ωn2

(13.23)

where ζ is the damping ratio, and ωn is the natural bandwidth. For the current regulator, ζ and ωn can be expressed as in (13.24) and (13.25), respectively:  ωn =

kic , kpc

(13.24)

ξ=

R + kpc . 2Ls ωn

(13.25)

To achieve the desired dynamic and statistic responses, the values of ζ and ωn , are always set according to the fundamental frequency and switching frequency. Then proportional and integral gain of the current regulator can be calculated.

13.3.1.2 DC link control and flux weakening control After designing the current controller, we discuss the dqcurrent references, which are generated from flux weakening controller and DC-link controller as mentioned earlier. The flux-weakening controller remains active during generator mode to maintain the stator voltage (|V|) at the required level and avoid overmodulation of the converter. When the machine rotates beyond the base speed, the flux-weakening operation is activated, and a negative d-axis current id∗ is injected based on the error between the reference voltage and the voltage limit set by the converter. The q-axis current reference is set by the outer DC power loop when the system operates in the generation mode. During the flight, the system operates in the generation mode, and the qaxis current reference iq∗ is set by the DC-link current or voltage demands dictated by the specific power sharing method. For instance, the current reference should be implemented when the power sharing method is voltage feedback-based droop control (current-mode droop control method). In contrast, voltage reference should be implemented when the power sharing method is

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

Figure 13.17. ESS controller.

current feedback-based droop control (voltage-mode droop control method). In Fig. 13.16 the control value of iq∗ is determined by the fluxweakening controller due to the function of the dynamic limiter prioritising FW control signals.

13.3.2 ESS control The controller for the bidirectional DC-DC converter is as shown in Fig. 13.17. Two PI controllers are connected in cascaded form and used to obtain the duty cycle d for switches. In practice, the DC-link voltage controller can be eliminated when the voltage regulation is carried out by other sources.

13.3.3 Power sharing control After discussing each local controller (LC) in the electric aircraft DC grid, it is essential to manage the power sharing among multiple generators. When a proper power sharing control method is implemented, system-level optimization, such as efficiency improvement and power management, can be achieved/realized. Based on communication usage, multisource control can be divided into three categories: centralized, distributed, and decentralized controls. These methods are briefly discussed as follows.

13.3.3.1 Centralized control As shown in Fig. 13.18, centralized control can be implemented in a distributed generation (DG) based DC grid by employing a centralized controller. Data from DG are collected in a centralized aggregator and then processed, and feedback commands are sent back to them via a digital communication network shown as dash lines in Fig. 13.18. It is easy to apply a centralized controller in a small scale grid system. Fig. 13.19 shows a typical its scheme. The centralized controller regulates the DC bus; thereafter current references for each

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Figure 13.18. Operation principle of centralized control.

Figure 13.19. Control scheme of centralized control.

power module are generated by splitting the total current reference according to the rated power ratio of sources. In some cases, master–slave control can be implemented as an alternative to centralized control. It is realized by a module chosen as a “master”, and the remaining modules as “slaves”. Normally, the module that has the largest capacity or is the most reliable can be chosen as a “master”. The master module regulates the DClink voltage and generates the current references for the remaining “slave” modules. The typical block diagram of master–slave control is illustrated in Fig. 13.20.

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

Figure 13.20. Control scheme of master–slave control.

Compared to the centralized controller, master–slave control is more difficult to apply. However, it minimizes the use of an extra centralized controller, which helps reduce the system cost. Although centralized control has much better voltage regulation and current sharing performance, the requirement of communication restricts its application area, so that it cannot be used in large-scale systems.

13.3.3.2 Distributed control As shown in Fig. 13.21, there is no central control unit in a distributed control system. Communication lines only exist between the neighboring modules. The main advantage of this approach is that the system can maintain full functionality, even if the failure of some communication links occurs. Therefore distributed control is immune to a single point of failure. However, differently from centralized control, the information directly exchanged between the local controllers can contain only locally available variables. In other words, if the two units are not connected directly by the communication link, then they do not have direct access to each other’s data, and their observation of the whole system is limited.

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

Figure 13.21. Operation principle of distributed control.

Figure 13.22. Operation principle of decentralized control.

13.3.3.3 Decentralized control Without using external communication among local controllers, decentralized control uses local measurement to implement local regulation, as shown in Fig. 13.22. If the failure of one module occurs, then the remaining modules can still contribute to power sharing according to their local droop settings. Thus system reliability is increased. Since communication links among the sources and an additional centralized controller are not needed, each parallel module can work independently relying on the local measurements and controllers. Droop control was firstly employed for AC systems as decentralized control due to no need for communication lines. Now it has been widely accepted for DC systems similarly. It utilizes a “virtual resistance” to achieve current sharing. In the DC system a relationship between current and voltage is built to realize the “virtual resistance” character. Generally, droop control can be classified into voltage (including V-I and V-P strategy) and current mode (including I-V and I-P strategy), which are shown in Fig. 13.23(a–d), respectively. However, the power of the converter is almost proportional to the output current when small voltage errors are considered. Hence, for voltage and current mode droop control, in this section, we will discuss only V-I and I-V strategies.

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

Figure 13.23. Droop characteristic. (a) V-I droop. (b) V-P droop. (c) I-V droop. (d) P-V droop.

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

Figure 13.24. Power sharing characters of V-I droop control curve.

Voltage-mode approach Voltage-mode droop control uses measured branch current to generate voltage reference. The calculation of reference for the voltage controller is as follows: ∗ ∗ = Vdc − kdi Idci , Vdci

(13.26)

∗ is the calwhere i represents the index of each converter, Vdci ∗ is the rated culated voltage reference for the ith converter, Vdc voltage, kdi is the droop coefficients in voltage-mode droop controllers of the ith converter, and Idci is the output current of the ith converter. Fig. 13.24 shows the diagram of the V-I droop control curve when three converters share the same DC bus voltage Vdci are considered. Then the power sharing ratio among three converters as obtained from Fig. 13.24 is expressed as

Idc1 : Idc2 : Idc3 =

1 1 1 : : . kd1 kd2 kd3

(13.27)

However, when considering the voltage drop on cables, the DC bus voltage in steady state can be expressed as follows: ∗ − (kdi + Ri ) Idci , Vb = Vdci − Idci Ri = Vdc

(13.28)

where Vb is the voltage of the common DC bus. Ri is the cable resistance of converter i. Then the current sharing ratio among sources can be rewritten as Idc1 : Idc2 : Idc3 =

1 1 1 : : . kd1 + R1 kd2 + R2 kd3 + R3

(13.29)

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

Figure 13.25. Power sharing characters of I-V droop control curve.

We can infer that both droop gain and line resistance will influence the power sharing ratio, that is, upgrading the droop gain or line resistance will decrease the power output of the source. Current-mode approach For current-mode droop control, the output current reference is obtained by the local voltage measurement and is expressed as ∗ Idci =

∗ −V Vdc dci . kdi

(13.30)

Fig. 13.25 shows the current sharing concept when voltage drop on cable is ignored and the current sharing ratio among the sources is expressed as Idc1 : Idc2 : Idc3 =

1 1 1 : : , kd1 kd2 kd3

(13.31)

where kdi is the droop gain for the ith local controller. When voltage drop on cable is considered, current sharing among three subsystems can be written as Idc1 : Idc2 : Idc3 =

1 1 1 : : . kd1 + R1 kd2 + R2 kd3 + R3

(13.32)

Based on the aforementioned analysis, in both current-mode and voltage-mode droop-controlled systems the ratio of the source powers is not as desired due to cable resistances.

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As expressed in (13.29) and (13.32), increasing the droop coefficients (kd1 , kd2 , and kd3 ) is the way to eliminate power sharing error. However, the voltage regulation performance is poor with this approach, that is, the voltage drop is high under high droop gains.

13.4 Summary A detailed modal analysis of the DC grid onboard aircraft has been performed in this chapter. The PMSG-based generation system provides the electrical power from the engine shaft to the DC bus. ESS provides extra power when an emergency occurs. Resistive and constant power load are also introduced. After modal analysis of each component, the control method is discussed. For each subsystem, such as PMSG and ESS, the local controller is firstly analyzed. Then power sharing control is introduced to coordinate the aircraft electrical power system (EPS). Centralized control and distributed control show better performance because of proper coordination and leadership in small- or medium-scale system. To avoid failure due to communication, decentralized control can achieve high reliability and modularity and only depends on the local variables. However, the droop coefficient should be carefully chosen to balance the trade-off between voltage drop and power sharing accuracy.

References [1] K.J. Karimi, The role of power electronics in more-electric airplanes (MEA). Presented at the Workshop on Computers in Power Electronics, 2006. [2] E.Q. Richard Jr., More Electric Aircraft, 1993. [3] I. Moir, A. Seabridge, Aircraft Systems: Mechanical, Electrical, and Avionics Subsystems Integration, 3rd ed., John Wiley & Sons, 2008. [4] M. Sinnett, 787 no-bleed systems: saving fuel and enhancing operational efficiencies, Boeing Aero Mag. 4 (2007) 6–11. [5] P. Wheeler, S. Bozhko, The more electric aircraft: technology and challenges, IEEE Electrification Magazine 2 (4) (Dec. 2014) 6–12. [6] B.H. Nya, J. Brombach, D. Schulz, Benefits of higher voltage levels in aircraft electrical power systems, in: Proc. Electrical Systems for Aircraft, Railway and Ship Propulsion (ESARS), Bologna, Oct. 2012, pp. 1–5, 230. [7] N. Mohan, W.P. Robbins, T.M. Undeland, R. Nilssen, O. Mo, Simulation of power electronic and motion control systems – an overview, Proceedings of the IEEE 82 (1994) 1287–1302. [8] S.V. Bozhko, T. Wu, C.I. Hill, G.M. Asher, Accelerated simulation of complex aircraft electrical power system under normal and faulty operational scenarios, in: IECON 2010 – 36th Annual Conference on IEEE Industrial Electronics Society, 2010, pp. 333–338.

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

[9] Martin R. Kuhn, Martin Otter, Loic Raulin, A multi level approach for aircraft electrical system design. Presented at the International Modelica Conference, 2008. [10] S. Chiniforoosh, J. Jatskevich, A. Yazdani, V. Sood, V. Dinavahi, J.A. Martinez, A. Ramirez, Definitions and applications of dynamic average models for analysis of power systems, IEEE Transactions on Power Delivery 25 (2010) 2655–2669. [11] S. Jian, Unified averaged switch models for stability analysis of large distributed power systems, in: Applied Power Electronics Conference and Exposition, 2000. APEC 2000. Fifteenth Annual IEEE, 2000, pp. 249–255. [12] B. Lehman, R.M. Bass, Switching frequency dependent averaged models for PWM DC-DC converters, IEEE Transactions on Power Electronics 11 (1996) 89–98. [13] P.C. Krause, O. Wasynczuk, S.D. Sudhoff, Analysis of Electric Machinery and Dirve Systems, Wiley InterScience, 2002. [14] S.V. Bozhko, T. Wu, T. Yang, G.M. Asher, More-electric aircraft electrical power system accelerated functional modeling, in: Power Electronics and Motion Control Conference (EPE/PEMC), 2010 14th International, 2010, pp. T9-7–T9-14. [15] T. Wu, S. Bozhko, G. Asher, P. Wheeler, Fast Reduced Functional Models of Electromechanical Actuators for More-Electric Aircraft Power System Study, SAE Technical Paper 2008-01-2859, 2008. [16] T. Wu, S.V. Bozhko, G.M. Asher, D.W.P. Thomas, Accelerated functional modeling of aircraft electrical power systems including fault scenarios, in: Industrial Electronics, 2009. IECON’09. 35th Annual Conference of IEEE, 2009, pp. 2537–2544. [17] Cleansky, AEGART project summary, available: http:// cordis.europa.eu/result/rcn/143557_en.html. (Accessed 11 November 2015). [18] G. Buticchi, L. Costa, M. Liserre, Improving system efficiency for the more electric aircraft: a look at dc/dc converters for the avionic onboard dc microgrid, IEEE Industrial Electronics Magazine 11 (3) (2017) 26–36. [19] R. Huggins, Energy Storage: Fundamentals, Materials and Applications, second edition, Springer International Publishing, Switzerland, 2016, p. 509. [20] H. Chen, X. Zhang, J. Liu, C. Tan, Compressed air energy storage, in: Energy Storage-Technologies and Applications, InTech, 2013, pp. 101–112. [21] T. Zhang, J. Mao, X. Liu, M. Xuan, K. Bi, X.L. Zhang, J. Hu, J. Fan, S. Chen, G. Shao, Pinecone biomass-derived hard carbon anodes for high-performance sodium-ion batteries, Royal Society of Chemistry Advances 7 (66) (2017) 41504–41511. [22] J. Töpler, J. Lehmann, Hydrogen and Fuel Cell, Springer, Berlin/Heidelberg, Germany, 2016. [23] C.A. Ramos-Paja, A. Romero, R. Giral, J. Calvente, L. Martinez-Salamero, Mathematical analysis of hybrid topologies efficiency for PEM fuel cell power systems design, International Journal of Electrical Power & Energy Systems 32 (9) (2010) 1049–1061. [24] J. Carroquino, R. Dufo-López, J.L. Bernal-Agustín, Sizing of off-grid renewable energy systems for drip irrigation in Mediterranean crops, Renewable Energy 76 (2015) 566–574. [25] H. Zhao, Y. Hou, Y. Zhu, L. Chen, S. Chen, Experimental study on the performance of an aircraft environmental control system, Applied Thermal Engineering 29 (16) (2009) 3284–3288. [26] Hongsheng Jiang, Sujun Dong, Helin Zhang, Energy efficiency analysis of electric and conventional environmental control system on commercial

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aircraft, in: 2016 IEEE International Conference on Aircraft Utility Systems (AUS), 2016, pp. 973–978. [27] D. Barater, et al., Multistress characterization of fault mechanisms in aerospace electric actuators, IEEE Transactions on Industry Applications 53 (2) (March–April 2017) 1106–1115. [28] S.S. Yeoh, M. Rashed, M. Sanders, S. Bozhko, Variable-voltage bus concept for aircraft electrical power system, IEEE Transactions on Industrial Electronics 66 (7) (July 2019) 5634–5643.

Index

367

Index A AC/AC converter, 4 AC/DC converter, 4, 346 Active-front-end converter (AFE), 346–348 Adaptive control, 300, 304, 306 Adaptive controller, 302, 304 Advanced control technologies, 325, 337 Aggregated arm model (AAM), 103 Aircraft systems, 3, 7 Alternating current (AC), 289 Approximate Bayesian computation (ABC), 183 Artificial neural networks, 291 Assignable equilibrium point, 296, 297 Asynchronous AC systems, 96 Automotive hybrid powertrains, 249 traction systems, 247

B Base injected power, 184, 187 power, 249 values, 85–87, 98 Battery bank, 350, 351 charging/discharging performance, 223 converter, 227, 232 dynamics, 223, 224 energy, 236 lifetime, 241 pack, 247 system, 218, 223, 233, 247 voltage, 227, 228, 350 Battery electric vehicle (BEV), 246 Battery storage system (BESS), 132, 135, 142

Bayesian approximation (BAp), 184–187 Bhattacharyya distance (BD), 186–188 Bidirectional DC/DC converter, 340, 349, 350 power flow, 290, 326 Boost converter, 248, 251, 253, 255, 258–261, 263, 265, 267–269, 271–275, 279, 280, 290, 294, 295, 304, 322, 323 Buck converter, 251, 253, 254, 257–261, 264–266, 268, 269, 271, 273–275, 279, 280, 290, 291, 293, 295, 302–304 Buck mode, 264 Bus voltage, 196, 331, 351 DC, 324, 330, 346, 362 offsets, 331

C Cable capacitance, 107 characteristics, 49 dynamics, 88, 149, 151 extremities, 88 faulted, 39 full model, 171 inductor, 30 interconnections, 90, 113 model, 84, 85, 88, 107, 118, 149, 152, 154, 155, 157 modeling, 111, 150, 169 parameters, 96 resistance, 30, 363 subsystem models, 110 terminal, 41, 49 Capacitance cable, 107 contribution, 83 equivalent, 42, 43, 83, 84, 89, 106, 108 values, 254

Capacitor DC, 30, 42, 44, 47, 54, 137, 138, 230, 232 DC link, 318 discharge current, 61 discharge phase, 30 energy, 134, 135 voltage, 30, 73, 84, 103–105, 253, 283, 293, 302, 316, 321, 322, 324, 332 amplitude, 321 dynamics, 105 Cell terminal variables, 342 Centralized control, 357–359, 364 Centralized controller, 357, 359, 360 Characteristic admittance, 150, 167–169 Circulating currents, 33, 43, 103–106 Communication systems, 142, 143 Constant power load (CPL), 217, 315, 319, 340, 354 Constant voltage, 7, 196, 201, 304 Control applications, 290 complexity, 215, 328, 329 converter, 76, 82 DC faults, 57 DC link, 356 DC voltage, 107, 115 delay, 327, 330, 332 design, 33, 292, 298, 299, 301, 307, 308, 354 development, 354 droop, 33, 134, 226, 313, 322, 360 for multiterminal DC networks, 216 frequency, 136, 313 gains, 239

368

Index

hysteresis, 330, 332 increment, 324 inertia, 330, 332 input, 217, 257, 293, 296, 297, 299, 300, 307 law, 258, 260, 291, 292, 296, 300, 307 levels, 33, 219 loops, 76, 83, 96, 106, 355 method, 142, 357, 364 microgrids, 331 objectives, 33, 97, 107, 115, 289, 313, 321, 328, 332 optimal, 216, 330 output, 290 PI, 20, 291 plant, 355 platform, 222 power, 14, 107, 115, 330 power flow, 32 reactive power, 313 scheme, 134, 217, 218, 312, 324, 330, 346, 354 secondary, 241, 290, 329 signals, 357 specifications, 222 strategy, 7, 19, 33, 61, 96, 130, 133, 134, 136, 256, 283, 315, 325, 332 structure, 105, 137 supervisor, 233, 235 surfaces, 258, 352 system, 33, 70, 73, 85, 97, 101, 103, 105, 106, 113, 132, 135, 139, 204, 215, 218–220, 230, 232, 238, 340 theory, 81, 290 variables, 255, 257, 258, 293 voltages, 3, 315 Controllable current source, 354 loads, 289, 312 variable, 328 Controller DC, 217 DC voltage, 119, 123 optimal, 222, 322 parameters, 70, 81, 113, 124, 238, 241

PI, 220, 229, 259, 299, 355, 357 secondary, 216, 218, 219, 222, 234, 235 supervisor, 216, 222 tertiary, 219, 222, 232, 241 tuning, 81 Converter AC/AC, 4 AC/DC, 4, 346 arbitrarily sets, 219 arm inductance, 32 battery, 227, 232 bridge, 30 cable resistance, 362 control, 76, 82 control loops, 70 controllers, 76 currents, 76 DC/DC, 4, 258, 340, 342, 349, 350 failure, 41 in wind turbines, 19 inductor, 351 inductor current, 258 lifetime, 253 model, 99, 316 operation mode, 330 output current, 350 output voltage, 225, 227 power, 31, 32, 53, 56, 232, 247, 296, 313–316, 324, 328, 330–332 stations, 113 switches, 315 switching behavior, 341 terminals, 30, 71, 82–87, 89, 90, 96–99, 103, 107, 111, 113, 115, 119, 121, 124 topology, 96, 97, 342 units, 70 valves, 135 Cost function (CF), 315, 316, 318, 327 Countervoltage, 60 Crossover frequency (CF), 265 Current source converter (CSC), 325, 326

D DC/DC converter, 4, 258, 340, 342, 349, 350 Deadbeat predictive control, 325, 328, 332 Decentralized control, 357, 360, 364 Decentralized model predictive controller (DMPC), 322 Direct current (DC) breakers, 53–56 bus, 251, 313, 331, 340, 351, 357, 362, 364 bus capacitor, 83 bus voltage, 324, 330, 346, 362 cable, 29, 34, 82, 107, 351 cable faults, 29 capacitor, 30, 42, 44, 47, 54, 137, 138, 230, 232 controller, 217, 355 fault, 28–31, 33, 41, 47, 53–55, 57–61 currents, 49, 56, 57, 59 detection, 28 protection, 28, 53 grid, 2, 5, 30–32, 34, 49, 51, 56, 83, 94, 96, 110, 124, 175–179, 181, 183, 201, 206, 290, 344, 354, 357, 364 grid configuration, 41, 49, 90 grid hierarchical control, 3 inductor, 317 inductor current, 317 line faulted, 28, 54 link, 135, 218, 224, 230, 232, 315, 319, 351 capacitor, 318 control, 356 dynamics, 317 voltage, 313, 317, 318 load voltage, 326 microgrid, 82, 175, 180, 181, 184, 188, 189, 195, 217–219, 223, 224, 231, 232, 238, 313–315, 318, 319, 322, 324, 325, 327–331 control level, 332 predictive control, 314

Index

secondary control, 329 systems, 314, 332 networks, 195, 196, 203, 289–291 node, 83–90, 96–99, 105, 107, 108, 113 interfaces, 89 voltages, 87, 118 power, 73, 196, 339, 346, 356 converters, 346 grids, 176 network, 218, 230 systems, 69 reactor voltage, 53 side, 136, 138, 141, 220, 317, 320 side voltage, 136, 321, 332 systems, 32, 64, 195, 325, 360 transmission cable, 30 voltage, 30, 33, 35, 37, 53, 55, 83, 84, 86, 103, 107, 115, 121, 124, 135–141, 217, 219, 221, 222, 235, 236, 239, 312, 317, 330 control, 107, 115 controller, 119, 123 regulation, 238 Discontinuous conduction mode (DCM), 323 Discrete-time sliding-mode current control (DSMCC), 258, 268, 269, 272, 273, 279 Distributed generation (DG), 312, 357 Double fed induction generator (DFIG), 15 Double thyristors, 59 Droop coefficients, 238, 241, 362, 364 control, 33, 134, 226, 313, 322, 360 controllers, 218, 221, 222, 241 Duty cycle, 255, 258, 261, 262, 264, 293, 302, 304, 305, 342 Dynamics battery, 223, 224 cable, 88, 149, 151 capacitor voltage, 105

DC link, 317 linear, 78 local, 121 voltage, 89

E Electric aircraft DC grid, 357 grid, 341 loads, 353 power, 17, 31, 175, 339, 351, 352 vehicle powertrain, 253 Electric motor (EM), 247 Electric power system (EPS), 339 Electric vehicle (EV), 1, 7, 246, 247, 251, 281 Electrical grid, 196, 231 loads, 353 networks, 130, 204, 289, 292 power, 3, 311, 339, 340, 346, 364 grid, 70 inertia systems, 130 subsystems, 325 systems, 72, 129, 312, 315 Electrical power system (EPS), 129, 311, 312, 364 Electro-mechanical actuator (EMA), 346, 352, 353 Electronic loads, 315, 319 power converters, 296, 342 power interfaces, 133 Energy storage, 72, 133, 196, 216, 223, 296, 340, 349 Energy storage device (ESD), 6, 7, 196, 198, 290, 296, 349 Energy storage system (ESS), 195, 219, 289, 312, 313, 330, 339, 340, 349, 357, 364 Environmental control system (ECS), 351, 352 Equilibrium point, 21, 76, 77, 79, 91, 92, 112, 123, 254, 255, 298

369

Equivalent capacitance, 42, 43, 83, 84, 89, 106, 108 capacitive energy storage, 106 energy storage dynamics, 106

F Fault clearing process, 28 current, 28, 30, 43, 54, 55, 59–64 detection, 28, 31, 49 detection algorithms, 28 distance, 52 generated surges, 52 location, 49, 52, 53 locator, 52 occurrence, 30 propagation, 28 resistance, 31, 47, 48 transient, 36 Fault current limiter (FCL), 61, 63 Faulted cable, 39 current, 29 DC, 28, 54 line, 28, 53–56 Field-oriented control (FOC), 346 Filter capacitor voltage, 220, 227 Finite control set model predictive control (FCS-MPC), 315, 325, 332 hybrid, 325 Flight controls, 351–353 Forward neural network (FNN), 179, 184, 186 Frequency balance, 133 changes, 138 control, 136, 313 deviation, 130, 134, 140, 141 domain, 150 excursions, 130 grid, 73, 225 nadir, 139, 143 operation, 302 regulation, 289, 312

370

Index

response, 132, 135, 136, 151 stability, 130, 139 synchronization, 313 variations, 135, 138 Fuel cell electric vehicle (FCEV), 246

G Global positing system (GPS), 52 Grid electric, 341 electrical, 196, 231 electrical power, 70 frequency, 73, 225 frequency changes, 136 hybrid, 216 inertia control improvement, 132 multiterminal HVDC, 216 operation, 290 system, 357 voltage, 84

H High-voltage direct current (HVDC) applications, 41, 64 cable, 51, 71, 84, 96, 107, 115, 123, 124, 149, 150, 154, 155, 164, 172, 174 cable in VHDL, 167 circuit breakers, 59 converter terminal, 105 distribution, 339 grids, 31, 71, 82, 90, 103, 107, 187 interfaces, 129 line, 20, 23, 52 multiterminal, 5, 27, 125 systems, 5, 19, 20, 23, 27, 29, 70, 71, 82, 86, 87, 124, 134–136 transmission, 2, 5, 13, 14, 18, 19, 59, 64, 69, 134, 135, 216, 339

transmission lines stability, 19 transmission systems, 71, 83, 118 VSC, 30, 70, 130, 135, 137–140, 142, 143 Hybrid control approach, 329 FCS-MPC, 325 grid, 216 HVDC circuit breaker, 59, 60 systems, 250 Hybrid electric vehicle (HEV), 246, 247 Hydraulic powered actuators, 346 Hyperbolic linearization, 202, 203

I Inductor cable, 30 converter, 351 current, 253–255, 259, 260, 263, 264, 267, 269, 275, 281, 282, 293, 302–304, 316, 319, 321–325 ripple, 255 situation, 323 waveform, 258 DC, 317 terminations, 53 Inertia control, 330, 332 emulation, 2, 134, 136, 137 emulation control, 133 HVSC, 139 reduction, 132 service, 137 value, 135 Inertial response, 134, 332 Injected powers, 182, 186, 198 Instantaneous power balance, 97, 99 Insulated Gate Bipolar Transistor (IGBT), 4, 19, 30, 41, 55, 57, 60, 342 Intermittent renewable energy sources, 330

Internal combustion engine vehicle (ICEV), 246 Islanded network, 232, 235 Isolated Gate Bipolar Transistor (IGBT), 133

J Jacobian approximate Bayesian computation (JABC), 184–188

L LC circuit, 59, 60, 62, 103 Linear controllers, 332 dynamics, 78 systems, 70, 71 Linearization point, 78, 92, 112, 113, 200, 202 Loads electric, 353 electrical, 353 power, 196, 198, 201, 315, 319, 322, 325, 340, 354, 364 variables, 329 Local control functions, 313 dynamics, 121 operation, 315 variables, 87, 364 Local controller (LC), 216, 219, 222, 314, 354, 357, 359, 360, 364 Lumped inductor model, 108

M Maximum permitted voltages, 198 power, 17, 135, 290 voltage, 345 Maximum Power Point Tracking (MPPT), 134 Mechanical power, 325, 337 switch, 59, 60 Meshed MTDC systems, 27

Index

Microgrid (MG), 312 components, 329 control, 331 control system, 219 description, 218 level, 313 nature, 330 net cost, 216 operation, 232, 236 system, 324 Model predictive control (MPC), 3, 216, 315 Model predictive controller, 232, 235, 236 Modular multilevel converter (MMC), 2, 19, 32, 33, 40, 49, 69, 102, 133, 135 Modulated model predictive control, 319, 322, 332 Modulation index (MI), 247 Monte Carlo simulation (MCS), 175, 177, 184, 185 More Open Electrical Technologies (MOET) aircraft, 343 More-electric aircraft (MEA), 325, 337, 351–353 Multiple cascaded control loops, 70 control methodologies, 290 control objectives, 315 operation blocks, 167 Multiterminal HVDC, 5, 27, 125 grid, 216 grid operation, 70 networks, 2 systems, 1, 114, 124 transmission system, 82, 124 systems, 2, 86 Multiterminal DC (MTDC), 27, 28, 34, 48, 49, 51, 53, 133, 139, 140 grid, 71, 85, 94 grid share power imbalances, 140 grid stability, 217 links, 130, 134

multiterminal systems, 135 network, 140 protection system, 28 systems, 27, 28, 49, 53, 134 VSC, 142

N Nested control loops, 253, 282 Network DC power, 218, 230 MTDC, 140 voltage, 219 Neutral point clamped (NPC) converter, 346 Nodal voltage, 4, 176, 177, 182, 189 voltage variables, 198 Noncontrolled variable, 300 Noncritical loads, 331 Nonlinear control, 296 dynamics, 17, 315 Nonlocal variables, 170, 171

O Offshore renewable power generation, 70 wind farms, 2, 5, 13, 32, 34, 186, 217 Onboard loads, 340 power demand, 349 Onshore network frequency deviations, 137 Operation costs, 219 frequency, 302 grid, 290 local, 315 microgrids, 232, 236 type, 7 Operational condition, 249 cost, 219 limits, 236 modes, 323

371

Optimal control, 216, 330 controller, 222, 322 state switching, 324 switching, 324, 327 Optimal power flow (OPF), 196, 197, 209 analysis, 196 problem, 196, 199, 200, 202, 204, 205, 207 problem in DC networks, 196

P Parallel converters, 313, 315, 354 Parametric sensitivity, 72, 81, 123, 124 Parametric sensitivity analysis, 114 Participation factors, 72, 81, 115, 119, 123, 217 Passivity-based controller (PBC), 291, 292 Permanent magnet synchronous generator (PMSG), 15, 339, 344, 349, 364 Permanent magnet synchronous machine (PMSM), 340, 352, 355 Phase Locked Loop (PLL), 133 Photovoltaic systems, 133, 195 PI control, 20, 291 control output, 137 control structure, 291 controller, 220, 229, 259, 299, 355, 357 PI current control (PICC), 225, 227, 273, 275 Plug-in hybrid electric vehicle (PHEV), 246, 247 Point estimate method (PEM), 177, 178, 184 Point of common coupling (PCC), 312 voltage, 220, 226 Point-of-load (POL) converter, 315

372

Index

Port-controlled Hamiltonian (PCH) model, 292 model bilinear, 292 representation, 292, 306 structure, 291, 296, 308 structure bilinear, 296 system, 297 system bilinear, 296, 297 Power balance, 106, 197, 348 constraints, 198 equations, 196, 198, 199, 201, 325 base, 249 cables, 339 consumption, 208, 351 control, 14, 107, 115, 330 control electronic converters, 292 conversion, 97 converter, 31, 32, 53, 56, 232, 247, 296, 313–316, 324, 328, 330–332 DC, 73, 196, 339, 346, 356 DC systems, 178 demand, 198 demand consumption, 218 electric, 17, 31, 175, 339, 351, 352 electrical, 3, 311, 339, 340, 346, 364 electronics, 1–3, 27, 129, 143, 256, 325, 337, 351–353 devices, 55, 256 devices duty cycle, 350 interfaces, 130, 143, 313 systems, 315 flow, 2, 3, 115, 177–179, 183, 189, 200, 202, 208, 210, 331, 332, 340 analysis, 33, 74, 177 computation, 177, 179 for DC grids, 196 for DC networks, 196 model, 178 problems, 186, 189, 201 fluctuation, 330, 331 generation, 344

generation control, 313 grids, 3, 175 imbalance, 131, 330 injections, 133, 179–181, 183 inverter, 247 levels, 69, 350 line, 313 loads, 196, 198, 201, 315, 319, 322, 325, 340, 354, 364 losses, 57, 61, 62, 196, 197, 205, 210, 293, 331, 348 management, 357 maximum, 17, 135, 290 microsources, 330 outputs, 140, 208, 209, 363 plants, 70, 136, 296 rated, 232, 239, 358 ratings, 87, 115, 346 requirement, 339 sharing, 322, 356, 360, 364 sharing control, 357, 364 sharing ratio, 363 source, 326 stations, 42 system, 1, 17, 70, 117, 176, 178, 204, 217, 218, 223, 246, 341, 355 system context, 70 trading, 27 transfer, 27, 28, 70 transmission, 27 transmission cables, 339 Power Electronic Converter (PEC), 3, 5, 16, 18, 74, 195, 196, 289, 290, 315, 340 Powertrain configurations, 247 topologies, 251, 281 Prediction horizon, 217, 218, 223, 233, 235–237, 324, 325 Predictive control, 140, 241, 256, 314, 322, 324, 325, 329–332 control schemes, 320 controller, 222, 234, 322, 331 controller supervisor, 235 Predictive digital current programmed control (PACC), 260

Probabilistic power flow (PPF), 175 problem, 176–178, 183, 187 solutions, 176 Proportional–integral (PI), 291 Proportional–integral–derivative (PID) design, 291 Protection system, 28, 30, 53–55, 64 for HVDC applications, 28 for MTDC plays, 28 MTDC, 28 Proton exchange membrane fuel cell (PEMFC), 248, 249 Pulse Width Modulation (PWM), 4, 101, 133 Pulse-width-modulated current-source converter (PWM-CSC), 19

R Railway networks, 216 system, 218, 219, 229, 232, 233, 235 Random variables, 177–180, 185, 187 Rate of change of frequency (RoCoF), 131, 132, 134, 138 Rated phase voltage, 248 power, 232, 239, 358 voltage, 362 Reactive power, 3, 33, 86, 133, 195, 226, 289, 290, 339 compensation, 4 control, 313 controllers, 119 flows, 312, 313 Regulation DC voltage, 238 error, 256, 257 frequency, 289, 312 passive output, 257 voltage, 198, 312, 315, 357, 359, 364 Relative error (RE), 186 Renewable energy sources, 5, 6, 33, 215, 216, 289, 313, 330

Index

Renewable energy technologies (RET), 311 Resistive load, 198, 223, 254, 255, 291, 292, 340, 354

S Sampling frequency, 52 time, 28, 52, 166, 315, 318, 323, 325, 327, 328 Secondary control, 196, 241, 290, 329 controller, 216, 218, 219, 222, 234, 235 level control, 332 Short circuit ratio (SCR), 46–48 Shunt capacitance, 82 Single PMSG control, 354 Singular value decomposition (SVD), 217, 218, 239 Smart grids, 216, 290, 331 loads, 331 Smart Energy Integration Lab (SEIL), 232 Source frequency, 329 powers, 363 Stability analysis, 2, 14, 70, 238, 241, 291 conditions, 5 enhancement, 289 frequency, 130, 139 limit, 80, 124, 218 margins, 124, 222 problems, 71, 81 properties, 78, 238, 298 voltage, 140 State equations, 73, 82, 86, 88, 90, 99, 105, 106, 108, 110, 111, 342 feedback control law, 217 variables, 20, 72, 73, 77, 80, 81, 85, 92, 97, 98, 106, 108, 112, 114, 152, 175, 176, 178, 183, 290, 296, 324, 330

State of charge (SOC), 350 Stator voltage, 345, 356 Stepup converter, 255 Submarine cables, 30 Subsystems electrical, 325 models, 71, 83, 85, 87, 90, 92, 93, 96, 111, 113, 114 Supercapacitors, 132, 249, 290 Supergrid, 70 Supervisor control, 233, 235 controller, 216, 222 predictive controller, 235 Switch configurations, 316 position, 254, 323 Switched controlled systems regulation, 256 Symbolic linearization, 111 Synchronous AC grid, 96 generators, 129, 130, 132, 138, 140 generators conventional physical inertia, 132 Synchronous reference frame (SRF), 220 Synchronously rotating reference frame (SRRF), 74 Synthetic inertia, 135, 143 Systems DC, 32, 64, 195, 325, 360 DC microgrid, 314, 332 DC power, 69 electrical, 72, 129, 312, 315 electrical power, 129, 311, 312 energy storage, 195, 219, 289, 312, 313, 330, 339, 340, 349 HVDC, 19, 20, 23, 27, 29, 70, 71, 82, 86, 87, 124, 134–136 hybrid, 250 linear, 70, 71 MTDC, 27, 28, 49, 53, 134 multiterminal, 2, 86 multiterminal HVDC, 1, 114, 124

373

power, 1, 17, 70, 117, 176, 178, 204, 217, 218, 223, 246, 341, 355 power electronics, 315 VSC HVDC, 130, 133, 135

T Terminal converter, 30, 71, 82–87, 89, 90, 96–99, 103, 107, 111, 113, 115, 119, 121, 124 HVDC grids, 184, 186 Tertiary control, 196, 241 controller, 219, 222, 232, 241 Thermal control, 352 inertia, 135 Thyristors, 19, 58, 59 Time-weighted absolute error (ITAE), 302 Transportation systems, 2, 3, 7 Tuned LC circuit, 62

U Ultracapacitors, 349 Undervoltage, 30 DC voltage, 51 protections, 51 Unidirectional power flow, 326 Unilinear power flow, 311 Universal line model (ULM), 107

V Variables control, 257, 258 loads, 329 local, 87, 364 speed operation, 14 state, 20, 72, 73, 77, 80, 81, 85, 92, 97, 98, 106, 108, 112, 114, 152, 175, 176, 178, 183, 290, 296, 324, 330 voltage, 198, 201 Vector control, 133, 330

374

Index

VHDL, 149, 150, 154, 155 conceptual design, 154 description, 150 language, 151 library, 155 package, 154 Virtual inertia, 130 inertial delivery, 130 Voltage amplitudes, 346 battery, 227, 228, 350 bias, 3 capacitor, 30, 84, 103–105, 253, 283, 293, 302, 316, 321, 322, 324, 332 characteristics, 249 control, 315 controllers, 220, 221, 230, 362 DC, 30, 33, 35, 37, 53, 55, 83, 84, 86, 103, 107, 115, 121, 124, 135–141, 217, 219, 221, 222, 235, 236, 239, 312, 317, 330 DC link, 313, 317, 318 drop, 30, 37, 62, 362–364 dynamics, 89

error, 325 fluctuations, 331 grid, 84 input, 87 level, 218 limit set, 356 loop, 251, 265 magnitudes, 53, 177, 183 maximum, 345 measurement, 363 mode, 253, 283 output, 291 profiles, 195, 196, 199, 201, 203, 210, 290 rated, 362 rating, 87, 98 regulation, 198, 312, 315, 357, 359, 364 regulators, 218 ripples, 254, 255 sags, 330 source, 103, 262, 269, 275, 317, 350 stability, 140 stability issues, 216 transformers, 52 variables, 198, 201 variation, 51

variation range, 135 vectors, 217, 315 Voltage-source converter (VSC), 2, 4, 19, 27, 28, 53, 54, 69, 97, 218–221, 230–232, 239, 316, 330 capacitors, 139 controls, 134, 137 HVDC, 30, 70, 130, 135, 137–140, 142, 143 control approaches for inertia emulation, 132 converters, 69 systems, 130, 133, 135 transmission stability analysis, 70 MTDC, 142 terminal, 89, 115, 117

W Wind power, 13, 188, 189 power plant, 134, 136, 137 turbine, 14–17, 130, 134–136, 291, 313, 314, 330 Wing ice protection system (WIPS), 353