Control of Large Wind Energy Systems: Theory and Methods for the User (Advances in Industrial Control) 3030848949, 9783030848941

Table of contents :
Series Editor’s Foreword
Preface
Acknowledgements
Contents
Notation
Variables, Functions and Parameters
Polynomial and Transfer Functions
Greek Variables
Meaning of Subscripts
Abbreviations
1 Overview
1.1 Introduction
1.2 Control of Modern Wind Turbines
1.3 Motivation
1.4 Scope of the Book
1.4.1 Part I: Modelling Wind Turbines for Control Purposes
1.4.2 Part II: Control System Design
1.4.3 Part III: Implementation and Case Study
References
Part I Modelling of Wind Turbines
2 Fundamentals of System Modelling
2.1 Introduction
2.2 Modelling Approaches
2.2.1 Classification by Model Classes
2.2.2 Classification by Modelling Approaches
2.2.3 Classification by the Available Knowledge
2.3 Dynamic Modelling of a System for Control Purposes
2.3.1 Definition of the System Topology
2.3.2 System Decomposition
2.3.3 Degrees of Freedom
2.4 Modelling Aspects of Wind Turbines
References
3 Modelling the Aerodynamic Subsystem
3.1 Introduction
3.2 Actuator Disc Model
3.3 Determination of the Aerodynamic Coefficients
3.3.1 Exponential Approximation
3.3.2 Trigonometric Approximation
3.3.3 Polynomial Approximation
3.4 Neural Network Approach
3.4.1 Artificial Neural Network Formulation
3.4.2 Aerodynamic Coefficients of Wind Turbines Adjusted by an ANN
3.5 Blade Element Momentum Theory
3.5.1 Classic Approach of the BEM Theory
3.5.2 Corrections to Improve the Model
3.5.3 Final Algorithms
References
4 Modelling the Rotating Subsystem
4.1 Introduction
4.2 Modelling the Rotor as Single-Mass System
4.2.1 Three-Mass Rotating Subsystem
4.2.2 Two-Mass Rotating Subsystem
4.2.3 One-Mass Rotating Subsystem
4.3 Modelling the Rotor as Multi-mass System
4.3.1 Two-Mass Rotor Representation
4.3.2 Three-Mass Rotor Representation
4.3.3 Four-Mass Rotor Representation
4.4 Modelling the Gearbox
4.5 Modelling with the Tip-Speed Ratio as State Variable
4.6 Blade Stiffening Due to Centrifugal Forces
References
5 Modelling the Tower Subsystem
5.1 Introduction
5.2 Simplified Modelling of the Tower
5.3 Modelling the Fore–Aft Dynamics Using Modal Analysis
5.4 Modelling the Blade–Tower Interaction
5.4.1 Approach Based on the Two-Mass Rotor Model
5.4.2 Approach Proposed in [4]
5.4.3 Approach Inspired in the Inverted Pendulum
5.5 Forces Actuating on the System
References
6 Modelling of Actuators and Generators
6.1 Introduction
6.2 Simplified Modelling of Pitch Actuators
6.2.1 First-Order Pitch Actuator
6.2.2 Second-Order Pitch Actuator
6.2.3 Actuators Based on Pitch Rate
6.3 Physical Modelling of Pitch Actuators
6.3.1 Hydraulic Pitch Actuators
6.3.2 Electric Pitch Actuators
6.4 Modelling the Yaw System
6.5 Electric Modelling of the Generator Subsystem
6.5.1 Simplified Generator Model
6.5.2 Generator Model Based on Physical Laws
References
Part II Control System Design
7 Overview of Wind Turbine Control
7.1 Introduction
7.2 Control Objectives
7.2.1 Maximization of Power Capture
7.2.2 Power Quality
7.2.3 Profit Maximization
7.2.4 Reduction of Vibrations
7.2.5 Improvement of Dependability
7.2.6 Reduction of Loads
7.3 Systemic Formulation of the Wind Converter System
7.4 Hierarchical Decomposition of the Control System
7.5 Control Strategies
7.5.1 Basic Control Strategies
7.5.2 Strategies for Variable Speed Variable Pitch Wind Turbines
References
8 Supervisory Control
8.1 Introduction
8.2 Description of Operational States of Wind Turbines
8.3 Design of a Simple Supervisory Control System
8.3.1 Operation in the Production States
8.3.2 Operation in the Shutdown State
8.4 Detailed Supervisory Control Scheme
8.5 Transition Between Operational States
References
9 Control in Partial Load Operation
9.1 Introduction
9.2 Indirect Power Control
9.2.1 Basic Approach of Optimal Torque Control
9.2.2 Optimal Torque Control with Acceleration Feedback
9.2.3 Optimal Torque Control with Electromagnetic Torque Feedback
9.2.4 Control with Tip-Speed Ratio Feedback
9.2.5 Control with Estimation of Aerodynamic Torque and Tip-Speed Ratio
9.3 Direct Power Control
9.3.1 Perturbation and Observation Control Technique (POC)
9.3.2 Power Signal Feedback Con-Trol (PSFC)
9.4 Damping of Vibrations
9.4.1 Increase the Damping by Filtering
9.4.2 Control with Increased Damping Coefficient
9.5 Control in the Transition Regions (I1/2 and II1/2)
9.5.1 Open-Loop Control Law
9.5.2 Closed-Loop Control Law
9.5.3 Control with Low Tip-Speed Constraint
9.6 Peak Shaving Control
9.7 Estimation of the Effective Wind Speed
9.7.1 Observer-Based Estimation with Unknown Input
9.7.2 Interval Observer-Based Estimation
9.7.3 Averaged Effective Wind Speed with Forgetting Factor
References
10 Control in Full Load Operation
10.1 Introduction
10.2 Control for Power Regulation
10.2.1 Collective Pitch Control
10.2.2 Controllers for the Implementation of CPC
10.2.3 Collective Pitch Control with Aerodynamic Power Limiter
10.2.4 Anti-windup Mechanism for the Collective Pitch Control
10.2.5 Adaptive Collective Pitch Control (Gain-Scheduling)
10.3 Active Damping Control of Vibrations
10.3.1 Active Tower Damping Control
10.3.2 Active Blade Damping Control
10.4 Control for Load Reduction
10.4.1 Individual Pitch Control with Coleman Transformation
10.4.2 Individual Pitch Control with Clarke Transformation
10.4.3 Mutivariable Pitch Control with Coleman Transformation
10.5 Control Improvement by Using Feedforward Control
References
11 Yaw Control and Shutdown Control
11.1 Introduction
11.2 Yaw Control
11.2.1 Approach Based on Direct Measurement of Wind Direction
11.2.2 Methods Based on Estimation of the Wind Direction
11.2.3 Methods Without Measurement of Wind Direction
11.2.4 Yaw Control for Load Reduction
11.3 Shutdown Control
11.3.1 Conventional Open-Loop Shutdown Control
11.3.2 Closed-Loop Shutdown Control
References
Part III Control Implementation
12 Parametrization and Reference Wind Turbines
12.1 Introduction
12.2 Parametrization of the Aerodynamic Subsystem
12.2.1 Aerodynamic Coefficients
12.2.2 Sensitivity Functions
12.3 Parametrization of the Rotating Subsystem
12.3.1 Estimation of the Rotor Moment of Inertia
12.3.2 Blades
12.3.3 Gearbox and Shafts
12.3.4 The Tower
12.4 Parametrization by Using the Steady-State Model
12.5 Reference Wind Turbines
12.5.1 General Specifications of the Reference Wind Turbines
12.5.2 Rotor Parameters and Characteristics
12.5.3 Blade Properties of the Reference Wind Turbines
12.5.4 Nacelle Properties of the Reference Wind Turbines
12.5.5 Drivetrain Properties of the Reference Wind Turbines
12.5.6 Tower Properties of the Reference Wind Turbines
References
13 Control Performance Evaluation
13.1 Introduction
13.2 Evaluation by Using Performance Indices
13.2.1 Classic Performance Indices
13.2.2 Performance Indices for Time-Limited Problems
13.2.3 Fractional Order Performance Indices
13.2.4 Objective Functions for Variables of Other Nature
13.3 Specific Control Evaluation for Wind Turbines
13.3.1 Performance Evaluation of Power Extraction Control Objective
13.3.2 Performance Evaluation for Load Reduction Control Objective
13.4 Performance Evaluation for the Protection of Actuators
References
14 Real-Time Control and Simulation
14.1 Introduction
14.2 Real-Time Systems
14.2.1 Definitions and General Aspects
14.2.2 Real-Time Operating Systems (RTOSs)
14.2.3 Real-Time Scheduling
14.2.4 Intertask Communication and Synchronization
14.3 Controller Realization
14.4 Filter Design and Implementation
14.5 Real-Time Simulation
14.6 Hardware in the Loop
References
15 Case Study: 20-MW Reference Wind Turbine
15.1 Introduction
15.2 Modelling the Wind Turbine
15.2.1 Aerodynamic Coefficients
15.2.2 Sensibility Functions
15.2.3 Rotor and Drivetrain
15.2.4 Tower
15.3 Observer Design
15.4 Experiments for the Simulation Environment
15.5 Control in Partial Load Region
15.5.1 Control Law for Region II
15.5.2 Control Law Region II1/2
15.5.3 Simulation Results
15.6 Collective Pitch Control Using a FO-NPI Controller
15.7 Individual Pitch Control Using FO-NPI Controller
15.7.1 Performance Evaluation
15.7.2 Simulation Results
References
Index

Citation preview

Advances in Industrial Control

Adrian Gambier

Control of Large Wind Energy Systems Theory and Methods for the User

Advances in Industrial Control Series Editors Michael J. Grimble, Industrial Control Centre, University of Strathclyde, Glasgow, UK Antonella Ferrara, Department of Electrical, Computer and Biomedical Engineering, University of Pavia, Pavia, Italy Editorial Board Graham Goodwin, School of Electrical Engineering and Computing, University of Newcastle, Callaghan, NSW, Australia Thomas J. Harris, Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada Tong Heng Lee , Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Om P. Malik, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada Kim-Fung Man, City University Hong Kong, Kowloon, Hong Kong Gustaf Olsson, Department of Industrial Electrical Engineering and Automation, Lund Institute of Technology, Lund, Sweden Asok Ray, Department of Mechanical Engineering, Pennsylvania State University, University Park, PA, USA Sebastian Engell, Lehrstuhl für Systemdynamik und Prozessführung, Technische Universität Dortmund, Dortmund, Germany Ikuo Yamamoto, Graduate School of Engineering, University of Nagasaki, Nagasaki, Japan

Advances in Industrial Control is a series of monographs and contributed titles focusing on the applications of advanced and novel control methods within applied settings. This series has worldwide distribution to engineers, researchers and libraries. The series promotes the exchange of information between academia and industry, to which end the books all demonstrate some theoretical aspect of an advanced or new control method and show how it can be applied either in a pilot plant or in some real industrial situation. The books are distinguished by the combination of the type of theory used and the type of application exemplified. Note that “industrial” here has a very broad interpretation; it applies not merely to the processes employed in industrial plants but to systems such as avionics and automotive brakes and drivetrain. This series complements the theoretical and more mathematical approach of Communications and Control Engineering. Indexed by SCOPUS and Engineering Index. Proposals for this series, composed of a proposal form downloaded from this page, a draft Contents, at least two sample chapters and an author cv (with a synopsis of the whole project, if possible) can be submitted to either of the: Series Editors Professor Michael J. Grimble Department of Electronic and Electrical Engineering, Royal College Building, 204 George Street, Glasgow G1 1XW, United Kingdom e-mail: [email protected] Professor Antonella Ferrara Department of Electrical, Computer and Biomedical Engineering, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy e-mail: [email protected] or the In-house Editor Mr. Oliver Jackson Springer London, 4 Crinan Street, London, N1 9XW, United Kingdom e-mail: [email protected] Proposals are peer-reviewed. Publishing Ethics Researchers should conduct their research from research proposal to publication in line with best practices and codes of conduct of relevant professional bodies and/or national and international regulatory bodies. For more details on individual ethics matters please see: https://www.springer.com/gp/authors-editors/journal-author/journal-author-helpdesk/ publishing-ethics/14214

More information about this series at https://link.springer.com/bookseries/1412

Adrian Gambier

Control of Large Wind Energy Systems Theory and Methods for the User

Adrian Gambier Fraunhofer Institute for Wind Energy Systems Bremerhaven, Bremen, Germany

ISSN 1430-9491 ISSN 2193-1577 (electronic) Advances in Industrial Control ISBN 978-3-030-84894-1 ISBN 978-3-030-84895-8 (eBook) https://doi.org/10.1007/978-3-030-84895-8 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of my parents, Emma and Mariano To Andrés and Nancy

Series Editor’s Foreword

Control engineering is a wide-ranging topic that is viewed rather differently by researchers and those that must commission and maintain control systems. The former often develop general algorithms with a strong underlying mathematical basis, whilst the latter have more immediate concerns over the limits of equipment, quality of control, safety and security, and plant downtime. The series Advances in Industrial Control attempts to bridge this divide and hopes to encourage the adoption of advanced control techniques if they are likely to be beneficial. The rapid development of new control theory and technology has an impact on all areas of engineering and applications. Many applications concern industrial manufacturing and process control problems but there are of course applications in bioengineering, medical systems, all areas of transport and many others. This monograph series has a focus on such applications since the rate of technological development continually provides new challenges. There is also a gradual change away from the traditional view of control engineering design into the more computer-focused subject of systems engineering. These developments require new solutions and stimulate the development of new control algorithms. The focus on applications is also desirable if the different aspects of the “control design” problem are to be explored with the same dedication that “control synthesis” problems have received in the past. The series provides an opportunity for researchers to present new work on the solution of industrial control and applications problems. It raises awareness of the substantial benefits that advanced control can provide without ignoring the difficulties that can sometimes arise. This monograph is concerned with the very important and popular topic of wind turbine modelling and control. It provides a very complete exposition of the subject, and it is the result of teaching the Control of Wind Turbines at Leibniz University Hannover where the pedagogical style was honed. The first part of the text is concerned with the modelling of wind turbine systems in some detail, and this is needed for the simulation and design studies that follow. The second part begins with a very useful overview of wind turbine control in Chap. 7. This covers the main control methods but also mentions problems like vibration- and load-reduction that are important design issues. Chapter 8 then looks at supervisory control issues that vii

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have perhaps received less attention in the literature but also have a critical role. Control during partial load operation is dealt with in Chapter 9 and in full load operation in Chap. 10. In the first case the ideal is to maximise the power generated and in the second the power must be limited to the rated value if components are not to be overstressed and damaged. The control approaches required are very different and practical solutions are described for both cases. Chapter 11 deals with topics that are also not so popular in academic papers but are significant, namely yaw control (alignment of the rotational axis with the wind direction) and shut-down control. The third part of the text considers the implementation of the control laws from a practical viewpoint. Chapter 12 concerns the modelling and parameterization aspects, since the simplest models that lead to reasonable design are desirable and the way these are parameterized determines the effectiveness of the resulting “design” model. Benchmarking and comparing performance is often very useful when the output of a system must be “the best” in some sense. Since wind turbines are built in large numbers it is worthwhile to look at control performance evaluation, considered in Chap. 13, which also deals with load reduction, fatigue, and damage. Real-time control and simulation including hardware in the loop testing are treated in Chap. 14. Finally, the case study in Chap. 15 provides typical results and illustrates the ideas presented. The text covers a wide range of topics in just sufficient depth to provide an intuitive understanding of the design problems. The author has therefore provided a very easy-to-read text that should be of great value to students or researchers entering the subject and for engineers in the wind energy industry. The book satisfies a real need and is a very timely addition to the Advances in Industrial Control series. Glasgow, UK July, 2021

Michael J. Grimble

Preface

The integration of renewable energy sources into power grids is a top goal in global strategic road maps for clean energy development. Wind energy converters, in particular, are nowadays major contributors to the renewable energy quota shared in the total energy mix of providers. Furthermore, they have a lot of potential for the future. However, this future will be characterized by rising energy demand, which will necessitate in turn larger wind energy converters. In this sense, the discussion today is about very large turbines, also known as multi-megawatt machines. In a simple walk through the history of the development of wind turbines, one can see the wealth of different ways proposed to extract energy from the wind. However, in the case of such large machines, it seems to be the case that the upwind, three-bladed, horizontal-axis variable-speed and variable-pitch systems have become established. An important contributor to the success of these machines is control engineering, since such systems will not be able to operate at all without an efficient and high-performance control system. Rapid technological advancement should be accompanied by constantly updated, specialized literature. Thus, there exists profuse literature in practically all fields of wind turbines, where there are essential works on wind turbine fundamentals, wind turbine design and environmental aspects. Books dedicated to general fundamentals include some chapters about essential aspects related to control but are insufficient for a concrete implementation of a control system. There are also some specialized books for the control of wind turbines, normally with a direct focus on a particular problem or on a particular control approach. This is so because technical and scientific work that would include all available developments and methods would be immensely extensive and would have the scope of an encyclopaedia. Hence, the present book has a more modest aim, which is to present the multifaceted aspects of the control field associated only with the upwind, three-bladed, horizontal-axis variable-speed and variable-pitch systems. A detailed examination of currents works reveals that a high number of studies fall into one of three categories: contributions from the wind energy discipline that need a “good” controller in order to be able to do the same studies, e.g., load calculation, contributions from the control systems community, which have developed a ix

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new control approach and want to demonstrate the controller performance by using a “good” application, i.e., a very simple model of a wind turbine and finally, contributions that intend to solve control problems of wind turbines by using advanced and innovative control approaches, which sometimes require a highly sophisticated mathematical background. In this sense, the present book can be placed in the third group. In this sense, the present book can be placed in the third group. However, approaches that need advanced mathematical tools, like MIMO control using linear parameter variant (LPV), H∞ and Model Predictive Control (MPC) that are normally studied in academic research, are not included. The idea is to provide the content in a theoretical manner while keeping a practical and implementation perspective in mind. Hence, it should be positioned in between a research book and a textbook because some of the material can be used in control courses, but the interest is also to offer a reference for students working on implementations or thesis projects. It should be a theoretic assistance for practitioners as well. The contents are organized into three main parts. The first one is dedicated to dynamic modelling of the wind turbines, and the second part treats the control laws and the control system design. The last part is reserved for implementation aspects, like parametrization, filtering, real-time implementation and numerical examples. The building of dynamic models is based on the decomposition and coordination principle, which is used as a common thread through the different chapters. It started with the aerodynamic subsystem. Because of the 3D nonlinear dynamics with distributed parameters, aerodynamics is a very complex problem. It can be presented at three different degrees of depth. The deepest one is the field of aerodynamic physics. It is based on 3D Computational Fluid Dynamics (CFD) calculations. However, this is not useful for control because of the extensive computational burden. The second level uses the blade element momentum theory (BEM). This is an interesting approximation based on two important assumptions: the validity of the “actuator disc” approach and a steady flow. Aerodynamic forces and moments are then computed iteratively, assuming homogeneity around the disc. The process is non-deterministic (in the sense of an undetermined time of convergence), inaccurate and time-consuming. It could be used for control, but it is not recommendable. The third level uses empirical formulas adjusted by data. However, the validity of the formulas is very limited. This last level is normally used for control. However, it is clear that for the control system there are no satisfactory solutions. An alternative is to use neural networks as nonlinear approximators to improve the quality of the third kind of approach. Thus, despite its importance, aerodynamics is presented as simply as possible for the sake of completeness. Models of other subsystems are presented at various levels of detail. In general, the model should be chosen as simple as possible. However, the model complexity is strongly related to the control objectives. For example, if the goal is to control the rotational speed, a two-mass model of the drivetrain will most likely be of sufficient complexity. If the control objective is blade vibration control, the model has to provide the blade tip deflections, which are not included in the two- or three-mass models. Thus, simple models are not useful for such control applications.

Preface

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In the second part devoted to control, the common thread through the different chapters is the structure of the supervisory control as well as the different operation regions that characterize the wind turbine. The main topics of this part are generator control and pitch control, for which numerous ways are provided, each with its own set of advantages and disadvantages. However, other less common approaches in books, such as shutdown control, yaw control, feedforward control, peak shaving control, pitch control with maximum power limitation and hill climb search control based on deep learning algorithms, are also presented. In addition, the control approaches are illustrated with the corresponding block diagrams. Because the supervisory control as well as many control approaches normally require the unknown effective wind speed, observer-based estimation algorithms are included. The third part of the book intends to clarify some aspects related to the practical implementation. The first one is associated with the parametrization of a lowresolution model from the data of high-resolution models. To this end, the freely available high reference models, i.e., the so-called reference wind turbines, are summarized. Another pertinent topic is the development of a control system for multitasking real-time operations, where real-time control should not be confused with real-time simulation. Principles for Hardware-in-the-Loop tools (HiL) to test control algorithms are considered as well. Wind turbine control also uses different kinds of filters. Therefore, a short summary of the filter design is added. A study case concludes the book. In the practical implementation of the control system of a wind turbine, one always comes up against the obstacle of how the controller, the control loop or any other related aspects should be implemented. These facts motivate the provision of some new materials covering implementation aspects, some control approaches that are novel in comparison to traditional control but simpler than advanced control methods, and adapted approaches that are available in control theory but not directly applicable to wind turbines. To conclude, some material is well-known but dispersed in the literature and, therefore, is included and summarized for the sake of completeness. Bremerhaven, Germany 2021

Adrian Gambier

Acknowledgements

This book is the result of many years of teaching the course Control of Wind Turbines at the Leibniz University Hannover as an external lecturer. The author thanks Prof. Andreas Reuter for the opportunity to teach the course at his institute. Moving from process control to wind turbine control was possible, thanks to the opportunity and the thrust provided by Fabian Vorpahl and Michel Strobel. Their decision marked the beginning of an amazing journey into the world of wind energy. Several colleagues and friends have also contributed to the journey by engaging in lengthy technical talks and assisting me in different ways and contexts. Thus, a particular recognition goes to Cristian Gebhardt, Andrea Rivarola, Claudio Balzani, Mario Rotea, Adam Zuga, Fanzhong Meng and Qinmin Yang. Many students have contributed to the work with motivating discussions, comments and ideas. Therefore, the author thanks Bilal Amjad, Anshu Behera, Sören Behnsen, Suryans Chamoli, Issofa Mfegnam, Sami Ur Rehman Khan, David Kenko, Lukas Katzmann, Zhuowei Li, Chris Pilz, Driss Slaoui, Xin Wang, Xiong Xiao and Yibo Xia. Book writing is a time-consuming activity and, since the day is constrained to 24 h, one needs to take time from other activities to write a book. Thus, the first source of additional time for the book was the family. My deepest gratitude to my wife Nancy and my son Andrés because they selflessly contributed to the book project with their precious family time. In addition, it is imperative to recognize the touching sacrifice and immeasurable love of parents when it comes to supporting and encouraging their children. To conclude, an important part of the manuscript was written in the very difficult worldwide pandemic times. In such a situation, Springer, in particular, Oliver Jackson and his team, were very considerate, patient and helpful.

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Contents

1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Control of Modern Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Part I: Modelling Wind Turbines for Control Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Part II: Control System Design . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Part III: Implementation and Case Study . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

1 1 2 3 4 4 5 6 7

Modelling of Wind Turbines

2

Fundamentals of System Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Modelling Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Classification by Model Classes . . . . . . . . . . . . . . . . . . . . . 2.2.2 Classification by Modelling Approaches . . . . . . . . . . . . . 2.2.3 Classification by the Available Knowledge . . . . . . . . . . . 2.3 Dynamic Modelling of a System for Control Purposes . . . . . . . . . 2.3.1 Definition of the System Topology . . . . . . . . . . . . . . . . . . 2.3.2 System Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Modelling Aspects of Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 12 13 14 14 15 15 16 18 19

3

Modelling the Aerodynamic Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Actuator Disc Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Determination of the Aerodynamic Coefficients . . . . . . . . . . . . . . . 3.3.1 Exponential Approximation . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Trigonometric Approximation . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Polynomial Approximation . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22 27 27 28 29 xv

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Contents

3.4

Neural Network Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Artificial Neural Network Formulation . . . . . . . . . . . . . . . 3.4.2 Aerodynamic Coefficients of Wind Turbines Adjusted by an ANN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Blade Element Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Classic Approach of the BEM Theory . . . . . . . . . . . . . . . 3.5.2 Corrections to Improve the Model . . . . . . . . . . . . . . . . . . . 3.5.3 Final Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 30 32 32 33 36 38 40

4

Modelling the Rotating Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Modelling the Rotor as Single-Mass System . . . . . . . . . . . . . . . . . . 4.2.1 Three-Mass Rotating Subsystem . . . . . . . . . . . . . . . . . . . . 4.2.2 Two-Mass Rotating Subsystem . . . . . . . . . . . . . . . . . . . . . 4.2.3 One-Mass Rotating Subsystem . . . . . . . . . . . . . . . . . . . . . 4.3 Modelling the Rotor as Multi-mass System . . . . . . . . . . . . . . . . . . 4.3.1 Two-Mass Rotor Representation . . . . . . . . . . . . . . . . . . . . 4.3.2 Three-Mass Rotor Representation . . . . . . . . . . . . . . . . . . . 4.3.3 Four-Mass Rotor Representation . . . . . . . . . . . . . . . . . . . . 4.4 Modelling the Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Modelling with the Tip-Speed Ratio as State Variable . . . . . . . . . . 4.6 Blade Stiffening Due to Centrifugal Forces . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 45 45 48 49 49 50 52 54 61 61 63 63

5

Modelling the Tower Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Simplified Modelling of the Tower . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Modelling the Fore–Aft Dynamics Using Modal Analysis . . . . . . 5.4 Modelling the Blade–Tower Interaction . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Approach Based on the Two-Mass Rotor Model . . . . . . . 5.4.2 Approach Proposed in [4] . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Approach Inspired in the Inverted Pendulum . . . . . . . . . . 5.5 Forces Actuating on the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 67 69 69 71 72 74 76

6

Modelling of Actuators and Generators . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Simplified Modelling of Pitch Actuators . . . . . . . . . . . . . . . . . . . . . 6.2.1 First-Order Pitch Actuator . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Second-Order Pitch Actuator . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Actuators Based on Pitch Rate . . . . . . . . . . . . . . . . . . . . . . 6.3 Physical Modelling of Pitch Actuators . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Hydraulic Pitch Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Electric Pitch Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Modelling the Yaw System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Electric Modelling of the Generator Subsystem . . . . . . . . . . . . . . . 6.5.1 Simplified Generator Model . . . . . . . . . . . . . . . . . . . . . . . .

77 77 78 79 80 81 81 82 84 86 88 88

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6.5.2

Generator Model Based on Physical Laws . . . . . . . . . . . . 6.5.2.1 Double Feed Induction Generator (DFIG) . . . . 6.5.2.2 Permanent Magnet Synchronous Generator (PMSG) . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II 7

88 89 90 91

Control System Design

Overview of Wind Turbine Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Maximization of Power Capture . . . . . . . . . . . . . . . . . . . . 7.2.2 Power Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Reduction of Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Improvement of Dependability . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Reduction of Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6.1 Reduction of Deterministic Loads . . . . . . . . . . . 7.2.6.2 Reduction of Stochastic Loads . . . . . . . . . . . . . . 7.2.6.3 Reduction of Several Harmonics . . . . . . . . . . . . 7.2.6.4 Load Reduction by Using Intelligent Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Systemic Formulation of the Wind Converter System . . . . . . . . . . 7.4 Hierarchical Decomposition of the Control System . . . . . . . . . . . . 7.5 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Basic Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1.1 Constant Generator Speed . . . . . . . . . . . . . . . . . . 7.5.1.2 Constant Tip-Speed Ratio . . . . . . . . . . . . . . . . . . 7.5.2 Strategies for Variable Speed Variable Pitch Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 96 96 96 97 97 97 98 98 99 99 99 100 101 102 103 103 103 104 105

8

Supervisory Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Description of Operational States of Wind Turbines . . . . . . . . . . . 8.3 Design of a Simple Supervisory Control System . . . . . . . . . . . . . . 8.3.1 Operation in the Production States . . . . . . . . . . . . . . . . . . 8.3.2 Operation in the Shutdown State . . . . . . . . . . . . . . . . . . . . 8.4 Detailed Supervisory Control Scheme . . . . . . . . . . . . . . . . . . . . . . . 8.5 Transition Between Operational States . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 109 111 112 113 114 114 116

9

Control in Partial Load Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Indirect Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Basic Approach of Optimal Torque Control . . . . . . . . . . .

119 119 122 123

xviii

Contents

9.2.2

Optimal Torque Control with Acceleration Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Optimal Torque Control with Electromagnetic Torque Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Control with Tip-Speed Ratio Feedback . . . . . . . . . . . . . . 9.2.5 Control with Estimation of Aerodynamic Torque and Tip-Speed Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Direct Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Perturbation and Observation Control Technique (POC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Power Signal Feedback Con-Trol (PSFC) . . . . . . . . . . . . 9.4 Damping of Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Increase the Damping by Filtering . . . . . . . . . . . . . . . . . . . 9.4.2 Control with Increased Damping Coefficient . . . . . . . . . . 9.5 Control in the Transition Regions (I1/2 and II1/2 ) . . . . . . . . . . . . . . 9.5.1 Open-Loop Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Closed-Loop Control Law . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Control with Low Tip-Speed Constraint . . . . . . . . . . . . . . 9.6 Peak Shaving Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Estimation of the Effective Wind Speed . . . . . . . . . . . . . . . . . . . . . 9.7.1 Observer-Based Estimation with Unknown Input . . . . . . 9.7.2 Interval Observer-Based Estimation . . . . . . . . . . . . . . . . . 9.7.3 Averaged Effective Wind Speed with Forgetting Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Control in Full Load Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Control for Power Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Collective Pitch Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Controllers for the Implementation of CPC . . . . . . . . . . . 10.2.3 Collective Pitch Control with Aerodynamic Power Limiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Anti-windup Mechanism for the Collective Pitch Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.5 Adaptive Collective Pitch Control (Gain-Scheduling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Active Damping Control of Vibrations . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Active Tower Damping Control . . . . . . . . . . . . . . . . . . . . . 10.3.2 Active Blade Damping Control . . . . . . . . . . . . . . . . . . . . . 10.4 Control for Load Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Individual Pitch Control with Coleman Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Individual Pitch Control with Clarke Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Mutivariable Pitch Control with Coleman Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124 126 127 127 128 128 138 139 139 140 141 143 144 144 145 147 148 149 152 152 155 155 157 157 160 164 164 167 172 172 176 180 182 191 193

Contents

xix

10.5 Control Improvement by Using Feedforward Control . . . . . . . . . . 193 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 11 Yaw Control and Shutdown Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Yaw Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Approach Based on Direct Measurement of Wind Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Methods Based on Estimation of the Wind Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Methods Without Measurement of Wind Direction . . . . . 11.2.4 Yaw Control for Load Reduction . . . . . . . . . . . . . . . . . . . . 11.3 Shutdown Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Conventional Open-Loop Shutdown Control . . . . . . . . . . 11.3.2 Closed-Loop Shutdown Control . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 202 204 206 206 207 208 209 210 210

Part III Control Implementation 12 Parametrization and Reference Wind Turbines . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Parametrization of the Aerodynamic Subsystem . . . . . . . . . . . . . . 12.2.1 Aerodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Sensitivity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Parametrization of the Rotating Subsystem . . . . . . . . . . . . . . . . . . . 12.3.1 Estimation of the Rotor Moment of Inertia . . . . . . . . . . . 12.3.2 Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Gearbox and Shafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 The Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Parametrization by Using the Steady-State Model . . . . . . . . . . . . . 12.5 Reference Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 General Specifications of the Reference Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Rotor Parameters and Characteristics . . . . . . . . . . . . . . . . 12.5.3 Blade Properties of the Reference Wind Turbines . . . . . . 12.5.4 Nacelle Properties of the Reference Wind Turbines . . . . 12.5.5 Drivetrain Properties of the Reference Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.6 Tower Properties of the Reference Wind Turbines . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215 215 216 217 217 218 219 220 220 223 223 225

13 Control Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Evaluation by Using Performance Indices . . . . . . . . . . . . . . . . . . . . 13.2.1 Classic Performance Indices . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Performance Indices for Time-Limited Problems . . . . . .

233 233 234 234 235

226 227 227 227 230 230 231

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Contents

13.2.3 Fractional Order Performance Indices . . . . . . . . . . . . . . . . 13.2.4 Objective Functions for Variables of Other Nature . . . . . 13.3 Specific Control Evaluation for Wind Turbines . . . . . . . . . . . . . . . 13.3.1 Performance Evaluation of Power Extraction Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Performance Evaluation for Load Reduction Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Performance Evaluation for the Protection of Actuators . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 236 237

238 241 243

14 Real-Time Control and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Real-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Definitions and General Aspects . . . . . . . . . . . . . . . . . . . . 14.2.2 Real-Time Operating Systems (RTOSs) . . . . . . . . . . . . . . 14.2.3 Real-Time Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Intertask Communication and Synchronization . . . . . . . . 14.3 Controller Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Filter Design and Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Real-Time Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Hardware in the Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 247 248 248 248 249 251 252 254 257 258 261

15 Case Study: 20-MW Reference Wind Turbine . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Modelling the Wind Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Aerodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Sensibility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Rotor and Drivetrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.4 Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Experiments for the Simulation Environment . . . . . . . . . . . . . . . . . 15.5 Control in Partial Load Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 Control Law for Region II . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2 Control Law Region II1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Collective Pitch Control Using a FO-NPI Controller . . . . . . . . . . . 15.7 Individual Pitch Control Using FO-NPI Controller . . . . . . . . . . . . 15.7.1 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263 263 264 265 266 266 269 269 270 270 271 271 272 273 274 274 275 277

237

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Notation

Variables, Functions and Parameters A B C Cd Cl Cl Cp Cq Ct D DC DEL Ek Ft F tip F root J K K Nb Ne Nf Nm Ns Nv P Pa

Cross section Viscous friction of bearing Execution time (Chap. 14) Drag coefficient Lift coefficient Thrust coefficient Power factor, power coefficient Rotor torque coefficient Thrust force coefficient Damping coefficient, damage (Chap. 13), deadline (Chap. 14) Duty cycle Damage equivalent loads Kinetic energy Thrust force Prandtl’s tip loss factor Prandtl’s root loss factor Mass moment of inertia, also performance index, depending on the subscript Stiffness coefficient (associated with the subscripts b, lss, hss, t and s) Gain of a controller (associated with the subscripts p, i, d and opt) Number of blades Number of independent differential and algebraic equations Degrees of freedom Number of manipulated variables Variables specified by the environment Total number of variables including inputs and outputs Power Mechanical power

xxi

xxii

Pm Pw R T U V V a a c1 ...c13 e j l lb m n nx pd− , pd+ ri s vwe vw wij t x u v vtip vw vew y f[·] g[·] F[·] L p x(t) ∈ Rn u(t) ∈ Rl y(t) ∈ Rm A B C D

Notation

Mechanical power Power contained in the wind Rotor radius Torque, task period (Chap. 14) Process utilization Volume Stationary wind speed Angular induction factor Axial induction factor Fitting coefficients Control error Imaginary unit Length Blade length Number of outputs Number of state variables Gearbox ratio Pressures immediately before and after the actuator disc Distance of the blade section i to the rotational axis Laplace variable Effective wind speed Wind speed Weight coefficients of a neural network Time State variable of a dynamic system Input of a dynamic system Flow velocity Tip speed Wind speed Effective wind speed Output of a dynamic system Vector field to represent the state equation of a nonlinear system Vector field to represent the output equation of a nonlinear system Vector-valued functional relationship (Chap. 2) Laplace transform Vector of parameters Vector of state variables (dimension n) Vector of input variables (dimension l) Vector of output variables (dimension m) System matrix of the state-space representation Input matrix of the state-space representation Output matrix of the state-space representation Direct transfer matrix of the state-space representation

Notation

xxiii

Polynomial and Transfer Functions P(s) Q(s) T (s) A(s) B(s) G(s) Gf (s) G(s)

Denominator controller polyynomial Feedback controller polynomial Feedforward controller polynomial Denominator system polynomial Numerator system polynomial Transfer function Transfer function of a filter Transfer matrix

Greek Variables α β γ δ ∂ Δ ε ζ η θ k λ, λ* μ u v ξ π ρ, ρ a σ τ ϕ ϕ w w0

Angle of attack of blade section Pitch angle XE Yaw angle Fractional order Partial derivative Difference between two variables Small increment Damping factor Efficiency Angles of a rotational axis Variable used in BEM algorithms (Chap. 3), exponent (Chap. 10), Lagrange multiplicator (Chap. 10) Tip-speed ratio, value of λ, for C p,max Fractional exponent, variable used in the optimization algorithms Wind speed divided by the rotor radius Fractional exponent Section twist angle Pi number Density, air density Activation function of neural networks (Chap. 3), switching function (Chap. 8), stress (Chap. 13) Time constant (Chap. 6), latency time (Chap. 14) Inflow angle Angle between blades (Chap. 4), tip deflection (Chap. 5) Rotational speed (time derivative of θ ) Resonant frequency of a filter (Chap. 14)

xxiv

Notation

Meaning of Subscripts 123 α, β β a b c d, q dt e g h hss i, j, k lss min, max opt p r root s t tip v w, we, wr x

Vector of three components Coordinate in the αβ-coordinate system Used regarding the pitch angle in the controller Aerodynamic, actuator output Blade Controller Coordinates in the dq-coordinate system Drivetrain Depending on the variable equivalent or effective Generator Hub High-speed shaft Used to index variables Low-speed shaft Indicate the minimum or maximum value of a variable Optimum Pitching Rotor Root of a blade Shaft Tower, but in case of F t is thrust force Depending on the variable, tip of a blade or tip speed Valve Wind, effective wind, relative wind Gearbox

Abbreviations

A/D AB4 ABDC ABM4 AdaGrad Adam AIE ANN ATDC BEM CFD CPC CPU Crone D/A DEL DFIG DLC DMS DOF DORFC DSA DTU EDF ESC FO-NPI FO-NPID FO-PI FO-PID FOTF FPS

Analog/Digital Adams–Bashforth of fourth order Active Blade Damping Control Adams–Bashforth–Moulton of fourth order Adaptive Gradient Adaptive moment Absolute Integral Error Neural Network Approach Active Tower Damping Control Blade Element Momentum Computational Fluid Dynamics Collective Pitch Control Central Processing Unit Commande robuste d’ordre non-entier Digital/Analog Damage Equivalent Load Double Fed Induction Generator Design Load Cases Deadline Monotonic Scheduling Degree Of Freedom Direct Online Rainflow Counting Deferrable Server Algorithm Technical University of Denmark Earliest Deadline First Extremum Seeking Control Fractional Order Nonlinear Proportional–Integral Fractional Order Nonlinear Proportional–Integral Derivative Fractional Order Proportional–Integral Fractional Order Proportional–Integral Derivative Toolbox for Fractional Order Transfer Functions Fixed-Priority Scheduling xxv

xxvi

HCS HHC HiL HSS IADU IBC IEA IFC IPC ISE ISTSE ITSE LiDAR LLF LPV LQG LQR LSF LSS LynxOS MA MIMO MLF MOOC MPC MPPT MSC MUF NAG NMPC NPI NPID NREL OTC PD PID PLC PMSG POC PR IPC PSC PSFC RMS RMS RMSE

Abbreviations

Hill Climb Searching Higher Harmonics Control Hardware-in-the-Loop High-Speed Shaft Integrated Absolute Derivative of U Individual Blade Control International Energy Agency Individual Flap Control Individual Pitch Control Integral Square Error Integral Square Time-weighted Squared Error Integral Time-weighted Squared Error Laser Imaging Detection and Ranging Least Laxity First Linear Parameter Variant Linear Quadratic Gaussian Linear Quadratic Regulator Least Slack-time First Low-Speed Shaft Lynx Operating System Momentum Algorithm Multi-Input Multi-Output Maximum Laxity First Multi-Objective Optimal Control Model-based Predictive Control Maximum Power Point Tracking Mode Sliding Control Maximum Urgency First Nesterov Accelerated Gradient Nonlinear Model Predictive Control Nonlinear Proportional–Integral Nonlinear Proportional Derivative National Renewable Energy Laboratory Optimal Torque Control Proportional Derivative Proportional–Integral Derivative Programmable Logic Controller Permanent Magnet Synchro-Generator Perturbation and Observation Control Proportional Resonant IPC Peak Shaving Control Power Signal Feedback Control Rate Monotonic Scheduling Root Mean Square Root Mean Square Error

Abbreviations

RMSProp RT-Linux RTOS SAA SISO SSA SSE TSRC

xxvii

Root Mean Square Propagation Real-Time Linux Real-Time Operating System Steepest Ascent Algorithm Single-Input Single-Output Sporadic Server Algorithm Sum of Square Errors Tip-Speed Ratio Feedback Control

Chapter 1

Overview

Summary Wind turbines, also known as wind energy systems or wind energy converters, have gained significant attention in the last two decades as the primary source of renewable energy. Such machines grow in size and importance, where control is an essential component. This chapter is thus an overview of the book, its structure, motivation, objectives, scope and a brief description of the contents of each chapter.

1.1 Introduction The integration of renewable energy sources into the power grids is an important priority in the strategic road maps around the world. In particular, wind energy systems are central contributors to renewable energy today and, moreover, they also have significant potential for the future. Wind energy converters are very complex machines that have to be operated optimally and safely for a very long-time horizon in order to be economically practicable. On the other hand, the energy needs are continuously growing, forcing a large increment in the capacity of the wind turbines and consequently in their sizes. This fact is only possible with the development of new methodologies as well as the incorporation of new technologies. Thus, control engineering now has a critical role to play in the dominance of such machines. This requires, in turn, the continuous update of the specialized literature.

© Springer Nature Switzerland AG 2022 A. Gambier, Control of Large Wind Energy Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-84895-8_1

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1 Overview

1.2 Control of Modern Wind Turbines There are many concepts for wind turbines, which can be classified from different points of view: (a) the size (very large, large, middle, small and very small); (b) the structure (vertical or horizontal axis), (c) the wind direction (upstream and downstream); (d) the number of blades (two or three are the most common); (e) the application (onshore and offshore); (f) the control strategy (fixed-speed fixed-pitch, variablespeed fixed-pitch, fixed-speed variable-pitch, variable-speed variable-pitch), etc. A basic and incomplete representation of the broad spectrum of types of wind turbines is given in Fig. 1.1 In the last 20 years, the three-bladed horizontal-axis upstream variable-speed variable-pitch wind turbines with double fed induction generator (DFIG) or permanent magnet synchro-generator (PMSG) have prevailed in the area of multimegawatts wind turbines. This group is called in the book modern wind turbines and is highlighted in Fig. 1.1. Thus, the present book is devoted only to the control of this kind of machines taking into account several aspects, for instance: • Incorporation of state-of-the-art approaches (e.g., multi-objective control tuning, nonlinear PID, fractional order PID, artificial neural networks, extremum seeking control and receding horizon observers for wind estimation). Wind Energy Converters Vertical Axis Wind Turbines

Horizontal Axis Wind Turbines

Savonious Darrieus Gorlov Giromills Machine Machine Machine Machine

Upwind Wind Turbines

Downwind Wind Turbines

Three Bladed Two Bladed Three Bladed Two Bladed Wind Turbines Wind Turbines Wind Turbines Wind Turbines Stall Regulated Pitch Regulated Wind Turbine Wind Turbine Fixed Pitch Fixed Speed

Variable Pitch Variable Pitch Fixed Pitch Fixed Speed Variable Speed Variable Speed DFIG Generator

Fig. 1.1 Types of wind turbines

PMSG Generator

1.2 Control of Modern Wind Turbines

3

• Improvement of the control performance without radical changes in the control system configurations. • Description and implementation of a supervisory control system (e.g., practical aspects of the controller switching). • Concrete implementation ways, i.e., examples about how to implement particular approaches (e.g., notch filter design, smooth transition functions and controllers) • Evaluation of the control performance.

1.3 Motivation Related to wind energy systems, there exists profuse literature in practically all fields of wind turbines. Wind turbine fundamentals can be found in essential works, for example, [1–3]. Wind turbine design is the subject of [4, 5]. A focus on the environmental aspects is given in [6]. Books dedicated to general fundamentals, like [1, 3], include some chapters about the control of wind turbines, which contribute with crucial information but are not sufficient for a concrete implementation of a control system. Specialized books for the control of wind turbines are, e.g., [7–11]. In [8], the emphasis is the application of linear parameter variant control (LPV) to wind turbines. An optimal control global approach is presented in [9] and many control strategies are described in [10]. Electrical control is the main topic of [11]. There are also many doctoral theses with important contributions to the pitch control of wind turbines. Two of them that can be mentioned are [12, 13]. There are also contributed books with particular chapters about specific subjects, for instance, [14, 15]. A thorough examination of the available material reveals that the majority of works fall into three categories: contributions from the wind energy discipline that need a “good” controller in order to be able to do the same studies, contributions from the control systems community, which have developed a new control approach and want to demonstrate the controller performance by using a “good” application, i.e., a very simple model of a wind turbine and finally, contributions that intend to solve control problems of wind turbines by using advanced and innovative control approaches, which require a highly sophisticated mathematical background. On the other hand, in the practical implementation of a control system of the wind turbine, one always comes up against the obstacle of how the controller, the control loop or any other related aspects should be implemented. However, large wind turbines are available working in the field. This indicates that they have a functioning control system inside. The problem is that the information is unknown and protected. Furthermore, standard industrial control procedures are simple, including many heuristics. These facts motivate the provision of some new material covering implementation aspects, some control approaches that are novel in comparison to traditional control but simpler than advanced control methods, and adapted approaches that are available

4

1 Overview

in control theory but not directly applicable to wind turbines. Finally, some material is well-known but disperse in the literature and therefore, it is included and summarized for the sake of completeness.

1.4 Scope of the Book The book is composed of 15 chapters organized into three parts. After this introductory chapter, it follows Part II, which is devoted to the dynamic modelling of wind turbines. Control approaches for the different operative states of the machine are condensed in Part II. Finally, implementation aspects and practical procedures for the model parametrization, real-time control, filtering and reference wind turbines are studied in Part III.

1.4.1 Part I: Modelling Wind Turbines for Control Purposes The first part of the book is devoted to the dynamic modelling aspects of wind turbines from the point of view of control system design. Wind energy converters are very complex machines that involve nonlinearities, distributed parameters and many interactions, which cannot be described mathematically in a precise way. Thus, it is important to find a modelling compromise between acceptable accuracy and usefulness for control purposes. Part I consist of six chapters. Chapter 2 introduces some modelling background and principles in order to rationalize procedures used in the other modelling chapters. The cornerstone is the system decomposition, which is used later for the separated modelling of all subsystems. The aerodynamic subsystem is presented in Chap. 3. This is a very complex problem because of the 3D nonlinear dynamics with distributed parameters. It can be presented in three different degrees of depth: according to aerodynamic physics, from the wind energy viewpoint and also for control and electrical purposes. The first approach uses 3D computational fluid dynamics (CFD). This is very important but useless in control because of the very high computational burden. The second group uses the blade element momentum theory (BEM). This is an interesting approximation based on two important assumptions: the validity of the “actuator disc” approach and a steady flow. Aerodynamic forces and moments are then computed iteratively assuming homogeneity around the disc. The process is non-deterministic (in the sense of undetermined time of convergence), inaccurate and time-consuming. It is used for control in some circumstances. The third approach is based on empiric formulas adjusted from data. However, the validity of the formulas is very limited. In the end, there are no good solutions for control.

1.4 Scope of the Book

5

Chapter 4 is devoted to the modelling of the rotating subsystem, which consists of the rotor, the low-speed shaft, the gearbox, the high-speed shaft and the mechanical part of the generator. The complexity of the subsystem depends on the level of depth used to describe the blades. Standard approaches assume the drivetrain rigid as a two-mass system. This abstraction normally works fine for very simple control objectives. However, these models do not allow model-based control including flexible blades, vibration control or blade tip deflection damping control. Therefore, models with a higher level of complexity are progressively included in the chapter. Precise modelling of a flexible rotating blade needs the finite element approach. However, this is not suitable for control and therefore not included. The tower and the tower–blade interaction is presented in Chap. 5. The tower presents similar flexible characteristics as the blade and would also require finite elements for an accurate mathematical representation. Instead of this, a high level of abstraction is used to represent the tower by using only the first natural frequency in each degree of freedom (DOF). The tower–blade interaction in the fore–aft/flapwise direction is modelled in the literature considering only the worst case, i.e., when the blade is in the highest vertical position. Finally, Part I concludes with Chap. 6, which covers actuators and the electric side of the generator. Pitch actuators based on hydraulic device and an electric synchronous motor are modelled. For the yaw actuator, only an electric drive is considered. Only two types of generators are studied, namely the doubly fed induction generator (DFIG) and the permanent magnet synchronous generator (PMSG).

1.4.2 Part II: Control System Design The control of large wind turbines is based on a combined multi-strategy concept depending on the operating points, which are managed by the values of the effective wind speed. Thus, depending on the wind speed, several operating regions with corresponding control strategies can be defined. Part II is organized into six chapters, the first of which is Chap. 7. It consists of an introduction to the general problem of control of wind turbines, explaining the control objectives, the hierarchical control topology, the concept of operational states and most of the control strategies. Chapter 8 presents the supervisory control system. It begins with the idea of hybrid automaton, which is used as a formalism in the supervisory control system, followed by a brief literature review, as this subject is rarely addressed in the literature. The chapter concludes with a mechanism for control switching in the sense that the machine is moving from one operational state to another. Wind energy converters have two distinctive operational states because they are only the states in which the machine delivers energy into the grid, namely the partial load state and the full load state. Thus, the following two chapters are dedicated to control in these states: Chap. 9 is for the partial load and Chap. 10 for the full load.

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1 Overview

In Chap. 9, not only the generator control to maximize the power extraction in the partial load region, i.e., maximum power point tracking (MPPT) control, is presented but also the control in the transitions entering and abandoning the region are included. Many MPPT algorithms are described and particular filtering aspects are studied. In the transition to the full load region, several additional control algorithms are considered including peak shaving and control under constrained tip speed. The longest chapter, Chap. 10, is devoted to the control of a wind turbine operated in overrated wind. Performance control and control for load and vibration reduction are studied. Several problems are considered, for instance, the integrator windup, adaption by gain-scheduling and advanced algorithm for the PID control like fractional order controllers and nonlinear PID controllers, different approaches for the individual pitch control, damping control for blades and tower and power limiter control. However, multivariable control approaches are not included. The used method is the multi-loop control system with multiple controllers and the controller tuning is based on multi-objective optimal parameter tuning. Chapter 11 deals briefly with the regulation in other operating states, namely Shutdown and Grid connection/disconnection. Part II concludes with Chap. 12, which is devoted to the evaluation of the control performance, not only from the classic perspective by using performance indices but also including objective function, which are specific for wind turbines.

1.4.3 Part III: Implementation and Case Study The last part of the book consists of three chapters, which intend to show approaches and methods from a practical point of view. Thus, Chap. 13 introduced the most common reference wind turbines and the model parametrization for control purposes. Unfortunately, there are no public parameter sets of real turbines. Hence, reference wind turbines, i.e., high-definition models with a large number of parameters that are simulated by using an aeroelastic simulation code, are at present the only way to have a simulation setup. This parametrized code is used as a virtual turbine. There are several complementary procedures to obtain parameters of a real turbine for a dynamic model, which is going to be used in the control system design. The start point is the information provided by the manufacturer and the information that can be obtained by direct measurement or observation, e.g., geometric data. The rest of the parameters must be obtained by some parameter estimation methods. If the model should be obtained for a reference wind turbine, the problem is how to carry out a parametric reduction from the high-resolution model to a model for control purposes. This is the central subject of Chap. 13. Because large complex wind turbines are not available for testing and experiments, and small-scaled systems behave dynamically in a completely different manner, realistic simulation tools, such as hardware-in-the-loop (HiL) simulators, are an acceptable alternative. Thus, the description of a concept for the implementation of an industrial programmable logic controller (PLC) for real-time control in a real-time

1.4 Scope of the Book

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simulation environment, the implementation of control algorithms and the real-time testing in the simulator is the subject of Chap. 14. Finally, Chap. 15 is devoted to presenting a case study by using a reference turbine.

References 1. Burton, T., Jenkins, N., Sharpe, D., Bossanyi, E. (2011). Handbook of wind energy. Chichester, UK: Wiley. 2. Gasch, R., & Twele, J. (2012). Wind power plants. Germany: Springer. 3. Manwell, J., McGowan, J., Rogers, A. (2009). Wind energy explained. Theory, design and applications. Chichester, UK: Wiley. 4. Tong, W. (2010). Wind power generation and wind turbine design. Chichester, UK: WIT Press. 5. Jamieson, P. (2011). Innovation in wind turbine design. Chichester, UK: Wiley. 6. Nelson, V. (2009). Wind energy: Renewable energy and the environment. Boca Raton, USA: CRC Press. 7. Rivkin, D., Anderson, L., & Silk, L. (2012). Wind turbine control systems. Burlington, USA: Jones and Bartlett Publishers. 8. Bianchi, F., de Battista, H., Mantz, R. (2007). Wind turbine control systems. London, UK: Springer. 9. Munteanu, I., Bractu, A., Nicolaos-Antonio, C., & Ceanga, E. (2008). Optimal control of wind turbines. London, UK: Springer. 10. Garcia-Sanz, M., & Houpis, C. (2012). Wind energy systems. Boca Raton, USA: CRC Press. 11. Anaya-Lara, O., Jenkins, N., Ekanayake, J., Cartwright, P., Hughes, M. (2009). Wind energy generation: Modelling and control. Chichester, UK: Wiley. 12. Lio, W. (2018). Blade-pitch control for wind turbine load reductions. Cham, Switzerland: Springer. 13. Shan, M. (2018). Load reducing control for wind turbines. Stuttgart, Germany: Fraunhofer Verlag. 14. Luo, N., Vidal, Y., & Acho, L. (2014). Wind turbine control and monitoring. Cham, Switzerland: Springer. 15. Wu, Q., Sun, Y., (Eds.). (2018). Modeling and modern control of wind power. Chichester, UK: IEEE Press (Wiley).

Part I

Modelling of Wind Turbines

Outline The Part I of the book is dedicated to presenting the mathematical modelling aspects of the wind turbine as a dynamic system under the objective of the control system design. After the general introductory Chap. 2, the following four chapters are devoted to the modelling of the most important subsystems in which the wind turbine is decomposed. Thus, the modelling of the aerodynamic is the focus of Chap. 3. The rotating subsystem is treated in Chap. 4. The tower subsystem and the tower–blade interaction are presented in Chap. 5. Finally, Chap. 6 is committed to the modelling of actuators as well as the electrical part of the generator.

Chapter 2

Fundamentals of System Modelling

Summary A mathematical model is today an essential tool for systems engineering in general and control engineering in particular. Nowadays, it is unimaginable to conceive an engineering development without the help of a mathematical model, which describes the dynamical behaviour of the technical system under consideration. On the other hand, the development of a model requires the confluence of knowledge from different sources as well as the mastery of modelling principles and techniques. Depending on the discipline, modelling and its main concepts can be interpreted differently. Therefore, the present chapter is a short introduction to the fundamentals of mathematical models of a dynamic system in order to establish a common basis before the model building of different wind turbine components is undertaken in the following chapters.

2.1 Introduction In the literature, it is possible to find many definitions of a model. Here, modelling is understood as the process to formulate a mathematical representation of the system, which captures and reproduces the dynamic responses of a system through the outputs when it is excited by external disturbances or by manipulating the inputs. Thus, the model is a set of differential and algebraic equations with the corresponding variables and parameters. The inputs are the independent variables, while the dependent variables are the outputs. External disturbances cannot be modified, but their random changes affect the system behaviour. It is important to remark that a model is only an abstraction of reality and therefore, the statement given in [1], “all models are wrong but some of them are useful”, is justified. Thus, dynamic models can be useful in the understanding of the system behaviour by using computer simulation. They can also be used for the optimization © Springer Nature Switzerland AG 2022 A. Gambier, Control of Large Wind Energy Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-84895-8_2

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of operation conditions, for model-based control system design and for the development of new control strategies. Hence, a model of different complexity can be developed depending on the needs and the purpose or application. For example, a high-resolution model, i.e., a model with a high level of detail, is necessary for a detailed analysis of the system behaviour and an accurate simulation. Contrarily, a model for control purposes has to be able to capture the essential dynamics that has to be under control, but it should be as simple as possible in order to facilitate the control system design. Models for wind turbines control are, in principle, simpler than models for other purposes. Nevertheless, dynamic modelling is a very complicated task and much more complex than the control itself. However, a dynamic model is absolutely indispensable for the development of a powerful control system. Although mathematical modelling of technical systems necessitates scientific knowledge and a detailed understanding of the system under modelling, it is essentially an art since no complete set of guides and methods exist, depends on developed skills gained by practice and requires inspiration as well as creativity. On the other hand, a progressive lack of practice will gradually erode the modelling abilities. In any case, confidence in basic rules is necessary and the well-established principles should be known a priori in order to begin a modelling task successfully. This is then the objective of the present introductory chapter.

2.2 Modelling Approaches As already mentioned, a dynamic model is a set of differential/difference and algebraic equations with the corresponding parameters. Hence, models can be organized using a variety of criteria, following either an equation or a parameter perspective. Another way to classify models is given by the procedure to develop the model.

2.2.1 Classification by Model Classes For control purposes, a dynamic model needs an input–output representation. For the purposes of this book, dynamic models consist of differential equations (i.e., continuous time models), which can be linear or nonlinear, with lumped parameters. The model representation can be direct, i.e., there is a direct functional relationship between inputs and outputs y(t) = F[p, u(t)],

(2.1)

where F [·] represents the real vector-valued functional relationship between the vector of l inputs u(t) ∈ Rl with a parametric vector p and the vector of m outputs y(t) ∈ Rm , or indirect through internal state variables

2.2 Modelling Approaches

13

x˙ (t) = f[t, x(t), u(t), px ] , y(t) = g[t, x(t), u(t), po ]

(2.2)

where f[·] and g[·] are real vector fields with the dimensions of x and y, respectively. The vector x(t) ∈ Rn is the vector of state variables. pi and po are parameter vectors. Vector-valued functions F [·], f[·]and g[·] belong to a particular class of functions. A class can be, for instance, the class of linear functions. In the case of nonlinear functions, there are many subclasses as can be the bilinear class or the quadratic class. Thus, the class of a model is defined by the class to which the functions belong. In the theoretic modelling approach, the class is a result of the modelling process, while in the experimental approach the class is specified at the beginning. Stationary or steady-state models consist of algebraic equations and have an important role in the system design. They are used to determine constants and thresholds, dimension the system, optimize the system and validate the dynamic model for t → ∞.

2.2.2 Classification by Modelling Approaches Basically, there are two approaches to attack a modelling task: the theoretical approach and the experimental approach. Both procedures have the same common aspects, for instance, the assumptions about the system, abstraction level, operation conditions, range of validity and available knowledge. The theoretical approach bases the model derivation in first principles as the conservation laws and, in particular, energy (including the Lagrange equation) and momentum balances, phenomenological equations and special laws like Newton for mechanical subsystems and Kirchhoff for electrical components. The advantage of the approach is the fact that the model maintains a connection with the physical meaning of the variables, balance equations provide linear relationships between variables and the model is analytic and therefore, with a wide valid operative range. The model class is obtained as a consequence of the modelling process. As a drawback, the intricate derivation process, a very complex resulting model that requires simplifications to be suitable for control and sometimes model infeasibility can be mentioned. The specific literature is not ample (see [2, 3]). In the experimental approach, the model is obtained from measured data of input and output variables. The procedure requires defining a priori a model class fixing, thus, the equations and the parameters are obtained by optimization with the objective to reduce the differences between measured and model outputs for the same measured inputs. Benefits of this procedure are, for instance, good accuracy for the range of data validity, unnecessary deep knowledge of the physical system and minor modelling effort. The major disadvantages are an abstract model without connection with the system physics and the range of validity is limited to the data validity. The experimental approach is also known as system identification and there are a very high number of methods and bibliography (see, e.g., [4–10]).

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2.2.3 Classification by the Available Knowledge In the previous subsection, the modelling procedure is separated into theoretic and experimental procedures. A theoretic approach requires much information about the system, equations are derived from first principles and parameters are either known from design or directly measured. This way is called white-box modelling. Contrarily, the experimental approach assumes a class to be known and parameters are estimated by system identification. Thus, no previous information is available and therefore, the procedure is known as black-box modelling. In reality, some laws as well as parameters are known a priori but many other parameters are unknown. Hence, missing parameters can be determined by parameter fitting, parameter identification or parameter to satisfy some conditions as can be steady-state values of the variables. This procedure of mixing both approaches is named grey-box modelling. This classification is illustrated in Fig. 2.1. Depending on the amount of available information, grey-box modelling can be divided into light-grey modelling and dark-grey modelling [9]. Models for wind turbines presented in the book follow the grey-box approach.

2.3 Dynamic Modelling of a System for Control Purposes The success in the implementation of a control system requires a suitable definition of the system topology and a powerful dynamic model that can be used to perform Modelling of Dynamic Systems Derivation of Model Equations

Model Parametrization

Derivation from Priorly Known Physical Laws Parameters White-box Modelling Priorly Known Empirical Laws Predetermination of a Class of Models Linear Models

Bilinear Models

Gray-box Modelling

Black-box Modelling A Class of Nonlinear Models

Fig. 2.1 Summary of modelling approaches

Parameter Fitting by Optimization Parameter Identification

2.3 Dynamic Modelling of a System for Control Purposes

15

the control system design, analysis and evaluation. Consequently, the reliability of the result depends on the validity of the system topology and the used model. Several topics have to be established at the beginning of the modelling process, namely the determination of the system topology and the system decomposition, the selection of the modelling approaches and the model class for each subsystem. The modelling approaches as well as the model classes have previously been presented. The next topic is the system topology, which includes the system decomposition and degrees of freedom.

2.3.1 Definition of the System Topology The system topology is defined here as a concept that includes the selected inputs, outputs and disturbance variables and the internal configuration of subsystems, in which the system is decomposed to undertake the modelling process. The number of possible variables that can be defined in a system is large and normally only a subset of it is actually necessary. This subset depends on the availability, either by means of measurements or through the estimation, and variables with the possibility to actuate on the system.

2.3.2 System Decomposition An important principle to study complex systems is to break down the system into subsystems. This assumes that the physical laws that describe the system are also valid for the application to the subsystems. This is verifiable for technical systems. Since wind turbines are complex systems in the sense that they have several clearly defined subsystems and many variables, they can be modelled using techniques for large-scale systems. These are based on the concepts of decomposition and coordination (e.g., [11–13]). The coordination is carried out by means of coordinating variables and the system decomposition can be done following one of three ways: the physical, horizontal or geographical decomposition, the functional, temporal or vertical decomposition and a combination of both that is called spatio-temporal decomposition. Example 2.3.1 There are currently no recommendations in the literature on how to decompose wind turbines. There is, however, an implicit consensus to use a spatiotemporal decomposition (see, among others, [4, 14–22]). As a result, the main system is decomposed physically, while the obtained subsystems are decomposed functionally. Following the spatio-temporal decomposition, an example for a possible system decomposition is proposed for a wind turbine. The example of decomposition is depicted in Fig. 2.2. It includes the coordinating

16

2 Fundamentals of System Modelling Tpitch Pitch Subsystem Ta β Tb

ωr Rotor ω xl

Drive Train

vw AerodyFt* namics

ωg Tg

Nacelle

γ

Mt

x&t y& t

Tower

P Generator Tg electric

Generator mechanics

Tyaw

Yaw Subsystem

Fig. 2.2 System decomposition of a wind turbine

variables that are involved, as well. This decomposition is founded on [4], and it will assist the modelling activities discussed in the coming chapters.

2.3.3 Degrees of Freedom The concept of degrees of freedom has, depending on the discipline, different meanings. Of particular interest here is the fact that two completely different meanings are applied to the same model. One definition is used for mechanical systems and the other is a general modelling concept that can also be applied to mechanical models. For mechanical systems, the degree of freedom is the number of independent motions of system components. This is normally noted as DOF. From the modelling point of view, the degree of freedom N f is defined as N f = Nv − Ne ,

(2.3)

where N v is the total number of variables including inputs and outputs, and N e is the number of independent differential and algebraic equations. In order to obtain a consistent mathematical model, it should be ensured that the model equations provide a unique relation between variables and equations, i.e., an exactly specified model. This is the same as requiring N f = 0. If N f < 0 (overspecified system), then additional independent model equations must be developed. If N f > 0 (underspecified system), then a number of sufficient variables have not been identified. On the other hand, an effect of the feedback control is the reduction of the degree of freedom of the system. Hence, in an underspecified system (N f > 0), the degrees of freedom can be used to select manipulated variables, adding control loops, as well as variables fixed by the system environment, such that N f = Nm + Ns ,

(2.4)

2.3 Dynamic Modelling of a System for Control Purposes

17

Fig. 2.3 Rotating two-mass system

where N m and N s are the number of manipulated variables and variables specified by the environment. The modelling process of the system for control purposes should provide some degree of freedom (N f > 0) such that feedback control loops may be added to reduce the degree of freedom to zero, resulting in a unique system behaviour. Example 2.3.2 The system of Fig. 2.3 consists of two masses with mass moments of inertia J 1 and J 2 connected by a gearbox with ratio nx and two shafts. The first one is flexible with damping Ds and stiffness K s . The second shaft is rigid. Both shafts are mounted on bearing with viscose frictions B1 , Bx 1 , Bx 2 and B2 , respectively. The first mass has a rotation speed ω1 and the second ω2 . The rotation is the reaction to the torque T. The flexible shaft experiments a torsion Δθ. The system is modelled by Eqs. (2.5), (2.6) and (2.7), respectively. ω˙ 1 = − ω˙ 2 =

B1 + Ds Ds Ks 1 ω1 + ω2 − θ + T, J1 n x J1 J1 J1

Ds Bx1 + Ds + n x (Bx2 + n x B2 ) Ks ω1 − ω2 + θ and n x J2 n 2x J2 n x J2 θ˙ = ω1 −

ω2 . nx

(2.5) (2.6) (2.7)

The degrees of freedom are analysed in the following. There are two masses and every mass has only one possible motion, the rotation in one axis. Therefore, the DOF is 2. On the other hand, the number of variables N v is 4 (ω1 , ω2 , θ, T ) and the number of equations N e is 3. Hence, the degree of freedom N f according to (2.3) is 1 and the model is underspecified (N f > 0). A control loop can be added (N m = 1, N s = 0) to obtain an exactly specified model.

18

2 Fundamentals of System Modelling

2.4 Modelling Aspects of Wind Turbines Large wind turbines are a complex system that is difficult to model. Moreover, such machines require multidisciplinary knowledge, and each discipline is dominated by different modelling formalisms. The disciplines involved in the study of wind turbines are summarized in Fig. 2.4. Notice that the scheme of Fig. 2.4 is done thinking in an offshore machine. In the case of onshore machines, water and ice layers have to be eliminated. Several modelling formalisms are used depending on the considered subsystem. The aerodynamic subsystem is very fast, highly nonlinear and spatially distributed. It is normally modelled assuming stationary behaviour and using algebraic equations [23]. The blade is a large and flexible component with spatially distributed parameters in a permanent rotation. The high-definition model is based on a finite element approach [24]. However, such kind of formalism is inadequate for control and therefore, it is modelled either by assuming rigidity or by inserting one or more hinge or breaking points. It has at least 3 DOF. The tower is treated similarly as a highresolution model but a different abstraction concept is used in the low-resolution case. The rotating subsystem including the rotor, the high-speed shaft, the gearbox, the Rotational Dynamics Electrodynamics

Gearbox

Power Electronics

G

Grid

Aerodynamics

Wind

Tower

Air Structural dynamics

Ice-Tower Interaction Hydrodynamics Soil dynamics

Stream Tower-Soil Interaction

Water Soil

Fig. 2.4 Disciplines involved in the model formulation of wind turbines

2.4 Modelling Aspects of Wind Turbines

19

low-speed shaft and the mechanical part of the generator are modelled as multibody systems [25]. The next chapters of Part I are dedicated to presenting models for the particular subsystems. However, the intention is not to build a final model for the whole machine for a specific goal, but to make available to the user a variety of models for different objectives.

References 1. Box, G. (1979). Robustness in the strategy of scientific model building. In: Robustness in statistics (pp. 201–236). New York, USA: Academic Press. 2. De Silva, C. (2017). Modeling of dynamic systems with engineering applications. Boca Raton, USA: CRC Press. 3. Fishwick, P. (2007). Handbook of dynamic system modeling. Boca Raton, USA: CRC Press. 4. Ljung, L., & Söderström, T. (1983). Theory and practice of recursive identification. Cambridge, UK: MIT Press. 5. Ljung, L. (1999). System identification: Theory for the user, 2nd edn. Upper Saddle River, USA: Prentice Hall International. 6. Keesman, K. (2011). System identification: An introduction. London, UK: Springer. 7. Nelles, O. (2021). Nonlinear system identification: From classical approaches to neural networks, fuzzy models, and gaussian processes, 2nd edn. Berlin, Germany: Springer. 8. Tangirala, A. (2014). Principles of system identification: Theory and practice. Boca Raton, USA: CRC Press. 9. Isermann, R., & Münchhof, M. (2011). Identification of dynamic systems. Berlin, Germany: Springer. 10. Katayama, T. (2010). Subspace methods for system identification. Springer, London, UK. 11. Chen, H., & Zhang, Y. (2016). Power system optimization: Large-scale complex systems approaches. Singapore, Singapore: Wiley. 12. Mahmoud, M., Hassan, M., & Darwish, M. (1985). Large-scale systems. Nueva York, USA: Marcel Dekker, Inc. 13. Leithead, W., de la Salle, S., Reardon, D., & Grimble, M. (1991). Wind turbine modelling and control. In Proceedings of the International Conference on Control’91 (pp. 1–6). Edinburgh, UK. 14. van der Tempel, J., & Molenaar, D.-P. (2002). Wind turbine structural dynamics-A review of the principles for modern power generation, onshore and offshore. Wind Engineering, 26(4), 211–220. 15. Mirzaei, M., Henriksen, L., Poulsen, N., Niemann, H., & Hansen, M. (2012). Individual pitch control using LIDAR measurements. In Proceedings of the 2012 IEEE International Conference on Control Applications (CCA) (pp. 1646–1651). Dubrovnik, Croacia. 16. Chen, W., Ding, S., Haghani, A., Naik, A., Khan, A., & Yin, S. (2011). Observer-based FDI schemes for wind turbine benchmark. In: Proceedings of the 18th IFAC World Congress (pp. 7073–7078). Milano, Italy. 17. Odgaard, P., Stoustrup, J., & Kinnaert, M. (2013). Fault-tolerant control of wind turbines: A benchmark model. IEEE Transactions on Control Systems Technology, 21(4), 1168–1182. 18. Pournaras, C., Riziotis, V., & Kladas, A. (2008). Wind turbine control strategy enabling mechanical stress reduction based on dynamic model including blade oscillation effects. In Proceedings of the 2008 International Conference on Electrical Machines (pp. 1–6). Vilamoura, Portugal. 19. Slootweg, J., de Haan, S., Polinder, H., & Kling, W. (2003). General model for representing variable speed wind turbines in power system dynamics simulations. IEEE Transactions on Power Systems, 18(1), 144–151.

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2 Fundamentals of System Modelling

20. Soliman, M., Malik, O., & Westwick, D. (2010). Multiple model MIMO predictive control for variable speed. In Proceedings of the 2010 American Control Conference (pp. 2778–2784). Baltimore, USA. 21. Pourmohammad, S., & Fekih, A. (2011). Fault-tolerant control of wind turbine systems-A Review. In Proceedings of the 2011 IEEE Green Technologies Conference (IEEE-Green) (pp. 1– 6). Baton Rouge, USA. 22. Burlibas, A., & Ceanga, E. (2014). Frequency domain design of gain scheduling control for large wind systems in full-load region. Energy Conversion and Management, 86, 204–215. 23. Gambier, A., & Gebhardt, C. (2017). Modelling the aerodynamics of wind turbines for realtime simulation and control purposes. In Proceedings of the 2017 Asian Control Conference (pp. 1432–1437). Gold Cost, Australia. 24. El Chazly, N. (1993). Static and dynamic analysis of wind turbine blades using the finite element method. Renewable Energy, 3(6–7), 705–724. 25. Asadi, S. (2018). Wind turbine drive train system dynamics. Ph.D. Thesis, ISBN: 978-91-7597747-8. Chalmers University of Technology, Göteborg, Sweden.

Chapter 3

Modelling the Aerodynamic Subsystem

Summary The aerodynamic subsystem is essential for the operation of a wind turbine but at the same time probably the most difficult to model in a satisfactory way. On the one hand, it has very fast dynamics such that it is possible to assume steady-state algebraic equations and, on the other hand, these equations include much heuristics and are unprecise. The reality shows that the dynamics is spatially distributed and unsteady. Hence, the use of computational fluid dynamics calculations (CFD) is the state of the art for the study of wind but is inadequate for control. The present chapter introduces the subject attempting a presentation without losing the control application perspective.

3.1 Introduction The aerodynamics of wind turbines describes the dynamic energy transfer from the wind to the rotor. This is a very complex process such that the task of describing it by means of equations leads to a particular compromise: simple models provide an imprecise and insufficient representation of the energy transfer and, sophisticated models are not suitable for control purposes. A complete understanding of the aerodynamic process requires a deep immersion in the fluid dynamics as well as in the interaction between solid bodies and moving air. This field is out of the scope of the present chapter but it can be studied, for example, in [1]. The objective here is to provide the necessary information in order to comprehend the procedures used in the control system design. Most approaches to model the aerodynamic of wind turbines for control purposes use simple nonlinear equations that are obtained from the so-called actuator disc model. These equations depend on the aerodynamic coefficients, which, in turn, are quantified by using some empiric functions. A more sophisticated modelling paradigm is based on the blade element momentum (BEM) theory. However, this © Springer Nature Switzerland AG 2022 A. Gambier, Control of Large Wind Energy Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-84895-8_3

21

22

3 Modelling the Aerodynamic Subsystem

Tpitch Pitch Subsystem Ta Tb β

vw Aerodynamics

ωr Rotor ω xl

Nacelle

ωg

Drive Train

γ

Mt

Ft* x&t y& t

P Generator Tg electric

Generator mechanics

Tg

Tyaw

Tower

Yaw Subsystem

Fig. 3.1 Aerodynamic subsystem in the wind turbine decomposition

approach does not provide an analytic formulation but a numeric code that has to be embedded in the controller software. Both approaches are summarized in this chapter. Finally, the aerodynamic subsystem is localized in the system decomposition of Fig. 3.1.

3.2 Actuator Disc Model The simplest model is obtained assuming the ideal case of a 1D, incompressible, time-invariant flow, where the rotor is an “actuator disc”, through which the wind passes at the same speed (see Fig. 3.2). The kinetic energy of the air mass contained in a volume element flowing through a cross section A is expressed by

Ao disc actuator

vo

vtip

V

Ft V

Ad

ωr

Ta R

vd

Ai V vi

Fig. 3.2 Airflow through a stream tube

3.2 Actuator Disc Model

23

Ek =

1 1 m a v2 = ρa Ax v2 , 2 2

(3.1)

where ρ a is the density of air, v is the flow velocity and V = A x is a volume element of air, respectively. Hence, the variation of the energy per unit time in the cross section is the power contained in the free-air stream, i.e., Pa =

1 d Ek = [ρa A(d x/dt) v2 + 2ρa Ax v v˙ ], dt 2

(3.2)

and assuming that the air is passing at a constant velocity, the expression reduces to Pa =

1 d Ek = ρa Av3 . dt 2

(3.3)

When the stream flow passes through the actuator disc, a force, Ft =

P 1 = ρa Ad vd2 , vd 2

(3.4)

appears, which is known as thrust force and is a consequence of the pressure drop across the actuator disc. Thus, the thrust force can also be expressed as Ft = ( pd+ − pd− ) Ad ,

(3.5)

where pd− and pd+ are the pressures immediately before and after the actuator disc. Applying Bernoulli’s equation before and after the actuator disc, i.e., 1 1 pi + ρa vi2 = pd+ + ρa vd2 2 2

(3.6)

1 1 pd− + ρa vd2 = po + ρa vo2 , 2 2

(3.7)

and

the pressure drop in the actuator disc can be written as pd+ − pd− =

1 ρa (vi2 − vo2 ), 2

(3.8)

assuming pi = po , which is true if the angular velocity ωd imparted to the flow stream crossing the actuator disc is small compared to the angular velocity ωr of the rotor. Hence, an angular induction factor a is defined as

24

3 Modelling the Aerodynamic Subsystem

a =

ωd . 2 ωr

(3.9)

In addition, the angular velocity of air relative to the rotor increases from ωr to ωr + ωd while vd remains constant as it is assumed in (3.6) and (3.7). Introducing (3.8) in (3.5), the expression Ft =

1 ρa Ad (vi2 − vo2 ) 2

(3.10)

for the thrust force is obtained. However, it can also be given by the variation of momentum in the air stream Ft = ρa Ad vd vi − ρa Ad vd vo = ρa Ad (vi − vo )vd ,

(3.11)

where ρ a Ad vd is the mass of air. Comparing (3.10) and (3.11), the relationship between velocities is obtained as ρa Ad (vi − vo ) vd =

1 1 ρa Ad (vi2 − vo2 ) = ρa Ad (vi − vo )(vi + vo ). 2 2

(3.12)

Eliminating terms, the final expression for the velocity at the actuator disc is found as the mean value of the velocities at the entrance and exit of the stream tube vd =

1 (v + vo ). 2 i

(3.13)

Finally, the extracted power is P = Ft vd =

1 ρa Ad (vi2 − vo2 )(vi + vo ). 4

(3.14)

From (3.3), it follows that the power contained in the stream flowing through the area Ad with velocity vi without any extraction is Pw =

1 ρa Ad vi3 . 2

(3.15)

Hence, the ratio between the extracted power and the power contained in the free-air is defined as power factor C p , i.e., Cp =

P . Pw

Introducing (3.14) and (3.15) in (3.16) and simplifying, it follows

(3.16)

3.2 Actuator Disc Model

25

ρa Ad (vi2 − vo2 ) (vi + vo ) (v2 − vo2 ) (vi + vo ) = i 3 2 ρa Ad vi 2vi3       v 1 v 2 v2 v 1 1+ o = 1+ o . 1 − o2 1− o = 2 vi 2 vi vi vi

Cp =

(3.17)

Notice that (3.13) can also be expressed as vd =

     vo vo 1 1 1 vi = 2 − vi . (vi + vo ) = 1+ 1− 2 2 vi 2 vi

(3.18)

Defining now the axial induction factor as   vo 1 , 1− a= 2 vi

(3.19)

the velocity at the actuator disc can be written as vd = (2 − a) vi

(3.20)

C p = 4 a (1 − a)2 .

(3.21)

and (3.17) as

From (3.15) and (3.16), it follows P=

1 ρa Ad C p vi3 . 2

(3.22)

The thrust force can also be rewritten taking into account (3.4) and (3.20) in the form Ft =

1 ρa Ad C p vi3 Cp P 1 1 vi2 = ρa Ad Ct vi2 . = 2 = ρa Ad vd (2 − a)vi 2 (2 − a) 2

(3.23)

Thus, the thrust coefficient is defined as Ct =

Cp . (2 − a)

(3.24)

The action of the air stream on the rotor yields the aerodynamic torque Ta =

P , ωr

(3.25)

26

3 Modelling the Aerodynamic Subsystem

where ωr is the rotational speed. The tangential speed at blade tip is vtip = R ωr and the tip-speed ratio λ is defined as λ=

R ωr . vi

(3.26)

From (3.26), the rotational speed is ωr = λ vi /R and introducing this and (3.22) in (3.25), the equation for the aerodynamic torque is obtained as Ta =

Cp 3 Cp 2 1 1 1 ρa R Ad v = ρa R Ad Cq vi2 , v = ρa R Ad 2 λvi i 2 λ i 2

(3.27)

where Cq =

Cp . λ

(3.28)

Finally, assuming that the area of the disc actuator is the rotor area Ad = π R2 and that the magnitude of vi is the effective wind speed vwe , the main relationships becomes P=

1 3 π ρa R 2 C p vwe , 2

(3.29)

Ta =

1 π ρa R 3 Cq vi2 , 2

(3.30)

Ft =

1 2 π ρa R 2 Ct vwe . 2

(3.31)

This leads to a model, which is very often used in the literature and it consists of the mean equations, which describe the extracted power, the total torque applied to the rotor and the thrust force, i.e., 1 3 π ρa R 2 C p (β, λ)vwe , 2

(3.32)

1 2 π ρa R 2 Cq (β, λ)vwe and 2

(3.33)

1 2 π ρa R 2 Ct (β, λ)vwe , 2

(3.34)

P= Ta =

Ft =

where ρ, R and vwe are the air density, the rotor radius and the effective wind speed, respectively. C q and C t are the rotor torque and thrust force coefficients, which in turn depends on the pitch angle β and the tip-speed ratio λ = Rωr /vwe . ωr is the rotor

3.2 Actuator Disc Model

27

speed. Moreover, C q is obtained dividing the power coefficient C p by λ, i.e., C q = C p /λ. An important drawback of the above-presented approach is given by the fact that no information is available about the forces and moments acting on each blade. This is, however, important, for example, in the case of individual pitch control. Thus, (3.33) is modified in [2] in order to represent each blade separately having its own pitch angle value. It is assumed that the torque of each blade is equal to a 1/N b , with N b as the number of blades, of the torque given by (3.33) Tai =

1 2 (π/Nb ) ρ R 2 Cq (βi , λi )vwe , i 2

(3.35)

where i is the ith blade, on which the wind speed vwei acts. The pitch angle for this blade is β i . Variable λi is the tip-speed ratio corresponding to the wind speed vwei . However, this simplification is only valid for very small differences between pitch angles.

3.3 Determination of the Aerodynamic Coefficients Aerodynamic coefficients C p , C q and C t are normally provided at least in part by blade manufactures in the form of 2D lookup tables. However, a model-based control system design requires an analytical and differentiable function for the aerodynamic coefficients. In order to compute C p , several empirical functions are available from the literature as a function of the pitch angle β and the tip-speed ratio λ. In the case of C q , it is obtained, in general, as C p /λ. Although it is possible to use a similar procedure, based on curve fitting, for C t as it will be described for C q , it has not been proposed in the literature. In the following, the three most used empiric functions for C p are presented.

3.3.1 Exponential Approximation The exponential function was proposed first in [3] and later in [4] as well as [5]. The equation is generalized in [6]. However, many variations can be found in the literature and therefore, the formula proposed in [6] has been modified here in order to include all other particular cases. Thus, the exponential approach is summarized as follows:   C p (β, λ) = c1 c2 /λi − c3 β − c4 β c5 − c6 e−c7 (1/λi −c8 ) + c9 λ, where

(3.36)

28

3 Modelling the Aerodynamic Subsystem

Table 3.1 Important references for the exponential approach References

Parameters of Eqs. (3.36) and (3.37) c2

c3

[3, 4]

1

0

[6]

c2 (λ, β)

c4

c5

c8

c9

c10

c11

c12

c13

0

0

1/R

0

0

0

c7 (λ, β)

0

0

1

1 1

2 –

[5–9]

c7

0

–

0

0

1

2.14

0

0

1

0

–

0

= 0

1

0

–

= 0

= 0

1

[17]

0

–

0

[18]

0

–

0

= 0

0

–

0

0

[10–12] [13–15] [16]

[19] [20]

0

R Cf

0

λi−1 =

–

0

0