Tài liệu David wood auth. small wind turbines analysis, design, and application

Green Energy and Technology For further volumes: http://www.springer.com/series/8059 David Wood Small Wind Turbines Analysis, Design, and Application 123 Dr. David Wood Department of Mechanical and Manufacturing Engineering University of Calgary University Dr NW 2500 Calgary, AB T2N 1N4 Canada e-mail: [email protected] Additional material to this book can be downloaded from http://extra.springer.com ISSN 1865-3529 ISBN 978-1-84996-174-5 DOI 10.1007/978-1-84996-175-2 e-ISSN 1865-3537 e-ISBN 978-1-84996-175-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library  Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: eStudio Calamar, Berlin/Figueres Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The IEC Standard for small wind turbine safety, IEC 61400-2, defines a small wind turbine as having a rotor swept area of less than 200 m2 which corresponds to a rated power of about 50 kW. This approximate definition will be used in this text, which, like the Standard, covers only horizontal-axis wind turbines. Until the beginning of the twentieth century, all wind turbines were small, at least in terms of power output, and were used for water pumping and milling rather than producing electricity. One of the earliest small turbines for electricity production is shown in Fig. P.1. It was built by English Brothers of Wisbech, England and designed by Edward Burne. Under circumstances that are not clear, one of Burne’s windmills was installed on a farm owned by Russell Grimwade near Frankston, Victoria, Australia, in 1924. Grimwade recorded: the electric mains are nowhere within reach. Artificial illumination must be provided and here we displayed our eccentricities to the full. A large [sic] Dutch-type windmill was set up on an attractive hardwood tower that housed the batteries in its base. For artistic effect it gained full marks – for the effective generation of electricity it hardly scored a point. … It was bad engineering that the mill should fail to come up to the wind so that it ran backwards until something broke. …I still believe that man [sic] will someday make use of the power of the wind for his own purpose, and I feel that I have contributed to that research by demonstrating that my method was not the way to do it.1 The aim of this book is to demonstrate that, a century later, small wind turbines can be designed and built to avoid many of the problems that faced Grimwade. This is not to say that small turbine technology is mature; there are still areas where it lags well behind current practice for large turbines. This lag is mirrored in the theme of this book which is to provide basic analysis and design guidelines to allow a group of, say, senior engineering undergraduates or junior engineers to design and build a small wind turbine. The approach follows the ‘‘Simple Load 1 pp 141–142 of Poynter JR (1967) Russell Grimwade, Melbourne University Press. Grimwade was technically literate, see for example http://adbonline.anu.edu.au/biogs/A090693b.htm. v vi Preface Fig. P.1 The Burne small wind turbine on Russell Grimwade’s property in the 1920s. Photograph courtesy of the University of Melbourne Archives Model’’ (SLM) of IEC 61400-2 which is shown in Chap. 9 to provide straightforward, but necessarily approximate, equations for the main turbine loads and component stresses. There is no equivalent to the SLM in the IEC standard for large turbines. There are at least five areas where a student or other design group would need additional specialist advice: • Finite element analysis (FEA) for detailed stress calculations of the critical components • Electrical engineering advice on the generator and rectifier and possibly the inverter and grid connection • Detailed dynamics analysis for more accurate stress calculations and fatigue analysis • Foundation design, and • Control engineering help in devising and implementing a control strategy. The first is easily met as FEA is now a standard engineering tool. Its use is highlighted in Chap. 10 on tower design and manufacture. For the second, it is assumed that the turbine’s generator will be selected rather than designed and built as part of the project, so the level of knowledge required can be gained from standard texts on the subject. The few issues specific to small turbines are discussed in Chaps. 1, 7 and 11. Detailed dynamics analysis based on ‘‘aero-elastic’’ modeling is still an immature subject for small wind turbines but will undoubtedly develop as more small turbines are built and tested. Some references for aeroelastic modeling are given in the further reading section of Chap. 9. Foundation Preface vii design is usually site-specific but straightforward once the forces and the base overturning moments are calculated as demonstrated in Chap. 10. There has been a rush of specialist books on wind turbine control and grid interfacing over the last few years, so it would be remiss for this mechanical engineer and aerodynamicist to attempt to match them. Many of the basic control issues are shared by large and small turbines and those that are not are highlighted in the relevant chapters. Small turbines differ significantly from large ones in blade design and manufacture. The main differences are: low operational Reynolds numbers (Re), the need for good low wind performance at even lower Re, and the structural requirements of more-rapidly rotating blades. These issues are covered in the first six Chapters and culminate in Chap. 7 on multi-dimensional blade optimisation and manufacture. Most small turbines use ‘‘free yaw’’ whereby a tail fin, rather than a mechanical yaw drive as on larger machines, is used to align the turbine with the wind direction. Yaw behavior and associated issues of tail fin design and aerodynamic over-speed protection are covered in Chap. 8. The text describes and lists a number of Matlab programs for wind turbine analysis and design. These and supplementary programs, referred to but not listed, can be downloaded from the online material (start at http://extras.springer.com) which also contains additional matter relating to small turbines and the solutions to the Exercises at the end of each chapter. The programs include blade element methods, Chap. 5, multi-dimensional optimisation methods for the design of blades, Chap. 7, and towers, Chap. 10. Excel spreadsheets are provided for noise estimation (Chap. 1) and the loads and component stresses under the IEC Simple Load Model (Chap. 9). All the programs and spreadsheets referred to in the book were written or re-written by the author and have been used for actual turbine analysis and design. The likelihood of errors in them is small but non-zero. They are provided without guarantee. The same applies to the supplementary programs some of which were written by others. This book is a distillation of more than twenty five years experience working in small wind turbine research, development, and commercialisation. Over the years, my work has been supported by the Australian Research Council, the NSW Renewable Energy Research and Development Fund, the NSW Renewable Energy Development Program, and the Asia-Pacific Partnership on Clean Development. A very important year spent at NASA Ames Research Center was funded by the U.S. National Research Council. There are also many, many people to thank for assistance over that time. I particularly acknowledge Professor Phil Clausen and Paul Peterson who shared much of that time with me. Paul and Sturt Wilson have also shared the vicissitudes of starting and developing a small wind turbine company, Aerogenesis Australia, which incidentally, had its first commercial installation on a farm in Victoria. Sturt Wilson and Phil Clausen provided the FEA of the monopole and lattice tower, respectively, in Chap. 10. Jason Brown wrote the initial version of the SLM spreadsheet in Chap. 9. My graduate students, starting with Phil Clausen and continuing down to Dr. Matthew Clifton-Smith as the last one to complete, have contributed enormously to my knowledge. Most of them appear as co-authors on publications referred to in the main text. I also thank viii Preface many other colleagues from around the world for providing specific information, answering my questions, listening to my thoughts developing, and correcting them when necessary. Earlier versions of some chapters were used for lecture notes at Newcastle University, where I spent most of those twenty five years, and for a short course at Kathmandu University organised by Dr. Peter Freere. The material was updated and expanded into this text during the first year of my tenure of the ENMAX/Schulich Chair of Renewable Energy at the University of Calgary. I thank the University and the ENMAX Corporation for their vision in supporting distributed generation, here in the form of small wind turbines. For specific help with this book I thank Peter Freere and Professor Ed Nowicki who co-authored Chap. 11. Phil Clausen and Sturt Wilson gave valuable comments on Chap. 10 and Sturt drew on his blade making skills to improve Chap. 7. Dr. Damien Leqlerq of Cyclopic Energy reviewed Chap. 12 and provided two of the figures. Jim Baxter, Colin Dumais, and Robert Falconer of the ENMAX Corporation provided photographs and information. Colin also brought to my attention several of the interesting web-sites referred to in the book. Mohamed Hammam read the entire manuscript, checked the programs and found and corrected a significant number of typographical errors. At this point it is customary for authors to thank their family for their supposed forbearance while the book was written. I will not do this because my children have left home and my partner Dr. Cassandra Arnold was working for Medecins Sans Frontieres in Africa for much of that time. However her influence, advice, and proofreading give me much to be thankful for. I also thank my daughter Katie who acquainted me with Burne and Grimwade. One of my great pleasures over the last twenty five years has been to meet people from around the world who are passionate about small wind turbine technology and its role in mitigating climate change and the huge imbalances in the distribution of wealth and health in this world. I dedicate this book to them and I hope that it will further their efforts. In this regard I acknowledge Springer’s generous and enthusiastic agreement to have a special price for the book in developing countries. All royalties from this book will be used to advance the cause of renewable energy in the developing world. Calgary, May 2011 David Wood Contents . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 6 8 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 15 17 22 24 25 28 2 Control Volume Analysis for Wind Turbines. . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conservation of Momentum. . . . . . . . . . . . . . . . . . . . . . 2.5 Conservation of Angular Momentum . . . . . . . . . . . . . . . 2.6 Conservation of Energy. . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Turbine Operating Parameters and Optimum Performance. 2.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 31 32 34 35 35 36 39 40 3 Blade Element Theory for Wind Turbines. . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Some Assumptions of Blade Element Theory . . . . . . . . . . . . 41 41 42 1 Introduction to Wind Turbine Technology . . . . . . . . . . . . . 1.1 How Much Energy is in the Wind? . . . . . . . . . . . . . . . 1.2 Examples of Wind Turbines . . . . . . . . . . . . . . . . . . . . 1.3 Wind Turbine Noise. . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Turbine Operating Parameters . . . . . . . . . . . . . . . . . . . 1.5 The Power Curve and the Performance Curve . . . . . . . . 1.6 The Variation in Wind Speed and Power Output with Height . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Turbulence and Wind Statistics . . . . . . . . . . . . . . . . . . 1.8 The Electrical and Mechanical Layout of Wind Turbines 1.9 The Size Dependence of Turbine Parameters . . . . . . . . . 1.9.1 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix x Contents 3.3 The Conservation Equations for Annular Streamtubes. 3.3.1 Conservation of Mass . . . . . . . . . . . . . . . . . 3.3.2 Conservation of Momentum. . . . . . . . . . . . . 3.3.3 Conservation of Angular Momentum . . . . . . 3.4 The Forces Acting on a Blade Element . . . . . . . . . . . 3.5 Combining the Equations for the Streamtube and the Blade Element . . . . . . . . . . . . . . . . . . . . . . 3.6 Matlab Programs for Blade Element Analysis . . . . . . 3.7 Some Consequences of the Blade Element Equations . 3.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 43 43 44 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 47 54 54 55 Aerofoils: Lift, Drag, and Circulation . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Geometry and Definition of Aerofoils . . . . . . . . . . . . . . . 4.3 Aerofoil Lift and Drag . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Aerofoil Lift and Drag at High Angles of Attack . . . . . . . 4.5 The Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Further Discussion on Reynolds Number, High Incidence, and Aspect Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Further Reading . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 57 57 59 64 66 . . . . . . . . . . . . 69 72 73 74 Element Calculations . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . Altering the Programs from Chap. 3 . Running the Programs. . . . . . . . . . . Changing the Aerofoil. . . . . . . . . . . Maximising Power Extraction . . . . . 5.5.1 Further Reading . . . . . . . . . 5.5.2 Exercises . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . 5 Blade 5.1 5.2 5.3 5.4 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 77 78 85 92 93 98 98 99 6 Starting and Low Wind Speed Performance 6.1 Introduction . . . . . . . . . . . . . . . . . . . . 6.2 Estimating the Starting Torque. . . . . . . 6.3 Analysis of Starting . . . . . . . . . . . . . . 6.4 Estimating the Rotor Inertia. . . . . . . . . 6.5 Matlab Programs for Starting . . . . . . . . 6.5.1 Exercises . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 101 105 108 111 112 116 117 Contents xi 7 Blade 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Design, Manufacture, and Testing . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . Optimisation Method . . . . . . . . . . . . . . . Matlab Programs for Optimisation . . . . . . Example Blade Design: A 750 W Turbine Blade Manufacture . . . . . . . . . . . . . . . . . Blade Testing. . . . . . . . . . . . . . . . . . . . . Forming the Rotor . . . . . . . . . . . . . . . . . 7.7.1 Exercises . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 119 119 121 126 134 139 141 142 143 8 The Unsteady Aerodynamics of Turbine Yaw and Over-Speed Protection . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 8.2 Fundamentals of Tail Fin Aerodynamics . . 8.3 Unsteady Aerodynamics of Tail Fins . . . . 8.4 Planform Effects on Tail Fin Performance. 8.5 Rotor Effects on Yaw Performance . . . . . 8.6 High Yaw Rates . . . . . . . . . . . . . . . . . . . 8.7 Aerodynamic Over-speed Protection . . . . . 8.7.1 Furling . . . . . . . . . . . . . . . . . . . 8.7.2 Pitching. . . . . . . . . . . . . . . . . . . 8.7.3 Exercises . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 145 146 149 155 157 158 159 160 162 164 166 Using the IEC Simple Load Model for Small Wind Turbines . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Simple Load Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Load Case A: Normal Operation . . . . . . . . . . . . . . . 9.2.2 Load Case B: Yawing. . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Load Case C: Yaw Error. . . . . . . . . . . . . . . . . . . . . 9.2.4 Load Case D: Maximum Thrust . . . . . . . . . . . . . . . . 9.2.5 Load Case E: Maximum Rotational Speed . . . . . . . . 9.2.6 Load Case F: Short at Load Connection . . . . . . . . . . 9.2.7 Load Case G: Shutdown (Braking). . . . . . . . . . . . . . 9.2.8 Load Case H: Parked Wind Loading. . . . . . . . . . . . . 9.2.9 Load Case I: Parked Wind Loading, Maximum Exposure . . . . . . . . . . . . . . . . . . . . . . . . 9.2.10 Load Case J: Transportation, Assembly, Maintenance and Repair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Stress Calculations and Safety Factors . . . . . . . . . . . . . . . . . 9.3.1 Equivalent Component Stresses . . . . . . . . . . . . . . . . 9.3.2 Partial Safety Factors . . . . . . . . . . . . . . . . . . . . . . . 169 169 171 173 174 175 176 176 176 177 177 9 179 179 180 180 181 xii Contents 9.3.3 Ultimate Stress Analysis . . . . . . . . . . . . . . . . . . 9.3.4 Fatigue Failure Analysis . . . . . . . . . . . . . . . . . . 9.4 Simple Load Model Analysis of 500 W Turbine . . . . . . . 9.4.1 Loads for Case A: Normal Operation . . . . . . . . . 9.4.2 Loads for Case B: Yawing . . . . . . . . . . . . . . . . 9.4.3 Loads for Case C: Yaw Error . . . . . . . . . . . . . . 9.4.4 Loads for Case D: Maximum Thrust. . . . . . . . . . 9.4.5 Loads for Case E: Maximum Rotational Speed . . 9.4.6 Loads for Case F: Short at Electrical Connection . 9.4.7 Loads for Case H: Parked Wind Loading . . . . . . 9.5 Equivalent Component Stresses and Ultimate Material Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Equivalent Stress for Case A: Normal Operation . 9.5.2 Equivalent Stress for Case B: Yawing . . . . . . . . 9.5.3 Equivalent Stress for Case C: Yaw Error . . . . . . 9.5.4 Equivalent Stress for Case D: Maximum Thrust. . 9.5.5 Equivalent Stress for Case E: Maximum Rotational Speed . . . . . . . . . . . . . . . . . . . . . . . 9.5.6 Equivalent Stress for Case F: Short at Electrical Connection . . . . . . . . . . . . . . . . . . . . 9.5.7 Equivalent Stress for Load Case H: Parked Wind Loading . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Spreadsheet for the Simple Load Model . . . . . . . . . . . . . 9.7 Further Test Requirements. . . . . . . . . . . . . . . . . . . . . . . 9.8 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.1 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Tower Design and Manufacture . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . 10.2 Monopole Towers. . . . . . . . . . . . . 10.3 Optimisation of Monopole Towers.. 10.4 Lattice Towers . . . . . . . . . . . . . . . 10.5 Guyed Towers . . . . . . . . . . . . . . . 10.5.1 Exercises . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Generator and Electrical System. . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . 11.2 Generators for Small Turbines . . . . . . . 11.3 Gearboxes.. . . . . . . . . . . . . . . . . . . . . 11.4 Rectifiers, Inverters, and Basic Control . 11.5 System Protection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 182 183 183 184 185 185 185 186 186 . . . . . . . . . . . . . . . 186 187 189 189 189 ... 189 ... 190 . . . . . . . . . . . . . . . . . . 190 190 192 194 196 198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 199 201 212 216 221 222 224 . . . . . . . . . . . . . . . . . . . . . . . . 227 227 228 233 234 240 Contents xiii 11.6 11.7 11.8 Manual Shutdown and Condition Electrical Wiring . . . . . . . . . . . Lightning Protection . . . . . . . . . 11.8.1 Further Reading . . . . . . 11.8.2 Exercises . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . Monitoring. ......... ......... ......... ......... ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 246 247 249 249 249 Site Assessment and Installation . . 12.1 Introduction . . . . . . . . . . . . . 12.2 Site Assessment.. . . . . . . . . . 12.3 Optimum Tower Height . . . . 12.4 Tower Raising and Lowering . 12.4.1 Exercises . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 251 252 255 259 262 263 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symbols and Abbreviations Because of the wide range of topics covered, a number of symbols have multiple meanings, such as R for blade tip radius and resistance in Ohms. A symbol that has a specific meaning for only one chapter is indicated by giving the chapter number. Many of the symbols used only in Chap. 9 and defined in IEC 61400-2 are not listed here. Table 9.1 lists the present symbols that are different from those in the standard. Symbols A A A A A AR a, a0 a a a a 0, a 1, a 2 B b b C C Ca C a0 CD Cd Cd0 CL Cl Cl,max CP CP CP,r CQ CT c c D Swept area of blades (m2) Coefficient in Eq. 4.6 Aerofoil cross-sectional area (m2), Chap. 6 Tail fin area (m2), Chap. 8 Cross-sectional area of structural member (m2), Chap. 10 Aspect ratio Axial and rotational induction factors respectively Weighting factor in evolutionary optimisation, e.g. Eq. 7.2, usually subscripted Side length of polygonal tower (m), Chap. 10 Duty cycle, Chap. 11 Coefficients in Eq. 10.6 Coefficient in Eq. 4.7 Tail fin span (m) Basis vector for evolutionary optimisation, Eq. 7.1 Cumulative probability density Eq. 1.18 Coefficient in Eq. 4.7 Axial force coefficient, Eq. 3.10 Tangential force coefficient, Eq. 3.11 Three-dimensional drag coefficient Two-dimensional drag coefficient Minimum drag coefficient Three-dimensional lift coefficient Two-dimensional lift coefficient Cl For maximum lift:drag Power coefficient Eq. 1.7 Aerofoil surface pressure coefficient, Chap. 4 Extracted power coefficient, Chap. 7 Torque coefficient Force (thrust) coefficient Eq. 1.13. Blade chord (m) Comparison vector of evolutionary optimisation Drag on three-dimensional body (N) xv xvi D d d d d d0 d1 dh E e F Fy f g H h hopt hr i I I I I1 , I 2 , I 3 Icp Iu J K K1, K2, K3 K p, K v k L LA Lp l l M M0 m mt mtt N Ncycles Nd NP Symbols and Abbreviations Drag per unit height on a tower (N/m), Chap. 10 Drag per unit span on two-dimensional body (N/m) Tower diameter (m), Chap. 10 Distance from turbine (m), Chap. 1 Distance from rotor to yaw axis (m), Chap. 8 Tower top diameter (m) Slope of linearly-tapered tower Tower base diameter (m) Young’s modulus (GPa) Eccentricity of rotor centre of mass (m), Chaps. 7 and 9 Prandtl tip loss factor, Eq. 5.1 Yield stress (MPa) Term in Prandtl tip loss factor, Eq. 5.2 Acceleration due to gravity = 9.81 m/s2 Effective turbine height (m), Chap. 12 Tower height (m) Optimum tower height (m), Chap. 12 Reference height (m), Eqs. 1.14 and 1.15 Indent on delta wing, Chap. 8 Moment of inertia about the yaw axis (kgm2), Chap. 8 Area moment of inertia (m2), Chap. 10 Current (amps), Chap. 11 Integrals in Eq. 6.10 Chord-pitch integral Eqs. 6.6, 6.7 Turbulence intensity, Eq. 1.17 Rotational inertia (kg m2) Lift-slope for a delta wing (1/rad) Unsteady slender body coefficients, Eq. 8.8 Polhamus coefficients for delta wing, Eq. 8.2 Numerical factor in Eq. 8.1 Lift on three-dimensional body (N) Noise level (dBA), Eq. 1.6 Sound power level (dB), Eq. 1.4 Lift per unit span on two-dimensional body (N/m) Distance from base of tower and turbine centre of mass (m), Chap. 12 Moment (Nm) Blade root bending moment (Nm) Exponent in power law, Eq. 1.4 Mass of tower (kg) Mass of turbine (tower top mass) (kg) Number of blades Number of fatigue cycles to failure Annual average number of lightning strikes, Chap. 11 Number of poles in generator Symbols and Abbreviations Ns n n1 Ns P P P 1, P 2 P p p Q Q Qr R R Re r r xvii T T T Td Ts t t t t U Up U? Vtip U10 U0 Us UT W x x Y1,Y2 z0 Number of tower sections Number of fatigue cycles Structural first natural frequency (Hz), Chap. 10 Synchronous generator speed (rpm), Chap. 11 Power (W) Aerofoil surface pressure (Pa),Chap. 4 Pressure on the upwind and downwind face of rotor (Pa), Chap. 2 Average power (W) Probability density function, Eq. 1.19 Vortex pitch, Chap. 6 Torque (Nm) Volume flow rate (m3/s), Chap. 2 Resistive torque (Nm) Blade tip radius (m) Resistance (Ohms), Chap. 11 Reynolds number Radial co-ordinate along blade (m) Distance from tail fin center of pressure to yaw axis (m) (Chap. 8 only) Turbine thrust (N) Temperature (C) Cable tension (N), Chap. 12 Turbine design lifetime Starting time (s) Time (s) Aerofoil thickness, Chap. 4 Tower thickness (m or mm), Chap. 10 Trial vector for evolutionary optimisation Wind speed (m/s) Wind speed for rated power (m/s) Wind speed in the far-wake (m/s) Blade tip circumferential velocity (m/s) Wind speed at 10 m (m/s) Wind speed at hub height (m/s) Wind speed for starting (m/s) Total velocity at blade element (m/s) Circumferential velocity (m/s) Distance along chord line (m), Chap. 4 Tail boom length (m), Chap. 8 Factors in Eq. 5.6 Roughness length (m) a a Coefficient of atmospheric absorption of sound (dB/m), Eq. 1.6 Angle of attack (rad) xviii Symbols and Abbreviations amax U f g h hp u k kf kp kr ks l m q r r ra rb X / / x xn Angle of attack for maximum lift:drag (rad) Circulation (m2/s) Damping ratio Efficiency Yaw angle (rad) Blade twist angle (rad) Wind direction (rad), Chap. 8 Tip speed ratio Eq. 1.10 Tip speed ratio at the end of starting Tip speed ratio for rated power Local tip speed ratio, Eq. 3.7 Tip speed ratio for starting Viscosity of air (kg/m/s) Kinematic viscosity of air (m2/s) Density (kg/m3) Blade element solidity, Eq. 3.14 Component stress (MPa), Chaps. 9 and 10 Axial stress (MPa), Chap. 10 Bending stress (MPa), Chap. 10 Blade speed (usually rad/s) Blade inflow angle (rad), Eq. 3.8 Azimuthal angle (rad), Chap. 8 Yaw rate (rad/s) Natural frequency Subscripts 0 1 2 ? b design LL max P s t tt tail overbar Well upstream of turbine (in undisturbed wind) Upwind face of rotor Downwind face of rotor Far-wake Blade Design value Line-to-line Maximum value Rated power Starting Tower Tower top Tail Time average Symbols and Abbreviations Abbreviations 1P AC AS ASCE BE BET CF CV DC IGBT IEC IEEE IG FEA GL KE MPPT NACA NREL ODE PD PMG RMS SCI SLM THD Blade Passing Frequency Alternating Current Australian Standard American Society of Civil Engineers Blade Element Blade Element Theory Capacity Factor Control Volume Direct Current Insulated Gate Bipolar Transistor International Electrotechnical Commission Institute of Electrical and Electronic Engineers Induction Generator Finite Element Analysis Germanischer Lloyd Kinetic Energy (J) Maximum Power Point Tracking U.S. National Advisory Committee on Aeronautics U.S. National Renewable Energy Laboratory Ordinary Differential Equation Power Density (W/m2) Permanent Magnet Generator Root Mean Square The Steel Construction Institute IEC Simple Load Model Total Harmonic Distortion xix