Ocean Engineering & Oceanography 7 Arthur Pecher Jens Peter Kofoed Editors Handbook of Ocean Wave Energy Ocean Engineering & Oceanography Volume 7 Series editors Manhar R. Dhanak, Florida Atlantic University, Boca Raton, USA Nikolas I. Xiros, New Orleans, USA More information about this series at http://www.springer.com/series/10524 Arthur Pecher • Jens Peter Kofoed Editors Handbook of Ocean Wave Energy Editors Arthur Pecher Wave Energy Research Group, Department of Civil Engineering Aalborg University Aalborg Denmark Jens Peter Kofoed Wave Energy Research Group, Department of Civil Engineering Aalborg University Aalborg Denmark ISSN 2194-6396 ISSN 2194-640X (electronic) Ocean Engineering & Oceanography ISBN 978-3-319-39888-4 ISBN 978-3-319-39889-1 (eBook) DOI 10.1007/978-3-319-39889-1 Library of Congress Control Number: 2016943821 © The Editor(s) (if applicable) and The Author(s) 2017. This book is published open access. 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All commercial rights are reserved by the Publisher, whether the whole or part of the material is concerned, speci fi cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro fi lms 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 publi- cation does not imply, even in the absence of a speci fi c 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland The original version of the book was revised: For detailed information please see erratum. The erratum to this book is available at DOI 10.1007/978-3-319-39889-1_11 Preface This Handbook for Ocean Wave Energy aims at providing a guide into the fi eld of ocean wave energy utilization. The handbook offers a concise yet comprehensive overview of the main aspects and disciplines involved in the development of wave energy converters (WECs). The idea for the book has been shaped by the devel- opment, research, and teaching that we have carried out at the Wave Energy Research Group at Aalborg University over the past decades. It is our belief and experience that it would be useful writing and compiling such a handbook in order to enhance the understanding of the sector for a wide variety of potential readers, from investors and developers to students and academics. At the Wave Energy Research Group, we have a wide range of wave energy related activities ranging from teaching at master and Ph.D. level, undertaking generic research projects and participating in speci fi c research and development projects together with WEC developers and other stakeholders. All these activities have created a solid background in terms of theoretical knowledge, experimental and numerical modeling skills as well as a scienti fi c network, which is why we found that the idea of putting this book together seemed realistic. With this as a starting point, we gathered a group of authors, each an expert within their speci fi c research topic. It was clear from the beginning that the ambition was to make a high-quality publication but still ensuring that it would have a high level of accessibility. Therefore, we wanted the book to be freely available in digital form. To make this happen, we sought and received funding from the Danish EUDP program (project no. 64015-0013), for which we are extreme thankful. The ten chapters of the handbook present a broad range of relevant rules of thumb and topics, such as the technical and economic development of a WEC, wave energy resource, wave energy economics, WEC hydrodynamics, power take-off systems, mooring systems as well as the experimental and numerical simulation of WECs. It covers the topic of wave energy conversion from different perspectives, providing the readers, who are experts in one particular topic, with a clear overview of the key aspects in other relevant topics in which they might be less specialized. vii We would especially like to thank our co-authors, who have contributed enthusiastically to the content and without whom we would never have been able to realize this handbook. We would also like to thank our colleagues at the Department of Civil Engineering for supporting us, especially Kim Nielsen who patiently helped us getting all the small fi nal details in place as well as reading through all the chapters for fi nal corrections and comments, and Vivi S ø ndergaard who gave the fi nal touch to the English language. Last but not least, we would like to thank our wives, Marie Isolde M ü ller and Kirsten Aalstrup Kofoed, for their endless patience and support. We have enjoyed working with you all and we are very grateful for each of your contribution. Aalborg, Denmark Arthur Pecher 2016 Jens Peter Kofoed viii Preface Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Arthur Pecher and Jens Peter Kofoed 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Successful Product Innovation. . . . . . . . . . . . . . . . . . . . . 2 1.3 Sketching WECs and Their Environment . . . . . . . . . . . . . . . . 3 1.4 Rules of Thumb for Wave Energy . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 The Essential Features of a WEC . . . . . . . . . . . . . . . . 5 1.4.2 Economic Rules of Thumb . . . . . . . . . . . . . . . . . . . . . 6 1.4.3 WEC Design Rules of Thumb. . . . . . . . . . . . . . . . . . . 9 1.4.4 Power Take-Off Rules of Thumb . . . . . . . . . . . . . . . . . 12 1.4.5 Environmental Rules of Thumb . . . . . . . . . . . . . . . . . . 13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 The Wave Energy Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Jens Peter Kofoed 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Potential of Wave Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Wave Energy Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Categorization of WEC ’ s . . . . . . . . . . . . . . . . . . . . . . 23 2.3.3 Examples of Various WEC Types . . . . . . . . . . . . . . . . 24 2.3.4 The Development of WECs . . . . . . . . . . . . . . . . . . . . 37 2.4 Test Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 The Wave Energy Resource . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Matt Folley 3.1 Introduction to Ocean Waves. . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.1 Origin of Ocean Waves . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.2 Overview of the Global Wave Energy Resource . . . . . . 45 ix 3.2 Water Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.1 De fi nition and Symbols . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.2 Dispersion Relationship . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.3 Water Particle Path and Wave Motions . . . . . . . . . . . . 48 3.3 Characterisation of Ocean Waves and the Wave Climate. . . . . . 50 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.2 Temporal, Directional and Spectral Characteristics of the Wave Climate . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.3 Spectral Representation of Ocean Waves . . . . . . . . . . . 55 3.3.4 Characterization Parameters . . . . . . . . . . . . . . . . . . . . 57 3.3.5 Challenges in Wave Climate Characterisation . . . . . . . . 60 3.3.6 Coastal Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.7 Case Study — Incident Wave Power . . . . . . . . . . . . . . . 66 3.4 Measurement of Ocean Waves. . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.2 Surface-Following Buoy . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.3 Sea-Bed Pressure Sensor. . . . . . . . . . . . . . . . . . . . . . . 69 3.4.4 Acoustic Current Pro fi ler . . . . . . . . . . . . . . . . . . . . . . 69 3.4.5 Land-Based and Satellite Radar . . . . . . . . . . . . . . . . . . 70 3.5 Modelling of Ocean Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5.2 General Spectral Wave Models . . . . . . . . . . . . . . . . . . 72 3.5.3 Third Generation Spectral Wave Models. . . . . . . . . . . . 74 3.5.4 Grid De fi nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4 Techno-Economic Development of WECs . . . . . . . . . . . . . . . . . . . 81 Arthur Pecher and Ronan Costello 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.1.1 Continuous Evaluation of the WEC Potential . . . . . . . . 81 4.1.2 Overview of the Techno-Economic Development. . . . . . 82 4.2 The WEC Development Stages . . . . . . . . . . . . . . . . . . . . . . . 83 4.3 Techno-Economic Development Evaluation . . . . . . . . . . . . . . . 85 4.3.1 The Technology Readiness and Performance Level . . . . 85 4.3.2 The WEC Development Stages and the TRL Scale . . . . 87 4.3.3 The TRL-TPL R&D Matrix . . . . . . . . . . . . . . . . . . . . 88 4.3.4 Uncertainty Related to the TRL-TPL Matrix . . . . . . . . . 90 4.3.5 Valuation of R&D Companies. . . . . . . . . . . . . . . . . . . 91 4.4 Techno-Economic Development Strategies . . . . . . . . . . . . . . . 92 4.4.1 R&D Strategy as TRL-TPL Trajectories . . . . . . . . . . . . 92 4.4.2 Extreme Cases of Techno-Economic Development Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4.3 Ef fi cient Techno-Economic Development . . . . . . . . . . . 95 x Contents 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.6 Overview of Some of the Leading WECs . . . . . . . . . . . . . . . . 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5 Economics of WECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Ronan Costello and Arthur Pecher 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Power Is Vanity — Energy Is Sanity . . . . . . . . . . . . . . . . . . . . 102 5.3 Economic Decision Making. . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3.1 Cash Flow Terminology . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.2 Time Value of Money (and Energy). . . . . . . . . . . . . . . 105 5.3.3 Economic Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.4 Effect of Depreciation on Discounting . . . . . . . . . . . . . 112 5.3.5 Effect of In fl ation on Discounting . . . . . . . . . . . . . . . . 112 5.3.6 Setting the Discount Rate . . . . . . . . . . . . . . . . . . . . . . 113 5.3.7 Economic Decision Making — Which Metric to Use? . . . . 114 5.3.8 Expert Oversight and Independent Review . . . . . . . . . . 116 5.4 Economic Analysis in Technology R&D. . . . . . . . . . . . . . . . . 117 5.5 Techno-Economic Assessment and Optimisation . . . . . . . . . . . 118 5.6 WEC Cost-of-Energy Estimation Based on Offshore Wind Energy Farm Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.6.2 De fi nition of the Categories . . . . . . . . . . . . . . . . . . . . 120 5.6.3 Wind Energy Project Case . . . . . . . . . . . . . . . . . . . . . 121 5.6.4 Wave Energy Case . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.6.5 Cost Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.6.6 Revenue and Energy Yield . . . . . . . . . . . . . . . . . . . . . 133 5.7 Strategic Support Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 133 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6 Hydrodynamics of WECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 J ø rgen Hals Todalshaug 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.1.1 Wave Energy Absorption is Wave Interference . . . . . . . 139 6.1.2 Hydrostatics: Buoyancy and Stability . . . . . . . . . . . . . . 140 6.1.3 Hydrodynamic Forces and Body Motions . . . . . . . . . . . 143 6.1.4 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.1.5 Oscillating Water Columns — Comments on Resonance Properties and Modelling . . . . . . . . . . . . . . . . . . . . . . 147 6.1.6 Hydrodynamic Design of a Wave Energy Converter. . . . . 149 6.1.7 Power Estimates and Limits to the Absorbed Power . . . . . 153 6.1.8 Controlled Motion and Maximisation of Output Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Contents xi 7 Mooring Design for WECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Lars Bergdahl 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.1.2 Mooring Design Development Overview . . . . . . . . . . . 160 7.1.3 Wave-Induced Forces on Structures . . . . . . . . . . . . . . . 162 7.1.4 Motions of a Moored Device in Waves . . . . . . . . . . . . 162 7.2 Metocean Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.2.1 Combinations of Environmental Conditions . . . . . . . . . 162 7.2.2 Design Wave Conditions . . . . . . . . . . . . . . . . . . . . . . 163 7.2.3 Environmental Data at DanWEC . . . . . . . . . . . . . . . . . 165 7.2.4 Example Design Conditions . . . . . . . . . . . . . . . . . . . . 166 7.3 Estimation of Environmental Forces . . . . . . . . . . . . . . . . . . . . 166 7.3.1 Overview and Example Floater Properties. . . . . . . . . . . 166 7.3.2 Mean Wind and Current Forces. . . . . . . . . . . . . . . . . . 167 7.3.3 Wave Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.3.4 Summary of Environmental Forces on Buoy . . . . . . . . . 178 7.4 Mooring System Static Properties. . . . . . . . . . . . . . . . . . . . . . 179 7.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.4.2 Catenary Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.4.3 Mean Excursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.5 Alternative Design Procedures . . . . . . . . . . . . . . . . . . . . . . . . 183 7.5.1 Quasi-Static Design . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.5.2 Dynamic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.5.3 Response-Based Analysis . . . . . . . . . . . . . . . . . . . . . . 188 7.6 Response Motion of the Moored Structure . . . . . . . . . . . . . . . 189 7.6.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.6.2 Free Vibration of a Floating Buoy in Surge . . . . . . . . . 190 7.6.3 Response to Harmonic Forces . . . . . . . . . . . . . . . . . . . 191 7.6.4 Response Motion in Irregular Waves . . . . . . . . . . . . . . 194 7.6.5 Equivalent Linearized Drag Damping . . . . . . . . . . . . . . 196 7.6.6 Second-Order Slowly Varying Motion . . . . . . . . . . . . . 197 7.6.7 Wave Drift Damping . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.6.8 Combined Maximum Excursions . . . . . . . . . . . . . . . . . 198 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 8 Power Take-Off Systems for WECs . . . . . . . . . . . . . . . . . . . . . . . 203 Am é lie T ê tu 8.1 Introduction, Importance and Challenges. . . . . . . . . . . . . . . . . 203 8.2 Types of Power Take-Off System. . . . . . . . . . . . . . . . . . . . . . 205 8.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.2.2 Air Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.2.3 Hydraulic Converters . . . . . . . . . . . . . . . . . . . . . . . . . 210 xii Contents 8.2.4 Hydro Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.2.5 Direct Mechanical Drive Systems . . . . . . . . . . . . . . . . 213 8.2.6 Direct Electrical Drive Systems . . . . . . . . . . . . . . . . . . 213 8.2.7 Alternative PTO Systems . . . . . . . . . . . . . . . . . . . . . . 214 8.3 Control Strategy of Power Take-Off System . . . . . . . . . . . . . . 214 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.3.2 Types of Control Strategy. . . . . . . . . . . . . . . . . . . . . . 215 8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 9 Experimental Testing and Evaluation of WECs . . . . . . . . . . . . . . . 221 Arthur Pecher 9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 9.2 Tank Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 9.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 9.2.2 Representative Sea States . . . . . . . . . . . . . . . . . . . . . . 224 9.2.3 Hydrodynamic Response . . . . . . . . . . . . . . . . . . . . . . 229 9.2.4 Power Performance Evaluation . . . . . . . . . . . . . . . . . . 234 9.2.5 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 9.2.6 Structural and Mooring Loads . . . . . . . . . . . . . . . . . . . 247 9.2.7 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.3 Sea Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 9.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 9.3.2 Performance Assessment of WECs Based on Sea Trials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 10 Wave-to-Wire Modelling of WECs . . . . . . . . . . . . . . . . . . . . . . . . 261 Marco Alves 10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 10.2 Wave-to-Wire Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.2.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.2.2 Excitation Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.2.3 Hydrostatic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 10.2.4 Mooring Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 10.2.5 Radiation Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 10.2.6 PTO Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10.2.7 End Stops Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 281 10.3 Benchmark Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.4 Radiation/Diffraction Codes. . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Erratum to: Handbook of Ocean Wave Energy . . . . . . . . . . . . . . . . . . E1 Arthur Pecher and Jens Peter Kofoed Contents xiii Abbreviations AEP Annual Energy Production BoP Balance of Plant CapEx Capital Expenditure CF Cash Flow or Capacity Factor CFD Computational Fluid Dynamics CHP Combined Heat and Power CoE Cost of Energy CRL Commercial Readiness Level CWR Capture Width Ratio Dec Decommissioning DevEx Development Expenditure DoF Degree of Freedom DPP Discounted Payback Period EMEC European Marine Energy Centre FCF Free Cash Flow FV Future Value IRF Impulse Response Function IRR Internal Rate of Return IW Irregular Waves LCoE Levelized Cost of Energy MAEP Mean Annual Energy Production NPV Net Present Value OpEx Operational Expenditure OWC Oscillating Water Columns PI Pro fi tability Index PLC Programmable Logic Controller PP Payback Period PTO Power Take-Off PV Present Value RADR Risk Adjusted Discount Rate xv RANSE Reynolds-Averaged Navier-Stokes Equation R&D Research and Development RAO Response Amplitude Operator RW Regular Waves SS Sea State TPL Technology Performance Level TRL Technology Readiness Level VoF Volume of Fluid WAB Wave Activated Body WACC Weighted Average Cost of Capital WEC Wave Energy Converter xvi Abbreviations Symbols a Amplitude of incident wave (m) A Cross-sectional area (m 2 ) A ( ω ) Frequency-dependent added mass (kg) A c Effective cross-sectional area of a pair of cylinders (m 2 ) A e Electric loading (A/m) AEP Annual energy production (MWh/year) a i Amplitudes at each frequency (m) A wp Water plane area (m 2 ) A ∞ Limiting added mass coef fi cient at in fi nite frequency (kg) B Air gap magnetic fl ux density (Tesla) or center of buoyancy (m) B e11 Equivalent damping coef fi cient (surge) ( – ) C Shape coef fi cient ( – ) c Wave celerity (m/s) c a Speed of sound in atmospheric conditions (m/s) C d Wave drift force coef fi cient ( – ) C D Drag coef fi cient ( – ) C g Group velocity (m/s) CI Con fi dence interval ( – ) C m Added mass coef fi cient ( – ) Contrib Contribution to the available wave power ( – ) D Damping coef fi cient (kg/s) D b Float draft below the water surface (m) D es Damping constant for the end stop mechanism (kg/s) D h Float height above surface (m) D t Turbine rotor diameter (m) E Electromotive force (V) F Fetch length (m) f Frequency (Hz) F Force (N) F 3 Restoring force (N) xvii F b Buoyancy force (N) F c Current force (N) F e Excitation force (N) f exc Excitation impulse response function ( – ) F exc Excitation force (N) F f Friction force (N) F hs Hydrostatic force (N) F m The Mooring force (N) F pto PTO force (N) F r Radiation force (N) f w (f) Wave force ratio ( – ) G Center of gravity (m) g Gravitational acceleration (m/s 2 ) G Hydrostatic matrix (N/m) ɣ Peak enhancement factor ( – ) G m Constant ( – ) GZ Righting arm (m) h Water depth (m) H Wave height (m), heaviside step function ( – ) or horizontal force (N) H 1/3 Signi fi cant wave height (m) H max Max wave height within a given duration of a sea state (m) Hp Horizontal pretension (N) H s Signi fi cant wave height (m) h ( t - τ ) Impulse-response function ( – ) H m 0 Signi fi cant Wave Height estimate from wave spectrum (m) I Current density in the conductor (A) I o Incident momentum ( – ) i,j Mode of motion ( – ). Translations: 1: Surge, 2: Sway, 3: Heave. Rotations: 4: Roll, 5: Pitch, 6: Yaw J Wave power fl ux or wave power level (equal to P wave ) (kW/m) K Roughness height (mm) k Spring coef fi cient or Stiffness (N/m) k Wave number (m − 1 ) k/D Relative roughness ( – ) K C Keulegan – Carpenter number ( – ) K es Spring constant for the end stop mechanism (N/m) K t Constant that depends only on turbine geometry ( – ) l Length (m) L l Leakage inductance (H) L m Main inductance (H) L p,0 Wave length based on peak wave period and deep water (m) L s Synchronous inductance (H) m Body mass (kg) m Mass (kg) M Mass matrix (kg) xviii Symbols MAEP Mean annual energy production (MWh/year) m 0 Variance of the wave spectra or ‘ zeroth ’ moment of the wave spectra (m 2 ) m n Spectral moment of the nth order (n = 0, 1, 2, ... ) (m 2 s − n ) m r Added mass (kg) m ̇ Mass fl ow rate of air through the turbine (kg/s) m n Spectrum moments ( – ) N Number of coil turns ( – ) N Number of harmonic wave components ( – ) N or ώ Rotational speed (radians per unit time) (rad/s) N c Number of pairs of cylinders ( – ) N L Length scaling factor ( – ) p Differential pressure in the pneumatic chamber (Pa) p a Atmospheric pressure (Pa) P abs Primary absorbed power from the waves (kW) P available Available power (kW) P el Generated electrical power (kW) P mech Available mechanical power (kW) Prob Probability of occurrence ( – ) P t Turbine power output (kW) P u Useful power (kW) P wave Wave power fl ux or wave power level (equal to J ) (kW/m) Q Volume fl ow rate of liquid displaced by the piston (m 3 /s) q Volume fl ow rate of air (m 3 /s) q 0 Mass per unit unstretched length (kg/m) Q m Flow rate (m 3 /s) r Amplitude of re fl ected wave (m) R Damping (kg/s) Re Reynolds number ( – ) R g Resistance inside the generator ( Ω ) R l Resistance ( Ω ) S Stiffness (N/m), spectral density function (m 2 /Hz) or scaling ratio (S) s Wave steepness or sample standard deviation ( – ) S f Spectral density at frequency component f (m 2 /Hz) S mbs Minimum breaking strength (N) S p,0 Wave steepness for the peak wave period and deep water ( – ) t Time or amplitude of transmitted wave (s) or (m) T Wave period or wave record with duration (s) T 0 Resonance period (s) T 01 Spectral wave period based on 0th and 1st moment (s) T 02 Spectral wave period based on 0th and 2nd moment (spectral estimate of T z ) (s) T B Breaking load (N) T e Wave energy period (s) T p Peak wave period (s) Symbols xix T QS Quasi static tension (N) T z Mean zero down crossing wave period (s) u Horizontal water velocity (m/s) or usage factor ( – ) U Velocity (m/s) U 10 Wind speed at a height of 10 m (m/s) U 10min,10m Mean wind speed over 10 min at 10 m height (m/s) U c Current speed (m/s) U f Full scale velocity (m/s) U m Model scale velocity (m/s) u max Maximum water velocity (m/s) u r Relative speed (m/s) V Volume (m 3 ) V s Available stroke volume (m 3 ) w Distance between the poles (m) x Horizontal position of the body (m) X c Quasi static line extension (m) ẋ Velocity of the body (m/s) ẍ Acceleration of the body (m/s 2 ) z Vertical displacement (m) ż Vertical velocity (m/s) λ Wave length (m) Δ f Frequency interval (Hz) Δ p c Pressure difference between the accumulators ( – ) Φ Flow coef fi cient ( – ) Π Power coef fi cient ( – ) Ψ Pressure coef fi cient ( – ) α i Phases of each frequency (Hz) β Wave direction (degree) φ Permanent magnet induced fl ux per pole or Constant for fi xed entropy (B) or ( – ) γ Speci fi c heat ratio for the gas ( – ) or peak enhancement factor ( – ) μ 0 Magnetic permeability (H m − 1 ) ν Speci fi c volume of gas (m 3 ) or kinematic viscosity (m 2 /s) ρ Density (kg/m 3 ) ρ cu Resistivity of the conductor material ( Ω ) ω Angular frequency (rad/s) ξ Acceleration vector (m/s 2 ) ∇ Submerged volume (m 3 ) ζ (z) Vertical displacement of the water particles (m) ξ (z) Horizontal displacement of the water particles (m) η Free surface elevation (m) or non-dimensional performance (also called CWR or ef fi ciency) ( – ) _ g Velocity of water surface (m/s) € g Acceleration (m/s 2 ) η 3 Body displacement (m) xx Symbols η i Position in mode (m) η lim Excursion limit for which end stop mechanism starts acting (m) η overall Overall non-dimensional performance (ef fi ciency) ( – ) η PTO PTO ef fi ciency ( – ) η ss Non-dimensional performance (ef fi ciency) in individual sea state ( – ) η w2w Wave-to-wire ef fi ciency ( – ) ε 0 Spectral bandwidth ( – ) Symbols xxi