Wind Energy Handbook, Notas de estudo de Engenharia Elétrica

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WIND ENERGY

HANDBOOK

Copyright # 2001 by John Wiley & Sons, Ltd Baffins Lane, Chichester West Sussex, PO19 1UD, England National 01243 779777 International (þ44) 1243 779777 e-mail (for orders and customer service enquiries): [email protected]

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Library of Congress Cataloguing-in-Publication Data

Handbook of wind energy / Tony Burton... [et al.]. p. cm. Includes bibliographical references and index. ISBN 0-471-48997-

1. Wind power — Handbooks, manuals, etc. I. Burton, Tony, 1947- TJ820.H35 2001 621.31’2136–dc21 2001024908 British Library Cataloguing in Publication Data

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

ISBN 0 471 48997 2

Typeset in 10/12pt Palatino by Keytec Typesetting Ltd, Bridport, Dorset Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry, in which at least two trees are planted for each one used for paper production.

Contents

Acknowledgements xv

• 1 Introduction List of symbols xvii
• 1.1 Historical Development
• 1.2 Modern Wind Turbines
• 1.3 Scope of the Book
• References
• Bibliography
• 2 The Wind Resource
• 2.1 The Nature of the Wind
• 2.2 Geographical Variation in the Wind Resource
• 2.3 Long-term Wind-speed Variations
• 2.4 Annual and Seasonal Variations
• 2.5 Synoptic and Diurnal Variations
• 2.6 Turbulence
  • 2.6.1 The nature of turbulence
  • 2.6.2 The boundary layer
  • 2.6.3 Turbulence intensity
  • 2.6.4 Turbulence spectra
  • 2.6.5 Length scales and other parameters
  • 2.6.6 Cross-spectra and coherence functions
• 2.7 Gust Wind Speeds
• 2.8 Extreme Wind Speeds
  • 2.8.1 Extreme winds in standards
• 2.9 Wind-speed Prediction and Forecasting
  • 2.9.1 Statistical methods
  • 2.9.2 Meteorological methods
• 2.10 Turbulence in Wakes and Wind Farms
• 2.11 Turbulence in Complex Terrain
• References
• 3 Aerodynamics of Horizontal-axis Wind Turbines
• 3.1 Introduction
• 3.2 The Actuator Disc Concept
  • 3.2.1 Momentum theory
  • 3.2.2 Power coefficient
  • 3.2.3 The Betz limit
  • 3.2.4 The thrust coefficient
• 3.3 Rotor Disc Theory
  • 3.3.1 Wake rotation
  • 3.3.2 Angular momentum theory
  • 3.3.3 Maximum power
  • 3.3.4 Wake structure
• 3.4 Vortex Cylinder Model of the Actuator Disc
  • 3.4.1 Introduction
  • 3.4.2 Vortex cylinder theory
  • 3.4.3 Relationship between bound circulation and the induced velocity
  • 3.4.4 Root vortex
  • 3.4.5 Torque and power
  • 3.4.6 Axial flow field
  • 3.4.7 Tangential flow field
  • 3.4.8 Radial flow field
  • 3.4.9 Conclusions
• 3.5 Rotor Blade Theory
  • 3.5.1 Introduction
  • 3.5.2 Blade element theory
  • 3.5.3 The blade element – momentum (BEM) theory
  • 3.5.4 Determination of rotor torque and power
• 3.6 Breakdown of the Momentum Theory
  • 3.6.1 Free-stream/wake mixing
  • 3.6.2 Modification of rotor thrust caused by flow separation
  • 3.6.3 Empirical determination of thrust coefficient
• 3.7 Blade Geometry
  • 3.7.1 Introduction
  • 3.7.2 Optimal design for variable-speed operation
  • 3.7.3 A practical blade design
  • 3.7.4 Effects of drag on optimal blade design
  • 3.7.5 Optimal blade design for constant-speed operation
• 3.8 The Effects of a Discrete Number of Blades
  • 3.8.1 Introduction
  • 3.8.2 Tip losses
  • 3.8.3 Prandtl’s approximation for the tip-loss factor
  • 3.8.4 Blade root losses
  • 3.8.5 Effect of tip loss on optimum blade design and power
  • 3.8.6 Incorporation of tip-loss for non-optimal operation
• 3.9 Calculated Results for an Actual Turbine
• 3.10 The Aerodynamics of a Wind Turbine in Steady Yaw
  • 3.10.1 Momentum theory for a turbine rotor in steady yaw
  • 3.10.2 Glauert’s momentum theory for the yawed rotor
  • 3.10.3 Vortex cylinder model of the yawed actuator disc
  • 3.10.4 Flow expansion
  • 3.10.5 Related theories
  • 3.10.6 Wake rotation for a turbine rotor in steady yaw
  • 3.10.7 The blade element theory for a turbine rotor in steady yaw
  • 3.10.8 The blade element–momentum theory for a rotor in steady yaw
  • 3.10.9 Calculated values of induced velocity
  • 3.10.10 Blade forces for a rotor in steady yaw
  • 3.10.11 Yawing and tilting moments in steady yaw
• 3.11 The Method of Acceleration Potential
  • 3.11.1 Introduction
  • 3.11.2 The general pressure distribution theory
  • 3.11.3 The axi-symmetric pressure distributions
  • 3.11.4 The anti-symmetric pressure distributions
  • 3.11.5 The Pitt and Peters model
  • 3.11.6 The general acceleration potential method
  • 3.11.7 Comparison of methods
• 3.12 Stall Delay
• 3.13 Unsteady Flow – Dynamic Inflow
  • 3.13.1 Introduction
  • 3.13.2 Adaptation of the acceleration potential method to unsteady flow
  • 3.13.3 Unsteady yawing and tilting moments
  • 3.13.4 Quasi-steady aerofoil aerodynamics
  • 3.13.5 Aerodynamic forces caused by aerofoil acceleration
  • 3.13.6 The effect of the wake on aerofoil aerodynamics in unsteady flow
• References
• Bibliography
• Appendix: Lift and Drag of Aerofoils
  • A3.1 Definition of Drag
  • A3.2 Drag Coefficient
  • A3.3 The Boundary Layer
  • A3.4 Boundary-layer Separation
  • A3.5 Laminar and Turbulent Boundary Layers
  • A3.6 Definition of Lift and its Relationship to Circulation
  • A3.7 The Stalled Aerofoil
  • A3.8 The Lift Coefficient
  • A3.9 Aerofoil Drag Characteristics
  • A3.10 Variation of Aerofoil Characteristics with Reynolds Number
  • A3.11 Cambered Aerofoils
• 4 Wind-turbine Performance
• 4.1 The Performance Curves
  • 4.1.1 The CP
    º performance curve
  • 4.1.2 The effect of solidity on performance
  • 4.1.3 The CQ
    º curve
  • 4.1.4 The CT
    º curve
• 4.2 Constant Rotational Speed Operation
  • 4.2.1 The KP
    1 =º curve
  • 4.2.2 Stall regulation
  • 4.2.3 Effect of rotational speed change
  • 4.2.4 Effect of blade pitch angle change
  • 4.2.5 Pitch regulation
  • 4.2.6 Pitching to stall
  • 4.2.7 Pitching to feather
• 4.3 Comparison of Measured with Theoretical Performance
• 4.4 Variable-speed Operation
• 4.5 Estimation of Energy Capture
• 4.6 Wind-turbine Field Testing
  • 4.6.1 Introduction
  • 4.6.2 Information sources for wind-turbine testing
• 4.7 Wind-turbine Performance Measurement
  • 4.7.1 Field testing methodology
  • 4.7.2 Wind-speed measurement
  • 4.7.3 Wind-direction measurement
  • 4.7.4 Air temperature and pressure measurement
  • 4.7.5 Power measurement
  • 4.7.6 Wind-turbine status
  • 4.7.7 Data acquisition system
  • 4.7.8 Data acquisition rate
• 4.8 Analysis of Test Data
• 4.9 Turbulence Effects
• 4.10 Aerodynamic Performance Assessment
• 4.11 Errors and Uncertainty
  • 4.11.1 Evaluation of uncertainty
  • 4.11.2 Sensitivity factors
  • 4.11.3 Estimating uncertainties
  • 4.11.4 Combining uncertainties
• References
• 5 Design Loads for Horizontal-axis Wind Turbines
• 5.1 National and International Standards
  • 5.1.1 Historical development
  • 5.1.2 IEC 61400-1
  • 5.1.3 Germanisher Lloyd rules for certification
  • 5.1.4 Danish Standard DS
• 5.2 Basis for Design Loads
  • 5.2.1 Sources of loading
  • 5.2.2 Ultimate loads
  • 5.2.3 Fatigue loads
  • 5.2.4 Partial safety factors for loads
  • 5.2.5 Functions of the control and safety systems
• 5.3 Turbulence and Wakes
• 5.4 Extreme Loads
  • 5.4.1 Non-operational load cases – normal machine state
  • 5.4.2 Non-operational load cases – machine fault state
  • 5.4.3 Operational load cases – normal machine state
  • 5.4.4 Operational load cases – loss of load
  • 5.4.5 Operational load cases – machine fault states
  • 5.4.6 Start-up and shut-down cases
  • 5.4.7 Blade/tower clearance
• 5.5 Fatigue Loading
  • 5.5.1 Synthesis of fatigue load spectrum
• 5.6 Stationary Blade Loading
  • 5.6.1 Lift and drag coefficients
  • 5.6.2 Critical configuration for different machine types
  • 5.6.3 Dynamic response
• 5.7 Blade Loads During Operation
  • 5.7.1 Deterministic and stochastic load components
  • 5.7.2 Deterministic aerodynamic loads
  • 5.7.3 Gravity loads
  • 5.7.4 Deterministic inertia loads
  • 5.7.5 Stochastic aerodynamic loads – analysis in the frequency domain
  • 5.7.6 Stochastic aerodynamic loads – analysis in the time domain
  • 5.7.7 Extreme loads
• 5.8 Blade Dynamic Response
  • 5.8.1 Modal analysis
  • 5.8.2 Mode shapes and frequencies
  • 5.8.3 Centrifugal stiffening
  • 5.8.4 Aerodynamic and structural damping
  • 5.8.5 Response to deterministic loads—step-by-step dynamic analysis
  • 5.8.6 Response to stochastic loads
  • 5.8.7 Response to simulated loads
  • 5.8.8 Teeter motion
  • 5.8.9 Tower coupling
  • 5.8.10 Wind turbine dynamic analysis codes
  • 5.8.11 Aeroelastic stability
• 5.9 Blade Fatigue Stresses
  • 5.9.1 Methodology for blade fatigue design
  • 5.9.2 Combination of deterministic and stochastic components
  • 5.9.3 Fatigue predictions in the frequency domain
  • 5.9.4 Wind simulation
  • 5.9.5 Fatigue cycle counting
• 5.10 Hub and Low-speed Shaft Loading
  • 5.10.1 Introduction
  • 5.10.2 Deterministic aerodynamic loads
  • 5.10.3 Stochastic aerodynamic loads
  • 5.10.4 Gravity loading
• 5.11 Nacelle Loading
  • 5.11.1 Loadings from rotor
  • 5.11.2 Cladding loads
• 5.12 Tower Loading
  • 5.12.1 Extreme loads
  • 5.12.2 Dynamic response to extreme loads
  • 5.12.3 Operational loads due to steady wind (deterministic component)
  • 5.12.4 Operational loads due to turbulence (stochastic component)
  • 5.12.5 Dynamic response to operational loads
  • 5.12.6 Fatigue loads and stresses
• References
• Appendix: Dynamic Response of Stationary Blade in Turbulent Wind
  • A5.1 Introduction
  • A5.2 Frequency Response Function
    • A5.2.1 Equation of motion
    • A5.2.2 Frequency response function
      • Blade A5.3 Resonant Displacement Response Ignoring Wind Variations along the
    • A5.3.1 Linearization of wind loading
    • A5.3.2 First mode displacement response
    • A5.3.3 Background and resonant response
      • Response A5.4 Effect of Ac-Wind Turbulence Distribution on Resonant Displacement
    • A5.4.1 Formula for normalized co-spectrum
  • A5.5 Resonant Root Bending Moment
  • A5.6 Root Bending Moment Background response
  • A5.7 Peak Response
  • A5.8 Bending Moments at Intermediate Blade Positions
    • A5.8.1 Background response
    • A5.8.2 Resonant response
• References
• 6 Conceptual Design of Horizontal Axis Wind Turbines
• 6.1 Introduction
• 6.2 Rotor Diameter
  • 6.2.1 Cost modelling
  • 6.2.2 Simplified cost model for machine size optimization—an illustration
• 6.3 Machine Rating - diameter 6.3.1 Simplified cost model for optimizing machine rating in relation to
  • 6.3.2 Relationship between optimum rated wind speed and annual mean
  • 6.3.3 Specific power of production machines
• 6.4 Rotational Speed
  • 6.4.1 Ideal relationship between rotational speed and solidity
  • 6.4.2 Influence of rotational speed on blade weight
  • 6.4.3 Optimum rotational speed
  • 6.4.4 Noise constraint on rotational speed
  • 6.4.5 Visual considerations
• 6.5 Number of Blades
  • 6.5.1 Overview
    • solidity 6.5.2 Ideal relationship between number of blades, rotational speed and
  • 6.5.3 Some performance and cost comparisons
  • 6.5.4 Effect of number of blades on loads
  • 6.5.5 Noise constraint on rotational speed
  • 6.5.6 Visual appearance
  • 6.5.7 Single-bladed turbines
• 6.6 Teetering
  • 6.6.1 Load relief benefits
  • 6.6.2 Limitation of large excursions
  • 6.6.3 Pitch–teeter coupling
  • 6.6.4 Teeter stability on stall-regulated machines
• 6.7 Power Control
  • 6.7.1 Passive stall control
  • 6.7.2 Active pitch control
  • 6.7.3 Passive pitch control
  • 6.7.4 Active stall control
  • 6.7.5 Yaw control
• 6.8 Braking Systems
  • 6.8.1 Independent braking systems—requirements of standards
  • 6.8.2 Aerodynamic brake options
  • 6.8.3 Mechanical brake options
  • 6.8.4 Parking versus idling
• 6.9 Fixed-speed, Two-speed or Variable-speed Operation
  • 6.9.1 Two-speed operation
  • 6.9.2 Variable-speed operation
  • 6.9.3 Variable-slip operation
  • 6.9.4 Other approaches to variable-speed operation
• 6.10 Type of Generator
  • 6.10.1 Historical attempts to use synchronous generators
  • 6.10.2 Direct-drive generators
• 6.11 Drive-train Mounting Arrangement Options
  • 6.11.1 Low-speed shaft mounting
  • 6.11.2 High-speed shaft and generator mounting
• 6.12 Drive-train Compliance
• 6.13 Rotor Position with Respect to Tower
  • 6.13.1 Upwind configuration
  • 6.13.2 Downwind configuration
• 6.14 Tower Stiffness
• 6.15 Personnel Safety and Access Issues
• References
• 7 Component Design
• 7.1 Blades
  • 7.1.1 Introduction
  • 7.1.2 Aerodynamic design
  • 7.1.3 Practical modifications to optimum design
  • 7.1.4 Form of blade structure
  • 7.1.5 Blade materials and properties
  • 7.1.6 Properties of glass/polyester and glass/epoxy composites
  • 7.1.7 Properties of wood laminates
  • 7.1.8 Governing load cases
  • 7.1.9 Blade resonance
  • 7.1.10 Design against buckling
  • 7.1.11 Blade root fixings
• 7.2 Pitch Bearings
• 7.3 Rotor Hub
• 7.4 Gearbox
  • 7.4.1 Introduction
  • 7.4.2 Variable loads during operation
  • 7.4.3 Drive-train dynamics
  • 7.4.4 Braking loads
  • 7.4.5 Effect of variable loading on fatigue design of gear teeth
  • 7.4.6 Effect of variable loading on fatigue design of bearings and shafts
  • 7.4.7 Gear arrangements
  • 7.4.8 Gearbox noise
  • 7.4.9 Integrated gearboxes
  • 7.4.10 Lubrication and cooling
  • 7.4.11 Gearbox efficiency
• 7.5 Generator
  • 7.5.1 Induction generators
  • 7.5.2 Variable-speed generators
• 7.6 Mechanical Brake
  • 7.6.1 Brake duty
  • 7.6.2 Factors govnering brake design
  • 7.6.3 Calculation of brake disc temperature rise
  • 7.6.4 High-speed shaft brake design
  • 7.6.5 Two level braking
  • 7.6.6 Low-speed shaft brake design
• 7.7 Nacelle Bedplate
• 7.8 Yaw Drive
• 7.9 Tower
  • 7.9.1 Introduction
  • 7.9.2 Constraints on first-mode natural frequency
  • 7.9.3 Steel tubular towers
  • 7.9.4 Steel lattice towers
• 7.10 Foundations
  • 7.10.1 Slab foundations
  • 7.10.2 Multi-pile foundations
  • 7.10.3 Concrete mono-pile foundations
  • 7.10.4 Foundations for steel lattice towers
• References
• 8 The Controller
• 8.1 Functions of the Wind-turbine Controller
  • 8.1.1 Supervisory control
  • 8.1.2 Closed-loop control
  • 8.1.3 The safety system
• 8.2 Closed-loop Control: Issues and Objectives
  • 8.2.1 Pitch control
  • 8.2.2 Stall control
  • 8.2.3 Generator torque control
  • 8.2.4 Yaw control
  • 8.2.5 Influence of the controller on loads
  • 8.2.6 Defining controller objectives
  • 8.2.7 PI and PID controllers
• 8.3 Closed-loop Control: General Techniques
  • 8.3.1 Control of fixed-speed, pitch-regulated turbines
  • 8.3.2 Control of variable-speed pitch-regulated turbines
  • 8.3.3 Pitch control for variable-speed turbines
  • 8.3.4 Switching between torque and pitch control
  • 8.3.5 Control of tower vibration
  • 8.3.6 Control of drive train torsional vibration
  • 8.3.7 Variable-speed stall regulation
  • 8.3.8 Control of variable-slip turbines
  • 8.3.9 Individual pitch control
• 8.4 Closed-loop Control: Analytical Design Methods
  • 8.4.1 Classical design methods
  • 8.4.2 Gain scheduling for pitch controllers
  • 8.4.3 Adding more terms to the controller
  • 8.4.4 Other extensions to classical controllers
  • 8.4.5 Optimal feedback methods
  • 8.4.6 Other methods
• 8.5 Pitch Actuators
• 8.6 Control System Implementation
  • 8.6.1 Discretization
  • 8.6.2 Integrator desaturation
• References
• 9 Wind-turbine Installations and Wind Farms
• 9.1 Project Development
  • 9.1.1 Initial site selection
  • 9.1.2 Project feasibility assessment
  • 9.1.3 The measure–correlate–predict technique
  • 9.1.4 Micrositing
  • 9.1.5 Site investigations
  • 9.1.6 Public consultation
  • 9.1.7 Preparation and submission of the planning application
• 9.2 Visual and Landscape Assessment
  • 9.2.1 Landscape character assessment
  • 9.2.2 Design and mitigation
  • 9.2.3 Assessment of impact
  • 9.2.4 Shadow flicker
  • 9.2.5 Sociological aspects
• 9.3 Noise
  • 9.3.1 Terminology and basic concepts
  • 9.3.2 Wind-turbine noise
  • 9.3.3 Measurement, prediction and assessment of wind-farm noise
• 9.4 Electromagnetic Interference
  • 9.4.1 Modelling and prediction of EMI from wind turbines
• 9.5 Ecological Assessment
  • 9.5.1 Impact on birds
• 9.6 Finance
  • 9.6.1 Project appraisal
  • 9.6.2 Project finance
  • 9.6.3 Support mechanisms for wind energy
• References
• 10 Electrical Systems
• 10.1 Power-collection Systems
• 10.2 Earthing (Grounding) of Wind Farms
• 10.3 Lightning Protection
• 10.4 Embedded (Dispersed) Wind Generation
  • 10.4.1 The electric power system
  • 10.4.2 Embedded generation
  • 10.4.3 Electrical distribution networks
  • 10.4.4 The per-unit system
  • 10.4.5 Power flows, slow-voltage variations and network losses
  • 10.4.6 Connection of embedded wind generation
  • 10.4.7 Power system studies
• 10.5 Power Quality
  • 10.5.1 Voltage flicker
  • 10.5.2 Harmonics
    • grid-connected wind turbines 10.5.3 Measurement and assessment of power quality characteristics of
• 10.6 Electrical Protection
  • 10.6.1 Wind-farm and generator protection
  • 10.6.2 Islanding and self-excitation of induction generators
  • 10.6.3 Interface protection
• 10.7 Economic Aspects of Embedded Wind Generation
  • 10.7.1 Losses in distribution networks with embedded wind generation
  • 10.7.2 Reactive power charges and voltage control
  • 10.7.3 Connection charges ‘deep’ and ‘shallow’
  • 10.7.4 Use-of-system charges
  • 10.7.5 Impact on the generation system
• References
• Index

List of Symbols

Note: This list is not exhaustive, and omits many symbols that are unique to particular chapters

a axial flow induction factor a 9 tangential flow induction factor a (^9) t tangential flow induction factor at the blade tip a 0 two-dimensional lift curve slope, (dC 1 =dÆ) a 1 constant defining magnitude of structural damping A, AD rotor swept area A 1 , AW upstream and downstream stream-tube cross-sectional areas b face width of gear teeth c blade chord; Weibull scale parameter cc ^ damping coefficient per unit length ci generalized damping coefficient with respect to the ith mode C decay constant C(í) Theodorsen’s function, where í is the reduced frequency: C(í) ¼ F(í) þ iG(í) Cd sectional drag coefficient Cf sectional force coefficient (i.e., Cd or C 1 as appropriate) C 1 sectional lift coefficient C mn coefficient of a Kinner pressure distribution Cp pressure coefficient CP power coefficient CQ torque coefficient CT thrust coefficient; total cost of wind turbine CTB total cost of baseline wind turbine Cx coefficient of sectional blade element force normal to the rotor plane Cy coefficient of sectional blade element force parallel to the rotor plane C(˜r, n) coherence—i.e., normalized cross spectrum – for wind speed fluctuations at points separated by distance s measured in the across wind direction Cjk(n) coherence—i.e., normalized cross spectrum – for longitudinal wind speed fluctuations at points j and k d streamwise distance between vortex sheets in a wake d 1 pitch diameter of pinion gear

KP power coefficient based on tip speed KSMB size reduction factor accounting for the lack of correlation of wind fluctuations over structural element or elements KSx(n 1 ) size reduction factor accounting for the lack of correlation of wind fluctuations at resonant frequency over structural element or elements Kí( ) modified Bessel function of the second kind and order í K(÷) function determining the induced velocity normal to the plane of a yawed rotor L length scale for turbulence (subscripts and superscripts according to context); lift force L xu integral length scale for the along wind turbulence component, u, measured in the longitudinal direction, x m mass per unit length, integer m (^) i generalized mass with respect to the ith mode mT1 generalized mass of tower, nacelle and rotor with respect to tower first mode M moment; integer M mean bending moment MT teeter moment MX blade in-plane moment (i.e., moment causing bending in plane of rotation); tower side-to-side moment MY blade out-of-plane moment (i.e., moment causing bending out of plane of rotation); tower fore-aft moment MZ blade torsional moment; tower torsional moment M (^) YS low-speed shaft moment about rotating axis perpendicular to axis of blade 1 M (^) ZS low-speed shaft moment about rotating axis parallel to axis of blade 1 M (^) Y N moment exerted by low-speed shaft on nacelle about (horizontal) y- axis M (^) Z N moment exerted by low-speed shaft on nacelle about (vertical) z-axis n frequency (Hz); number of fatigue loading cycles; integer n 0 zero up-crossing frequency of quasistatic response n 1 frequency (Hz) of 1st mode of vibration N number of blades; number of time steps per revolution; integer N(r) centrifugal force N(S) number of fatigue cycles to failure at stress level S p static pressure P aerodynamic power; electrical real (active) power P mn ( ) associated Legrendre polynomial of the first kind q(r, t) fluctuating aerodynamic lift per unit length Q rotor torque; electrical reactive power Qa aerodynamic torque QQ^ _ rate of heat flow Q mean aerodynamic lift per unit length QD dynamic factor defined as ratio of extreme moment to gust quasistatic moment Qg load torque at generator

LIST OF SYMBOLS xix

QL loss torque Q mn ( ) associated Legrendre polynomial of the second kind Q 1 (t) generalized load, defined in relation to a cantilever blade by Equation (A5.13) r radius of blade element or point on blade; correlation coefficient between power and wind speed; radius of tubular tower r 9 radius of point on blade r 1 , r 2 radii of points on blade or blades R blade tip radius; ratio of minimum to maximum stress in fatigue load cycle; electrical resistance Re Reynold’s number Ru (n) normalized power spectral density, n:Su(n)=ó (^2) u, of longitudinal wind- speed fluctuations, u, at a fixed point s distance inboard from the blade tip; distance along the blade chord from the leading edge; separation between two points; Laplace operator; slip of induction machine s 1 separation between two points measured in the along-wind direction S wing area; autogyro disc area; fatigue stress range S electrical complex (apparent) power (bold indicates a complex quantity) S( ) uncertainty or error band Sjk(n) cross spectrum of longitudinal wind-speed fluctuations, u, at points j and k (single sided) SM (n) single-sided power spectrum of bending moment S (^) Q 1 (n) single-sided power spectrum of generalized load Su(n) single-sided power spectrum of longitudinal wind-speed fluctuations, u, at a fixed point S^0 u(n) single-sided power spectrum of longitudinal wind-speed fluctuations, u, as seen by a point on a rotating blade (also known as rotationally sampled spectrum) S^0 u(r 1 , r 2 , n) cross spectrum of longitudinal wind-speed fluctuations, u, as seen by points at radii r 1 and r 2 on a rotating blade or rotor (single sided) Sv(n) single-sided power spectrum of lateral wind speed fluctuations, v, at a fixed point Sw(n) single-sided power spectrum of vertical wind-speed fluctuations, w, at a fixed point t time; gear tooth thickness at critical root section; tower wall thickness T rotor thrust; duration of discrete gust; wind-speed averaging period u fluctuating component of wind speed in the x-direction; induced velocity in x-direction; in-plane plate deflection in x-direction; gear ratio u^ friction velocity in boundary layer U 1 free stream velocity U, U(t) instantaneous wind speed in the along-wind direction U mean component of wind speed in the along-wind direction – typically taken over a period of 10 min or 1 h Uave annual average wind speed at hub height

xx LIST OF SYMBOLS