Burton T. (et. al.) Wind energy Handbook

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streamwise veloci ty at the rotor disc

streamwise veloci ty in the far wake

extreme 3 s gust wind speed with 1 year return period

e50

extreme 3 s gust wind speed with 50 year return period

turbine upper cu t-out speed

turbine rated wind speed, deﬁned as the wind speed at which the

turbine’s rated power is reached

ref

reference wind speed deﬁned as 10 min mean wind speed at hub

height with 50 year return period

strain energy of plate ﬂexure

in-plane strain energy

v ﬂuctuating component of wind speed in the y-direction; induced

velocity in y-direction; in-plane plate deﬂection in y-direction

V airspeed of an autogyro; longitudinal air velocity at rotor disc,

(1  a) (Section 7.1.9); voltage (shown in bold when complex)

V(t) instantaneous lateral wind speed

VA electrical volt-amperes

ﬁbre volume fraction in composite material

blade tip speed

w ﬂuctuating component of wind speed in the z-direction; induced

velocity in z-direction; out-of-plane plate deﬂection

W wind velocity relative to a point on rotating blade; electrical power

loss

x downwind co-ordinate – ﬁxed and rotating axis systems; downwind

displacement

x(t) stochastic component of a variable

length of near wake region

ﬁrst-mode component of steady tip displacement

X electri cal inductive reactance

y lateral co-ordinate with respect to vertical axis (starboard positive) –

ﬁxed-axis system

y lateral co-ordinate with respect to blade axis – rotating-axis system

y lateral displacement

z vertical co-ordinate (upwards positive) – ﬁxed-axis system

z radial co-ordinate along blade axis – rotating-axis system

Z section modulus

Z electrical impedance (bold indicates a complex quantity)

ground roughness length

number of teeth on pinion gear

z(t) periodic component of a variable

Greek

Æ angle of attack – i.e., angle between air ﬂow incident on the blade and

the blade chord line; wind-shear power law exponent

 inclination of local blade chord to rotor plane (i.e., blade twist plus

pitch angle, if any)

LIST OF SYMBOLS xxi

 wake skew angle: angle betwee n the axis of the wake of a yawed rotor

and the axis of rotation of rotor



weighted mass ratio deﬁned in Section 5.8.6



logarithmic decrement of aerodynamic damping



logarithmic decrement of structural damping

 logarithmic decremen t of combined aerodynamic and structural

damping; width of tower shadow deﬁcit region



angle between axis of teeter hinge and the line perpendicular to both

the rotor axis and the low-speed shaft axis



, 

proportion of time in which a variable takes the maximum, mean or

minimum values in a three-level square wave

 ﬂow angle of resultant velocity W to rotor plane

ª yaw angle; Euler’s constant (¼ 0:5772)

load factor

partial safety factor for material fatigue strength

partial safety factor for material ultimate strength

ˆ blade circulation; vortex strength

ˆ( ) gamma function

 ellipsoidal parameter; shaft tilt; one eighth of Lock number (deﬁned in

Section 5.8.8)

j impact factor in DS 472, deﬁned as ratio of extreme moment to the

10 min mean

k von Karman’s constant

(s) cross-correlation function between velocity components at points in

space a distance s apart, in the direction parallel to the line joining

them

(s) cross-correlation function between velocity components at points in

space a distance s apart, in the direction perpendicular to the line

joining them

(r, ) auto-correlation function for alon g-wind velocity component at radius

r on stationary rotor

(r, ) auto-correlation function for along-wind velocity component as seen

by a point at radius r on a rotating rotor

, r

, ) cross-correlation function between along-wind velocity components

at radii r

and r

(not necessarily on same blade), for stationary rotor

, r

, ) cross-correlation function between along-wind velocity components

as seen by points (not necessarily on same blade) at radii r

and r

a rotating rotor

º tip speed ratio; latitude; ratio of longitudinal to transverse buckle half

wavelengths

tangential speed of blade element at radius r divided by wind speed:

local speed ratio

¸ yaw rate

 non-dimensional radial position, r=R; viscosity; coefﬁcient of friction



(r) mode shape of ith blade mode



(z) mode shape of ith tower mode



(z) tower ﬁrst mode shape

xxii LIST OF SYMBOLS



(r) normalized rigid body deﬂection of blade j resulting from excitation

of tower ﬁrst mode



mean value of variable z

 ellipsoidal co-ordinate; mean zero up-crossing frequency

Ł wind-speed direction change; random phase angle; cylindrical panel

co-ordinate; brake disc temperature

r air density

, r

, ) normalized cross-correlation function between along-wind velocity

components as seen by points (not necessarily on same blade) at radii

and r

on a rotating rotor (i.e., k

, r

, )=

)

 blade solidity; standard deviation; stress

 mean stress



standard deviation of bending moment



standard deviation of ﬁrst-mode resonant bending moment, at blade

root for blade resonance, and at tower base for tower resonance



standard deviation of quasistatic bending moment (or bending

moment background response)



standard deviation of hub dishing moment



standard deviation of teeter moment for rigidly mounted, two-bladed

rotor



standard deviation of mean of blade root bending moments for two-

bladed rotor



standard deviation of generalized load with respect to ﬁrst mode



rotor solidity



standard deviation of ﬂuctuating component of wind in along-wind

direction



standard deviation of wind speed in across-wind direction



standard deviation of wind speed in vertical direction



standard deviation of ﬁrst-mode resonant displacement, referred to

blade tip for blade resonance, and to nacelle for tower resonance

 time interval; non-dimensional time; shear stress

ı Poisson’s ratio

ø angular frequency (rad = s)

demanded generator rotational speed

natural frequency of ith mode (rad=s)

generator rotational speed

induction machine rotor rotational speed

induction machine stator ﬁeld rotational speed

 rotational speed of rotor; earth’s rotational speed

 damping ratio

ł angle subtended by cylindrical plate panel

(r, r9, n) real part of normalized cross spectrum (see Appendix 1, section A1.4)

 blade azimuth

 teeter angle

Subscripts

a aerodynamic

LIST OF SYMBOLS xxiii

B baseline

c compressive

d disc; drag; design

e1 extreme value with return period of 1 year

e50 extreme value with return period of 50 years

ext extreme

f ﬁbre

i mode i

j mode j

J blade J

k characteristic

l lift

m matrix

M moment

max maximum value of variable

min minimum value of variable

n value at end of nth time step

Q generalized load

R value at tip radius, R

s structural

t tensile

T thrust

u downwind; ultimate

v lateral

w vertical

w wake

x deﬂection in along-wind direction

Superscripts

0 rotationally sampled (applied to wind-speed spectra)

xxiv LIST OF SYMBOLS

z is radially outwards from the hub centre along the

blade axis

x is perpendicular to the blade axis, and in the plane

passing through the blade and shaft axis – in the

downwind direction

y is perpendicular to the blade and shaft axes, to give

a right-hand co-ordinate system

NB: Although shaft is tilt and rotor coning are not shown

on the sketch, the axis definitions given accommodate

both these variants

Axis definitions

Figure C1 Co-ordinate System for Blade Loads, Positions and Deﬂections (rotates with

blade) NB: Although shaft tilt and rotor coning are not shown on the sketch, the axis

deﬁnitions given accommodate both these variants.

x is along the shaft axis, in the downwind

direction

z is perpendicular to the shaft axis and in the

vertical plane passing through the shaft axis

y is perpendicular to the shaft axis and

horizontal to starboard, giving a right-hand

co-ordinate system

NB: Although shaft is tilt and rotor coning are not

shown on the sketch, the axis definitions given

accommodate both these variants

Axis Definitions

Figure C2 Fixed Co-ordinate System for Hub Loads and Deﬂections, and Positions with

Respect to Hub NB: Although shaft tilt and rotor coning are not shown on the sketch, the axis

deﬁnitions given accommodate both these variants.

Introduction

1.1 Historical Development

Windmills have been used for at least 3000 years, mainly for grinding grain or

pumping water, while in sailing ships the wind has been an essential source of

power for even longer. From as early as the thirteenth century, horizontal-axis

windmills were an integral part of the rural economy and only fell into disuse with

the advent of cheap fossil-fuelled engines and then the spread of rural electriﬁca-

tion. The use of windmills (or wind turbines) to generate electricity can be traced

back to the late nineteenth century with the 12 kW DC windmill generator

constructed by Brush in the USA and the research undertaken by LaCour in

Denmark. However, for much of the twentieth century there was little interest in

using wind energy other than for battery charging for remote dwellings and these

low-power systems were quickly replaced once access to the electricity grid became

available. One notable exception was the 1250 kW Smith–Putnam wind turbine

constructed in the USA in 1941. This remarkable machine had a steel rotor 53 m in

diameter, full-span pitch control and ﬂapping blades to reduce loads. Although a

blade spar failed catastrophically in 1945, it remained the largest wind turbine

constructed for some 40 years (Putnam, 1948).

Golding (1955) and Shepherd and Divone in Spera (1994) provide a fascinating

history of early wind turbine development. They record the 100 kW 30 m diameter

Balaclava wind turbine in the then USSR in 1931 and the Andrea Enﬁeld 100 kW

24 m diameter pneumatic design constructed in the UK in the early 1950s. In this

turbine hollow blades, open at the tip, were used to draw air up through the tower

where another turbine drove the generator. In Denmark the 200 kW 24 m diameter

Gedser machine was built in 1956 while Electricite

de France tested a 1.1 MW 35 m

diameter turbine in 1963. In Germany, Professor Hutter constructed a number of

innovative, lightweight turbines in the 1950s and 1960s. In spite of these technical

advances and the enthusiasm, among others, of Golding at the Electrical Research

Association in the UK there was little sustained interest in wind generation until

the price of oil rose dramatically in 1973.

The sudden increase in the price of oil stimulated a number of substantial

Government-funded programmes of research, development and demonstration. In

the USA this led to the construction of a series of prototype turbines starting with

the 38 m diameter 100 kW Mod-0 in 1975 and culminating in the 97.5 m diameter

2.5 MW Mod-5B in 1987. Similar programmes were pursued in the UK, Germany

and Sweden. There was considerable uncertainty as to which architecture might

prove most cost-effective and several innovative concepts were investigated at full

scale. In Canada, a 4 MW vertical-axis Darrieus wind turbine was constructed and

this concept was also investigated in the 34 m diameter Sandia Vertical Axis Test

Facility in the USA. In the UK, an alternative vertical-axis design using straight

blades to give an ‘H’ type rotor was proposed by Dr Peter Musgrove and a 500 kW

prototype constructed. In 1981 an innovative horizontal-axis 3 MW wind turbine

was built and teste d in the USA. This used hydraulic transmission and, as an

alternative to a yaw drive, the entire structure was orientated into the wind. The

best choice for the number of blades remained unclear for some while and large

turbines were co nstructed with one, two or three blades.

Much important scientiﬁc and engineering information was gained from these

Government-funded research programmes and the prototypes generally worke d as

designed. However, it has to be recognized that the problems of operating very

large wind turbines, unmanned and in difﬁcult wind climates were often under-

Figure 1.1 1.5 MW, 64 m diameter Wind Turbine (Reproduced by permission of NEG

MICON, www.neg-micon.dk)

INTRODUCTION

estimated and the reliability of the prototypes was not good. At the same time as

the multi-megawatt prototypes were being constructed private companies, often

with considerable state support, were constructing much smaller, often simpler,

turbines for commercial sale. In par ticular the ﬁnancial support mechanisms in

California in the mid-1980s resulted in the installation of a very large number of

quite small (, 100 kW) wind turbines. A number of these designs also suffered

from various problems but, being smaller, they were in general easier to repair and

modify. The so-called ‘Danish’ wind tu rbine concept emerged of a three-bladed,

stall-regulated rotor and a ﬁxed-speed, induction machine drive train. This decep-

tively simple architecture has proved to be remarkably successful and has now been

implemented on turbines as large as 60 m in diameter and at ratings of 1.5 MW. The

machines of Figures 1.1 and 1.2 are examples of this design. However, as the sizes

of commercially available turbines now approach that of the large prototypes of the

1980s it is interesting to see that the concepts investigated then of variable-speed

operation, full-span control of the blades, and advanced materials are being used

increasingly by designers. Figure 1.3 shows a wind farm of direct-drive, variable-

speed wind turbines. In this design, the synchronous generator is coupled directly

to the aerodynamic rotor so eliminating the requirement for a gearbox. Figure 1.4

shows a more conventional, variable-speed wind turbine that uses a gearbox, while

a small wind farm of pitch-regulated wind turbines, where full-span control of the

blades is used to regulate power, is shown in Figure 1.5.

Figure 1.2 750 kW, 48 m diameter Wind Turbine, Denmark (Reproduced by permission of

NEG MICON, www.neg-micon.dk)

HISTORICAL DEVELOPMENT 3

The stimulus for the development of wind energy in 1973 was the price of oil and

concern over limited fossil-fuel resources. Now, of course, the main driver for use

of wind turbines to generate electrical power is the very low CO

emissions (over

the entire life cycle of manufacture, installation, operation and de-commissioning)

Figure 1.3 Wind Farm of Variable-Speed Wind Turbines in Complex Terrain (Reproduced

by permission of Wind Prospect Ltd., www.windprospect.com)

Figure 1.4 1 MW Wind Turbine in Northern Ireland (Reproduced by permission of Renew-

able Energy Systems Ltd., www.res-ltd.com)

INTRODUCTION