WIND POWER
Wind is one of the oldest widely used sources of energy. Although its
use is many centuries old, it has not been a dominant factor in the energy
picture of developed countries for the past 50 years because of the
abundance of fossil fuels. Recently, the realization that fossil fuels are
in limited supply has awakened the need to develop wind power with
modern technology on a large scale. Consequently, there has been a
tremendous resurgence of effort in wind power in just the past few
years. The state of knowledge is rapidly increasing, and the reader is
referred to the current literature and the NREL Internet address cited
above for information on the latest technology. Wind energy is one of
the lowest-cost forms of renewable energy. In 1995, more than
1,700 MW of wind energy capacity was operating in California, generating
enough energy to supply a city the size of San Francisco with all
its energy needs. European capacity was almost the same. For the latest
status on worldwide use of wind energy, the reader is referred to the
American Wind Energy Association (AWEA) at the Internet address
cited above. The fundamental principles of wind power technology do
not change and are discussed here.
Wind Turbines The essential ingredient in a wind energy conversion
system (WECS) is the wind turbine, traditionally called the windmill.
The predominant configurations are horizontal-axis propeller turbines
(HAWTs) and vertical-axis wind turbines (VAWTs), the latter most often
termed Darrieus rotors. In the performance analysis of wind turbines, the
propeller devices were studied first, and their analysis set the current
conventions for the evaluation of all turbines.
General Momentum Theory for Horizontal-Axis Turbines Conventional
analysis of horizontal-axis turbines begins with an axial momentum
balance originated by Rankine using the control volume depicted in
Fig. 9.1.1. The turbine is represented by a porous disk of area A which
extracts energy from the air passing through it by reducing its pressure:
on the upstream side the pressure has been raised above atmospheric by
the slowing airstream; on the downstream side pressure is lower, and
atmospheric pressure will be recovered by further slowing of the
stream. V is original wind speed, decelerated to V(1 2 a) at the turbine disk, and to V(1 2 2a) in the wake of the turbine (a is called the
interference factor). Momentum analysis predicts the axial thrust on the
turbine of radius R to be
T 5 2pR2rV2a(1 2 a) (9.1.1)
where air density, r, equals 0.00237 lbf ? s2/ft4 (or 1.221 kg/m3) at sealevel
standard-atmosphere conditions.
Fig. 9.1.1 Control volume.
Application of the mechanical energy equation to the control volume
depicted in Fig. 9.1.1 yields the prediction of power to the turbine of
P 5 2pR2rV3a(1 2 a)2 (9.1.2)
This power can be nondimensionalized with the energy flux E in the
upstream wind covering an area equal to the rotor disk, i.e.,
E 5 1⁄2rV3pR2 (9.1.3)
The resulting power coefficient is
Cp 5
P
E
5 4a(1 2 a)2 (9.1.4)
This power coefficient has a theoretical maximum at a 5 1⁄3 of Cp 5
0.593. This result was first predicted by Betz and shows that the load
placed on a windmill must be optimized to obtain the best power output:
If the load is too small (small a), too much of the power is carried off
with the wake; if the load is too large (large a), the flow is excessively
obstructed and most of the approaching wind passes around the turbine.
This derivation includes some important assumptions which limit its
accuracy and applicability. In particular, the portion of the kinetic energy
in the swirl component of the wake is neglected. Partial accounting
for the rotation in the wake has been included in the analysis of Glauert
with the resulting prediction of ideal power coefficient as a function of
turbine tip speed ratio X5VR/V (where V is the angular velocity of the
turbine) shown in Fig. 9.1.2. Clearly, the swirl is made up of wasted
kinetic energy and is largest for a high-torque, low-speed turbine. Actual
farm, multiblade, and two- or three-bladed turbines show somewhat
lower than ideal performance because of drag effects neglected in ideal
flow analysis, but the high-speed two- or three-bladed turbines do tend
to yield higher efficiency than low-speed multiblade windmills.
Fig. 9.1.2 Performance curves for wind turbines.
Blade Element Theory for Horizontal-Axis Turbines Wilson and
Eggleston describe blade element theory as a mechanism for analyzing
the relationship between the individual airfoil properties and the interference
factor a, the power produced P, and the axial thrust T of the
turbine. Rather than the stream tube of Fig. 9.1.1, the control volume
consists of the annular ring bounded by streamlines depicted in Fig.
9.1.3. It is assumed that the flow in each annular ring is independent of
the flow in all other rings.
Fig. 9.1.3 Annular ring control volume.
A schematic of the velocity and force vector diagrams is given in Fig.
9.1.4. The turbine is defined by the number B of its blades, by the
variation of chord c, by the variation in blade angle u, and by the shape
of blade sections a9 5v/(2V), where v is the angular velocity of the air
just behind the turbine and V is the turbine angular velocity. Also W is
the velocity of the wind relative to the airfoil. Note that the angle f will
be different for each blade element, since the velocity of the blade is a
function of the radius. In order to keep the local flow angle of attack a 5
u 2 f at a suitable value, it will generally be necessary to construct
twisted blades, varying u with the radius. The sectional lift and drag
coefficients CL and CD are obtained from empirical airfoil data and are
unique functions of the local flow angle of attack a 5 u 2 w and the
local Reynolds number of the flow. The entire calculation requires trialand-
error procedures to obtain the axial interference factor a and the
angular velocity fraction a9. It can, however, be reduced to programs for
small computers.
Fig. 9.1.4 Velocity and force vector diagrams.
A typical solution for steady-state operation of a two- or three-bladed
wind axis turbine is shown in Fig. 9.1.2. When optimized, these turbines
run at high tip speed ratios. The curve shown in Fig. 9.1.2 for the two- or
three-bladed wind turbine is for constant blade pitch angle. These turbines
typically have pitch change mechanisms which are used to feather
the blades in extreme wind conditions. In some instances the blade pitch
is continuously controlled to assist the turbine to maintain constant
speed and appropriate output. Turbines with continuous pitch control
typically have flatter, more desirable operating curves than the one
depicted in Fig. 9.1.2.
The traditional U.S. farm windmill has a large number of blades with a
high solidity ratio s. (s is the ratio of area of the blades to swept area of
the turbine pR2.) It operates at slower speed with a lower power coefficient
than high-speed turbines and is primarily designed for good starting
torque.
high- and low-speed wind axis turbines are theoretically predicted performance
curves which have been experimentally confirmed.
Vertical-Axis Turbines The Darrieus rotor looks somewhat like an
eggbeater (Fig. 9.1.5). The blades are high-performance symmetric airfoils
formed into a gentle curve to minimize the bending stresses in the
blades. There are usually two or three blades in a turbine, and as shown
in Fig. 9.1.2, the turbines operate efficiently at high speed. Wilson
shows that VAWT performance analysis also takes advantage of the
same momentum principles as the horizontal axis wind turbines. However,
the blade element momentum analysis becomes much more complicated
(see Touryan et al.).
Fig. 9.1.5 Darrieus rotor.
Care must be taken not to overemphasize the aerodynamic efficiency
of wind turbine configurations. The most important criterion in evaluating
WECSs is the power produced on a per-unit-cost basis.
Drag Devices Rotors utilizing drag rather than lift have been constructed
since antiquity, even though they are bulky and limited to low
coefficients of performance. The Savonius rotor is a modern variation
of these devices; in practice it is limited to small sizes. Eldridge describes
the history and theory of this type of windmill.
Augmentation Occasionally, the use of structures designed to concentrate
and equalize the wind at the turbine is proposed. For its size, the
most effective of these has been a short diffuser (hollow cone) placed
around and downwind of a wind turbine. The disadvantage of such
augmentation devices is the cost of the bulky static structures required.
Rotor Configuration Trends Hansen and Butterfield describe some
trends in turbine configurations which have developed from 1975 to
1995. Although no single configuration has emerged which is clearly
superior, HAWTs have been more widely used than VAWTs. Only
about 3 percent of turbines installed to date are VAWTs.
HAWT rotors are generally classified according to rotor orientation
(upwind or downwind of the tower), blade articulation (rigid or teetering),
and number of blades (generally two or three). Downwind turbines
were favored initially in the United States, but the trend has been toward
greater use of upwind turbines with a current split between upwind (55
percent) and downwind (45 percent) configurations.
Downwind orientation allows blades to deflect away from the tower
when thrust loading increases. Coning can also be easily introduced to
decrease mean blade loads by balancing aerodynamic loads with centrifugal
loads. Figure 9.1.6 shows typical upwind and downwind configurations
along with definitions for blade coning and yaw orientation.
Free yaw, or passive orientation with the wind direction, is also possible
with downwind configurations, but yaw must be actively controlled
with upwind configurations. Free-yaw systems rely on rotor thrust loads
and blade moments to orient the turbine. Net yaw moments for rigid
rotors are sensitive to inflow asymmetry caused by turbulence, wind
shear, and vertical wind. These are in addition to the moments caused by
changes in wind direction which are commonly, though often incorrectly,
considered the dominant cause of yaw loads.
Coning angle
Wind
Wind
Upwind Downwind
Teeter axis
Teeter motion
Fig. 9.1.6 HAWT configurations. (Courtesy of Atlantic Orient Corp.)
Some early downwind turbine designs developed a reputation for
generating subaudible noise as the blades passed through the tower
shadow (tower wake). Most downwind turbines operating today have
greater tower clearances and lower tip speeds, which result in negligible
infrasound emissions (Kelley and McKenna, 1985).
Blade Articulation Several different rotor blade articulations have
been tested. Only two have survived—the three-blade, rigid rotor and
the two-blade, teetered rotor. The rigid, three-blade rotor attaches the
blade to a hub by using a stiff cantilevered joint. The first bending
natural frequency of such a blade is typically greater than twice the rotor
rotation speed 2p. Cyclic loads on rigid blades are generally higher than
on teetering blades of the same diameter. Richardson and McNerney
describe a 33-m, 300-kW turbine currently under development which
reflects a mature version of this configuration.
Teetered, two-blade rotors use relatively stiff blades rigidly connected
to a hub, but the hub is attached to the main drive shaft through a
teeter hinge. This type rotor is commonly used in tail rotors and some
main rotors on helicopters. Two-blade rotors usually require teeter
hinges or flexible root connections to reduce dynamic loading resulting
from nonaxisymmetric mass moments of inertia. In normal operation,
the cyclic loads on the teetering rotor are low, but there is risk of teeterstop
bumping (‘‘mast bumping’’ in helicopter terminology) that can
greatly increase dynamic loads in unusual situations.
Number of Blades Most two-blade rotors operating today use teetering
hinges, but all three-blade rotors use rigid root connections. For
small turbines (smaller than 50-ft diameter) rigid, three-blade rotors are
inexpensive and simple and have the lowest system cost. As the turbines
become larger, blade weight (and hence cost) increases in proportion to
the third power of the rotor diameter, whereas power output increases
only as the square of the diameter. This makes it cost-effective to reduce
the number of blades to two and to add the complexity of a teeter hinge
or flex beams to reduce blade loads. In the midscale rotor size (15 to
30 m), it is difficult to determine whether three rigidly mounted blades
or two teetered blades are more cost-effective. In many cases, the choice
between two- and three-blade rotors has been driven by designers’ lack
of experience and the potential risk of high development cost rather than
by technical and economic merit. Currently 10 percent of the turbines
installed are two-bladed, yet approximately 60 percent of all new designs
being considered in the United States are two-blade, teetered
rotors.
Design Problems A key design consideration is survival in severe
storms. Various systems for furling the rotor, feathering the blades, or
braking the shaft have been employed; failure of these systems in a high
wind has been known to cause severe damage to the turbine.
A different, but related, consideration is the control of the turbine
after a loss of electrical load, which also could cause severe overspeeding
and catastrophic failure.

The other major cause of mechanical failure is the high level of
vibration and alternating stresses. Loosening of inappropriately chosen
fasteners is common. Fatigue considerations must be taken into account,
especially at the rotor blade root.
Resonant oscillations are also possible if exciting frequencies and
structural frequencies coincide. The dominant exciting frequency tends
to be the blade passage frequency, which is equal to the number of
blades times the revolutions per second. An important structural frequency
in HAWTs is the natural frequency of the tower. One design
approach is to make the tower so stiff that the exciting frequency is
always below the lowest natural frequency of the tower. Another is to
permit the tower to be more flexible, but manage the speed of the
turbine such that the exciting frequency is never at a structural frequency
for any significant length of time.
Use of Wind Energy Conversion Systems Historically, wind energy
conversion systems were first used for milling grain and for pumping
water. These tasks were ideally suited for wind power sources,
since the intermittent nature of the wind did not adversely affect the
operation.
The largest impact of wind power on the energy picture in the developed
countries of the world is expected to be in the generation of electric
power. In most cases, this involves feeding power into the power
grid, and requires induction or synchronous generators. These generators
require that the rotor turn at a constant speed. Wind turbines operate
more efficiently (aerodynamically speaking) if they turn at an optimum
ratio of tip speed to wind speed. Thus the use of variable-speed operation,
using power electronics to obtain constant-frequency utility-grade
ac power, has become attractive. Richardson describes the modern use
of variable speed in wind turbines.
Gipe explains that in remote locations, where the power grid is not
accessible and the first few units of electric energy may be very valuable,
dc generation with storage and/or wind and diesel ‘‘village power
systems’’ have been used. These systems are now being optimized to
supply stable, constant-frequency ac electric energy.
Power in the Wind Since wind is air in motion, the power in wind
can be expressed as
Pw 5 1⁄2rV3A (9.1.5)
where Pw 5 power, W; A 5 reference area, m2 ; V 5 wind speed, m/s;
r 5 air density, kg/m3. Since V appears to the third power, the wind
speed is clearly very important. Figure 9.1.7 is a map of the United
States showing regions of annual average available wind power.
The wind speed at a location is random; thus it can be modeled as a
continuous random variable in terms of a density function f (v) or a
distribution function F(v). The Weibull distribution is commonly used
to model wind:
F(v) 5 1 2 exp [2(v/a)b ] (9.1.6)
f (v) 5 b(vb21/ab ) exp [2(v/a)b ] (9.1.7)
In Eqs. (9.1.6) and (9.1.7), a and b are two parameters which can be
adjusted to fit available data over the study period, typically one month.
They can be calculated from the sample mean mv and the sample variance
s2v
using the following equations:The electrical equipment needed for wind-to-electric conversion depends,
above all, on whether the aeroturbine is operated in the constantspeed,
nearly constant-speed, or variable-speed mode. With constantspeed
and nearly constant-speed operation, the power coefficient Cp in
Eq. (9.1.15) becomes a function of wind speed. If variable-speed mode
is used, it is possible to operate the turbine at a constant optimum Cp
over a range of wind speeds, thus extracting a larger fraction of the
energy in the wind.
Synchronous and induction generators are ideally suited for constantspeed
and nearly constant-speed operation, respectively. Variablespeed
operation requires special and/or additional electrical hardware if
constant-frequency utility-grade ac power output is desired. Most of the
early prototypes employed constant-speed operation and synchronous
generators. However, power oscillations due to tower interference and
wind shear effects can be nearly eliminated by operating the turbine and
the generator in variable-speed mode over at least some limited range of
speeds. It appears likely that large (greater than 100-kW) wind-toelectric
systems may employ some kind of a variable-speed constantfrequency
power generation scheme in the future.
Several options are available for obtaining constant frequency
utility-grade ac output from wind-to-electric systems operated in the
variable-speed mode. Some of the schemes suggested are: permanent
magnet alternator with output rectification and inversion, dc generator
feeding a line commutated (synchronous) or force commutated inverter,
ac commutator generator, ac-dc-ac link, field modulated generator system,
and slip ring induction machines operated as generators with rotor
power conditioning. The last type is also known as a double output
induction generator or simply a doubly fed machine. In general, the
simpler the electrical generation scheme, the poorer the quality of the
constant-frequency ac output. For example, synchronous inverters are
very simple, are economical, and have been popular in small (less than
50 kW) commercial units; however, they have power quality and harmonic
injection problems, and they absorb (on the average) more reactive
voltamperes from the utility line than the watts they deliver. The latter is
also a problem with simple induction generators. Schemes such as the
field modulated generator system and doubly fed machine deliver excellent
power quality, but at a higher cost for the hardware.
Economics Costs of wind energy systems are often divided into
two categories: annualized fixed costs and operation and maintenance
(O&M) costs. Annualized fixed costs are comprised primarily of the
cost of capital required to purchase and install the turbines. In addition,
they include certain fixed costs such as taxes and insurance. O&M costs
include scheduled and unscheduled maintenance and the levelized cost
for major equipment overhauls.
The initial capital cost of a wind turbine system includes the cost of
the turbine, installation, and balance of plant. Turbine costs are often
expressed in terms of nameplate rating ($/kW). In 1995, utility-grade
turbines cost on the order of $800 per kilowatt. Installation and balance
of plant costs add approximately 20 percent. The cost of capital varies,
but (in 1995) was often estimated as 8 percent per annum for wind
energy projects. Other fixed costs were estimated at around 3 percent of
the installed turbine cost. The fixed charge rate (FCR), combined capital
and other fixed costs, was approximately 11 percent per annum. O&M
costs in modern wind farms are around $0.01 per kilowatthour (1995).
In addition to capital and O&M costs, an economic assessment of
wind energy systems must account for system performance. A commonly
used parameter that describes the production of useful energy by
wind and other energy systems is the capacity factor C, also called the
plant factor or load factor. It is the ratio of the annual energy produced
(AEP) to the energy that would be produced if the turbine operated at
full-rated output throughout the year:
Cf 5
AEP
8,760 PR
(9.1.16)
where AEP is in kWh, 8,760 is the number of hours in 1 year, and PR is
the unit’s nameplate rating in kW.
In order of decreasing importance, Cf is affected by the average
power available in the wind, speed vs. duration curve of the wind regime,
efficiency of the turbine, and reliability of the turbine. Variablespeed
turbines which tend to have low cut-in speeds and high efficiency
in low winds exhibit better capacity factors than constant-speed turbines.
Modern utility-grade turbines at good sites (class 4) can achieve
capacity factors in the range of 25 to 30 percent.
The combination of cost and performance can be used to calculate the
cost of energy (COE) as follows:
COE 5
FCR 3 ICC
8,760Cf
1 (O&M) (9.1.17)
where FCR is the fixed charge rate for the cost of capital and for other
fixed charges such as taxes and insurance, ICC is the installed capital
cost of the turbine and balance of plant in dollars per kilowatt. This
method is useful to estimate the cost of energy for different technologies
or sites. However, for investment decisions, more detailed analyses that
include the effects of various investment strategies, tax incentives, and
environmental factors should be performed. Ramakumar et al. discuss
the economic aspects of advanced energy technologies, including wind
energy systems.

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