wind energy_part_2_power in the wind(1)

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Renewable and Green Energy

ADEL A. A. ELGAMMAL

ASSOCIATE PROFESSORADEL ELGAMMAL

THE UNIVERSITY OF TRINIDAD AND TOBAGO UTT

WIND ENERGYPart - 2

ADEL ELGAMMAL


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Available Power in the Wind

ADEL ELGAMMAL THE UNIVERSITY OF TRINIDAD AND TOBAGO UTT


• How much energy is in the wind? And how

much of that energy can a wind turbine

"catch"?

• Or how much of the wind's energy can a wind

turbine convert into useful electrical energy?


3

Available Power in the Wind• The following figure shows the "tube" of wind

energy that goes into a wind turbine.

• The "D" represents the diameter of the turbineblades.

• The blue oval is to show that the circular areaof wind swept by the blades is the areaavailable for producing power.




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Air density• The Air Density, symbolized by the Greek letter (rho), is an

important parameter to know in wind power applications.

• Air density is the mass of air per unit volume:

where

• is the air density, in kilograms per cubic meter (kg/m3)[pounds mass per cubic foot (lbm/ft3)].

• m is the mass of air, in kilograms (kg) [pounds mass (lbm)].

• V is the volume, in cubic meters (m3) [cubic feet (ft3)].


Air density• The air density varies with atmospheric pressure,

temperature, humidity, and altitude:• In S.I. units, is equal to 1.225 kg/m3 under standard

sea level conditions, which are: a temperature of 15.5°C,an atmospheric pressure of 101.325 kPa, and a relativehumidity of 36%.

• In U.S. customary units, is equal to 0.076 lbm/ft3 understandard (sea level) conditions, which are: a temperatureof 60°F, an atmospheric pressure of 14.7 psia or 0 psig,and a relative humidity of 36%.

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Kinetic energy in the wind• Any object or fluid in motion has kinetic

energy. For example, wind, which is a mass of

air in motion, has kinetic energy.

• The faster the speed of the wind, the higher the

kinetic energy of the wind.

POWER IN THE WIND

Consider a “packet” of air with mass m moving at a speed v. Its kinetic energy K.E., is given by the familiar relationship:

(2)


2.

2mvEK

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Kinetic energy in the wind• where K.E is the kinetic energy, in joules (J) [feet-

pound force (ft·lbf)].

• m is the mass of air, in kilograms (kg) [pounds mass(lbm)].

• v is the velocity of the mass of air, in meters persecond (m/s) [feet per second (ft/s)].

• 2 is a constant. When working in U.S. customaryunits, this constant must be multiplied by thegravitational constant, g (32.174 lbm·ft/lbf·s2).


Kinetic energy in the wind• The gravitational constant, g must be used to change

from pounds mass (lbm) to pounds force (lbf). Theequation for calculating kinetic energy is, therefore:

• Where g is equal to 32.174 lbm·ft/lbf·s2.

• Note that the term wind speed is also used todesignate the wind velocity, v.

g

mvEK

2.

2

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Kinetic energy in the wind• Kinetic energy is a function of mass and

velocity. So to know the energy in the wind wehave to know the wind's velocity as well as thedensity of the air.

• The density is the mass per unit volume(kilograms per cubic meter [kg/m3], or poundsmass per cubic foot [lbm/ft3] are a couple oftypical examples of units of density).


Kinetic energy in the wind• Of course, what we are really interested in as

engineers and producers or users of electricity is thepower we can get out of the wind.

• Power is how fast we are producing or converting aquantity energy.

• Power has units of energy divided by time.

• A Watt is a unit of power. It represents one joule ofenergy transformed every second. A 60 Watt lightbulb converts 60 joules of energy every second intolight and heat (mostly heat - don't touch the bulb).


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POWER IN THE WINDSince power is energy per unit time, the power represented by a mass of air moving at velocity v through area A will be

(2)


POWER IN THE WINDThe mass flow rate ṁ, through area A, is the product of air density ρ, speed v, and cross-sectional area A:


3

2

( / )

( / )

sec ( )

( / )

m A v kg s

density kg m

A cross tional area m

v fluid velocity m s

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Calculating wind power• The formula below shows the variables that

determine the power in the wind going into thewind turbine (not the power obtainable,because we can't get it all):

2

2

1vmPavail


POWER IN THE WIND

Pavail (Watts) = power in the wind

ρ (kg/m3)= air density (1.225kg/m3 at 15˚C and 1 atm)

A (m2)= the cross-sectional area that wind passes through

v (m/s)= wind speed normal to A (1 m/s = 2.237 mph)


3

2

1vAρPavail

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Calculating wind power• Notice that the power in the wind going into a

specific wind turbine, depends on 3 variables,

1) the density of the air,

2) the diameter of the turbine bladessquared (D times D), and

3) the velocity of the wind to the thirdpower (v times v times v).


Calculating wind power

Density = P/(RxT)P - pressure (Pa)R - specific gas constant (287 J/kgK)T - air temperature (K)

Area = r2 Instantaneous Speed

(not mean speed)

kg/m3 m2 m/s


3

2

1vAρPavail

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Importance of Wind Speed• No other factor is more important to the amount

of power available in the wind than the speed ofthe wind

• Power is a cubic function of wind speedv X v X v

• 20% increase in wind speed means 73% morepower

• Doubling wind speed means 8 times more power.• Energy in 1 hour of 20 mph winds is the same as

energy in 8 hours of 10 mph winds• Nonlinear, so we cannot use average wind speed.• Faster is better, and bigger is better (if you can

afford it and can build it strong enough).• Hence, selecting the right site play a major role

in the success of a wind power projects.


Importance of Wind Speed• Of course, the wind doesn't blow all the time in most

places and when it blows too hard the turbine bladescan break or spin so fast they break off (not good wheneach blade can weigh several tons).

• In that case, the blades are usually "feathered" to reducestresses on them and to slow them down. This meanswe can't take advantage of really high wind speeds.


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• For a conventional HAWT, A = (π/4)D2, so wind power isproportional to the blade diameter squared

• Doubling the diameter increases the power available by afactor of four. That simple observation helps explain theeconomies of scale that go with larger wind turbines. Thecost of a turbine increases roughly in proportion to bladediameter, but power is proportional to diameter squared,so bigger machines have proven to be more costeffective.

• Cost is roughly proportional to blade diameter.

Power in the wind is proportional to the swept area


Power in the wind, per square meter of cross section, at 15◦C and 1 atm.


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Example



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Example• Calculate how much more power is available

at a site where the wind speed is 12 mph thanwhere it is 10 mph

• P ~ V3

• P2/P1 = (V2/V1)3

– P2 = (12/10)3P1 = 1.73 P1

1.7 x the power (almost a factor of 2 increase), with only 2 mph increase in wind speed!


V = 5 meters (m) per second (s) m/sρ = 1.0 kg/m3

R = .2 m >>>> A = .125 m2

Power in the Wind = ½ρAV3

= (.5)(1.0)(.125)(5)3

= 7.85 WattsUnits = (kg/m3)x (m2)x (m3/s3)

= (kg-m)/s2 x m/s= N-m/s = Watt

Power in the Wind = ½ρAv3

(kg-m)/s2 = Newton


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Maximum Efficiency & Betz

LimitADEL ELGAMMAL


Turbine Efficiency

• If the turbine could convert all the wind'spower to mechanical power we would say itwas 100% efficient.

• But as you probably know, the real world isnever so generous.

• To even achieve 50% is unlikely, and would bea very efficient machine.

• A 50% efficient turbine would convert half ofthe power in the wind to mechanical power.


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Turbine Efficiency

Two extreme cases, and neither makessense-

1) Downwind velocity is zero – turbineextracted all of the power

2) Downwind velocity is the same as theupwind velocity – turbine extracted nopower.


Turbine Efficiency• Imagine a wind energy extraction machine of

100% efficiency that could take all of thekinetic energy out of the wind.

• That would mean the velocity on the "out" or"leaving" or "exit" side of the turbine bladeswould be zero, nothing. No kinetic energy leftin the wind.


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Turbine Efficiency• If the velocity leaving the blades is zero then

the air wouldn't be leaving at all.

• There would be no air movement, meaning theair after the blades isn't getting out of the wayof the air coming in, which would mean thefresh air couldn't come in, which would meanthere is no air flowing through the turbineblades, which would mean no power.


Turbine Efficiency• In order to at least keep the wind moving

through the turbine there has to be somevelocity or energy in the air after goingthrough the blades so that the air can get out ofthe way of the air coming through next.

• Just to keep the machine running at all theefficiency has to be less than 100%.


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Betz Limit• Albert Betz, 1919, pointed this out and then

proceeded to prove, with solid physics andmath, that there must be some ideal slowing ofthe wind so that the turbine extracts themaximum power.

• Albert Betz figured out that the best that couldbe achieved by a wind turbine is around 59%of the power in the wind.


Betz Limit• In other words, a perfect best-possible wind

turbine would be able to convert almost 59%of the power in the wind into mechanicalrotating power.

• But we can't achieve perfection.


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In Fig., the upwind velocity of the undisturbed wind is v, thevelocity of the wind through the plane of the rotor blades is vb,and the downwind velocity is vd . The mass flow rate of air withinthe stream tube is everywhere the same, call it ṁ.

Approaching wind slows andexpands as a portion of itskinetic energy is extracted bythe wind turbine, forming thestream tube shown.

The power extracted by the blades Pb is equal to the difference inkinetic energy between the upwind and downwind air flows:

• ṁ = mass flow rate of air within stream tube

• v = upwind undisturbed wind speed

• vd = downwind wind speedADEL ELGAMMAL


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The easiest spot to determine mass flow rate ṁ is at the plane ofthe rotor where we know the cross-sectional area is just the sweptarea of the rotor A.

The mass flow rate is thus

If we now make the assumption that the velocity of thewind through the plane of the rotor is just the average ofthe upwind and downwind speeds (Betz’s derivationactually does not depend on this assumption), then wecan write

Determining Mass Flow Rate


Determining Mass Flow Rate

• Assume the velocity through the rotor vb is theaverage of upwind velocity v and downwindvelocity vd:

=2

db

v vv

2dv v

m A


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Power Extracted by the Bladesthen we can write:

To help keep the algebra simple, let us define the ratio of downstream to upstream wind speed to be λ:


Power Extracted by the Blades

PW = Power in the wind

2 2 21 (6.22)

2 2b

v vP A v v

3 2 3 3 3 3

2 2 2 = - + - 2 2 2 2 2

v v v v v vv v

3

2 = 1 - 12

v

3

2 = 1 12

v

3 21 11 1 (6.22)

2 2bP Av

CP = Rotor efficiencyADEL ELGAMMAL


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Maximum Rotor Efficiency• Find the speed wind speed ratio λ which maximizes

the rotor efficiency, CP

• From the previous slide

2 3

21 11 1 = - + -

2 2 2 2 2PC

2=-2 1 3 0PC

2=3 2 1 0PC

= 3 1 1 0PC

1

3

maximizes rotor efficiency

Set the derivative of rotor efficiency to zero and solve for λ:


Maximum Rotor Efficiency

• Plug the optimal value for λ back into CP tofind the maximum rotor efficiency:

2

1 1 1 161 1 = 59.3% (6.26)

2 3 3 27PC

• The maximum efficiency of 59.3% occurs when air is slowed to 1/3 of its upstream rate

• Called the “Betz efficiency” or “Betz’ law”


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Maximum Rotor Efficiency

Rotor efficiency CP

vs. windspeed ratio λ

The blade efficiency reaches a maximum when the wind is slowed to one-third of its upstream value.


Power Coefficient vs Tip Speed Ratio

• Power Coefficient Varies with Tip Speed Ratio

• Characterized by Cp vs Tip Speed Ratio Curve

0.4

0.3

0.2

0.1

0.0

Cp

12108642 0 Tip Speed Ratio


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Betz Limit

Cp is called the power coefficient. Cp is the percentage of power in the wind that is converted

into mechanical energy. What is the maximum amount of energy that can be

extracted from the wind?

Betz Limit: 5926.27

16max, pC


3

2

1vAρCP PRotorWind

Betz Limit• All wind power cannot be captured by rotor or air would be

completely still behind rotor and not allow more wind to passthrough.

• Theoretical limit of rotor efficiency is 59%• Most modern wind turbines are in the 35 – 45% range


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Power from a Wind Turbine Rotor = Cp½ρAV3


Betz Limit

Tip-Speed RatioTip-speed ratio is the ratio ofthe speed of the rotatingblade tip to the speed of thefree stream wind.

Rv

=

R

R

Where, = rotational speed in radians /secR = Rotor Radiusv = Free Stream Velocity


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Tip-Speed Ratio (TSR)• Efficiency is a function of how fast the rotor turns

• Tip-Speed Ratio (TSR) is the speed of the outer tip of theblade divided by wind speed

• D = rotor diameter (m)

• v = upwind undisturbed wind speed (m/s)

• rpm = rotor speed, (revolutions/min)

v

Dn

v

Rn

v

RTSR

60602


Performance co-efficient and Betz criterion

The proportion of the power inthe wind that the rotor canextract is termed thecoefficient of performance (orpower coefficient or efficiency;symbol Cp) and its variation asa function of tip speed ratio iscommonly used to characterizedifferent types of rotor.


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A plot of typical efficiency for various rotor types versusTSR is given in Fig. The American multi-blade spinsrelatively slowly, with an optimal TSR of less than 1 andmaximum efficiency just over 30%. The two- and three-blade rotors spin much faster, with optimum TSR in the 4–6 range and maximum efficiencies of roughly 40–50%.Also shown is a line corresponding to an “ideal efficiency,”which approaches the Betz limit as the rotor speedincreases. The curvature in the maximum efficiency linereflects the fact that a slowly turning rotor does notintercept all of the wind, which reduces the maximumpossible efficiency to something below the Betz limit.


Tip-Speed Ratio (TSR)

Rotors with fewer blades reach their maximum efficiency at higher tip-speed ratios


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Example40-m wind turbine, three-blades, 600 kW, windspeed is 14m/s, air density is 1.225 kg/m3

a. Find the rpm of the rotor if it operates at a TSR of 4.0b. Find the tip speed of the rotorc. What gear ratio is needed to match the rotor speed to

the generator speed if the generator must turn at 1800rpm?

d. What is the efficiency of the wind turbine under theseconditions?


Examplea. Find the rpm of the rotor if it operates at a TSR of 4.0,

We can also express this as seconds per revolution:

Tip-Speed-Ratio (TSR) 60rpm

D

v

4.0 60sec/min 14m/srpm = 26.7 rev/min

40m/rev

26.7 rev/minrpm = 0.445 rev/sec or 2.24 sec/rev

60 sec/min


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Exampleb. Tip speed

c. Gear Ratio

rpm DRotor tip speed=

60 sec/min

Rotor tip speed = (rev/sec) D

Rotor tip speed = 0.445 rev/sec 40 m/rev = 55.92 m/s

Generator rpm 1800Gear Ratio = = = 67.4

Rotor rpm 26.7ADEL ELGAMMAL


Exampled. Efficiency of the complete wind turbine (blades, gear

box, generator) under these conditions

From (6.4):

Overall efficiency:

3 2 31 1P A = 1.225 40 14 2112 kW

2 2 4W v

600 kW28.4%

2112 kW


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• Depends on rotor, gearbox, generator, tower, controls,terrain, and the wind

• Overall conversion efficiency (Cp·ηg) is around 30%

Estimates of Wind Turbine Energy

WPBP EP

Power in the Wind

Power Extracted by Blades

Power to ElectricityPC

Rotor Gearbox & Generator

g


KineticEnergy

Mechanical Energy

Electrical Energy

Overall: 42 – 50% Efficient Today… Theoretical Maximum is 59.3% (no losses)

Component Rotor Gearbox Generator Converter

Efficiency 45-52% 95-97% 97-98% 96-99%


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Turbine Power Curve


Turbine Power Curve

0

1 000

2 000

3 000

4 000

5 000

6 000

7 000

8 000

9 000

10 000

0 5 10 15 20 25

Wind Speed [m/s]

Po

wer

[k

W]

Wind Power

Turbine Power

Cut-in Wind speed Rated Wind

speed

Cut-out or Furling Wind

speed

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Turbine Power Curve• No operation until wind velocity reaches a

minimum called the cut-in velocity• Then operate at full turbine output power until

turbine output is greater than generator canaccept

• Limit turbine output power to full generatorpower at high wind speeds

• No operation above maximum velocity calledcut-out velocity


Turbine Power CurveCut –in wind speed, rated wind speed, cut-out wind speed

Figure 6.32

Idealized power curve. No power is generated at wind speeds belowVC; at Wind speeds between VR and VF , the output is equal to therated power of the generator; above VF the turbine is shut down.


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The figure shows a sketch a how the power output from a wind turbine varies with steady

wind speed.


Cut-in Wind speed• At very low wind speeds, there is insufficient torque exerted by the wind on

the turbine blades to make them rotate.

• The speed at which the turbine first starts to rotate and generate power iscalled the cut-in speed and is typically between 3 and 4 meters per second.

• The cut-in wind speed VC is the minimum needed to generate net power.


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Idealized Power Curve• Cut-in Wind speed. Low-speed winds may not

have enough power to overcome friction in thedrive train of the turbine and, even if it do rotating,the electrical power generated may not be enough tooffset the power required by the generator fieldwindings.

• The cut-in wind speed VC is the minimum needed togenerate net power. Since no power is generated atwind speeds below VC, that portion of the wind’senergy is wasted. Fortunately, there isn’t much energyin those low-speed winds anyway, so usually notmuch is lost.


Idealized Power Curve• Rated Wind speed. As velocity increases above the

cut-in wind speed, the power delivered by thegenerator tends to rise as the cube of wind speed.When winds reach the rated wind speed VR, thegenerator is delivering as much power as it isdesigned for.

• Above VR, there must be some way to shed some ofthe wind’s power or else the generator may bedamaged. Three approaches are common on largemachines: an active pitch-control system, a passivestall-control design, and a combination of the two.


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Rated Wind speed• As velocity increases above the cut-in wind speed, the power delivered by the

generator tends to rise as the cube of wind speed.

• However, typically somewhere between 12 and 17 meters per second, the poweroutput reaches the limit that the electrical generator is capable of.

• This limit to the generator output is called the rated power output and the windspeed at which it is reached is called the rated output wind speed.


Rated Wind speed• As velocity increases above the cut-in wind speed, the power

delivered by the generator tends to rise as the cube of wind speed.

• When winds reach the rated wind speed VR, the generator isdelivering as much power as it is designed for.


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Rated Wind speed• Above VR, there must be some way to shed some of

the wind’s power or else the generator may bedamaged.


Rated Wind speed

• At higher wind speeds, the design of theturbine is arranged to limit the power to thismaximum level and there is no further rise inthe output power.

• How this is done varies from design to designbut typically with large turbines, it is done byadjusting the blade angles so as to keep thepower at the constant level.


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Rated Wind speed

• Three common approaches to shed excess wind

– Pitch control – physically adjust blade pitchto reduce angle of attack

– Passive Stall control – blades are designed toautomatically reduce efficiency in high winds

– Active stall control – physically adjust bladepitch to create stall


Idealized Power Curve• Cut-out or Furling Wind speed. At some

point the wind is so strong that there is realdanger to the wind turbine. At this wind speedVF , called the cutout wind speed or the furlingwind speed (“furling” is the term used insailing to describe the practice of folding upthe sails when winds are too strong), themachine must be shut down. Above VF , outputpower obviously is zero.


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Cut-out or Furling Wind speed

• At some point the wind is so strong thatthere is real danger to the wind turbine.

• At this wind speed VF , called the cut-out windspeed or the furling wind speed (“furling” isthe term used in sailing to describe the practiceof folding up the sails when winds are toostrong), the machine must be shut down.

• Above VF , output power obviously is zero.



• As the speed increases above the rate outputwind speed, the forces on the turbine structurecontinue to rise and, at some point, there is arisk of damage to the rotor.

• As a result, a braking system is employed tobring the rotor to a standstill.

• This is called the cut-out speed and is usuallyaround 25 metres per second.


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• Above cut-out or furling wind speed, thewind is too strong to operate the turbinesafely, machine is shut down, output power iszero

• Rotor can be stopped by rotating the blades topurposely create a stall

• Once the rotor is stopped, a mechanical brakelocks the rotor shaft in place


Power output

0

1 000

2 000

3 000

4 000

5 000

6 000

7 000

8 000

9 000

10 000

0 5 10 15 20 25

Wind Speed [m/s]

Po

wer

[kW

]

Wind Power

Turbine Power


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How much energy can we get?


How do you site a turbine?• Wind resource• Current land use• Environmental impacts• Government regulations• Cost of wind farm • Economic payback• Community opinion –

aesthetics• Noise and flicker issues• Transmission lines• Spacing of turbines• Much, much more!

Mountaineer Wind Energy Center, WV www.communityenergy.com


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Wind turbine power and torque


Wind turbine power and torque• The wind turbine cannot extract the available

power completely from the wind.

• When the wind stream passes the turbine, a partof its kinetic energy is transferred to the rotor andthe air leaving the turbine carries the rest away.

• Actual power produced by a rotor would thus bedecided by the efficiency with which this energytransfer from wind to the rotor takes place.


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Wind turbine power and torque• This efficiency is usually termed as the power coefficient (Cp).

• Thus, the power coefficient of the rotor can be defined as the

ratio of actual power developed by the rotor to the theoretical

power available in the wind. Hence,

• where PT is the power developed by the turbine.


Wind turbine power and torque• The power coefficient of a turbine depends on many

factors such as the profile of the rotor blades, bladearrangement and setting etc.

• A designer would try to fix these parameters at itsoptimum level so as to attain maximum Cp at a widerange of wind velocities.

• The thrust force experienced by the rotor (F) can beexpressed as


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Wind turbine power and torque• Hence we can represent the rotor torque (T) as

• where R is the radius of the rotor. This is themaximum theoretical torque and in practice therotor shaft can develop only a fraction of thismaximum limit.


Wind turbine power and torque• The ratio between the actual torque developed by the

rotor and the theoretical torque is termed as the torquecoefficient (CT). Thus, the torque coefficient is givenby

• where TT is the actual torque developed by the rotor.


44

Wind turbine power and torque• The power developed by a rotor at a certain wind speed greatly depends

on the relative velocity between the rotor tip and the wind.

• For example, consider a situation in which the rotor is rotating at a verylow speed and the wind is approaching the rotor with a very high velocity.Under this condition, as the blades are moving slow, a portion of the airstream approaching the rotor may pass through it without interacting withthe blades and thus without energy transfer.

• Similarly if the rotor is rotating fast and the wind velocity is low, the windstream may be deflected from the turbine and the energy may be lost dueto turbulence and vortex shedding.

• In both the above cases, the interaction between the rotor and thewindstream is not efficient and thus would result in poor powercoefficient.


Wind turbine power and torque• The ratio between the velocity of the rotor tip and the wind

velocity is termed as the tip speed ratio (). Thus,

• where is the angular velocity and N is the rotational speedof the rotor. The power coefficient and torque coefficient of arotor vary with the tip speed ratio.

• There is an optimum for a given rotor at which the energytransfer is most efficient and thus the power coefficient is themaximum (CP max).


45

Wind turbine power and torque• Now, let us consider the relationship between the

power coefficient and the tip speed ratio.


Wind turbine power and torque• Thus, the tip speed ratio is given by the ratio

between the power coefficient and torquecoefficient of the rotor.


46

Typical torque versus speed curve at the wind turbine rotor

• Figure shows a typical torque-versus-speed curve at the rotorof a wind turbine obtained for a given wind speed.

• As the rotor speed increases, the torque produced at the rotorincreases until a point is reached, beyond which the torquegradually decreases to zero.

• Consequently, the mechanical power produced at the rotor alsoincreases up to a certain maximum value, and then graduallydecreases to zero, as Figure shows.

• The point at which the mechanical power is maximum isreferred to as the maximum power point (MPP). The rotorspeed and torque at the MPP are commonly referred to as theoptimum speed and optimum torque, respectively.



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• A wind turbine must be operated as close as possibleto the optimum speed to maximize the mechanicalpower developed at the rotor and thus obtain themaximum amount of electrical power.

• This is performed by setting the rotor torque to theoptimum value, through adjustment of the currentdrawn by the electrical load at the wind turbinegenerator output.



• Figure shows a set of typical curves at the rotor of a windturbine, for different wind speeds: the torque-versus-speedcurves (Figure a) and the mechanical power-versus-speedcurves (Figure b).

• On each torque-versus-speed curve in Figure a, a diamond-shaped marker indicates the optimum rotor torque and speed atwhich the maximum amount of mechanical power is producedat the wind turbine rotor.

• The maximum power point (MPP) is also indicated by adiamond-shaped marker on each of the correspondingmechanical power curves in Figure b.


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• Note that the rotor speed at which the maximumamount of mechanical power is produced at the rotorof a wind turbine varies with the wind speed.

• Therefore, to operate the wind turbine at themaximum power point (MPP) and maximize theenergy produced at any wind speed, the rotor speedmust be continuously monitored and kept at theoptimum value, through adjustment of the rotortorque when necessary. This is generally performedautomatically by a controller in the wind turbine.


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• The following conclusions can be drawn fromexamination of the family of curves in Figure.

• Figure a shows that higher speeds and torquesare reached when the wind speed increases.

• Consequently, higher amounts of mechanicalpower are produced at the rotor when the windspeed increases, as Figure b shows.



• When the maximum power points on the variousmechanical power curves in Figure b are connectedtogether, they form a curve which increasesexponentially (see dashed line in Figure b. In fact, themechanical power at the MPP’s increases by eightwhenever the wind speed doubles.

• This occurs because the power in the wind varieswith the cube (the third power) of the wind speed.


50

Current versus voltage and electrical power versus speed curves at the wind turbine generator output for different

wind speeds

• Figure shows a set of typical curves related to theoutput of a wind turbine generator, for different windspeeds: the current-versus-voltage curves of thegenerator output (figure a) and the correspondingelectrical power-versus speed curves (figure b).

• The following conclusions can be drawn bycomparing the family of curves in this Figure with thefamily of curves in previous Figure:


51

Current versus voltage and electrical power versus speed curves at the wind turbine generator output for different

wind speeds

• The voltage and current at the output of the wind turbine generatorare proportional to the speed and torque at the wind turbine rotor,respectively.

• Consequently the current-versus-voltage curves of the wind turbinegenerator (Figure a) are similar to the torque-versus-speed curves atthe wind turbine rotor (shown in Figure a).

• Also, the electrical power-versus-speed curves of the wind turbinegenerator (Figure b) are similar to the mechanical power-versus-speed curves at the wind turbine rotor (shown in Figure b).

• Through proper control of the electrical load applied to the wind-turbine generator output, the rotor speed and torque can be adjustedin order to keep the generator operating at the maximum powerpoint (MPP) at any wind speed.


ExampleConsider a wind turbine with 5 m diameter rotor.Speed of the rotor at 10 m/s wind velocity is 130r/min and its power coefficient at this point is0.35.a) Calculate the tip speed ratio and torque

coefficient of the turbine.b) What will be the torque available at the rotor

shaft? Assume the density of air to be 1.24kg/m3.


52


Example• Area of the rotor is:

• As the speed of the rotor is 130 r/min, its angularvelocity is


Example

53

Temperature Correction for

Air DensityADEL ELGAMMAL


Temperature Correction for Air Density

When wind power data are presented, it is often assumedthat the air density is 1.225 kg/m3; that is, it is assumed thatair temperature is 15◦C (59◦ F) and pressure is 1atmosphere. Using the ideal gas law, we can easilydetermine the air density under other conditions.


54

where

P is the absolute pressure (atm),

V is the volume (m3),

n is the mass (mol),

R is the ideal gas constant = 8.2056 × 10−5 m3 · atm · K−1

· mol−1, and

T is the absolute temperature (K),where K = ◦C + 273.15.

One atmosphere of pressure equals 101.325 kPa (Pa is theabbreviation for pascals, where 1 Pa = 1 newton/m2). Oneatmosphere is also equal to 14.7 pounds per square inch(psi), so 1 psi = 6.89 kPa. Finally, 100 kPa is called a barand 100 Pa is a millibar, which is the unit of pressure oftenused in meteorology work.




The following observations can be made from the above equation:

• Any change in the temperature of the air, atmospheric pressure, orrelative humidity causes the air density to change, causing thewind power to change in the exact same way (for given wind speedand cross-sectional area).– Humid climates have greater air density than dry climates– Lower elevations have greater air density than higher elevations

• For instance, when the air density increases by 5%, the windpower PW also increases by 5%.

3

2

1vAρCP PRotorWind

= P/(RxT)P - pressure (Pa)R - specific gas constant (287 J/kgK)T - air temperature (K)

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• The density of airdecreases with theincrease in sitetemperature as illustratedin Fig.

• The air density may betaken as 1.225 for most ofthe practical cases.

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If we let M.W. stand for the molecular weight of the gas(g/mol), we can write the following expression for airdensity, ρ:

• Air density is greater at lower temperatures

Combining (5) and (6) gives us the following expression:

(6)

(7)



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Altitude Correction for

Air DensityADEL ELGAMMAL


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Effect of elevation on air density• The density of air

decreases with theincrease in siteelevation as illustratedin Fig.

• The air density maybe taken as 1.225 formost of the practicalcases.

Effect of elevation on air density–Wind energy increases with height to

the 1/7 power–2X the height translates into 10.4%

more electricity


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RECENT CAPACITY ENHANCEMENTS

20031.8 MW350’

2000850 kW265’

20065 MW600’


Velocity with Height


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Wind Speed & Height

Higher means stronger,

smoother wind


Altitude Correction for Air DensityAir density, and hence power in the wind, depends onatmospheric pressure as well as temperature. Since airpressure is a function of altitude, it is useful to have acorrection factor to help estimate wind power at sites abovesea level.

Consider a static column of air with cross section A, asshown in Fig. A horizontal slice of air in that column ofthickness dz and density ρ will have mass ρA dz. If thepressure at the top of the slice due to the weight of the airabove it is P(z + dz), then the pressure at the bottom of theslice, P(z), will be P(z + dz) plus the added weight per unitarea of the slice itself:


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A column of air in static equilibrium used to determine the relationship between air pressure and altitude.


where g is the gravitational constant, 9.806 m/s2. Thus we can write the incremental pressure dP for an incremental change in elevation, dz as

That is,


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The air density ρ given in (10) is itself a function of pressure as described in (7), so we can now write

To further complicate things, temperature throughout theair column is itself changing with altitude, typically at therate of about 6.5◦C drop per kilometer of increasingelevation. If, however, we make the simplifyingassumption that T is a constant throughout the air column,we can easily solve (11) while introducing only a slighterror. Plugging in the constants and conversion factors,while assuming 15◦C, gives


which has solution,

where P0 is the reference pressure of 1 atm and H is inmeters.


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A simple way to combine the temperature and pressurecorrections for density is as follows:

the correction factors KT for temperature and KA foraltitude are tabulated in Tables 6.1 and 6.2.


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Impact of Elevation and Earth’s Roughness

on Wind speed


How to calculate wind speed increase with height

• Conservative Approximation:

V2 = (H2/H1)αV1• α is the Roughness exponent

– Smooth terrain value (water or ice): 0.10

– Rough terrain value (suburb woodlands): 0.25

– Grasslands: 0.14


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Example• Consider doubling the height of your tower from 10 m to 20 m.

V2 = (H2/H1)V1 = (20/10)

.14 V1 = 1.1V1

• The power available increases to:

P2 = (H2/H1)P1 = (2)

P1 = 1.34 P1• If you multiply height by a factor of 5:

P2 = (H2/H1)P1 = (5)

P1 = 1.97 P1


Example• You live in a forested area. Calculate how much more power you can get from a turbine at 87meters than a turbine at 30meters.

V2 = (H2/H1)V1 = (87/30)

.25 V1 = 1.3V1

• The power available increases to:

P2 = (H2/H1)P1 = (2.9)

P1 = 2.22 P1


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Impact of Elevation and Earth’s Roughness on Wind speed

• Since power increases like the cube of wind speed, wecan expect a significant economic impact from even amoderate increase in wind speed

• There is a lot of friction in the first few hundred metersabove ground – smooth surfaces (like water) are better

• Wind speeds are greater at higher elevations – tall towersare better

• Forests and buildings slow the wind down a lot

• Can characterize the impact of rough surfaces and heighton wind speed


One expression that is often used to characterize the impact of the roughness of the earth’s surface on wind speed is the following:

• α = friction coefficient – given in Table 6.3

• v = wind speed at height H

• v0 = wind speed at height H0 (H0 is usually 10 m)

• Typical value of α in open terrain is 1/7

• For a large city, α = 0.4; for calm water, α = 0.1ADEL ELGAMMAL


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There is another approach that is common in Europe. The alternative formulation is

where z is called the roughness length

Note that both equations are just approximations of the variation in wind speed due to elevation and roughness–the best thing is to have actual measurements



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Figure a shows the impact of friction coefficient on windspeed assuming a reference height of 10 m, which is acommonly used standard elevation for an anemometer. Ascan be seen from the figure, for a smooth surface (α = 0.1),the wind at 100 m is only about 25% higher than at 10 m,while for a site in a “small town” (α = 0.3), the wind at 100m is estimated to be twice that at 10 m. The impact ofheight on power is even more impressive as shown in

Fig. b.


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Since power in the wind varies as the cube of wind speed,we can rewrite (15) to indicate the relative power of thewind at height H versus the power at the reference heightof H0:


In Figure b, the ratio of wind power at otherelevations to that at 10 m shows the dramaticimpact of the cubic relationship between windspeed and power. Even for a smooth groundsurface—for instance, for an offshore site—thepower doubles when the height increases from 10 mto 100 m. For a rougher surface, with frictioncoefficient α = 0.3, the power doubles when theheight is raised to just 22 m, and it is quadrupledwhen the height is raised to 47 m.


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Increasing (a) windspeed and (b) power ratios with height for variousfriction coefficients α using a reference height of 10 m. For α = 0.2(hedges and crops) at 50 m, windspeed increases by a factor of almost1.4 and wind power increases by about 2.6.



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Example 6.6 illustrates an important point about the variation in wind speedand power across the face of a spinning rotor. For large machines, when ablade is at its high point, it can be exposed to much higher wind forces thanwhen it is at the bottom of its arc. This variation in stress as the blade movesthrough a complete revolution is compounded by the impact of the toweritself on wind speed—especially for downwind machines, which have asignificant amount of wind “shadowing” as the blades pass behind the tower.The resulting flexing of a blade can increase the noise generated by the windturbine and may contribute to blade fatigue, which can ultimately cause bladefailure.


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Impact of Blade Swept Area


Wind Energy Natural Characteristics

• Blade swept area– Wind energy increases proportionally with

swept area of the blades• Blades are shaped like airplane wings

– 10% increase in swept diameter translatesinto 21% greater swept area

– Longest blades up to 413 feet in diameter• Resulting in 600 foot total height


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Swept Area

• Power in the wind is also proportional to the swept area

A = R2

• Increase the radius from 10 m to 12 m:

A2 = (R2/R1)2 A1

A2 = (12/10)2 A1 = 1.44A1

Nothing tells you more about a wind turbine’s potential than the rotor radius.


Importance of Rotor Diameter• Swept are is proportional to square of the rotor diameter

• 20% increase in rotor diameter increases area by 44%

• Doubling diameter increases area 4 times


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Rotor diameter


Blade Plan form - SolidityBlade plan form is the shape of the flat wise blade surface

Solidity is the ratio of total rotor plan form area to total swept area

Low solidity (0.10) = high speed, low torque

High solidity (>0.80) = low speed, high torque

R

A

a

Solidity = 3a/A


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Solidity and Tip speed ratio


Questions?


wind energy_part_2_power in the wind(1)

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