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Wind 1. How Lift Based Turbines Extract Energy from Fluid. Bernoulli’s Principle - air pressure on top is lower than air pressure on bottom because it has further to travel, creates lift. Airfoil – could be the wing of an airplane or the blade of a wind turbine.

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how lift based turbines extract energy from fluid
How Lift Based Turbines Extract Energy from Fluid

Bernoulli’s Principle - air pressure on top is lower than air pressure on bottom because it has further to travel, creates lift

Airfoil – could be the wing of an airplane or the blade of a wind turbine

angle of attack lift and drag
Angle of Attack, Lift, and Drag

Increasing angle of attack increases lift, but it also increases drag

When angle of attack is too great, “stall” occurs where turbulence destroys the lift

wind turbines
Wind Turbines

“Windmill”- used to grind grain into flour (or pump water in Holland)

Can have be horizontal axis wind turbines (HAWT) or vertical axis wind turbines (VAWT)

Groups of wind turbines are located in what is called either a “wind farm” or a “wind park”

Important to note: very fast “energy payback” – it takes a few months for a wind turbine to generate (i.e. convert) as much energy as it took to manufacture it!

vertical axis wind turbines
Vertical Axis Wind Turbines

Darrieus rotor - the only vertical axis machine with any commercial success

Wind hitting the vertical blades (airfoils) generates lift to create rotation

  • Advantages
  • No yaw (rotation about vertical axis) control needed to keep facing into wind
  • Heavy machinery located on the ground
  • Disadvantage
  • Blades are closer to ground where windspeeds are lower
horizontal axis wind turbines
Horizontal Axis Wind Turbines

“Downwind” HAWT – a turbine with the blades behind (downwind from) the tower

No yaw control needed- they naturally orient themselves in line with the wind

Shadowing effect – when a blade swings behind the tower, the wind it encounters is briefly reduced and the blade flexes

-Also causes noise

horizontal axis wind turbines1
Horizontal Axis Wind Turbines

“Upwind” HAWT – blades are in front of (upwind of) the tower

Most modern wind turbines are this type

Because blades are “upwind” of the tower

  • Require active yaw control to keep facing into wind
  • Operate more smoothly and deliver more power
power in the wind
Power in the Wind

Consider the kinetic energy of a “packet” of air with mass m moving at velocity v

Divide by time and get power

The mass flow rate is

power in the wind1
Power in the Wind

Combining we get

P (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)= windspeed normal to A (1 m/s = 2.237 mph)

power in the wind2
Power in the Wind

Power increases as (wind speed)3

Doubling the wind speed increases the power by eight

1h x 20mph wind is same energy as 8h x 10 mph wind…

-i.e., most power from a turbine is produced at high wind speed for a short time…

us wind resources
US Wind Resources

power in the wind cont
Power in the Wind (cont.)

Power in the wind is also proportional to A

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

Cost is roughly proportional to blade diameter

How do you think cost of wind power scales with turbine diameter?

power curve for turbine
Power Curve for Turbine


Generator maxed out

Cut out speed

Park turbine to avoid damage

Cut in speed

Not enough energy to justify O&M costs

maximum rotor efficiency
Maximum Rotor Efficiency

At the extremes:

  • Downwind velocity is zero – turbine extracted all of the energy (for zero time…)
  • Downwind velocity is the same as the upwind velocity – turbine extracted no energy…

Albert Betz 1919

Q: What is the ideal extraction of KE from wind so that the turbine extracts the maximum power

maximum rotor efficiency1
Maximum Rotor Efficiency

Consider wind passing though turbine: as energy extracted, air slows down

ṁ = mass flow rate of air within stream tube

v = upwind undisturbed windspeed

vd = downwind windspeed

mass flow rate
Mass Flow Rate

At the rotor with area A and, mass flow rate is

If velocity through the rotor vb is the average of upwind velocity v and downwind velocity vd

power extracted by the blades
Power Extracted by the Blades

Then power relationship at the rotor could be

Define new parameter l such that

We can rewrite the power relationship as

power extracted by the blades1
Power Extracted by the Blades

Power in the wind

Rotor efficiency (CP)

maximum rotor efficiency2
Maximum Rotor Efficiency

So what is the windspeed ratio λ which maximizes the rotor efficiency, CP ?

  • Plug into CP to find the maximum rotor efficiency:

Maximum efficiency of 59.3% when air is slowed to 1/3 of its upstream speed!

“Betz limit”

number of rotating blades
Number of Rotating Blades

Windmills have multiple blades

  • need to provide high starting torque to overcome weight of the pumping rod
  • must be able to operate at low windspeeds to provide nearly continuous water pumping
  • a larger area of the rotor faces the wind

Turbines with many blades must operate at lower rotational speeds – as speed increases, turbulence caused by one blade impacts other blades

Most modern wind turbines have two or three blades

tip speed ratio tsr
Tip-Speed Ratio (TSR)

Efficiency is a function of how fast the rotor turns

Define “Tip-Speed Ratio” (TSR) as ratio of speed of tip of blade to windspeed

D = rotor diameter (m)

v = upwind undisturbed windspeed (m/s)

rpm = rotor speed, (revolutions/min)


Air moved this far

Airfoil interacted with

this much air, call it Xs

optimal tip speed ratio
Optimal Tip Speed Ratio

If ts<<tw then wind turbine is interacting with disturbed air → low efficiency

If ts>>tw then turbine does not get to all useful air… → low efficiency

Optimal is if ts≈tw

optimal tip speed ratio1
Optimal Tip Speed Ratio

Then for a three bladed turbine,

And for a two bladed turbine

tip speed ratio tsr1
Tip-Speed Ratio (TSR)

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

impact of terrain on windspeed
Impact of Terrain on Windspeed

We saw power depends on cube of windspeed: small change of wind speed can have large impact

System design must consider effect of terrain friction on wind speed

  • Important in first few hundred meters above ground level
  • Smooth surfaces (like water) are better
  • Windspeeds are greater at higher elevations – tall towers are better
  • Forests and buildings slow the wind down a lot

Can we quantify impact of terrain and height on wind speed?

wind speed losses as function of terrain
Wind Speed Losses as Function of Terrain

v = windspeed at height H

v0 = windspeed at height H0 (H0 is usually 10 m)

α = friction coefficient

Open terrain, α ≈ 1/7 (0.147)

City, α= 0.4;

Calm water, α= 0.1

Note this is just an approximation, others exist (ex. von Karman’s log velocity profile)

impact of terrain on wind power
Impact of Terrain on Wind Power

Remember wind power goes as third power of wind speed.

impact of terrain on wind power1
Impact of Terrain on Wind Power

In a town (a≈0.3), windspeed at 100 m is twice that at 10 m

Areas with smoother surfaces have less variation with height

rotor stress
Rotor Stress

Let’s calculate ratio of power at highest point to lowest point on wind turbine with hub at 50m, 30m diameter rotor, α = 0.2

65 m

50 m

35 m

  • Power in the wind at the top of the blades is 45% higher!
  • Can cause significant stress (failure)

Picture may not be to scale

wind farms
Wind Farms

It makes sense to install a large number of wind turbines in a wind farm or a wind park


Able to get the most use out of a good wind site

Reduced development costs

Simplified connections to the transmission system

Centralized access for operations and maintenance

How many turbines should be installed at a site?

What is a sufficient distance between wind turbines so that windspeed has recovered enough before it reaches the next turbine?

wind farms1
Wind Farms

For closely spaced towers, efficiency of array becomes worse as more wind turbines are added

wind farms2
Wind Farms

Previous figure considered square arrays

(but square arrays don’t make much sense)

Rectangular arrays with only a few long rows are better

Recommended spacing is 3-5 rotor diameters between towers in a row and 5-9 diameters between rows

Offsetting or staggering the rows is common

Sites commonly have a prevailing wind direction

average power in the wind
Average Power in the Wind

How much energy can we expect from a wind turbine?

Remember, power goes as cube of wind speed

Therefore we need to know the average of the cube of wind speed…

I.e., we can’t use average windspeed to find the average power in the wind

windspeed probability density function pdf
Windspeed probability density function (pdf)

If we had a function f(v) that gave windspeed we could calculate average power in wind…

People have examined statistics of windspeed over various locations

A reasonable approximations is the Weibull distribution

# of hours/year that the wind is between two windspeeds:

weibull p d f
Weibull p.d.f.

k=2 looks reasonable for wind

wind probability density functions
Wind Probability Density Functions

Windspeed probability density function (p.d.f)

Values between 0 and 1

Area under the curve is equal to 1

weibull p d f1
Weibull p.d.f.

Weibull with k=2 has shape similar to windspeed distribution

Often used as first guess when little is known about a particular site

Fairly realistic for a wind turbine site: winds are mostly pretty strong, but there are some periods of low wind and high wind

k = shape parameter

c = scale parameter

rayleigh p d f weibull with k 2
Rayleigh p.d.f. (Weibull with k=2)

Higher c implies higher average windspeeds

real data vs rayleigh statistics
Real Data vs. Rayleigh Statistics

(It is important to gather as much real wind data as possible!)

average windspeed using p d f
Average Windspeed using p.d.f.

Now that we have a function that approximates wind speed…

And for average v3

average windspeed from rayleigh p d f
Average windspeed from Rayleigh p.d.f.

For a Rayleigh p.d.f., there is a direct relationship between average wind speed v and scale parameter c (not surprising really)

You can, of course, use this to extract a c for your site

rayleigh statistics average power in wind
Rayleigh Statistics – Average Power in Wind

Remember, to find average power in the wind, we needed (v3)avg

If we are still assuming wind speed distribution has a Rayleigh distribution

Then we can put (v3)avg in terms of vavg!

rayleigh statistics average power in wind1
Rayleigh Statistics – Average Power in Wind

Using the expression for (v3)avg in terms of vavg (from Rayleigh distribution assumption), average power in wind is

estimates of wind turbine energy

Not all of the power in the wind is retained - the rotor spills high-speed winds and low-speed winds are too slow to overcome losses (see power curve)

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

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

Estimates of Wind Turbine Energy



Gearbox & Generator

Power in the Wind

Power Extracted by Turbine

Electric Power

time variation of wind
Time Variation of Wind

Need to consider when wind blows with respect to the electric load

  • In the Midwest the wind tends to blow the strongest when the electric load is the lowest…

Wind patterns vary with geography

  • Wind can change drastically within hours…
  • Coastal and mountain regions have steadier winds
wind power variability and integration
Wind Power Variability and Integration

Currently wind is a small fraction of generation

  • Impact of grid operations is small
  • As wind power grows it will have larger impact
  • Impacts expected to range from transient stability (seconds) to steady-state (power flow)

Because wind turbine output varies as cube of wind speed, small changes in wind speed can have large impact

  • Current perception is that at the 10%-15% penetration level, wind may cause system instability
  • BPA balancing region is ca. 10GW, has signed LGIA for 4GW wind
wind power variability and integration1
Wind Power Variability and Integration

The key constraint:

Total power system generation must match the total load plus losses

Sudden generation shortfalls dealt with by maintaining sufficient “spinning reserve” to account for the loss of the largest single generator in region

Spinning reserve: generation that is on-line but not fully used and can be brought into production in very short time period

Capital Cost = $200/kW, $350/kWh Cycle Cost = $0.46/kW·h·cycle

Operating Cost = $5/kW·yr Replacement Cost = 0.20∗CapitalCost∗Cycle/2000

environmental aspects of wind energy
Environmental Aspects of Wind Energy

US National Academies 2007 report:

Wind systems emit no air pollution and no carbon dioxide; have essentially no water requirements

Wind serves to displace energy production from mainly fossil fuel burning: net decrease in emissions

Other impacts of wind energy are on animals (primarily birds and bats) and humans

Large bird (raptor) mortality is about 0.04 bird/MW/year

environmental aspects of wind birds and bats
Environmental Aspects of Wind: Birds and Bats

Wind turbines kill birds and bats!

But so do lots of other things

Windows kill between 100 and 900 million per year

Estimated Causes of Bird Fatalities, per 10,000

Source: Erickson,, 2002. Summary of Anthropogenic Causes of Bird Mortality

environmental issues human aesthetics
Environmental Issues: Human Aesthetics

Aesthetics is the primary human concern about wind energy projects (beauty is in the eye of the beholder);

Night lighting (aircraft collision warning) can also be an issue

environmental issues offshore wind and aesthetics
Environmental Issues: Offshore Wind and Aesthetics

Remember, terrain effect is smallest over water…

Capacity factors are much better off-shore

Offshore wind currently needs to be in shallow water; maximum distance from shore depends on the seabed

Image Source: National Renewable Energy Laboratory

environmental human well being
Environmental: Human Well-Being

Some people living near turbines may be affected by noise and shadow flicker

Noise comes from gearbox/generator

and aerodynamic interaction of the

blades with the wind

Noise impact is moderate:

50-60 dB up close (40m)

35-45 dB at 300m

Shadow flicker appears to be issue

in high latitude regions

(lower sun casts long shadows)

environmental human well being1
Environmental: Human Well-Being

Variables related to annoyance by wind turbine noise

  • Stress related to turbine noise
  • Daily hassles
  • Visual intrusion of wind turbines in the landscape
  • Age of turbine site
  • The longer system in operation, the less the annoyance

Anthony L. Rogers

Renewable Energy Research Laboratory, U. Mass. Amherst