4 m/s as our estimated windspeed, we must scale our estimate down, mul-

tiplying it by (4/6)^{3} ≅ 0.3. (Remember, wind power scales as wind-speed

cubed.)

On the other hand, to estimate the typical power, we shouldn’t take the

mean wind speed and cube it; rather, we should find the mean cube of the

windspeed. The average of the cube is bigger than the cube of the average.

But if we start getting into these details, things get even more complicated,

because real wind turbines don’t actually deliver a power proportional to

wind-speed cubed. Rather, they typically have just a range of wind-speeds

within which they deliver the ideal power; at higher or lower speeds real

wind turbines deliver less than the ideal power.

### Variation of wind speed with height

Taller windmills see higher wind speeds. The way that wind speed increases

with height is complicated and depends on the roughness of the

surrounding terrain and on the time of day. As a ballpark figure, doubling

the height typically increases wind-speed by 10% and thus increases the

power of the wind by 30%.

Some standard formulae for speed *v* as a function of height *z* are:

- According to the wind shear formula from NREL [ydt7uk], the speed

varies as a power of the height:

where *v*_{10} is the speed at 10 m, and a typical value of the exponent α

is 0.143 or 1/7. The one-seventh law (*v*(*z*) is proportional to *z*^{1/7}) is

used by Elliott et al. (1991), for example.

- The wind shear formula from the Danish Wind Industry Association

[yaoonz] is

where *z*_{0} is a parameter called the roughness length, and *v*_{ref} is the

speed at a reference height *z*_{ref} such as 10 m. The roughness length

for typical countryside (agricultural land with some houses and shel-

tering hedgerows with some 500-m intervals – “roughness class 2”)

is *z*_{0} = 0.1 m.

In practice, these two wind shear formulae give similar numerical answers.

That’s not to say that they are accurate at all times however. Van den Berg

(2004) suggests that different wind profiles often hold at night.