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:

1. According to the wind shear formula from NREL [ydt7uk], the speed
   varies as a power of the height:
   
   
   
   where v₁₀ 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.
   
2. The wind shear formula from the Danish Wind Industry Association
   [yaoonz] is
   
   
   
   where z₀ 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 z₀ = 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.

Figure B.7. Top: Two models of wind speed and wind power as a function of height. DWIA = Danish Wind Industry Association; NREL = National Renewable Energy Laboratory. For each model the speed at 10 m has been fixed to 6 m/s. For the Danish Wind model, the roughness length is set to z₀ = 0.1 m. Bottom: The power density (the power per unit of upright area) according to each of these models.