Why Wind Speed Matters So Much

A wind turbine's power grows with the cube of the wind speed. The power rule turns v³ into 3v², so each extra meter per second adds more power the windier it already is. That's why turbines are built on ridges and offshore.

How it works

Differentiation formulas: the power rule

In one sentence

For any real $n$ , $\frac{d}{dx}x^n = nx^{n-1}$ ; derivatives are linear, so $\frac{d}{dx}\left(c\,f(x)\right) = c\,f'(x)$ and $\frac{d}{dx}(f + g) = f' + g'$ .

General formula

{\htmlData{sym=0}{P(v)}} = \tfrac12{\htmlData{sym=1}{\rho}} {\htmlData{sym=2}{A}} {\htmlData{sym=3}{C_p}}\,v^3 \quad\Rightarrow\quad {\htmlData{sym=4}{P'(v)}} = \tfrac32{\htmlData{sym=1}{\rho}} {\htmlData{sym=2}{A}} {\htmlData{sym=3}{C_p}}\,v^2

Grow $v$ a little and $v^3$ , a cube with side $v$ , gains three thin slabs of area $v^2$ each: $3v^2\,dv$ .

What the symbols mean

$P(v)$
Power
Watts the turbine produces at wind speed v.
$\rho$
Air density
About 1.225 kg/m³ at sea level.
$A$
Swept area
πr², the circle the blades sweep.
$C_p$
Power coefficient
Fraction of the wind's power captured, about 0.4 (at most 0.59).
$P'(v)$
Sensitivity
Extra power per extra m/s of wind.

The wind's kinetic energy per second through the rotor is $\frac12\rho A v^3$ ; a turbine captures a fraction $C_p$ of it (at most 59%, the Betz limit).
Power rule: $\frac{d}{dv}v^n = nv^{n-1}$ , and constants come along: $P'(v) = \frac32\rho AC_p v^2$ .
$P'(v)$ is the extra power per extra m/s of wind, and it grows like $v^2$ .
Doubling the wind speed multiplies power by 8 and the sensitivity $P'$ by 4.

Learn the math

Differentiation FormulasCalc I · Lesson 16, Derivatives