Wind power varies as cube of velocity

Power varies as cube of wind velocity

Consider air of density $\rho$ moving at speed $v$ through the swept area $A$ of the rotor.

Mass of air passing per second:

$$\dot{m} = \rho A v$$

Kinetic energy carried per second (power):

$$P = \frac{1}{2}\dot{m}v^2 = \frac{1}{2}(\rho A v)v^2 = \frac{1}{2}\rho A v^3$$

The turbine extracts a fraction $C_p$ (power coefficient):

$$P_{out} = \frac{1}{2}\rho A v^3 C_p \;\Rightarrow\; P_{out} \propto v^3$$

So doubling wind speed gives $2^3 = 8$ times the power, and 10% higher speed gives $1.1^3 = 1.33$ times the power.

Other factors that guide the output of a wind project

1. Swept area / blade length – $A = \pi r^2$, so power $\propto r^2$.
2. Air density – lower at high altitude and high temperature; at about 3000 m density is roughly 25–30% less than at sea level.
3. Power coefficient – blade design, tip-speed ratio, pitch control; limited by Betz limit (0.593).
4. Hub height – wind speed increases with height above ground (wind shear).
5. Wind speed distribution – mean speed, variability and duration (Weibull distribution), turbulence.
6. Cut-in, rated and cut-out speeds of the turbine.
7. Mechanical and electrical efficiency – gearbox, generator, converter, transformer losses.
8. Availability and wake losses – downtime for maintenance; turbines in a wind farm shading each other.
9. Grid availability and curtailment.