Power/Mean power of wind

The power of the wind: cube of wind speed
Power content of wind         The wind speed is extremely important for the amount of energy a wind turbine can convert to electricity: the energy content of the wind varies with the cube (the third power) of the average wind speed, e.g. if the wind speed is twice as high it contains 2 x 2 x 2 = eight times as much energy.
        Why does the energy in the wind vary with the third power of wind speed? Well, from everyday knowledge it comes that if the speed of a car is doubled, it takes four times as much energy to brake it down to a standstill, essentially this is Newton's second law of motion. In the case of the wind turbine the energy from braking the wind is used and if the wind speed is doubled, we get twice as many slices of wind moving through the rotor every second and each of those slices contains four times as much energy, as was shown from the example of braking a car.
        The graph shows that at a wind speed of 8 m/s we get a power, that is the amount of energy per second, of 314 W/m2 exposed to the wind (the wind is coming from a direction perpendicular to the swept rotor area). At 16 m/s we get eight times as much power, i.e. 2509 W/m2.

The average bottle fallacy         
        What is the average energy content of the wind at a wind turbine site? Most people who are new to wind energy think they could easily live without the Weibull distribution. After all, if the average wind speed is known, then the average power of the wind is known also, or no? So, can't we just use the power (or energy) at the mean wind speed to figure out how much power (or energy) will hit the wind turbine? In other words, couldn't we just say, that with an average wind speed of 7 m/s we get an average power input of 210 W/m2 of rotor area?         
        The answer is no! We would underestimate wind resources by almost 100%. If we did that, we would be victims of what we could call the Average Bottle Fallacy: Look at the smallest and largest bottle in the picture. Both have exactly the same shape. One is 0.24 m tall, the other is 0.76 m tall. How tall is the average bottle? If you answer 0.5 m tall, you are a victim of the Average Bottle Fallacy. Bottles are interesting because of their volume, of course. But the volume varies with the third power of their size. So, even though the largest bottle is only 3.17 times larger than the small bottle, its volume is actually 3.173=32 times larger than the small bottle. The average volume is therefore 16.5 times that of the small bottle. This means that a bottle with an average volume would have to be 2.55 times the height of the small bottle, i.e. 0.61 m tall.

The mean/average power of wind (balancing the power distribution)
Diurnal wind variations         The reason why we care about wind speeds is their energy content, just like with the bottles before: We cared about their content in terms of volume. Now, the volume of a bottle varies with the cube of the size, just like wind power varies with the cube of the wind speed.
        Let us take the Weibull distribution of wind speeds and for each speed we place a bottle on a shelf each time we have a 1% probability of getting that wind speed. The size of each bottle corresponds to the wind speed, so the weight of each bottle corresponds to the amount of energy in the wind. To the right, at 17 m/s we have some really heavy bottles, which weigh almost 5000 times as much as the bottles at 1 m/s (at 1 m/s the wind has a power of 0.61 W/m2. At 17 m/s its power is 3009 W/m2). Finding the wind speed at which we get the mean of the power distribution is equivalent to balancing the bookshelves. In this case, as you can see, although high winds are rare, they weigh in with a lot of energy. So, in this case with an average wind speed of 7 m/s, the power weighted average of wind speeds is 8.7 m/s. At that wind speed the power of the wind is 402 W/m2, which is almost twice as much as we figured out in our naive calculation on the previous paragraph.

Video frames
- Wind speed, energy and power (A): general equations
- Wind speed, energy and power (B): more equations, notes
- The mean power of the wind: general explanation