14. Calculating the mean power.

Due to the non-linear variation of power with steady wind speed, the mean power obtained over time in a variable wind with a mean velocity U_m is not the same as the power obtained in a steady wind of the same speed.

The sketch below shows a power output curve W(u) for a Vestas 90 metre 2 megawatt turbine in a steady wind of speed u. Also shown is the probability density distribution p(u) for a particular mean speed U_m of 6 metres/second.

The final mean power at a mean wind speed U_m is the steady power W(u) multiplied by the probability density distribution p(u) and summed (i.e. integrated) over all the range of wind speeds. Thus, the mean power P_m(U_m) at a mean speed U_m is given by

In the WindPower programme, this integral is evaluated over a range of mean wind speeds from 5 metres/second through to 10 metres/second in 0.2 metre/second steps. This encompasses the range of mean wind speeds likely to be encountered in the vast majority of cases. The programme also allows the standard deviation of the variable part of the wind speed to be altered although the default value of a standard deviation of 62% of the mean wind speed is likely to be representative of most wind turbine sites.

It should be noted that the above equation does not take into account the dynamics of the turbine in responding to rapid changes in wind speed. For very large turbines, this is probably not too significant but for small turbines with various mechanical devices used to avoid overspeeding and damage at high wind speeds, the neglect of the rotor dynamics might make estimates of mean power at higher wind speeds a process of uncertain accuracy.

The sketch below shows a comparison between the mean power output in a variable wind compared with the steady wind power curve. The influence of the unsteady component of the wind on the mean power output is rather typical of most turbines in that at mean speeds below around 8 metres/second, the mean power output is greater than the steady power output values. This is due to the weighting from the upper end of the speed variations on the steep part of the power curve. Above this speed, the situation reverses because the high speed components of the wind speed are now reaching the flat part of the power curve and may even reach as far as the cut-out part of the power curve.

It should be noted that there is never a practical circumstance where the mean power output reaches anything like the rated power output. It is therefore a very misleading practice when the rated output of a wind turbine is quoted as if this was the available power from an installation. It has caused great confusion in discussions about the power contributions that wind turbines can make.

The ratio of the mean power produced at a particular mean speed to the so-called rated power output is called the capacity factor. This is itself a function of mean wind speed and so has no more significance than the mean power outputs themselves. As can be seen from the results above, it ranges from about 0.2 at a mean wind speed of 5 metres/second to about 0.5 at 10 metres/second.