Wind Energy Handbook. Michael Barton Graham

3.63 for a turbine designed with an optimum tip speed ratio of 7. As an example, assume that this turbine is stall regulated and operates at a fixed rotational speed at a site where the average wind speed is 6 m/s and the Weibull shape factor k = 1.8, then, from Eq. (2.2), the scale factor c = 6.75 m/s.

Figure 3.64 shows the K_P ‐1/λ curve for the turbine: from inspection of that curve the tip speed ratio at which stall (maximum power) occurs is 3.7, and the corresponding C_P is 0.22.

      The required maximum electrical power of the machine is 500 kW, the transmission loss is 10 kW, the mean generator efficiency is 90%, and the availability of the turbine (amount of time for which it is available to operate when maintenance and repair time is taken into account) is 98%.

Figure 3.63 C_P ‐λ curve for a design tip speed ratio of 7 at 7 m/s.

Figure 3.64 K_P ‐1/λ curve for a fixed‐speed, stall‐regulated turbine.

      The maximum rotor shaft power (aerodynamic power) is then

      (3.97)

      The wind speed at which maximum power is developed (where dC_P/dλ = 3C_P/λ for fixed speed) is 13 m/s, therefore the rotor swept area must be, assuming an air density of 1.225 kg/m³,

      The rotor radius is therefore 24.6 m.

The tip speed of the rotor will be 3.7 × 13 m/s = 48.1 m/s, and so the rotational speed will be

      The power vs wind speed curve for the turbine can then be obtained from Figure 3.64.

      (3.98)

since wind speed = 48.1 m/s / λ, and these are shown in Figure 3.65.

      To determine the energy capture of the turbine over a time period T, the product of the power characteristic P(u) with the probability f(u) is integrated with respect to time over T. This can be converted to an integral with respect to wind speed u over the wind speed range, since f(u) is the proportion of time T spent at wind speed u, and therefore:

      (3.99)

      with

      (3.100)

P(u)f(u) can be plotted against u as in Figure 3.66 and then integrated over the operational wind speed range of the turbine to give the total energy capture.

      The operational speed range will be between the cut‐in speed and the cut‐out speed. The cut‐in speed is determined by the transmission losses: at what wind speed does the turbine begin to generate power? The cut‐in speed is usually chosen to be somewhat higher than the zero power speed, in the present case, say 4 m/s.

Figure 3.65 Power vs wind speed.

The cut‐out speed is chosen to protect the turbine from high loads, usually about 25 m/s.

      The total energy captured (E) by the turbine in a time period T is

      (3.101)

      which is the area under the curve of Figure 3.66 times the time T. Unfortunately, the integral does not have a closed mathematical form in general, and so a numerical integration is required, such as the trapezoidal rule or, for better accuracy, Simpson's rule.

For a time period of one year, the energy capture can be calculated numerically as indicated in Figure 3.67 to be

      (3.102)

Figure 3.66 Energy capture curve.

Figure 3.67 Energy capture curve for numerical integration.

Figure 3.68 Power vs wind speed for variable‐speed turbine.

      The