## Weibull Probability Distribution

The variation in wind speed are best described by the Weibull probability distribution function 'h' with two parameters, the shape parameter 'k', and the scale parameter 'c'. The probability of wind speed being v during any time interval is given by the following:

In the probability distribution chart, h is plotted against v over a chosen time period, where:

h = fraction of time wind speed is between v and (v + Av) (4 20)

By definition of the probability function, probability that the wind speed will be between zero and infinity during that period is unity, i.e.:

If we choose the time period of one year, then express the probability function in terms of the number of hours in the year, such that:

h = number of hours the wind is between v and (v + Av) (4 22)

The unit of 'h' is hours per year per meter/second, and the integral (4-21) becomes 8,760 (the total number of hours in the year) instead of unity.

Figure 4-7 is the plot of h versus v for three different values of k. The curve on the left with k = 1 has a heavy bias to the left, where most days are windless (v=0). The curve on the right with k = 3 looks more like a normal bell shape distribution, where some days have high wind and equal number

FIGURE 4-7

Weibull probability distribution function with scale parameter c=10 and shape parameters k = 1, 2 and 3.

FIGURE 4-7

Weibull probability distribution function with scale parameter c=10 and shape parameters k = 1, 2 and 3.

of days have low wind. The curve in the middle with k = 2 is a typical wind distribution found at most sites. In this distribution, more days have lower than the mean speed, while few days have high wind. The value of k determines the shape of the curve, hence is called the 'shape parameter'.

The Weibull distribution with k = 1 is called the exponential distribution which is generally used in the reliability studies. For k>3, it approaches the normal distribution, often called the Gaussian or the bell-shape distribution.

Figure 4-8 shows the distribution curves corresponding to k = 2 with different values of c ranging from 8 to 16 mph (1 mph = 0.446 m/s). For greater values of c, the curves shift right to the higher wind speeds. That is, the higher the c, the more number of days have high winds. Since this shifts the distribution of hours at a higher speed scale, the c is called the scale parameter.

At most sites the wind speed has the Weibull distribution with k = 2, which is specifically known as the Rayleigh distribution. The actual measurement data taken at most sites compare well with the Rayleigh distribution, as seen in Figure 4-9. The Rayleigh distribution is then a simple and accurate enough representation of the wind speed with just one parameter, the scale parameter "c".

Summarizing the characteristics of the Weibull probability distribution function:

Time in Percent

Wind Speed (mph)

FIGURE 4-8

Weibull probability distribution with shape parameter k = 2 and the scale parameters ranging from 8 to 16 miles per hour (mph).

Wind Speed (mph)

FIGURE 4-8

Weibull probability distribution with shape parameter k = 2 and the scale parameters ranging from 8 to 16 miles per hour (mph).

lindspeed (mph)

FIGURE 4-9

Rayleigh distribution of hours/year compared with measured wind-speed distribution at St. Ann's Head, England.

k = 1 makes it the exponential distribution, h = X •e XV where X = 1c k = 2 makes it the Rayleigh distribution, h = 2X2 • V • e~(XV' , and k > 3 makes it approach a normal bell-shape distribution.

FIGURE 4-10

Weibull distributions of hours/year with three different shape parameters k = 1.5, 2, and 3.

FIGURE 4-10

Weibull distributions of hours/year with three different shape parameters k = 1.5, 2, and 3.

Since most wind sites would have the scale parameter ranging from 10 to 20 miles per hour (about 5 to 10 m/s), and the shape parameter ranging from 1.5 to 2.5 (rarely 3.0), our discussion in the following sections will center around those ranges of c and k.

Figure 4-10 displays the number of hours on the vertical axis versus the wind speed on the horizontal axis with distributions of different scale parameters c = 10, 15, and 20 mph and shape parameters k = 1.5, 2, and 3. The values of h in all three sets of curves are the number of hours in a year in the speed interval v + Av divided by Av. Figure 4-11 depicts the same plots in the three-dimensional h-v-k space. It shows the effect of k in shifting the shape from the bell shape in the front right hand side (k = 3) to the Rayleigh and flatter shapes as the value of k decreases from 3.0 to 1.5. It is also observed from these plots that as c increases, the distribution shifts to the higher speed values.

FIGURE 4-11

Three-dimensional h-v-k plots with c ranging from 10 to 20 mph and k ranging from 1.5 to 3.0.

FIGURE 4-11

Three-dimensional h-v-k plots with c ranging from 10 to 20 mph and k ranging from 1.5 to 3.0.

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Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable.

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