Page 101 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

94 Fundamentals of Ocean Renewable Energy

4.3.2 Temporal Distribution: Probability Density Function
of Wind Speed

Here, we focus more on the temporal variations of wind speed at a site. As
mentioned, the wind speed varies over a range of timescales: decades, years,
seasons, days, hours, and seconds. There is still much uncertainty to understand
and predict these variations. For instance, it is not possible to predict how the
wind speed varies between years. However, seasonal variations are easier to
understand and predict. Very small fluctuations in wind speed at the scale of
seconds give rise to turbulence, which is not important for resource assessment;
however, they should be considered in the design of blades. Turbulence causes
cyclic loading on turbine blades and can lead to damage.
Temporal variations in the wind speed at scales of hours, seasons, and years
that are important in wind energy resource assessments can be represented by
a probability density function (PDF). PDFs and their properties are usually
discussed in probability and statistics books, so only a short review is pro-
vided here.
If we consider the wind speed as a continuous random variable, the
probability of wind speed falling within a specified range can be evaluated using
a PDF. The PDF of wind speed at a site can be constructed using historical wind
data, as we will discuss later. By definition, if we denote f(u) as the PDF of
wind speed, the probability of having a wind speed within u 1 and u 2 can be
evaluated as

u 2
Pr [u 1 ≤ u ≤ u 2 ]= f(u)du (4.9)
u 1
Fig. 4.10 shows a sample PDF that represents the variation in wind speed at
a site. As we can see, the probability of wind speed falling between 5 and
10 m/s (the area between 5 m/s and 10 m/s; A1) is relatively high (42%). In
other words, 42% of the time, the wind speed is between 5 and 10 m/s. However,
the probability of having a wind speed greater than 15 m/s (i.e. the area under
the curve for wind speeds greater than 15 m/s; A2) is relatively low (10%).
Further, it is clear that the chance of having wind speeds greater than 0 is 100%.
Therefore, the total area under a PDF is 1.
∞

Pr [0 ≤ u ≤∞]= f(u)du = 1 (4.10)
0
The average (or expected value) of wind speed is the first moment of the
PDF, that is,
∞

E(u) =¯u = uf(u)du (4.11)
0
Another useful function that can be derived from a PDF is the cumulative
distribution function. It simply evaluates the probability of wind speed being
greater than a specified value. Therefore,
∞

F(u) = Pr [x ≥ u]= f(x)dx (4.12)
u