Page 52 - Fluid Power Engineering

30 Chapter Three

Computation of A and k given mean and variance requires
approximations. 2

σ

−1.086
k = (3-5)
¯ v
¯ v
A = 1 (3-6)
1 +
k
Equation (3-5) is a good approximation when 1 ≤ k ≤ 10.

Power Density
In order to understand the impact on power generation of statistical
distribution of wind speed, consider the impact on power density.
Power density is defined as:
Power 1 3 W
PD = = ρv , units are 2 (3-7)
Area 2 m
If the statistical distribution of wind is ignored and it is assumed
that there is no variation in wind speed, then the power density is
incorrectly computed (see column 2 in Table 3-3).
1 3
Incorrect Power Density = ρ(¯v) (3-8)
2
where ¯v is the average wind speed.
However, if the energy density is computed correctly while taking
intoaccountprobabilitydensityofwindspeed,thenthepowerdensity
numbers are very different. (See column 3 in Table 3-2.)

∞
1 3
Correct Power Density = ρ v pd(v)dv (3-9)
0 2
where pd(v) is the Weibull probability density function in Eq. (3-1).
As Table 3-2 illustrates, the power density of rotor is underesti-
mated if computed based on average wind speed in Eq. (3-8).
Comparison of the probability density of wind speed and power
density is illuminating. As an illustration, a wind speed profile with
A = 8 m/s and k = 2 is chosen, (see Fig. 3-5). The mean wind speed
is 7.09 m/s and mode is 5.657 m/s. The power contained in this wind
profile peaks at approximately 11 m/s. Reason is power delivered at
3
a particular value of v is ρv pd(v) dv/2; although pd(v) is decreasing
3
above 5.657 m/s, ρν /2 keeps rising; the product of the two quanti-
ties, which is power delivered at a particular wind speed, peaks at
11 m/s.
In most cases, during prospecting for wind projects, the only data
available is mean wind speed. This is because actual measurements