WIND STRUCTURE & STATISTICS
E. SREEVALSAN
WIND STRUCTURE & STATISTICS
E. SREEVALSAN
Centre for Wind Energy Technology, Chennai, India
1. Winds in Atmospheric Boundary Layer
The atmospheric boundary layer {ABL) is the lowest layer of the troposphere where wind
is influenced by friction. This layer is particularly characterized by well developed mixing
(turbulence) generated by frictional drag as the atmosphere moves across the rough and rigid
surface of the earth and by the bubbling up of the air parcels from the heated surface. It is in
this layer where windmills are to be placed to extract the wind energy. The characteristics of the
turbulent atmospheric boundary layer are prime importance in wind turbine design and
operations. Schema of the ABL is given in the fig 1.
Fig. 1.Schema of the ABL
1.1 Variation of Wind speed with Height
Wind shear is the variation of wind speed with height. The rate of increase with height
strongly depends upon the roughness of the terrain and the changes in this roughness. The
variation also depends on the atmospheric stability conditions. Even within the course of 24
hours, the wind profile will change between day and night, dawn and dusk This can be described
by the so called logarithmic wind profile with stability correction. This expression, which is well
supported by theoretical considerations, is written

u  z
( 1)
u z  = x  I L   
k  z0

Where u* is the friction velocity, k the von Karman constant, z0 the roughness length, and  a
stability dependent function, positive for unstable condition, zero for neutral and negative for
stable conditions. Conditions and negative for stable conditions. The wind speed gradient is
diminished in unstable conditions (heating of the surface, increased vertical mixing) and
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Wind Resource Assessment Unit
Centre for Wind Energy Technology, Chennai
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WIND STRUCTURE & STATISTICS
E. SREEVALSAN
increased during stable conditions (cooling of the surface, suppressed vertical mixing), See
figure 2.
Fig 2. The wind speed gradient in unstable, stable and neutral conditions.
Another option is power law approximation. The expression is as follows.

z 
u z1
=  1 
uz2
 z2 
(2 )
Where uz1 and uz2 are the wind speed at heights z1 and z2 respectively and p is the power law
exponent, with a typical value of 0.14 for most of the homogeneous site. A serious problem with
this approach is that p varies with height, surface roughness and stability, which means this
equation, is of quite limited applications.
1.2 Obstacles on the ground
A changing orography of the earth, varying vegetation in the landscape, buildings, natural
or man made obstacles have a local effect on the wind speed profile. This reduces the wind
speed distribution and generates turbulence. Usually, the effects of surface obstacles on the
leeward side have to be gathered empirically, in spite of using basic theory. Windward, it can be
assumed that a cluster of the trees of a height it causes disturbances of the air stream five times
the height, H in downward direction, disturbances of the air stream can reach fifteen times the
height H( See fig 2). Figure 3 estimates changes in speed and turbulence in the wake of sloped
roof building.
Fig. 2. Disturbance of the air stream by a cluster of trees
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Wind Resource Assessment Unit
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WIND STRUCTURE & STATISTICS
E. SREEVALSAN
Fig. 3 Changes in speed and turbulence in the wake of sloped roof building.
1.3 Turbulence
The turbulent variations of the wind speed are typically expressed in terms of the
standard deviation,
, of velocity fluctuations measured over 10 to 60 minutes, normalized by
the friction velocity or by the wind speed. The variation in these ratios is caused by a large
natural variability, but also to some extent because they are sensitive to the averaging time and
the frequency response of the sensor used. In horizontally homogeneous terrain, the turbulence
intensity
, is a function of height and roughness length in addition to stability,
whereas _
not too far from the ground, may be considered a function of stability only. A
typical value for neutral conditions is
2:5 for homogeneous at terrain, often larger for
inhomogeneous terrain, but with very large local variations.
The turbulence intensity is a widely used measure, and for neutral conditions with a logarithmic
wind profile over at terrain, we find
Typical values of
for neutral conditions
in different terrains are: Flat open grassland: 13% Sea: 8% Complex terrain: 20% or more
2.0.Wind speed statistics.
It varies with the time of day, season, height above ground, and type of terrain. An area's
surface roughness and obstacles are also important determinants in wind resource. High surface
roughness and larger obstacles in the path of the wind result in slowing the wind by creating
turbulence.
These fluctuations are immediately apparent (instantaneous wind speed values) from an
anemometer recording of wind speed. But precise information about wind speed is very
important in assessing wind resource of a site. The way out is making it desirable to describe the
wind by statistical methods.
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WIND STRUCTURE & STATISTICS
2.1.
E. SREEVALSAN
Annual average wind speed.
A first rough judgment about the wind energy potential can be made on the basis of the
annual average wind speed, i.e. the average of all measured data, including calm over a period of
one year. The annual average should be calculated for as many years as possible.
Suppose we have a set of measured wind speed values ui, , then the mean of the set is defined
as
= 1
u
n
n
i 1
(3)
uI
The sample size or the number of measured value is n.
2.2.Standard deviation
In general, the standard deviation is a summary measure of the difference of each
observation from the mean. Here, the standard deviations of wind speed and direction are
pointers of the turbulence level and atmospheric stability. Standard deviation is also useful in
detecting suspect or erroneous data when validating average values. The standard deviation is
defined as
=
1
n
n  1
i 1
2
u  u 
(4)
i
2.3.Time distribution
Generally an Electronic Wind data logger records a large number of hourly wind speed,
standard deviation and direction values. This is known as time series of wind data and it is a
sequence of observations that are ordered in time. This may be a continuous record of hourly or
10- minute average of wind speed and direction for one year. The simplest transformation is to
rearrange the time series of data into a monthly table of 24 columns for each hour of the day and
30 or 31 rows for each day of month. Averaging of these data in different way will help us to
know the diurnal, monthly annual wind pattern of the site. A sample of time distribution is given
in the table 1.
------------------------------ Data Information ---------------------------Spd1, SD1, Dir1, Spd2, SD2, Dir2, Spd3, SD3, Anlg, Gust1, Time, Date
8.38, 1.07, 343, 12.39, 1.01,
89, 14.68, 1.28,
0,
12, 0000, 020102
8.01, 1.01, 343, 11.53, 1.01,
87, 13.29, 1.01,
0,
11, 0100, 020102
7.74, 0.91, 343, 11.42, 1.01,
90, 13.35, 1.17,
0,
10, 0200, 020102
6.35, 1.07, 343, 9.56, 1.28,
89, 11.53, 1.39,
0,
9, 0300, 020102
6.78, 1.71, 343, 10.89, 2.30, 100, 12.87, 2.56,
0,
13, 0400, 020102
7.58, 1.28, 343, 12.28, 1.17, 111, 14.57, 1.07,
0,
12, 0500, 020102
5.98, 1.28, 343, 10.57, 1.39, 115, 14.15, 1.12,
0,
11, 0600, 020102
7.26, 1.44, 345, 10.84, 1.49, 127, 13.77, 1.28,
0,
12, 0700, 020102
9.02, 2.03, 345, 11.26, 2.08, 129, 12.97, 1.76,
0,
15, 0800, 020102
10.46, 2.19, 345, 12.33, 2.08, 131, 13.24, 1.87,
0,
17, 0900, 020102
10.14, 1.98, 345, 11.48, 1.82, 124, 12.12, 1.76,
0,
16, 1000, 020102
Table 1. Time distribution data
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WIND STRUCTURE & STATISTICS
E. SREEVALSAN
2.4. Frequency distribution.
Apart from the time distribution of wind speed and direction, it is important to know the
frequency distribution of wind speed over a wind regime i.e. the number of occurrences of each
wind speed range are counted, or binned, and then expressed as a fraction of the total number of
U
0
30
60
90
120
150
180
210
240
270
300
330
Total
1.00
57
65
15
8
23
47
99
117
61
14
10
66
22
2.00
80
49
17
9
21
48
86
75
67
8
12
55
20
3.00
106
103
24
15
33
63
139
126
92
14
21
111
32
4.00
169
144
45
23
52
97
169
186
148
29
38
126
51
5.00
138
140
56
36
80
130
202
187
172
48
58
167
69
6.00
131
121
70
57
110
95
88
85
143
56
63
124
72
7.00
78
94
74
74
144
115
36
52
105
60
72
108
79
8.00
81
85
72
117
186
147
52
51
91
68
83
98
96
9.00
61
83
77
135
183
125
48
47
73
73
103
75
102
10.00
52
54
84
142
96
84
24
52
30
94
105
35
98
11.00
27
38
121
135
43
26
10
14
8
110
102
14
93
12.00
8
12
113
105
19
9
12
5
3
112
89
8
81
13.00
4
8
101
71
7
5
10
3
4
104
81
5
68
14.00
6
1
66
44
1
4
13
0
0
88
72
3
53
15.00
1
1
27
18
1
1
10
0
1
63
46
3
33
16.00
2
1
22
7
0
0
0
0
1
36
24
2
18
17.00
0
0
10
1
0
0
1
0
0
16
12
0
8
18.00
0
1
5
0
0
0
0
0
0
6
5
0
3
19.00
0
0
1
0
0
0
0
0
0
2
2
0
1
20.00
0
1
0
0
0
0
0
0
0
1
2
0
1
Table 2. An Example of frequency distribution
wind speed occurrences in all bins. This is a statistical description of the wind climate at any site.
For the best statistical representation, the winds should be measured over a period of many years
as possible. A similar frequency distribution can be made for wind direction for a specific period
of time. A joint frequency distribution can also be prepared from the raw data. When the number
of hours in each interval is plotted against the wind speed, the frequency distribution emerges as
a histogram. The shape of the frequency distribution characterizes the wind regime: steady wind
regimes
have
a
nearly
symmetrical
frequency
distribution
(approaching
a Gauss or normal distribution) whereas unsteady wind regimes have an asymmetrical
distribution, which in many cases can be approached by a so called Weibull distribution.
2.5. Probability density function
The probability density function is the continuous counterpart to the histogram. The area
under the density function is unity, which is shown by the equation given below.

 f u du = 1
(5)
0
The cumulative distribution function F(u) is given by
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WIND STRUCTURE & STATISTICS
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
 f x dx
(6)
0
The variable x inside the integral is just a dummy variable representing wind speed for
the purpose of integration. Both of the integration above start a zero because , the wind speed
cannot be negative. When the wind speed is considered as a continuous random variable the
cumulative distribution function has the properties F(0) = 0 and F() = 1. The quantity F(0) will
not necessarily be zero in the discrete case.
There are several density functions that can be used to describe the wind speed frequency
curve. The three most common are
1. Gaussian (Normal) distribution
2. Rayleigh distribution and
3. Weibull distribution
4.
2.5.1.Gaussian (Normal ) distribution.
The wind speed u is distributed as the Gaussian (Normal) distribution if its probability
density function is
 u u 2

= ( 1  2 )exp 
(7)
f u 
2
 2 
Where u is the mean and is the standard deviation.


2.5.2.Rayleigh distribution
The Rayleigh probability density function is given by
f u 
2

u 
=( u 2u )exp   4   
 u  

2
(8)
2.5.3.Weibull distribution
The Weibull probability density function is described as follows.
f u  =
k u
 
cc
k 1
  u k 
exp    
  c  
(k >0, u >0, c >1)
(9)
This is a two-parameter distribution where c and k are the scale parameter and shape
parameter respectively. The influences of the shape parameter on the shape of the function f(u).
For k> 1 the function has a maximum away from the origin, while k <1 it is monotonically
decreasing. For k =1 the distribution is exponential, k =2 gives the Rayleigh distribution and k
= 3.5 gives an approximation to the normal distribution. The wind speed distributions are
generally found to have a k value between 1.5 and 3.0 and the value is often close to 2.0. Weibull
curve gives maximum fit to the histogram compared to other distributions.
The accumulated Weibull distribution F(u) which gives the probability of having wind
speed equal to or less than u is obtained by integrating equation with the result.
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WIND STRUCTURE & STATISTICS
E. SREEVALSAN
2.5.3.1. Estimation of Weibull parameters from given data
There are several methods available for determining the Weibull parameter c and k.
These include,
1. Least squares fit method
2. The Maximum likelihood method
3. Mean Wind Speed and Standard deviation analysis.
Determining c and k by least squares fit is explained below.
2.5.3.1.1. Least square fit method
If F(ui) is a cumulative distribution function defined as the probability that a measured
wind speed will be less than or equal to ui , then,
i
F(vi) =  p(uj)
(10)
j=1
The cumulative distribution function is represented by the Weibull parameters as given
below
F(u)=1- exp [-(u/c)k]
(11)
F(u) contains an exponential term and, in general, exponentials are linearised by taking the
logarithm. Then
ln [-ln(1-F(u))] = k ln u- k ln c
(12)
Eqn (4) is in the form of an equation of a straight line. So,
y = ax + b
(13)
Where x and y are variables, a is the slope, and b is the intercept of the line on the y axis. Also,
y= ln [-ln(1-f(u))]
a=k
x=ln u
b=-k ln c
(14)
It is shown [1] that the proper values for a and b are:
w
p2(ui)(xi –x)(yi –y)
I=1
a
=
(15)
w
p2(ui)(xi –x)2
i=1
1
b =
w
w
 yi -I=1
a
w
w
 xi
( 16)
I-1
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WIND STRUCTURE & STATISTICS
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where x and y are the mean values of xi and yi respectively and w is the total number of pairs
of values available.
Then the Weibull parameters are,
k=a
(17)
and,
c = exp(-b/k)
(18)
It should be emphasized that the actual histograms of wind speeds may be difficult to fit
by any mathematical function, especially if the period of time is short.
Of course, if the actual distribution is available, then the use of these distributions is
preferred. In all those cases, however, where the monthly averages over several years are
available and only one year of detailed measurements, then the twelve monthly k values (in case
of Weibull distribution) can be used to regenerate the distributions of the years before. The
Weibull distribution also shows its usefulness when the wind data of one reference station are
being used to predict the wind regime in the surroundings of that station. The idea is that only
monthly average wind speeds are sufficient to predict the complete frequency distribution of the
year or the month. Figure 4 shows actual distribution as well as Weibull distribution
Fig. 4 Actual and Weibull distribution of wind speed
Wind rose
Figure 5 shows Wind rose diagram
Fig. 5 Wind rose diagram
Wind rose is a diagram that indicates frequency of occurrence of winds shown in each
direction sectors and different wind speed classes for a given location for a given time.
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WIND STRUCTURE & STATISTICS
E. SREEVALSAN
4.0.Power in the Wind
Wind as already mentioned is merely air in motion. The air has mass-though its density is
low and when this mass has velocity the resulting wind has kinetic energy which is proportional
to 0.5[mass x velocity2]
Kinetic energy passing through the area in unit time is the power in the wind and is given
by
P = ½ Au x u2
( 19)
P = ½ Au3
( 20)
Where  = mass per unit volume of air
u = velocity of wind and
A = an area through which the wind passes normally.
The above expression gives the total power available in the wind, for extraction by a wind driven
machine; only a fraction of which can be actually extracted. A. Betz of Gottingen showed in
1927 that the maximum fraction of power in the wind that could be extracted by an ideal aero
motor was 16/27 or 0.593.
The power density is a flow of air through a unit area at right angles to the surface of the
earth is given by
Pd =½u3 Watts/ m2.
(21 )
5.0.Annual Energy Production & Capacity Factor.
To estimate the annual energy production from a given machine at a site, power
curve method can be used since this method gives most realistic results. The wind speed
frequency distribution will be used to estimate the annual energy production of a wind turbine by
multiplying the number of hours in each interval with the power output that the windmill
generates at that wind speed interval. If the frequency distribution of wind speed at the hub
height is not available, the wind speed at the hub height level is to be generated by the power law
equation.
Capacity factor is one element in measuring the productivity of a wind turbine or any
other power production facility. It compares the plant's actual production over a given period of
time with the amount of power the plant would have produced if it had run at full capacity for the
same amount of time . A reasonable capacity factor would be 0.25 to 0.30. A very good capacity
factor would be 0.40.
Example: If a 600 kW turbine produces 1.5 million kWh in a year, its capacity factor
is = 1500000 / ( 365* 24 * 600 ) = 1500000 / 5259600 = 0.285 = 28.5 per cent.
References:
1. Gary L. Johnson, ”Wind Energy Systems”1985, Prentice-Hall, Inc., Englewood Cliffs, New Jersey
07632.
2 E.H. Lysen, “Introduction to Wind Energy” Basic and Advance Introduction to Wind
Energy
With Emphasis on Water Pumping Windmills, CWD-Consultancy Services Wind Energy Developing
Countries, P.O.Box 85-3800 AB Amersfoort – The Netherlands, 1983
3. E.L .Peterson, N.G.Mortenson, Lars Landberg. Jorgen Hojstrup and Helmut P. Frank.
"Wind
Power Meteorology", Riso National Laboratory, Roskilde, Denmark, December 1997.
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