ECE 7800: Renewable Energy
Systems
Topic 11: Wind Power System Design
Spring 2010
© Pritpal Singh, 2010
Wind Resource
The average power in the wind is given by:
Pavg
 
1
1
3
  Av   A v 3
2
 avg 2
avg
We therefore need to determine the average
of the value of the cube of the wind velocity.
We assume a statistical variation of the wind
velocity designated by a windspeed
probability density function as shown below:
Wind Resource Statistics
The average value of the cube of the wind
velocity is given by:

(v 3 ) avg   v 3 . f (v)dv
0
where f(v) is the probability density function.
There are two common types of probability
density functions that are used to describe
the statistical variation of the wind resource
– Weibull statistics and Rayleigh statistics.
Weibull Probability Density Function
The Weibull probability density function is
given by:
k 1
k

k v
v 
f (v)    exp    
c c
  c  
where k is called the shape parameter and c
the scale parameter. The probability density
function with k=1,2, and 3 are shown below:
Rayleigh Probability Density Function
The Rayleigh probability density function is
given by:
2

2v
v 
f (v)  2 exp    
c
  c  
The Rayleigh probability density function
for different values of c is shown below:
Average Power in Rayleigh Distribution
The average power in the wind for a
Rayleigh distribution can be calculated as
follows:
2

3 3
v 
3
3 2v
  v f (v)dv   v 2 exp     dv  c 
c
4
  c  
0
0

(v 3 ) avg

This can also be expressed as:
(v ) avg 
3
6

v 3  1.91v 3
where v is the average wind speed. Thus,
6 1
P  . Av 3
 2
Average Power in Rayleigh Distribution
(cont’d)
Example 6.10
Wind Resource Data
Some actual wind data for Altamont
Pass, CA together with a Rayleigh
probability distribution function with
the same average wind speed is
shown in the figure below:
Wind Power Classifications
The procedure demonstrated in
example 6.10 is commonly used to
estimate average wind power density
(W/m2) in a region. Winds are
classified by average wind speed as
shown in the below table.
Wind Resource Maps
Wind resource maps, as shown below,
are available from NREL.
Wind Resource Potential
The potential for wind energy, based
on class 3 winds or higher, are shown
by states in the table below:
Simple Estimation of Wind Energy
How much of the wind energy can be
converted to electrical energy?
Average wind turbine efficiency can
be used to estimate wind energy
delivered on an annual basis.
However, this information has limited
utility in terms of detailed wind energy
planning.
Example 6.11
Wind Farms
Wind turbines arranged in wind farms
must be designed so that upwind
turbines do not interfere with downwind turbines. Theoretical studies on
square arrays of wind turbines have
been performed to determine the
effect of interference on array output.
We can define a quantity called the
“array efficiency” which is the output
of the array divided by the output of
the array without interference.
Wind Farm Optimization
The array efficiency of a square array
of wind turbines as a function of
tower spacing is shown below:
Wind Farm Optimization
The optimum arrangement of a wind
farm is shown below:
Energy Potential for a Wind Farm
Example 6.12
Wind Turbine Performance Calculations
The calculations until now have
assumed an average wind turbine
efficiency. However, we now want to
look at more detailed/accurate
calculations.
Aerodynamics of Blades
A conventional airfoil achieves lift
based on Bernoulli’s principle. Air
moving over the top of the airfoil has
a longer way to go than over the
bottom and so must move faster. This
results in lower pressure at the top
than at the bottom resulting in lift. In a
wind turbine additional lift on the
blade is created by the wind.
Aerodynamics of Blades (cont’d)
Increasing the angle of attack of the
blade increases the lift up to a point.
Beyond this point, the flow of the air
over the blade changes from laminar
to turbulent, resulting in loss of lift.
This is called “stalling” of the blade.
Wind Turbine Power Curve
The most important information for a
specific wind turbine is the power
curve which shows the power
delivered by a wind turbine as a
function of wind speed. An example
of a power curve is shown below:
Wind Turbine Power Curve (cont’d)
Features of the wind turbine power curve:
Cut-in Windspeed – no power generated for
windspeeds less than this value.
Rated windspeed – as velocity increases
above cut-in windspeed, power increases
as the cube of the windspeed up to the
rated windspeed at which point the
generator is delivering as much power as
it is designed for.
Cut-out windspeed – at high windspeeds
damage can be done to the wind turbine
and so the turbine is shut down. This
happens at the cut-out windspeed and the
power output is zero.
Wind Turbine Overspeed Control
The generator can be operated above
its rated speed without damage using
one of three control approaches:
1) Active Pitch Control
- an electronic system monitors the
generator output and if it exceeds rated
output, the pitch of the blades is
adjusted with a hydraulic system.
2) Passive Stall Control
- the blades are designed to automatically
reduce efficiency when winds are too
strong.
3) Active Stall Control
- the blades are designed to automatically
stall when winds are too strong.
Wind Turbine Optimization
Two ways can be used to optimize the
performance of a wind turbine – the
rotor diameter may be increased, and
the generator size may be increased.
Their effects are shown below:
Wind Speed Cumulative Distribution
Function
See text pp. 357-360
Real Power Curves
The power curves for three large wind
turbines is shown below:
Estimating Real Wind Energy Output
The power curves for the turbines can
be combined with wind speed vs. time
data to determine the energy output for
a wind system.
If wind data for a location are available,
this procedure may be used. However,
if complete data is not available, usually
Weibull statistics are used to estimate
the wind resource. If only average wind
speed is available, Rayleigh statistics
are used to estimate the wind resource.
Estimating Real Wind Energy Output (cont’d)
The wind resource probability distribution
may be discretized.
The probability that the wind speed is
between v-Δv/2 and v+Δv/2
v  v / 2
=
 f (v)dv  f (v)v
v  v / 2
Thus we can discretize a continuous
speed probability distribution function by
stating that the probability that the wind
blows at a speed V is f(V). This turns out
to be a reasonable assumption.
Estimating Real Wind Energy Output (cont’d)
Example 6.15
Estimating Real Wind Energy Output (cont’d)
Estimating Real Wind Energy Output (cont’d)
Estimating Real Wind Energy Output (cont’d)
Capacity Factor for Wind Energy Systems
The capacity factor for a wind energy
system is the fraction of the energy
generated by the wind energy system
compared to what it potentially could
produce if it operated at rated power
for every hour of the year. Thus, for
example 6.15,
CF = 2.8561 x 106 kWh/yr = 32.5%
1000kW x 8760h/yr
Capacity Factor for Wind Energy Systems
(cont’d)
Suppose we want to estimate the capacity
factor for a wind energy system when very
little is known about the site or the wind
turbine. The capacity factor for a NEG Micon
1000/60 wind turbine as a function of wind
speeds (assuming Rayleigh statistics) is
shown below:
Capacity Factor for Wind Energy Systems
(cont’d)
For wind speeds in the range 4-10 m/s
(9-22 mph) CF varies linearly with wind
speed. A linear curve fit to this portion
gives:
CF = 0.087 V – 0.278
For the NEGMicon 1000/60 (PR=1000kW;
D=60m), it turns out PR = 0.278 !
D2
Thus, for this particular turbine,
CF = 0.087 V – PR/D2
Capacity Factor for Wind Energy Systems
(cont’d)
It turns out that this equation works quite
well as a means of estimating the
capacity factor for many wind turbines.
Using this expression, for CF we can
estimate the energy delivered from a
turbine with diameter D (m) and rated
power PR (kW) in Rayleigh winds with
average speed V (m/s) by:
Annual energy (kWh/yr)

PR (kW ) 
= 8760.PR (kW )0.087V (m / s) 
2
D(m) 

Capacity Factor for Wind Energy Systems
(cont’d)
Example 6.17