Wind farm turbine formations
Therefore the width or diameter of a rotationally symmetrical wind farm is a critically important factor and arranging the formation of wind turbines
to maximise the diameter of the wind farm is important.
Consider two different rotationally symmetrical wind turbine formations, I have called the "Ring formation" and the "Compact formation".
Let n be the number of wind turbines in the wind farm
Let s be the spacing between the wind turbines
Ring formation
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The circumference of the ring formation is simply n times s.
Circumference = n x s
The diameter of the ring formation is simply n times s divided by PI.
Diameter = n x s / PI
Compact formation
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The area of the compact formation, for large n, is n times s squared. This is slightly too big an area for small n.
Area = n x s^2 (for large n)
The diameter of the compact formation, for large n, is 2 times s times the square root of n divided by PI. This is slightly too big a diameter for
small n.
Diameter = 2 x s x SQRT(n/PI)
This is easily corrected for small n greater than 3 by adding a "compact area trim constant" (CATC) (which is a negative value so really it is a
subtraction) to the s-multiplier factor.
The CATC is 4 divided by PI minus 2 times the square root of 4 divided by PI.
CATC = 4/PI - 2 x SQRT(4/PI) = - 0.9835
This CATC correction was selected to ensure that the compact formation diameter equation for n=4 evaluates to the same value as does the ring
formation equation for n = 4, that being the largest n for which the ring and compact formations are indistinguishable.
The CATC works out to be minus 0.9835 which gives
Diameter = s x ( 2 x SQRT(n/PI) - 0.9835) (for n > 3)
Ratio of diameters
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It is of interest to compare the two formations of wind farm for the same n and s.
The diameter of the ring formation is larger by the ratio of diameter formulas in which the spacing s drops out.
Ring formation diameter : Compact formation diameter
n/PI : 2 x SQRT (n/PI) - 0.9835
This ratio can be evaluated for any n > 3 and here are some ratios with the compact value of the ratio normalised to 100% so that the ring value of
the ratio will give the ring formation diameter as a percentage of the equivalent compact formation diameter.
Here are some examples,
n = 4, 100 : 100
n = 10, 123 : 100
n = 18, 151 : 100
n = 40, 207 : 100
n =100, 309 : 100
n =180, 405 : 100
n =300, 514 : 100
n =500, 656 : 100
As we can see that for big wind farms, with more turbines, the ratio of diameters increases.
Since the Dow equation for the power and energy of a wind farm is proportional to the diameter of the wind farm then it predicts that the power and
energy of the ring formation wind farms will be increased compared to the compact formation wind farms by the same ratio.
In other words, the Dow equation predicts, for example, that a 100 turbine wind farm in the ring formation generates 3 times more power and energy
than they would in the compact formation, assuming the spacing is the same in each case.
Practical application when designing a wind farm
My recommendation would be to prefer to deploy wind turbines in a wind farm in the ring formation in preference to the compact formation all other
things being equal.
The compact formation can be improved up to the performance of a ring formation by increasing the turbine spacing so that the circumference is as big
as the ring but then if a greater turbine spacing is permitted then the ring formation may be allowed to get proportionally bigger as well keeping its
advantage, assuming more area for a larger wind farm is available.
The ring formation may be best if there is a large obstacle which can be encircled by the ring, such as a town or lake where it would not be possible
or cost effective to build turbines in the middle of it and so a compact formation with larger spacing may not be possible there.
Where it is not possible to install a complete ring formation then a partial ring formation shaped as an arc of a circle would do well
also.
edit on 25-3-2012 by Mr Peter Dow because: (no reason given)