Peter Dow
2012-04-01 14:33:28 UTC
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"Dow" Equation for the power and energy output of a wind farm
"The power and energy of a wind farm is proportional to (the square
root of the wind farm area) times the rotor diameter".
In his book which was mentioned to me on another forum and so I had a
look, David MacKay wrote that the power / energy of a wind farm was
independent of rotor size which didn't seem right to me considering
the trend to increasing wind turbine size.
Now I think the commercial wind-turbine manufacturing companies know
better and very possibly someone else has derived this equation
independently of me and long ago - in which case by all means step in
and tell me whose equation this is.
Derivation
Assume various simplifications like all turbine rotors are the same
size and height, flat ground and a rotationally symmetrical wind
turbine formation so that it doesn't matter what direction the wind is
coming from.
Consider that an efficient wind farm will have taken a significant
proportion of the theoretically usable power (at most the Betz Limit,
59.3%, apparently, but anyway assume a certain percent) of all the
wind flowing at rotor height out by the time the wind passes the last
turbine.
So assume the wind farm is efficient or at least that the power
extracted is proportional to the energy of all the wind flowing
through the wind farm at rotor height.
This defines a horizontal layer of wind which passes through the wind
farm of depth the same as the rotor diameter. The width of this layer
which flows through the wind farm is simply the width of the wind farm
which is proportional to the square root of the wind farm area.
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
Visit http://scot.tk/forum/viewtopic.php?t=603
for this topic with the essential diagrams embedded.
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.
Visit http://scot.tk/forum/viewtopic.php?t=603
for this topic with the essential diagrams embedded.
for this topic with the essential diagrams embedded.
"Dow" Equation for the power and energy output of a wind farm
"The power and energy of a wind farm is proportional to (the square
root of the wind farm area) times the rotor diameter".
In his book which was mentioned to me on another forum and so I had a
look, David MacKay wrote that the power / energy of a wind farm was
independent of rotor size which didn't seem right to me considering
the trend to increasing wind turbine size.
Now I think the commercial wind-turbine manufacturing companies know
better and very possibly someone else has derived this equation
independently of me and long ago - in which case by all means step in
and tell me whose equation this is.
Derivation
Assume various simplifications like all turbine rotors are the same
size and height, flat ground and a rotationally symmetrical wind
turbine formation so that it doesn't matter what direction the wind is
coming from.
Consider that an efficient wind farm will have taken a significant
proportion of the theoretically usable power (at most the Betz Limit,
59.3%, apparently, but anyway assume a certain percent) of all the
wind flowing at rotor height out by the time the wind passes the last
turbine.
So assume the wind farm is efficient or at least that the power
extracted is proportional to the energy of all the wind flowing
through the wind farm at rotor height.
This defines a horizontal layer of wind which passes through the wind
farm of depth the same as the rotor diameter. The width of this layer
which flows through the wind farm is simply the width of the wind farm
which is proportional to the square root of the wind farm area.
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
Visit http://scot.tk/forum/viewtopic.php?t=603
for this topic with the essential diagrams embedded.
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
Visit http://scot.tk/forum/viewtopic.php?t=603
for this topic with the essential diagrams embedded.
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
Visit http://scot.tk/forum/viewtopic.php?t=603
for this topic with the essential diagrams embedded.
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.
Visit http://scot.tk/forum/viewtopic.php?t=603
for this topic with the essential diagrams embedded.