Power in the Wind:
Making Statistical and Economic Project Comparisons
April 22, 2016
This work is licensed under a Creative Commons Attribution-
NonCommercial-ShareAlike 4.0 International License.
Where Does the Wind Come
From?
• Uneven heating of the Earth
– What if the Earth did not rotate?
CC BY-SA 3.0,
https://commons.wikimedia.org/w/index.php? http://scioly.org/wiki/index.php/
curid=2310777 Meteorology/Everyday_Weather
Where Does the Wind Come
From?
• Add rotation (and thus the Coriolis effect) and
what happens?
Global Winds
"Earth Global Circulation - en"
by Kaidor - Own work based on
File:Earth Global
Circulation.jpgThe picture of the
Earth is File:Lunar eclipse from
moon-2007Mar03.png
vectorized with Inkscape..
Licensed under CC BY-SA 3.0 via
Commons -
https://commons.wikimedia.org
/wiki/
File:Earth_Global_Circulation_-_
en.sv
... (link is external)
Where Does the Wind Come
From?
• Secondary and tertiary
circulations also occur around
the earth
– Secondary
• Hurricanes (tropical cyclones)
• Extratropical cyclones
– Tertiary
• Uneven heating due to terrain or
surface proprties
– Land/sea breezes
– Mountain/valley breezes
– Thunderstorms Land and sea breezes compared
– Tornadoes License: CC BY-NC 3.0
Source:
https://www.e-education.psu.edu/geog497i/node/
329
What Happens to the Wind as you Go Up in
Elevation?
A sample graph of wind speeds from 0 to 140 m
Source: NREL
Power in the Wind Foundations
mass m kg
• Density [ 3 ]
volume V m
• Volume area * length AL [m 3 ]
• Velocity
length L m
time
[ ]
t s
• Power
energy J
[ W ]
time s
• Energy 1 1
K .E . mass * ( velocity ) 2 mv 2 [ J ]
2 2
Come up with an equation for power as a function only of Area, Velocity and Density:
1 2 1 1
K .E. 2 mv ( V ) v 2
( AL ) v 2
1
Power 2 2 Av 3
t t t t 2
Power in the Wind
kg m
• Let’s check our units: N 2
s
1 J N m
PW Av 3
2
3 2
kg 2 m
kg m
3
m 3
m s s
kg m 2 N m J
s 3
s
s
Watt
Let's say we'd like to know how much power is available in the wind flowing through a
hula-hoop (D= 1 m) for a wind speed of 10 m/s:
A Boeing 747 has an approximate "diameter" of 65 m, but the rotor shown in the
figure below has a diameter of 80 m. How much power is available in the wind for a
turbine of this size at a velocity of 10 m/s?
Using the 747 example from above, note the differences in available power just by
increasing the velocity of the wind by a factor of two from 10 m/s to 20 m/s:
Influence of Air Density… Let’s Investigate
How Air Density Changes?
• Let’s look at the Ideal Gas Law
PV nRT
• P= pressure, V = volume, n = # moles, T =
temperature, R is the ideal gas constant 8.314
kJ/kg-K
• We are going to reconfigure this to solve for
density and apply it specifically to air.
Ideal Gas Law – mass basis
• First we need to move from moles to mass:
m
n
M
• M is the molecular weight of the substance you are
using, and m is the mass.
R
PV m T mRT
M
• Where R is now a constant specific to the gas used in
the equation.
Ideal Gas Law - rearranging
• We know that density is mass/Volume, so let’s
reorganize the terms to solve for this:
m P
V RT
• For air, R = 286.9 J/kg-K
– Mainly composed of 78% Nitrogen (R=296.8 J/kg-
K) & 21% Oxygen (R=259.8J/kg-K)
Ideal Gas Law Example
• Example:
– Calculate the air density for standard atmospheric
conditions, T = 15C, P = 101.325 kPa (Pa = N/m2):
kg P [ Pa ]
3
m J
R T [K ]
kg K
kg 101,325 [ Pa ] kg
3 1.225 3
m J m
287 (15 273.15)[ K ]
kg K
Density Conclusion
• Density increases with increasing pressure and
decreases with increasing temperature:
P
RT
• As air rises, it becomes cooler and the
pressure decreases, thus density will?
decrease
What Happens to Air Density in the Mile High City?
What Happens to Air Density in Cold Weather?
Behavior of the Wind
Weibull Distribution
(for your information)
Shape factor
k
Scale factor (m/s)
Wind Turbine Power Curves
1
Pturbine C p Av 3
2
Calculating Energy Production
from Wind Resource Data
Wind Data Source
• Eastern Wind Integration Data Set
– Model data generated for wind grid integration studies.
• Not real output, but predicted for actual locations.
– No gaps in data
• No QA needed
– Lat & Long provided
Capacity Factor 24 hr 365 days
# hr / yr 8760 hr / yr
day yr
• The amount of energy generated divided by
the amount which could be produced if the
turbine were running at its full capacity all of
the time.
– Typically considered over a year, but could be
measured over any timeframe.
• Example: If a 2 MW (2000 kW) wind turbine
generates 5,956,800 kWh of energy in a year,
what would this project’s
5,956,800 kWh / yr capacity factor be?
CF 0.34 34%
2000 kW 8760hr / yr
Wind Turbine Power Curves [kW]
Alstom Alstom Vestas
Alstom Eco Eco Eco Siemens Siemens Siemens Vestas V126-3.3 Vestas Vestas
Wind Speed 74/1670 Class 80/1670 80/2000 SWT-2.3- SWT-2.3- SWT-2.3- V126 - MW IEC V110-2.0 V90-2.0 GE 1.6-
[m/s] II Class II Class II 113 93 82 VS 3.0 MW IIIA MW MW 100
3 0 1 0 70 0 0 14 20 23 0 0
4 32 33 32 223 98 42 179 162 140 97 47
5 98 94 98 409 210 136 416 395 314 220 180
6 185 197 200 722 376 276 712 694 549 392 449
7 301 346 339 1074 608 470 1148 1060 900 616 745
8 460 538 522 1570 914 727 1713 1714 1347 927 1058
9 671 771 766 2009 1312 1043 2219 2432 1775 1275 1371
10 938 1033 1054 2191 1784 1394 2566 2999 1972 1622 1523
11 1232 1288 1370 2270 2164 1738 2858 3260 1999 1899 1585
12 1495 1489 1668 2298 2284 2015 3000 3300 2000 2000 1600
13 1634 1607 1894 2300 2299 2183 3000 3300 2000 2000 1600
14 1669 1655 1981 2300 2300 2260 3000 3300 2000 2000 1600
15 1670 1668 2000 2300 2300 2288 3000 3300 2000 2000 1600
16 1670 1670 2000 2300 2300 2297 3000 3300 2000 2000 1600
17 1670 1663 2000 2300 2300 2299 3000 3300 2000 2000 1600
18 1670 1638 2000 2300 2300 2300 3000 3300 2000 2000 1600
19 1670 1596 1982 2300 2300 2300 3000 3300 2000 2000 1600
20 1670 1549 1936 2300 2300 2300 3000 3300 2000 2000 1600
21 1670 1499 1876 2300 2300 2300 3000 3300 2000 2000 1600
22 1670 1449 1824 2300 2300 2300 3000 3300 2000 2000 1600
23 1670 1401 1772 2300 2300 2300 0 3300 2000 2000 1600
24 1670 1356 1712 2300 2300 2300 0 3300 2000 2000 1600
25 1670 1314 1660 2300 2300 2300 0 3300 2000 2000 1600
Procedures for the Competition
• Choose a location and its accompanying data for this exercise
• Create a histogram
• Select a Power Curve
• Calculate the energy produced from the selected turbine at
your site using the histogram and the power curve
• Calculate and record the Levelized cost of energy
• Calculate and record the capacity factor
• Iterate until you believe you have found the best turbine for
your location.
• Compare results with the class and see who came up with the
lowest LCOE and highest CF. Discuss!
Calculating Energy
• https://youtu.be/GVHc1zpLnXw Alstom Eco
Al
Wind Speed 74/1670 Class 80
[m/s] II Cl
3 0
4 32
25 m / s 5 98
Energy
v 0 m / s
[ Power time]at each velocity 6
7
8
185
301
460
9 671
10 938 1
11 1232 1
12 1495 1
13 1634 1
14 1669 1
15 1670 1
16 1670 1
17 1670 1
18 1670 1
19 1670 1
20 1670 1
21 1670 1
22 1670 1
23 1670 1
24 1670 1
25 1670 1
Wind Energy Economics
Levelized Cost of Energy
TIC rc
LCOE fcr
en en
• en = energy, to be calculated from wind resource data and
selected wind turbine power curve
• TIC = total installed cost for project (equation to be
provided for educational purposes)
• fcr = fixed charge rate - an annualized presentation of the
cost of financing a wind project
• rc = recurring charges. Typically provided per unit of
energy, e.g. $/kWh. For instance, operation and
maintenance costs.
LCOE Example Problem
• Under the following conditions for a wind turbine
project, calculate the levelized cost of energy for this
project:
• Total installed cost for the project is $30,000,000
• Average amount of energy produced at the project site
is 61,320,000 kWh/yr
• Fixed charge rate (fcr) = 0.09
• Operation & Maintenance charges (recurring charges
rc/en) = $0.01/kWhLCOE TIC fcr rc
en en
$30,000,000 * 0.09
LCOE $0.01 / kWh $0.054 / kWh
61,320,000 kWh
Competition
We are supplying wind data for several locations.
You will form teams and choose which data set you
would like to work with. You will then characterize
this wind resource with a histogram and calculate the
energy production by applying a power curve.
Several are provided to you. Cost data has also been
provided. Your objective is to minimize the LCOE of
your wind turbine installation by choosing an optimal
turbine for your location. You will also calculate the
resulting capacity factor of your design.
Levelized Cost of Energy
TIC rc
LCOE fcr
en en
Assumptions :
fcr 0.11
rc
$0.015 / kWh
en
• en = energy, to be calculated from wind resource data
• TIC = total installed cost, to be calculated from the
following equation which is a function of the turbine
size:
