wind energy systems mase 5705 spring 2014, feb. 11-13, l7 + l8
TRANSCRIPT
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Wind Energy Systems MASE 5705
Spring 2014, Feb. 11-13, L7 + L8
1. Key points of L5+L6
2. Turbulence ( pp. 39 – 52)
2.1 Atmospheric boundary layer (ABL) and
surface layer turbulence
2.2 Turbulence intensity (TI)
2.3 Autocorrelation and Power Spectral
Density (psd)
2.4 Preview of later lectures on turbulence
2.5 Recommended reading
2
L7 and L8
3. Ch. 3 (Aerodynamics of Wind Turbines)
3.1 One-dimensional momentum theory and Betz limit
4. Project 2
3
1. Key points of L5+L6
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1. Bin width, bin probability or frequency and the
corresponding pdf
For arbitrary values of bin width ΔU,
2. Estimation of AEO (Annual Energy Output) of a
proposed site
a) When bin database is available;
That is, (Umax, Umin, no. of hours in each bin)
b) When only is known U
Key Points of L5+L6
iUU
i f(U)pdfingcorrespondΔUwidthBin
ffrequency Bin
For case (2a), we use the method of bins.
For case (2b) , we create our own bins; then we
estimate the number of hours in each bin by the
Rayleigh estimation.
3. Estimation of AEO of a specific wind turbine in a
proposed site
a) Specific wind turbine ≡ a wind turbine with
specified Uci, Uco and power curve.
b) power curve ≡ an official document that gives
measured power output vs wind speed (other
details are similar to (2a) and (2b))
For example see next table and figure, and also
consider the method of bins:
7
Measurements of site Bin Data and WT power
output data for Mod-2 2.5 MW WT
* Reasons for such a wide bin width are not known. Lower values are more
likely than higher values. Take Ū18= 17.25 m/s
*
(Measured) Power Curve for Mod-2 2.5 MW HAWT
(Uci = 6.25 m/s , Uco = 22.4 m/s)
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BINBin Avg. Speed
(Ui)
Power output
(kW)
1 3.125 0
2 6.5 175
3 7 318
4 7.5 460
5 8 603
6 8.5 745
7 9 949
8 9.5 1153
9 10 1316
10 10.5 1479
11 11 1642
12 11.5 1805
13 12 1968
14 12.5 2120
15 13 2263
16 13.5 2385
17 14 2500
18 17.25 2500
0
500
1000
1500
2000
2500
3000
6 7 8 9 10 11 12 13 14 15 16 17 18
Po
we
r o
utp
ut
(kW
)
Wind Speed U
Power output (kW)
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Method of Bins
2. Turbulence (pp. 39-53)
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"Turbulence is a dangerous topic which is often at
the origin of serious fights in the scientific meetings
devoted to it since it represents extremely different
points of view, all of which have in common their
complexity, as well as an inability to solve the
problem. It is even difficult to agree on what
exactly is the problem to be solved.”
Marcel Lesieur, Turbulence in Fluids , Springer
2008, The Netherlands
11
One is usually at a loss to calculate with
greater accuracy, say ±50%"
Lumley, AIAA Journal, 1998.
Valid even today
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2.1 Atmospheric boundary layer (ABL) and
surface layer turbulence
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2.1 (Ambient Boundary Layer (ABL)
and Surface Layer Turbulence
Of concern to wind turbines is turbulence in the lowest
levels of the ABL, also referred to as surface-layer
turbulence. The ABL thickness is not a precisely defined
quantity; it varies “from a few hundred meters to several
kilometers.” For practical purposes, surface layer
encompasses operational heights up to 200 ft, say 10% of
the boundary layer, and the rest is called outside layer.
14
Surface turbulence has been an actively
researched area of the past 30 years by
meteorologists, and engineers associated with
dynamic loading on exposed structures and wind
turbines. As for modeling, “ classical turbulence
theory” is keyed to surface layer conditions on the
basis both of phenomenological and analytical
considerations as well as other guidelines such as
continually updated Engineering Science Data Unit
(ESDU) series.
15
Basically, the von Karman turbulence
model with empirically adjusted parameters
correlate well with test data and it is widely
used. Therefore in this lecture, we follow the
text and briefly describe turbulence intensity
and von Karman power spectral density
(PSD), which is a common frequency-
domain description of turbulence, Equation
(2.27), p. 43). This PSD is usually referred to
as von Karman turbulence model.
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2.2 Turbulence intensity (TI)
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In this lecture, we follow the text (Ch. 2):
Just mention the basic equations and apply
them. We address turbulence in some detail,
well beyond the text, after covering Ch. 3
(Aerodynamics of Wind Turbines) and Ch. 4
( Mechanics and Dynamics).
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U = Ū + u
Ū = wind speed (x-direction)
u = fluctuating (random) component in the x- direction (at present we neglect lateral and vertical components)
19
20
z(vertical)
x
disk plane
x
w)v,(u,)u,u,(uu
(U,0,0)U
uUU
321
In General,
For the present, we follow the text:
Wind speed in the x direction, which is
perpendicular to the disk (γ=0)
uUU
Turbulence Intensity TI
(2.23 p. 40)
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static loads in the system.
23
u
u
Ū Ū
Text,
p. 41
Sample wind data (fig, 2.14 , 41)
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U
25
(y, z) ≡ rotor plane
x ≡ +ve in the downwind direction
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Wind Fluctuations and Average Power
U = Ū + u
27
?
28
29
E(u) = 0
E(un) = 0 for n odd (why?)
E(u2) = σU2
30
31
32
33
34
35
(2.54) p. 58 and (2.63) p. 60 in 2nd Edition
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(2.23) p. 40 2nd Edition
No U-bar^2
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p. 61, 2nd Edition
Table 2.4
P. 61
k Ke
1.2 0.837 3.99
2 0.523 1.91
3 0.363 1.40
5 0.229 1.15
39
Variation of parameters with
Weibull k shape factor
Compare Ke = 1+3(TI)2 = 1+3(.837)2 = 3.10
1+3(.523)2 = 1.82
1+3(.363)2 = 1.40
1+3(.229)2 = 1.16
Random Data, Analysis of Measurement
Procedures (3rd Edition, 2000), Wiley
J.S. Bendat and A.G. Piersol
A thorough and easy-to-read account of
random processes.
41
For details, see
Each Ui(t) represents a unique set of
measurements (not likely to be repeated)
Ui(t) = sample function or a random signal
Ui(t) = wind speed (in our case)
{Ui(t)} ≡ {U(t)}
The ensemble {U(t)} describes a random or
stochastic process of wind speed, and the
sample function Ui(t) belongs to this
process.
42
43
UN(t)
U3(t)
U2(t)
U1(t)
2.3 Autocorrelation and Power Spectral Density
44
Autocorrelation
We consider a stochastic process {U(t)} and
consider two time instances ‘t’ and ‘t + ’.
Given {U(t)}, we cannot predict {U(t + )}.
But {U(t)} and {U(t + )} are ‘related’ or
correlated. This correlation is described by
the autocorrelation function RUU(t,)
RUU(t, t + ) = E[U(t) U(t + )]
45
46
Wind Speed measurements belong to a
Stationary random process.
That is,
Ū(t) = constant, E[U2(t)] = constant
Autocorrelation function t2 – t1 depends only
on time difference:
RUU(t1 , t2) = RUU(t2 – t1)
RUU() = E[U(t)U(t+)]
While the autocorrelation function
characterizes turbulence as a function of
time or time lag in the time domain, the
corresponding description in the frequency
domain leads us to the power spectral
density function.
That is, the power spectral density function
characterizes turbulence as a function of
frequency in the frequency domain.
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It is proved that (Wiener-Khinchine
relation) autocorrelation function and
power spectral density function are a
pair of Fourier transform.
(N. Wiener in the United States and
A.I. Khinchine in the USSR).
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49
UU
UU
UU
UU
50
For future reference we also define the
Gaussian pdf (2.25), p. 42 2nd Edition
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0
Turbulent Wind. Velocity about Mean -4
-1 1
Fig. 2.15
p. 41
4
52
53
Turbulence Modeling Two important parameters in turbulence
modeling
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p. 43, 2nd Edition
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(2.27) p. 43, 2nd Edition
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57
58
59
, Fig. 2.16, p. 42
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Power spectral density functions, Fig. 2.`17, p. 44
von Karman Turbulence Model
Based on first principles;
No empirical constants;
For above about 150 m (height z > 150 m);
good correlation with test data;
For z < 150 m, some terrain-sensitive
deficiencies.
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B. 2.13 (p. 620)
Power Spectral Density Estimation Similar to Equation 2.27 in the Text (p. 43) , the following empirical
expression has been used to determine the power spectral density
(psd) of the wind speed at a wind turbine site with a hub height of z. The frequency is f(Hz), and n (n = f z / Ū ) is a non-dimensional
frequency.
Where
Plot the power spectral density of the wind at a site where the surface
roughness is 0.05 m (z0) and the hub height is 30 m (z), and the mean wind speed Ū is 7.5 m/s.
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Table 2.2 (p. 46)
Values of surface roughness length for various types of terrain
Terrain Description Zo (mm)
Very smooth, ice or mud 0.01
Calm open sea 0.20
Blown sea 0.50
Snow surface 3.00
Lawn grass 8.00
Rough pasture 10.00
hallow field 30.00
Crops 50.00
Few trees 100.00
Many trees, hedges, few buildings 250.00
Forest and woodlands 500.00
Suburbs 1500.00
Centers of cities with tall buildings 3000.00
next:
b) Plot of S(f) vs f
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─ 10
─ 1
─ 0.1
─ 0.01
─ 0.001
─1
0
─1
─0
.1
─0
.01
─0
.00
1
f(hz)
S(f
) m
s2 /hz
(Observe the rapid decrease in turbulence energy with increasing
frequency.)
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bending
An example:
68
bending
Around 4 Hz
The turbulence has little energy!
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─ 100
─ 10
─ 1
─ 0.1
─ 0.01
─ 0.001
─1
0
─1
─0
.1
─0
.01
─0
.00
1
f(Hz)
S(f
) m
/s2 /H
z
fn= 4 Hz
b ) Why is it that blades are sensitive to gust
(turbulence) excitation?
The answer is keyed to the difference between
turbulence PSD in space-fixed non-rotating
coordinates (hub element) and turbulence psd in
blade-fixed rotating coordinates (blade element).
Hub element and blade element feel turbulence
differently.
See next section on preview of later lectures on
turbulence. We will study this difference in detail
under rotational sampling. pp. 143, 330.)
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2.4 Preview of later lectures on
turbulence
Ch. 3 (aerodynamics of wind turbines, p. 143)
and Ch. 7 (wind turbine design, p. 330)
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In later lectures on turbulence we cover the
following:
1.Classical theory of turbulence, that is,
theory of homogeneous and isotropic
turbulence.
2.Revisiting von Karman model and
extensions
3.Rotational sampling, that is, why
turbulence seen by the hub element differs
from that seen by a blade element.
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For later reference:
Homogeneous and isotropic
turbulence
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Turbulence is homogenous. That
is, its statistical properties do not
vary form point to point in the field,
and thus all the functions described
are independent of the location of
the origin in the field.
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The concept of isotropy simplifies
the description of turbulence even
further. If a turbulence field is
isotropic, its statistical properties
are independent of direction in the
field, and thus they do not change
with rotation of the coordinate axes.
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A Descriptive Account of
Rotational Sampling
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Predicted and Measured Longitudinal Turbulence PSD
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• Experiment
___ Von Karman
(Radius 24.5 m, Rotation Rate = 0.625 Hz)
Rotational-coordinates
Fixed-coordinates
This figure shows the measured and
predicted psd (von Karman) of longitudinal
turbulence in both space-fixed and blade-
fixed coordinates.
The model refers to a 35-m radius HAWT
and the blade element has a local radius of
24.5 m, 70% of radial location. Two sets of
predictions from von Karman model –with
and without rotational effects– are shown.
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Since the area under the PSD curve
represents the turbulence energy, which is
the same in both coordinates, the turbulence
energy is transferred from the low-frequency
region (P<1) to the high-frequency region
with PSD peaks at 1P, 2P, etc., where
P=0.625 Hz. The correlation in this graph
brings out two key points of a much broader
significance.
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First, the classical turbulence
theory that is judiciously modified to
account for the local conditions
provides a means of modeling
surface-layer turbulence seen by a
turbine blade element.
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Second, the von Karman model
in rotating coordinates correlates
qualitatively well with the
measurements.
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2.5 Recommended Reading
The Nature of Wind R.I. Harris
(It is available on pass-word-protected Telesys.)
Although some 40 years old, this 25 - page article gives a
through description of classical turbulence theory and its
adaptation to modeling turbulence for structural
applications.
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3. Ch. 3 (Aerodynamics of Wind Turbines)
3.1 One-dimensional momentum theory
and Betz limit
84
The assumptions are
1. Fluid flow is incompressible,
2. No frictional drag,
3. Uniform thrust over entire disk area,
4. Nonrotating wake,
5. Steady-state operation: constant wind
speed over the turbine disk and constant
rotational speed.
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Momentum or actuator disk
theory of wind turbines
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87
88
89
U (1)
U - vi U-w
(4) (3)
+ve direction
Force on
fluid
disk
F = force on the
disk = Thrust
= T
(2)
90
U U - vi U-w
+ve direction
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92
(1) (4)
(3) (2)
- Turbine does negative work on the fluid ≡ extracts energy from the fluid
- Fluid slows down
Tube expands
93
94
95
96
97
Limitations of Betz’s (Momentum)
Theory
Windmill or windmill brake state
98
99
Power is extracted from the fluid by the wind
turbine
Windmill State
Velocity slows down (tube expands) WT brakes
the fluid
Windmill brake state
100
101
(p. 93)
102
Operating parameters for a Betz turbine; U, velocity of undisturbed air; U4 , air
velocity behind the rotor, Cp power coefficient, CT thrust coefficient
(p. 95)
In L9 we will consider an extension of the
momentum theory with rotational effects on
the fluid; p. 96
103
Project 2: Not assigned in 2014
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B.2.11 Actual Data Analysis and Power
Prediction, P. 619
Based on the spreadsheet (MtTomData.xls) which contains
one month of data (mph) from Holoyke, MA. determine:
a) The average wind speed for the month
b) The standard deviation
c) A histogram of the velocity data (via the method of
bins- suggested bin width of 2 mph)
d) From the histogram data develop a velocity-duration
curve
e) From above develop a power-duration curve for a
given 25 kW Turbine at the Holyoke site.
For the wind turbine,
assume:
P = 0 kW
P = U3/ 625 kW
P = 25 kW
P = 0 kW
f) From the power duration curve, determine the energy that would
be produced during this month in kWh.
0<U<6mph)
6<U <25 (mph)
25<U<50 (mph)
50 <U(mph)
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Verify and explain:
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f) The total energy produced can be
determined from integrating the product of
the turbine power and the numbers of hours
of operation at that power level, yielding an
annual energy production of 2474 kWh.
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