Modeling Stochastic Wind Loads , on Vertical Axis Wind Turbines

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Modeling Stochastic Wind Loads , on Vertical Axis Wind Turbines
SANDIA REPORT
Printed
September
SAND83–1909
●
Unlimited
Release
1984
Modeling Stochastic Wind Loads
, on Vertical Axis Wind Turbines
Prepared
by
Sandia National Laboratories
Albuquerque,
New Mexico 87185
for the United States Department
under
Contract
CIE-AC04-76DPO0789
and Livermore,
of Energy
California
94550
●
UC–60
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MODELING STOCHASTIC WIND LOADS
ON VERTICAL AXIS WIND TURBINES
Paul S. Veers
Applied Mechanics Division 1524
Sandia National Laboratories
Albuquerque, New Mexico
Abstract
The Vertical AXIS Wind Turbine (VAWT) is a machine which extracts energy
Since random turbulence is always present, the effect of
from the wind.
This
this turbulence on the wind turbine fatigue life must be evaluated.
problem 1s approached by numerically simulating the turbulence and calcu–
latlng, In the time domain, the aerodynamic loads on the turbine blades.
These loads are reduced to the form of power and cross spectral densities
The relawhich can be used In standard linear structural analysls codes.
tive Importance of the turbulence on blade loads 1s determined.
The most common des]gn for Vertical Axis Wind
Turb]nes (VAWT’S) was first patented in 1931 by
Th]s “egg beater” shaped
Darrleus, a Frenchman.
machine consists of one or more blades with air–
foil cross sections attached to the top and bottom
of a central shaft, or tower.
To m]nlmize bending
stresses In the blades while the turbine rotates,
the blades usually have a characteristic
tropos–
Torque is
klen, or “splnnlng rope,” shape.
produced when the turb]ne rotor turns In the wind.
Fig. 1 shows the 17 meter research VAWT at Sandla
National Laboratories, Albuquerque, NM
This
turb]ne has the most common configuration
of
VAWT’S currently being built, two blades w]th a
central tower and guy cables supporting the top of
the rotor.
The aerodynamic analysls of the VAWT 1s compli–
Since the VAWT blade
cated by several factors
rotates through 360 degrees relatlve to the lncl–
dent wind during each rotation, the rate of change
much greater than
of angle of attack 1s often
But
experienced
]n other airfoil applications.
because the speed of the blade 1s normally much
greater than the w]nd speed, the angle of attack
varies between posltlve and negative values around
zero degrees.
The higher the w]nd speed, the
greater the angle of attack excursions.
When the
winds become h]gh enough, the angles of attack
become large enough that the alrfoll begins to
stall dynamically.
Research In VAWT aerodynamics
1s currently lnvestlgatlng both dynamic stall and
Aerodynrunlc analys]s using
p]tch rate effects.
the streamtube momentum balance approach was the
earnest
and slm lest approach used to est]mate
Y
VAWT performance
The vortex llftlng llne ap–
preach has produced some better results, in an
average sense, over a wider range of wlnd2conexpensive
d]tlons, but 1s computatlonally
Both methods suffer from the lack of a valldated
method of predicting dynemlc stall.
Except for
emplrlcal llft and drag curves from wind tunnel
testing, accurate pltchlng airfoil, dynamic stall
models do not exist.
Most VAWT’S experience stall
over a s]gnlflcant portion of the normal operat]ng
The loss of llft due to stall in high
range
winds 1s considered benef]clal because It llmlts
the maximum power that the drive train must trans–
mlt, thereby holding down the turbine cost.
The structural analysls of the VAWT must deal with
Corlolls
the fact that the structure is rotating.
and centrifugal effects make the modes of vlbra–
tlon complex, but the system remains l]near.
Free v]bratlon analysls and aero lastlc analysis
5
are well developed and valldated
The forced
v]bratlon analysis of the VAWT rotor ]s developed,
but agreement w]th field data from operating
turbines 1s sometimes poor, especlall~ In the high
w]nd speed, significant stall, regime
It is not
clear whether the source of the error 1s In
yet
the structural analysls or ]n the calculation of
The relatively good agree–
the aerodynem]c
loads
ment of the structural analysis to the data gathered at low wind speeds would seem to indicate
that the problem may l]e in the aerodynamics.
2
Fig.
1
Darrleus Vertical AXIS Wind Turb]ne
at Sand]a National Laboratories,
Albuquerque, NM
(VAWT)
Another possible source of error In previous load
calculations
1s the assumption of a steady w]nd.
Until now, all the aerodynamic loads have been
The
calculated based on a constant emblent wind.
loads which result from this calculation are shown
In Fig. 2. Tangential forces are def]ned as the
component In the dlrect]on tangent to the path of
the blade as It rotates and normal forces are
normal to this path.
The tangential forces pro–
duce a net torque about the central tower while
When a
the normal forces produce no net energy.
constant wind lS assumed, the forces repeat exThe frequency
actly for each rotor revolution.
content of these forces 1s therefore limlted to
Integer mult;ples of the turbine rotational speed
(abbreviated as “per rev” frequencies).
This
approach will produce zero excltat]on at any
frequency other than the per rev frequencies.
This lS not totally unreasonable because the per
rev frequency content of the loads 1s produced by
the mean wind and the rotation of the rotor, and
]s therefore large compared to the turbulence
However, data collectinduced, stochastic loads
ed from VAWT’S operat]ng in high winds have shown
slgnlf]cant response at the structural resonant
frequencies as well as at the per rev frequencies.
The spectral content of the blade stress response
3 could never be predicted with
shown in Flg
loads calculated assuming e steady wind.
By calculating the aerodynamic loads due to a
turbulent wind, the loads will contain all fre–
These
quencles, not only the per rev frequencies.
stochastic loads are described by their power
spectral densities (psd’s) and cross spectral
Because the relationship
densltles (csd’s).
between incident wind speed and blade forces is
nonlinear, even before the onset of stall, the
blade loads are calculated in a time marching
manner
But since the structural response to the
stochastic loads is linear, the psd’s and csd’s
will be sufficient input for structural analysis.
J’urbulent Wind Model
-1.64
00
\
I
30
20
10
TURBINE
49
The first step in a time domain calculation of the
the stochastic blade loads IS to produce a numeriThe
cal model of the atmospheric turbulence.
turbulence model must posses frequencies of interest for structural analysls (O – 10 per rev).
In a NASA summary of a mospheric environments by
k
Frost, Long and Turner , the followlng fOrm for
the psd of atmospheric turbulence is suggested;
REVOLUTIONS
clfi[ln(]O/zo+
l)ln(h/zo+
1)]–1
(1)
S(Q) =
1 + c2[hti ln(lO/zo+
1)/~ ln(h/zo+
1)]
5/3
where
u
= frequency
h
= height
~
= mean wind
z
Lo
30
20
TURBINE
Fig. 2
BC032
1
roughness
coefficient
40
Clx
REVOLUTIONS
Dlmenslonless
Blade Loads calculated
assuming a steady wind, (a) normal to the
blade path, (b) tangential to the blade
path.
12(D
1-
= surface
speed at h = 10m
and c are constants which differ for each
c1
These constants
spatial ~omponent of turbulence.
are given as;
.oll~
0.0
0
above ground
CHANNEL
I
I
13
I
=i
Cly
= 12.3, C2X = 192.
=
Clz =
4.0,
c
2y
=
0“5’ C2Z =
70.
8.0
for the longitudinal
direction,
for the lateral direction
and
for the vert)cal
direction.
An example turbulence psd at a reference height of
30ft (lOm) with a surface roughness coeff]clent of
.1 1s shown :n Fig. 4. For the wind velocity a
lo’s
so
a
w
ld :
0.
Io”z
4{D
10-‘:
n
-i)mo
1.6
3.2
4.8
6.4
8.0
f
1
10-’
10°
FREQUENCY (HZ)
FREQUENCY
Fig. 3
Power spectral density (psd) of the stress
in a VAWT blade measured during operation
In high winds.
The turbine rotational
frequency 1s 0.8 Hz,
Fig. 4
[
Id
Psd of turbulence using Eq. (l);
h=lom,
Zo=0.1,V=15m/s.
3
Geussian time ser)es described b~ this psd can be
ct.talned by the followlng method”
Represent the
and
power In a narrow band, &@, of the psd by s,n~
cosine components, with random phase, .gt the
rcntral frequency and sum these Inputs over al 1
the frequencies of Interest.
Such a veloclty time
hlstor’y 1s g]ven by,
where
a
= decay coeff]c]ent
~
= frequvncv
l,r = distance
~
between
po]nts
1 and j
= mean wind speed
[1
x
V(l)=v,
(AJ S]Ilti,t+ B
COSU
(2)
~
t)
1
j=l
Ili
‘-”—~
1
Wht’re
r,
s
——
= magnitude
of the psd at frequency
J
$, = a uniformly distributed random
on the Interval zero to 2n,
L!
J
n o ;— ----—r—00
100
variable
300
400
60.0
TIME(sEcoNtIs)
(a)
The central llmlt. theorem guarantees convergence
Of ~(t) to a Gaussian form as the number of
slnllsoidal corrponents at different frequencies
with random phase becomes large,
This ]s equlv–
aler,t to band l]m] ted, f]ltered wh]te no]se
If
the’ frequency spacing is regular and beg]ns at
zero. the compuiatlon of the time series can be
tif<omp] l~l)ed w th 8 d]screte Inverse Fourier
-1
trensform
.#
of the complex series,
—-+
200
40(17
- ——-—--”~
I
I
..
v(t) = v +.x
-1
(Aj + }Bj)
(3)
x
j=l
If the three directional components of the turbu–
lence are assumed to be uncorrelated,
each com–
ponent can be Independently generated using the
above method
The usual approach to transforming
th]s temporal representation
Into a spatial var]a–
tlon IS to use Taylor’s frozen turbulence hypo–
thesis
Using this approach, It 1s assumed that
all turbulence IS propagated in the longltudlnaJ
dlrectlon at the mean wind speed
F]g
5 shows a single point wind simulation time
series where the horizontal
longltudlnal, and
lateral veloclty components have been resolved
Into magnitude and dlrect]on.
A s]mple approach to simulating a full three
dlmens]onal
field of w]nd ]s to assume that the
w)nd speed and dlrectlon In a plane normal to the
mean wind dlrectlon are constant.
This IS obvl–
OUSIV quite crude, but can be adequate If the
region of Interest In this plane 1s relatively
small and the w nd IS h)ghly correlated over that
i
region
Frost
has an est]mate of spat]al coher–
ence of the form,
7:J = exp(–a
u Ar/~)
(4)
4
-400 --–— -—T--—
00
100
(b)
F]g. 5
~o~o,o
TIME(SECONDS)
Example simulated, single point, wind
veloc]ty time series, (a) w]nd speed,
wind dlrectlon
(b)
The suggested value for the decay coefficient,
“a, “ 1s 7.5 for both lateral and vertical sepa–
rations.
This expression lndlcates that the size
of the reg]on of slgnlflcant correlation 1s a
For
function of frequency and mean wind speed.
the 17m diameter VAW’i’s, the most common built to
date , the coherence over the range of most ener–
get]c frequencies IS greater than .5. This
Implles that the correlation over most of the
swept area of the turbine rotor IS qu]te high.
For the larger rotors of the future, or the cur–
rent large d]ameter propeller type turbines, the
one d)mens]onal wind varlat]on may not be ade–
quate
A fully
can be
three–dimensional
spatially
simulated
by producing
time
varying
wind
series of the
wind speed at several points which lie in a plane
perpendicular
to the mean wind direction, as shown
schematically
in Fig. 6. The longitudinal varia–
tions are, as before, produced by Taylor’s frozen
turbulence hypotheses,
The velocity time history
at a particular point 1s reproduced at a down–
stream point at a later time.
The time shift is
equal to the distance between the points divided
by the mean wind speed.
The wind time series at
each point )s a random process which is correlated
to wind at every other point.
AS shown by Eq. 4,
the level of correlation depends on the distance
between the po:nts, the mean wind speed and its
frequency.
z
L
Y
x
The csd’s are, in general, complex fmctlons
of
frequency.
Eq. 5 relates only the magnitude of
the csd’s to the coherence and the psd’s and is
therefore insufficient to completely define the
spectral matrix.
Analytical expressions for the
csd’s have been suggested by Frost, but they are
very complicated functions Involving modified
Bessel functions of fractional order,
If the
csd’s are approximated as real functions, Eqs. 1,
4 and 5 can be used to define the spectral matrix.
Since there is usually no relat]ve phase between
the winds at two points that are at the same
height above the ground, the csd between these
points is real.
For points with some vertical
separation, there 1s a tendency for the winds at
the higher elevation to lead the lower elevation
winds , wh]ch results in a complex csd. However,
this tendency is not strong and the magnitude of
the lmaglnary part is usually much smaller than
the real part.
The csd’s are therefore asstuned 0
be entirely real and approximated by Eq, 5.
v~(t)
A
The method of converting the spectral matrix into
a vector of tlm
realizations of the random
Define a lower triangular
PrOcess fo~lows~ser’es
matrix of transfer functions, [H], such that
[H]*[H]T = [s]
Fig, 6
Schematic of a wind simulation with varla–
Correlated
tions in three dimensions.
single point time series are generated at
several Input points.
Physically,
[H] 1s a transfer matrix from uncorrelated white noise to the correlated spectra of
the wind at the Input po)nts.
In general, both
[s] and [H] are complex, but for the degenerate
case of a real [S], [H] is real also.
The general
case of a complex [s] is derived in Ref. 7,
Because [H] IS lower triangular, lt 1s completely
defined by Eq. 6. The elements of [H] can be
determined recursively.
H
1/2
11 = ’11
H ~. =
H
Since the wind speed at each point can be repre–
sented by Gaussian random processes, the total
input wind field can be represented as a random
process vector, {V]. The second order statistics
of {V} can be defined by the spectral matrix [s].
The element In the i–th row and j-th column of [s]
is the cross spectral density (csd) between the
The diagonal
wind speeds at input points i and j
The
terms are the psd’s of the Input wind speeds.
csd’s are related to the coherence function and
the psd”s by the expression,
s..
l]
2
= 7:, Sli s..
22
32
S
31
/H1l
= (S32-H31
.
Sii = psd at point
i.
H21)/H22
H
11
= (s,, -
H,12)1’2
x
j =1
j
(5)
–1
= (Slj -
H,k HJk)/Hj J
z
k=l
function
points
(7)
i-1
11
Sij = csd between
1/2
.
.
H
7;, = coherence
= (S22 - H212)
’31 =
H
S21/H11
JJ
where
(6)
i and j
Define [X] as a diagonal matrix of complex frequency domain representations
of Independent white
The
noise Gaussian signals with unit variance.
5
correlated vector of wind
then be calculated from
{v]
random processes
=L%-l([H][X]){lI
can
misrepresenting
physical
anced by the statistical
of this approach.
(8)
&dvnemic
where
{II IS a column vector
of
ModeL
ones
Multiplying
[H] by [X] gives a random phase at
each frequency to each column of [H]. The multi–
pllcatlon in (8) by the vector of ones has the
effect of summing the rows of the Inverse Fourier
transform of [H][X].
Once [H] has been calcu–
lated, new unique realizations of the process can
be formed simply by introducing a new [X] Into
Eq. 8.
The vector of wind turbulence components, {V~, has
zero mean and describes only the turbulent part of
the wind.
The mean wind, wh]ch may vary with
height to Include wind shear effects, 1s added to
the turbulent part.
The main dlff]culty in Implementing this procedure
ls the size of the matrices Involved.
For N Input
po]nts, the matrices [s], [H] and [X] are NxN.
Each element in the matrices 1s a function of
frequency wh]ch must be defined at each discrete
Input frequency.
If the number of Input points
and discrete frequencies 1s not kept to a mlnlmum,
the size of the problem can easily exceed the core
storage space on a main frame computer.
In spite
of the size of the problem, the use of the fast
Fourier transform makes the computation relatively
fast.
Example problems which would exceed the
core storage on a Cray were run on a VAX 11/780 in
a few minutes.
In summary, a full field of turbulent wind can be
simulated with varlat]ons In one to three spatial
The one dimensional variations are
directions.
produced by creating a wind time ser)es at a
point and assuming that the w:nd speed 1s constant
In a plane perpendicular
to the mean wind dlrec–
tlon
The longitudinal varlatlon 1s achieved by
aPPIYlng Taylor’s frozen turbulence hypothesis.
A
three dimensional wind field can be simulated by
producing an array of correlated single point time
The need for the three dimensional repre–
series.
sentatlon depends on the size of the wind field of
Larger fields have lower spatial cor–
Interest
turbu–
relatlon and require a multldlmenslonal
If the different components of the
lence model.
turbulence are uncorrelated,
they can each be
created Independently by repeating the procedures
outl]ned above.
The main difference between this turbulence slmu–
latlon and other methods currently being developed
(Ref. 8, 9, 10) 1s that in this simulation the
turbulence IS based solely on Its first and second
order statistical moments w)thout any reference to
There 1s no
the fluld dynamics of the wind.
guarantee that this simulated flow field will be
continuous , compatible or physically realizable.
However, by matching Its f]rst two stat]stlcal
moments , the turbulence properties which have the
most s:gnlflcant Impact on the blade loads and
therefore the structural response of the wind
The r]sk of
turb]ne are accurately represented.
6
flow properties 1s bal–
accuracy and slmpllclty
Various approaches to calculating the aerodynamic
forces on a VAWT blade have been developed in the
past
The earnest,
and simplest, ]nvolves
Since then,
streamtube momentum balance methods
vortex llftlng llne codes have been written which
are more ac urate over a wider range of operating
5
wind speeds
The cost of this generality IS a
tremendous Increase In computational time needed
to calculate the blade loads.
A random blade load
s]mulatlon necessitates a computatlonally
fast
method of converting instantaneous winds into
To reduce the statistical error, the
blade loads.
loads must be calculated repeatedly and the psd
and csd estimates averaged.
The streamtube momen–
tum balance method 1s by far the fastest
Subdlvldlng the swept area of the
available.
turbine rotor Into a bundle of streamtubes creates
a ready made computational grid.
Each streamtube
can be used as a locatlon to Input correlated
The wind
single point wind speed time series.
speed at each point In the streamtube can then be
tracked as lt passes through the rotor
The basic aerodynamic model used for this analysis
was developed by Strickland.
The method, which is
comprehensively
described in Ref. 11, WI1l be
outllned briefly here.
The Idea 1s to subdivide
the rotor Into streamtubes along the mean wind
direction.
As a blade passes through a stream–
tube , the flow IS retarded.
The amount of retardation ]s related to an “Interference factor.”
The retardation 1s related to the net stresmwlse
force by conservation of momentum wlthln the
streemtube
Because of the Interdependence of
streamwlse force and angle of attack, the force
Once the iteration
calculation
1s lteratlve
converges , the angle of attack lS found and the
coefficients of normal and tangential force are
Interpolated from a table of values.
The code
used In this study neglects both Reynolds number
and dynamic stall effects.
The results reported
here are for low wind speeds where these effects
are relatively small.
Fig. 7 shows a typ]cal streamtube as it passes
through a VAWT rotor.
As shown In the plan view,
each streamtube is divided Into three regions by
upstream,
Intersections with the blade path.
The wind speed
lnslde the rotor and downstream.
1s reduced In each reg]on by successive blade
Each streamtube has three mean wind
passages
The
speeds which correspond to the three regions.
mean wind speed Inside the rotor, ~ , 1s reduced
from the mean free stream veloclty,r~m, by the
Interference factor which is calculated by as–
sumlng a steady wind.
Although they propagate at
the reduced mean wind speed, Vr, the turbulence
components transverse to the mean wind dlrectlon
are assumed undiminished as they pass through the
rotor
The Instantaneous wind speed at a given blade
element 1s determined in the following manner.
,–
––L
.-L
L.
.._
---.2
-.
For each Streamtube , tne wlnas wnlcn nave IJM>SCU
through the plane just upwind of the rotor (see
The length
Fig. i’) in the recent past are stored.
of time, At, requ]red to traverse the distance
from the upwind plane to the current posltlon of
the blade element ]s calculated by,
...I.
--W,, c, c
a=
0<
R–rslnO
b=O
O<e<n
II
2r sln .5
(9)
VuPWIND PLANE
t
a
v
e < Zn
T<
8 < 2n
The wind speed that passed through the upwind
plane at a time, At, earner
]s then determined
For the downstream passage of the blade through
the streamtube, the upwind wind speed IS reduced
by twice the mean upwind streamtube interference
factor
An Instantaneous Interference factor and
angle of attack are calculated and the Instan–
taneous forces on the blade are determined.
I
R
i
b
BLADE ELEMENT
FLIGHT PATH
Ik
STREAMTUSE
PLAN
VIEW
Results
and Dlscusslon
If a steady wind is assumed, the calculated forces
on the blade are Identical for each rotation.
Plots of the normal and tangential dimensionless
forces near the turbine equator In a steady wind
are shown in Flg
2. The force components are
defined ]n a reference frame fixed to the VAWT
blade.
The normal force changes direction as the
blade passes through upwind and downwind orlen–
tatlons. The tangential force 1s almost always
posltlve, only going to zero, or sll,ghtly nega–
tlve, when the blade chord 1s al)gned with the
wind
The steady wind loads cons]st entirely of
frequencies which are Integer multlples of the
turbine rotating frequency (per rev frequencies).
Table 1 contains the sine and cosine coefficients
of a Fourier series representation of these steady
wind loads for a 50 ft (15m) dlmneter turbine in a
.?1mph wind.
VERTICAL
AXIS
TABLE
/
1
A
NORMAL
STREAMTUBE
TANGENTIAL
PER REV
FREQ
Cos
SIN
1
2
3
4
5
–2.21
1.37
216
–.036
– 073
18.8
2.15
1.32
.302
148
/
Clr
R
(h
v
DOWNSTREAM
Cos
SIN
–.001
– .926
–1 .58
–.043
–.192
–.008
– .342
.261
– .063
.014
‘RDTOR
EOUATOR
Table
VIEW
Fig. ‘7 The path of a typical streemtube with
The mean
respect to the VAWT rotor.
freestream wind speed, ~m, 1S reduced tO
~r by the first blade passage and reduced
again by the second blade passage.
1
Cosine and sine coefficients of a Fourier
series representation
of the VAWT loads
calculated assuming a steady wind.
(~= 21 mph).
Fig. 6 shows plots of the normal and tangential
components of the blade forces.
These are the
force components at the turbine equator calculated
using a one dimensional turbulent wind simulation.
The basic form of the steady loads 1s retained,
but a stochastic component caused by the turbu–
Ience has been added to the underlying deter–
However, due to the nonlinear
ministlc part.
relationship between the incident wind and blade
loads, this 1s not simply a superposition of the
loads due to the steady wind and the loads due to
the turbulent component of the wind.
7
Standard random data analysls can not be applled
to this data until the determ]nlstlc part has been
removed
S]nce the determ]nlstlc part can be
represented by a sum of frequency components at
the per rev frequencies, It can be removed by
The Buys-Bal ~~t
fllterlng those frequencies
filter 1s perfectly su]ted to this purpose
Fig. 9 contains psd’s of the normal and tangential
forces near the equator of a VAWT blade after the
per rev frequency content has been extracted with
The sharp downward spikes
the Buys–Ballot
filter.
In the spectra of Fig. 9 indicate where the per
rev components have been removed.
-1.6I
0.0
1
10
20
30
As may be seen by comparing Fig. 9 with Fig. 4,
the low frequency content of the tangential forces
1s a qua.s–static reflection of the energy present
The blade load psd’s have much
In the turbulence.
more high frequency content than would be expected
There are two
by examlnlng the turbulence psd.
F~rst, the motion of the blade
reasons for this
through the alr causes the frequency content of
the wind that the blade sees to shift to higher
This IS much llke the situation
frequencies.
where the frequency content of the loads on a car
travellng over a rough road depend on the speed of
The more energetic low frequencies of
the car
the wind are transformed Into higher frequency
loads
Second, relatively small changes in the
Incident wind at the blade can cause large changes
in the Induced angle of attack, and therefore
large changes In the load. This magnifies the low
energy, h]gh frequency part of the turbulence
spectrum
40
(a)
TURBINE
REVOLUTIONS
04,
I
I’A
-OIL—-------00
20
10
30
Table 2 shows the dimensionless forces due to a
turbulent wind which has been separated Into
determlnlstlc
and random parts by the Buys–Ballot
The deterrnln]stlc part 1s written in
filter
The
and cosine coefficients.
terms of sine
“ZRANDOM” IS the percentage of the total variance
of the blade load In a one per rev band around
each per rev frequency due to the random part.
At
low frequency, the random part 1s comparable to
the determlnlstlc
per rev’s, but lt dominates the
loads at the higher frequencies.
40
(b)
TURBINE
Fig. 8
REVOLUTIONS
Dlmenslonless
Blade Loads calculated using
a turbulent w,nd, (a) normal to the blade
path, (b,) tangent. to the blade path
TABLE
2
TANGENTIAL
NORMAL
_———
PER REV
FREQ
Cos
–2.24
1 24
260
1
2
3
4
004
– 084
5
Table
SIN
ZWNDOM
Cos
2
18.7
2 20
131
.225
.129
%RLNDOM
SIN
————— ———
—
17
67
?-i
98.
99.
029
–1 52
–.101
– 211
– 019
– 891
–.387
,212
– 004
031
69
45
89.
90
99
The frequency content of the blade loads, calculated using a tur–
bulent wind (~ = 21 mph), 1s dlv]ded Into one per rev wide bands
around each per rev frequency.
The determlnlstlc per rev fre–
quencles are written In terms of cosine and sine coefficients.
The
random part 1s written as a percentage of the total variance in each
frequency band.
10-’
TABLE
3
—
—
NORMAL
PER REV
FREQ
10-:
Cos
1
SIN
—————
–3.62
–1 90
153
1 69
237
2
3
4
5
10-2
TANGENTIAL
27.6
8.82
3 56
–1 24
.508
Cos
SIN
1.32
–1 .60
–2 17
– 931
–.569
–.508
.816
–1 .25
–.391
1.03
10-;
j
10-2
(
10-’
I
1
10-’
FREQUENCY
Fig. 9
Table 3
I
Id
(CYCLES PER REVOLUTION)
mEQuENcy
10-’
I
lcf
1
1
I&
Id
I
(HCLES PER REVOLUTION
Cosine and sine coefficients of a Fourier
series representation of the VAWT loads
calculated assuming a steady wind.
(~= 34mph)
the same mean wind speed as above 1s shown In
Table 4. The magnitudes of the components are
aPPrOxlmate Since the aerodynamic stall model IS
not exact.
However, the trend ]n the data ]s
apparent
Although the random part of the loads
1s not much different than It was for the case of
low w]nds without stall (higher at some per revs
and lower at others), the per rev components are
noticeably
lower for the turbulent wind case,
For
this mean wind speed, Table 5 shows the percentage
difference In the rms determlnlst]c
loads for the
two models
For high w]nds, the loads due to the
steady mean wind are not the same as the mean
loads due to a turbulent wind.
Structural
response calculations using the steady wind load
estimates have had a tendency to match field data
quite well at low wind speeds and deviate at
F]g
higher wind speeds.
10 1s a comparison of
measured VAWT blade stress and estimated stress
Power spectral dens it]es (psd’s) of the
blade loads calculated using a turbulent
wind
The per rev frequency content has
been removed so only the random part
remains, (a) normal force psd, (b) tangential force psd.
●
FFIVD
Predictions
+
Blade
1 Oata
x
Blade
$
J
●
2 Oata
+
$~
L*
An Important difference between the steady and the
i,urbulent wind loads 1s the change in the deter–
As indi–
minlstlc sine and cosine coefficients.
cated by Tables I and 2, this difference is
greatest for the tangential forces and at higher
If the relat)onshlp between wind and
frequencies
blade loads were llnear, the coefficients would be
The fact that they are
ldentlcal In both cases.
very close lnd]cates that, for this example, where
stall is not occurring (~ = 21 mph), the non–
IInearltles are relatively small.
If a higher wind case 1s examined (~ = 34 mph),
the change In the per rev components 1s more
The sine and cosine coefficients
for
dramatic.
loads calculated assuming a steady wind which
causes a slgnlflcant amount of stall are shown in
l’able 3. The results from a turbulent wind with
I
10
20
Iifndspeed
Fig.
10
30
40
(mph)
Comparison of measured VAWT stress
response and stresses calculated using
The analysls code is
steady wind loads.
“FFEVI).” Data were collected from both
blades of the turbine.
9
response using stea~y wind Input as reported by
The overpredictlon
of the
Lobltz and Sullivan
stress response in high winds IS consistent with
the overpredictlon
of blade forces which results
from assuming a steady wind.
The coherence is generally low at high frequency
A high coherence
and during changes in the phase
is generaly found near per rev frequencies
However, high coherence
tend to repeat at fre–
quency Intervals somewhat wider than one per rev.
The cause of this effect IS not clear, e \\hough 1 t
has been found Independently by Anderson
, using
a frequency domain approach.
The correlations and phase relationships between
the loads at different points on the rotor are
The complex csd can be
contained In the csd’s.
expressed in terms of its magnitude and phase.
The level of correlation ]s estimated using Eq. 5
by calculating the coherence function from the csd
The phase and coherence between ]den–
magnitude.
tlcal points on opposite blades of a two bladed
rotor are shown in Fig. 11. Fig. 12 shows the
phase and coherence between loads on the same
blade but separated by a distance of one fifth of
the rotor height.
Both Figs. 11 and 12 are for a
50ft (15m) dlemeter turbine using a one dlmen–
s]onal wind simulation at low wind speeds where
there is no stall.
The coherence funct]on ls a
measure of the correlation of the calculated
The coherence 1s also useful as an estl–
loads
mate of the statistical significance of the calcu–
A coherence near one Indicates a
lated phase.
consistent phase relat]onshlp between the loads at
that frequency while a coherence near zero indl–
cates that the calculated phase has no meaning.
Larger diameter turbines were also examined with
The only dlffer–
the one d]menslonal wind model.
ences were an overall reduction in the coherence
and a sl]ght increase in the percent random con–
The psd’ s and
trlbut]on for the larger turbines.
phase relationships remained essentially the same.
Another factor which Increases the random contri–
butlon IS the amount of atmospheric turbulence.
This 1s controlled In the model by ad]ustlng the
surface roughness coeff]clent (zo).
Summarv
and Conclusions
A method for simulating a single po]nt w]nd time
ser]es ]s outl]ned. This method produces a
Gauss Ian random process with the specified tur–
One dimensional
bulence power spectral dens]ty.
var]at]ons are obtained by using Taylor’s frozen
turbulence hypothesis and a single point wind time
TABLE 4
NORMAL
Cos
SIN
—
–3 40
–1 43
–.069
762
1
2
3
4
5
Table
TANGENTIAL
—
PER REV
FREQ
248
4
%RANOOM
Cos
SIN
7ZRANDOM
——
1.32
–1.36
–1.57
–.936
055
.202
– 607
– 776
– :348
.246
76
‘?3
85
81
97
———
1
10
0.35
3 33
– 111
.153
50
‘iZ
95
99.
z?
The frequency content of the blade loads, calculated using a tur–
bulent wind (~ = 34 mph), is divided Into one per rev wide bands
around each per rev frequency.
The determlnlstlc per rev fre–
quencies are written In terms of cosine and sine coefficients.
The
random part is written as a percentage of the total variance In each
frequency band.
TABLE
5
NORMAL
PER REV
FREQ
Steady
1
19 7
6 38
2.52
1.48
.396
2
3
4
5
Table
10
5
Turbulent
19 3
5 99
2.36
.545
.206
TANGENTIAL
Z dlff
–2.0
–6.1
–6.5
–63.2
–46 .0
Steady
1.00
1.27
1.77
.714
.632
Turbulent
—
.994
1.05
1.24
.706
.178
Z dlff
—
–5.6
–17.3
–30.0
–1 1
–78 .6
The deterministic
rms variation at each per rev frequency is shown
for loads calculated in a 34 mph mean wind speed assuming both
The percentage difference between them
steady and turbulent winds.
is also shown.
180.0
q
90.0
E
0.0
4
1
10-’
(a)
1
10°
1
I
Id
(a)
mmumcy
FRi3quENcy(CYCLESPER REV)
(cycLEsPER REV)
10 [
1.0
A
A
I
1
08
0.8
W
u
z
w
$
z
0
u
0.6
0,4
0.6
04
0.2
~
0.2
p,
:11 :’,L
FREQUENCY
Id
10°
10-’
(b)
(b)
(cYcIXs
PER REV)
rREfauENcY (CYCLES pER REV)
1800
1800
90,0
w
w 90.0
s
n.
0.0
0,0
1
10-’
(c)
1
10°
10-’
1
10’
(c)
10’
10°
FREQUENCY (cycLEspER REV)
rREqumfcY (CYCLES PFR REV)
l.o
0,80.6
0.60.4
0.4-
0.2
0.2-
~
,,,,,,,M
10-’
,!, L
10°
ld
FREQUENCY (CYCLES PER REV)
iwmumcy
(CYCLES pER REV)
Fig.
Fig.
11
Estimates of phase and coherence between
the loads at the equator of the two
blades of a VAWT rotor; (a) and (b) are
for the normal forces, (c) and (d) are
for the tangential forces.
12
Estimates of phase and coherence between
the loads at two different points,
separated by a distance of one fifth of
the rotor height, on the same blade; (a)
and (b) are for the normal forces, (c)
and (d) are for the tangential forces.
11
series
A three dimensional wind simulation 1s
produced by creating single point s]mulatlons at a
number of Input points.
These t]me series are
random and partially correlated In a way that
matches the emplrlcal coherence function suggested
by Frost
4
Lobltz. D. W., and Sulllvan, W N., “A
Comparison of Flnlte Element Predictions and
Experimental Data for the Forced Response of
the DOE 100kW Vertical AXIS W]nd Turbine, ”
SAND82-2584 (Albuquerque, NM.
Sandla
National Laboratories, February, 1984).
The simulated wind turbulence ]s tracked as lt
passes through the rotor by using a multlple
streamtube representation
of the flow. B1 &de
loads are calculated using a momentum balance
Before reducing
apprOach developed by Strickland.
the blade load time series to the form of psd’s
and csd’s, the random and determlnlst]c parts are
separated using the Buys–Ballot
filter,
5
Frost , Walter, Long, D. H. , and Turner,
R.E>
“Englneer]ng Handbook on the
Atmospheric Environment Guldellne for Use
Wind Turbine Generator Development, ” NASA
Technical Paper 1359, December, 1979
The load psd’s Ind]cate that the turbulent energy
In the wind ]s shifted to higher frequencies In
the blade loads
This ]s due both to the motion
of the blade through the turbulence and the magnl–
fylng effect of small variations In angle of
The percentage of the total variance of
attack
the load that 1s random lS comparable to the per
rev contr)butlon
at lower frequency, but lt doml–
nates the loads at h]gher frequency.
This random
percentage
Increases with turbine SIZ? and amount
of turbulence
The most Important result is that when steady
winds are assumed, the high wind per rev loads are
overpredlcied.
This IS consistent with the di
ference between actual measured stresses and
stresses calculated using the steady w]nd mode
The difference between the determlnlstlc per rev
components )n steady and turbulent winds IS due to
the nonl)nearlty
In the relationship between angle
of attack and blade load, especially in h]gh winds
where stall in occurring.
The actual magn]tude of
the difference has not been determined since the
aerodynamic stall model 1s approximate.
More
accurate estimates of the stochastic loads In high
winds require a valldated dynemlc stall model
References
12
1.
Templln, R. J , “Aerodynamic Performance
Theory for the NRC Vertical–Axis Wind
Turbine, ” National Research Council of Canada
Report LTR–LA–160, June, 1974.
2
Strickland, J H , Smith, T , and Sun, K
Vortex Model of the Darrleus Turbine.
An
Analytical and Experimental Study,”
Sandla
SAND81-7017
(Albuquerque, NM
National Laboratories,
June, 1981).
3.
Carrie, T. G., Lobltz. D W., Nerd, A. R.,
Watson, R. A., “F~nite Element Analysls and
Modal Testing of a Rotating Wind Turbine, ”
Sandla
SAND82-0345
(Albuquerque, Nh.
National Laboratories, October, 1982).
“A
In
6
Shlnozuka, M., “Simulation of Multlvarlate
and Multldlmenslonal
Random Processes,r’
Journal of the Acoustical Society of America,
Vol. 49, No. 1 (part 2), p. 357, 1971.
7
Smallwood, D. O., “Random Vibration Testing
of a Single Test Item with a Multlple Input
Control System,” Proceedings of the Institute
of Environmental
Sc]ences, April, 1982.
8
Thresher, R. W., Honey,
W E., Smith, C.
E., Jafarey, N., Lin, S.–R., “Modellng the
Reponse of Wind Turbines to Atmospheric
Turbulence, ” U. S. Department of Energy
Report DOE/ET/23144-81/Z,
August, 1961.
9
W E,, Hershberg,
Thresher, R. W, , Honey,
E. L , Lln, S.–R., “Response of the MOD–OA
Wind Turbine Rotor to Turbulent Atmospheric
Winds,’< U. S. Department of Energy Report
DOE/RL/10378–82/1,
October, 1983.
10
Flchtl> G. H., “COvariance Statistics of
Turbulence Veloclty Components for Wind
Energy Conversion System Design –
PNL–3499
Homogeneous , Isotropic Case,”
(Paclflc Northwest Laboratories, Rlchland,
WA) , September, 1983.
11
A
Strickland, J. H , “The Darr eus Turbine.
Performance Predlctlon Model Using Multlple
Streamtubes,”
SAND75-0431 (A buquerque, NM.
October, 1975).
Sandla National Laboratories
12
Koopmans, L. H. , The %ectral
.knalVSIS of
Time Series, Academic press, New York, 1974.
13
Anderson, M. B., Slr Robert McAlplne & Sons,
Ltd. , London, UK, private correspondence,
May, 1984.
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J. Savino
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P.O. BOX 2267
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Mexico
RANN, Inc.
260 Sheridan Ave., Suite 414
Palo Alto, CA
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Chairman of the Board
Renewable
190 south
Honolulu,
Attn:
G.
Energy Ventures
King Street, suite
96813
HI
W. Stricker
2460
The Resources Agency
Department of Water Resources
Energy Division
1416 9th street
P.o. BOX 388
95802
Sacramento,
CA
Attn:
R. G. Ferreira
Reynolds Metals Company
Mill Products Division
6601 West Broad Street
23261
Richmond, VA
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Dept.
A. Robb
Memorial University of Newfoundland
Faculty of Engineering
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St. John’s Newfoundland
CANADA AIC 5S7
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Rocky Flats Plant
P.O. BOX 464
Golden, CO
80401
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Dr . -Ing. Hans Ruscheweyh
Institut fur Leichbau
Technische Hochschule Aachen
Wullnerstrasse
7
GERMANY
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Establissement
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77, Rue de Serves
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Schreiner
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National Atomic Museum
87185
Albuquerque,
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Gwen
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Riso National Laboratory
DK-4000 Roskilde
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Technion-Israel
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Department of Aeronautical
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Haifa, ISRAEL
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Kingston Polytechnic
Canbury Park Road
Kingston, Surrey
UNITED KINGDOM
Kent
Texas Tech University
Mechanical Engineering
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79409
LuDbock, TX
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.J. W. Oler
J. Strickland
Smith
Technologico
Apartado 159 Cartago
COSTA RICA
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Rico
Tulane University
Dept. of Mechanical Engineering
70018
New Orleans, LA
R. G. Watts
Attn:
Solar Initiative
Citicorp Plaza, Suite 900
180
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94612
Oakland, CA
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L, ,.:.
Sarenson
,.
Rc,:,:ilde University
Ene[gy Group, Bldg.
~]:~:;~>.
?.0. BOX 260
DK-400 Roskilde
DENF?ARK
R. J. Templin (3)
Low speed Aerodynamics
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NRC-National
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Montreal Road
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650 Ford Street
Colorado Springs, CO
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Center
17.2
J, M. TUrner
Terrestrial
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Energy Conversion
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Aero Propulsion
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Wright Aeronautical
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Air FOrCe
45433
Wright-Patterson
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57701
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Southern California
Edison
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southern Illinois University
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The University of Reading
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Whiteknights,
Reading, RG6 2AY
ENGLAND
United Engineers and Constructors,
PO BoX 8223
Philadelphia,
PA
19101
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University of Alaska
Geophysical
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99701
Fairbanks, AK
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University of California
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92521
Riverside, CA
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Dr. P. J. Baum
University of Colorado
Dept . of Aerospace Engineering
80309
Boulder, CO
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J. D. Fock, Jr.
Sciences
Stanford University
Mechanical
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Design Division
94305
Stanford, CA
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University of Massachusetts
Mechanical anti Aerospace Engineering
01003
Amherst, MA
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Stanford University
(2)
Civil Engineering
Department
94305
Stanford, CA
C. A. Cornell
Attn:
S. Winterstien
University of New Mexico
Kew Mexico Engineering Research
campus P.o. Box 25
87131
Albuquerque,
NM
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G. G. Leigh
stanford university
and
Dept . of Aeronautics
Astronautics
Mechanical
94305
Stanford, CA
Holt Ashley
Attn:
16
Engineering
Inc.
University of Oklahoma
Aero Engineering
Department
73069
Norman, OK
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University
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Sherbrooke,
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8th Floor
101 South Webster street
Madison, WI
53702
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JIK 2Rl
R. Camerero
The University of Tennessee
Engineering
Dept . of Electrical
37916
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T. W. Recdoch
Robert Akins
Professor of Engineering
Washington and Lee University
Lexington, VA
24450
[JSDA, Agricultural
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:;outhwest Great Plains Research Center
79012
13ushland, TX
Attn:
Dr. R. N. Clark
[Jtah Power and Light
51 East Main Street
co.
p-o.
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84003
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UT
K. R. Rasmussen
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pullman, WA
99163
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F. K. Bechtel
WeSt Texas State University
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79015
Canyonr TX
Texas State University
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~>.o, Box 248
79016
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West
West Virginia University
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D .
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2129
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Sir Robert ?4cAlpine & Sons,
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