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Wind power potential and time response of wind energy machines
Solar Energy Vol. 37, No. I, pp. 15-23, 1986
0038-092X/86 $3.00 + .00 © 1986 Pergamon Journals Ltd.
Printed in the U.S.A.
W I N D P O W E R P O T E N T I A L A N D TIME R E S P O N S E OF WIND E N E R G Y M A C H I N E S A. S. BARKERJR. Trinity Western University, Langley, B.C. Canada and Energy Research Institute, Simon Fraser University, Burnaby, B.C. Canada V5A 1S6 (Received 20 November 1984; revision received 19 November 1985; accepted 17 December 1985)
AbstractnLow-cost digital wind speed histogram recorders were designed to survey the west coast of British Columbia. Results are presented for several shore and island locations in terms of an available power parameter. Additional short term measurements of autocorrelation and cross-correlation functions showed ten-second exponential correlation in velocity fluctuations and gave values for the root mean square fluctuation. A derivation is given of the response time ofa Darrieus wind energy converter, which has implications for the sampling time of any wind speed recorder, and for the power fluctuations to be expected from such a converter. appropriate, however, for use in predicting wind converter performance. For example, if the wind It is well known that the power available from a blows for one hour at 4 msec-1 and then one hour steady wind blowing with velocity ~ is proportional at 32 msec -J, the average speed is ~ = 18 msec -1 to the magnitude of this velocity cubed times the and (v3)1/3 = 25.4 m s e c - 1. Both of these means precrossectional area of wind captured[l, 2, 3]. dict full power output using eqn 2; however, the detailed wind information tells us that a wind enP~Area × v3 (1) ergy converter which obeys eqn 2 will give zero output for this two hour period. In practice, the power extracted from the wind and The above discussion shows that a useful wind converted to electric energy by the best wind enrecorder must measure and store more than just the ergy converters follows a far different law princimean speed. In the present study we have chosen pally because of limitation on blade design[2] at low to record a six 'bin' histogram of the wind speed v, and limitations on generator design at high v distribution. The recorder continuously measures Many wind energy devices in fact give a power outwindspeed and adds one count to the appropriate put P which is approximately constant over a range bin every 2 seconds. Once each quarter the data are of speedst; e.g. retrieved and scaled to convert to hours. The first bin gives the number of hours the wind had speed ~constant, for 14 < v < 30 meter sec - j between 0 and 4.5 m s e c - ~; the second bin between P = [ 0 , for v < 5 and v - 30 meter sec -1. 4.5 and 9.0 msec -1, etc. For the first five points (2) the histogram interval is 4.5 msec -~. For the sixth point the histogram interval is the number of hours Figure 1 shows a typical power curve of this type the wind blew between 22.5 and 40 msec -~. The and defines the cut-in speed Vo, and rated speed Z'r. transducer which measures v is a three-cup aneMost wind energy converters$ are shut down mometer which gives a signal proportional to wind above some windspeed vs to protect the blades. In speed, but gives no information on wind direction. our example this speed is vs = 30 m/s. Most wind energy converters will generate power Wind speed data is often quoted as a mean speed from a wind which comes from any direction or for a given location. Reduction of weather data to even from a wind which changes direction slowly the mean speed or even to the mean of v3 is not (i.e., over a period of several seconds). For this reason the extra complexity of recording wind dit Most of the measurements discussed in this paper rection has been avoided. are made by a device which measures wind speed v, not To accomplish a survey of a large area (e.g., the velocity ~, under conditions where the fluctuations in 3 (in west coast of Vancouver Island) many wind redirection and magnitude and the spatial derivatives of each) are small on a scale of 1 sec for time changes and cording sites are required. In our work we have de20 cm for spatial changes. This ignoring of the vector na- cided to design an instrument which has low cost ture is valid for all of the considerations in this paper, and (less than $200 parts cost) and low power drain so we use "speed" in most of our discussion. thatit will operate on dry-cell batteries. These conThis paper follows the modern practice of naming siderations led to the concept of separating the remachines which couple energy from the wind and convert it to mechanical or electrical energy, wind energy con- corder and the readout devices. The major circuits for generating the time base, for counting and for verters (rather than windmills). I. INTRODUCTION
15
16 POWER
A. S. BARKER JR.
proach is aimed at a simple analysis which will allow for practical assessments of wind power potential. Section 4 presents a 'one number' average power estimate for some wind sites and discusses the assumptions underlying such estimates. 2. WIND DATA--COASTOF B.C.
Wind recorders were developed as discussed above and mounted near high water mark at three sites along the west coast of Vancouver Island between Tofino and Uclulet (see Fig. 2). Data was gathered over various periods (usually 3 month integration periods) with the cup anemometers 30100 meters inland from the high water mark and 30Vo Vs 40 m above mean sea level (AMSL). Since low cost WINDSPEED was important for this study, a 4-m mast was atFig. 1. Wind machine power output as a function of wind tached to the top of the tallest available tree at each speed for a hypothetical, but typical, modem wind energy site, which generally put the anemometer 10-15 m converter. The cut-in speed, rated speed where the con- above all nearby obstructions and considerably verter reaches rated power Pr, and shutdown speed are higher above all obstructions to the seaward side. noted on the horizontal axis. The data showed close equality between the sites when identical time periods were analysed. The the storage registers have been designed on one cir- data from the three sites have been averaged tocuit board which can be mounted with a battery in gether and presented as a 6-point histogram in Fig. a waterproof case. This device is called the wind 2. Figure 2 also shows data collected from various recorder. A second circuit, the readout instrument, others sources converted to the same type of hiscontains a four digit liquid crystal display panel and togram[6, 7, 8, 9]. Table 1 lists the recorder sites, the logic for interrogating the storage registers in period of observation and other details. Note that the wind recorder. A facility for resetting all storage while the first bar in the histogram for Tofino-Ucluregisters to zero is also provided. For the present let is quite high, this region of low velocities would study five wind recorders and two readout instru- theoretically yield very little power because of v3 ments were constructed and tested. law (eqn 1). For a practical wind energy converter In addition to considerations of wind power in with the power curve shown in Fig. 1, there would steady winds, one must question the effect of a fast be no power generated in this region. A significant rising puff of wind on power output. Most small wind power potential must rely on a significant cup-type anemometers are capable of measuring a height for the second, third and fourth histogram wind velocity 'pulse' with risetime of 1 to 3 sec. bars. This is discussed more fully in Section 4. Hot-wire anemometers can extend the measuring Figure 3 shows some wind speed data taken over range down to 20 Ix sec. If the wind velocity v puffs a two-day period in April 1982 at the Christopher by A v and falls again within 1 second say, should Point site (Table 1). This site has a tower 30 m inthis (v + Av) be cubed in a wind power assessment? land from high water mark with 3-cup anemometers To answer, we obviously must know the dynamic re- at 20, 28, and 38 m AMSL. 40 m further inland is sponse time of the wind energy machine. We also a 50 kW Darrieus type wind energy machine with must know whether such puffs exist over the entire a 2-blade rotor 11.2 m wide by 16.8 m high. Labarea of the wind machine, i.e., what are the cor- oratory tests were conducted on the anemometers relations in wind fluctuations between points sep- to determine their time response. For step function arated in space. changes in wind speed the anemometer output In this paper the above topics are dealt with from changed first quickly, then more slowly towards the the point of view of practical recording of wind data value appropriate to the final wind speed. This dyfor wind power assessment. The theory of wind namic response is often characterized approxiflow and fluctuations over anything other than fiat mately by a distance "constant" Lo, i.e., the prodfeatureless terrain is in its infancy. For a recent re- uct of the time taken to change to within 37% of view of such theories, the interested reader is re- the final speed multiplied by the initial speed. For ferred to the proceedings of the Portland Confer- our 3-cup anemometer this distance constant is 6 ence on Wind Energy Characteristics[4] and texts 8 meters for decreasing speeds which start at values on atmospheric turbulence[5]. In Section 2 wind in the range 4-20 m/s with the larger value holding data gathered on the west coast of British Columbia at the higher speeds. Therefore, the time constant is presented in histogram form. Some data on fluc- varies from 0.4 to 1.5 seconds according to the retuations is also presented. In Section 3, the dynamic lation, T = Lo/v. This is sufficiently fast for the response of a wind machine is considered. The ap- purposes of the present study. Figure 3(a) shows a
17
Wind energy machines
/ Christopher Point
_k iCop s
55ON
A=.49 Weothership
~ I~
~A I.55 ~
Sea - Toc Airport
.i - -A. 1 1
"~'~-~~
50°N
1450
140°
135=
130°
125° W
Fig. 2. Annual wind speed distribution histograms for six sites on the west coast of British Columbia and Washington. The Tofino-Uclulet graph was constructed using a two second response time recorder which gives a more accurate assessment of wind potential as described in the text.
Table !. Wind data stations
of Ane~meter (ANSI.) of Record ( y e a r s )
Name
Location Lat(N)-Long(W)
To f Ino-Ue l u l e t ( t h i s study)
49°-125°40 '
Sea-Tat airport (Ref. 6)
#7°27 ' - 1 2 2 ° 1 9 '
137m 17Y
.11
Tatoosh Is.
4802 3 ' - 1 2 4 0 4 4 °
35m
.)4
(source of data)
(Ref. 6)
Height Time P e r i o d
30-40m 2Y
Annual Energy Avatlabitlty A
.14
17Y
Cape St. 3ames (Ref. 7)
51056 ' -131001 '
89m 25Y
.49
Weather Ship
50°-145 °
2~m
.55
(Ref. 8 ) Chrlstopher Point ( t h i s study & Ref. ? )
15Y 48°20'-121)°40 '
28m 1.5Y
.22
18
A. S. BARKERJR.
short portion of a wind speed recording designated v2. Samples like v2 were digitized using a 1 sec sampiing rate for runs of 500 seconds. To study the fluctuations in v2 we must calculate power spectra or autocorrelation functions[10]. F o r each run, the mean ~ is calculated and subtracted from every point. The autocorrelation function tO of the speed about the mean is then calculated as
+~(T)
= (v(t) -
~)(v(t + T) -
(3)
~).
F o r a statistically stationary variable, the result will be independent of t and symmetric in T[10]. Figure 3b shows this function for a portion of the v2 data. We find that to always has a fast decay from a finite value at T = 0. There are smaller peaks beginning near T --- 20, but these depend on where in the data
(m/s)
(a)
~V2V 2 (mZ/s z)
E.8
~
(b)
;~XX~--~ Urms= ~ - ~ o ) = 13
12 ,
5
I0
15
20
rrel. t i m e
1.34m/s
Ilsec.
0
25 t
(s)
(s)
V4 (m/s)
(C)
13
(d)
( m2/s z)
2.6
= 1.6 m/s
Urms = ~ o )
2.0
12 correl.time
9 sec.
1.0
II I0
I
I
5
I0
v
I
I
15
20
v
I 25
o 2O
t (s)
T
(e)
30
,./FN, I i 40 ~,~60 ~
I
j /T'~.IOOX~-
T(s)
(f)
50
,,/3
= 21 kw
I/~Prms
20 --
= 7.0 kw
25 I0-I 15
\
20
0
25
30
35
(st )
20
40
60 ~ 8 0
~I00/ T(s)
Fig. 3. Short portions of various recordings of wind speed and wind machine output at Christopher Point. Because of the short (500 second) data records the correlation functions yield most accurate information over the first 20-40 sec.
19
Wind energy machines we begin the correlation calculation (i.e., on t). If we ignore these weak peaks, the data show that is approximately exponential with a correlation time for decay to 1/e for the fluctuations of 11 sec. The value of ~ at T = 0 yields the root mean square deviation about the mean. This value is shown in the figure. Figure 3(c) shows number 4 run from another day. Here the mean velocity 3 4 is slightly less than 32, but the fluctuations are larger and faster as shown by the autocorrelation in Fig. 3(d). We have taken Fourier transforms of ~ to find the power spectra of the fluctuations[ 10]. These spectra show a falling offat frequencies near and above 0.01 Hz as would be expected for the correlation times noted in Fig. 3. The peaks in ~ cause, however, extensive peaks in the power spectrum which make the latter of little value for analysis. Kaimal has taken extremely large data sets from the high tower at Boulder Colorado[ 11 ]. Using Fourier transforms, he found an f-5/s fall off for frequencies f greater than about 0.01 Hz for the horizontal components (the Kolmogorov law). He has also correlated spectral features with height of anemometer and with the depth of the convective boundary layer. These results for very large data sets confirm our methods using rather restricted sets as long as we restrict our interest to fluctuations with frequencies below a b o u t . 1 Hz. In this paper we are concerned mostly with power generation, and not as much with rotor stress. As will be shown, high frequency variations in windspeed are smoothed out by the response of the wind machine. They are not important for power generation, but may be very important as driving forces for high frequency mechanical stresses which can lead to machine breakdown. We do not analyze such stresses here. Figure 3 shows that two parameters may readily be extracted from the data. One of them is the rms fluctuation in velocity. Since we employ a single vertical axis anemometer our measurement yields a value appropriate to the longitudinal plus transverse horizontal components. These are usually denoted by u and v. Several studies have related the velocity fluctuations to the roughness of the surface over which the wind approaches the measurement point. We are interested principally in winds which approach a site over the sea so that the surface roughness which is appropriate is the wave height. Runs v2 and U4 yield t/~ms/32
=
~vlv z
(m2/s2)
(o)
1.0
~
o/ / I
J (o)
= 0.95
e o r r e l . time = IO.Os
;o
-20
T(s) (b)
~Jv4p 3
I0.0
/~~"~
~"--'----'------~ s h i f t 9 . 6 s
(o)= 9.e kWmI,
5.0
\f..~..--correl.
t i m e = iO.
s
0.10
u]m~4 = 0.13. F o r these runs wave heights were roughly 0.5 m and there were whitecaps on some wave crests. The preceding afternoon, however, during the initial set up of the instruments, there were somewhat slower but very steady winds (35 = 9.2 msec-~). At that time the sea was quite smooth and we measured U~ms/~ 5 =
While a detailed study of the dependence of the above ratio on sea surface conditions has not been attempted, the two higher values are consistent with values obtained on the coast of Denmark[12]. Figures 3(e) and 3(13 show power output from the Darrieus wind machine with a 3-phase induction motor feeding the local power grid. The nameplate rating of the motor (which acts as generator at sufficiently high wind speeds) is 50 kW. Again, 500 second records were taken for analysis. Figure 3(e) shows a short portion of run P3 containing a large fluctuation. This run is concurrent with run v4 (Fig. 3(c)), and the cause of the power fluctuation is seen immediately as the velocity fluctuation recorded 10 seconds earlier at the anemometer (which is 40 m upstream). The autocorrelation function of the P3 data clearly shows an exponential decay characteristic with a decay time of 10.5 sec. As with the speed data, the small peaks and structure in t~ beyond T = 30 sec are not stable features. They change for subsequent portions of the run while the exponential decay characteristic remains constant. Finally in Fig. 4 the spatial characteristics of the
.02
P
T(s)
Fig. 4. (a) Cross correlation for two anemometers spaced vertically by 10 meters. Peaks beyond T = 20 sec should be ignored, however the value of the correlation time is precise to better than 5%. (b) Cross correlation between wind speed and power generated by the wind machine located 40 meters downstream from the wind speed anemometer.
20
A. S. BARKER JR.
wind fluctuations are studied by calculating the cross correlation function between data records vl and v2. vl is not shown in Fig. 3 but is recorded 10 m higher on the same instrument mast. We thus obtain a measure of the correlation over a distance of 10 m which is approximately the diameter of the wind machine rotor. Figure 4 shows that the fluctuations are strongly coupled over this distance. The correlation time is the same as for ~)2 and O(0) has dropped by only 50%. While these data apply to a vertical correlation, we can anticipate the horizontal correlations will be somewhat similar. We conclude therefore that under the conditions at this type of test site a substantial portion of the fluctuations in the low frequency range affect the entire wind machine rotor and drive related fluctuations in power output. This result is confirmed by Fig. 4(b) where the crosscorrelation of v4 and P3 is calculated. Again a strong peak is obtained with, however, a significant delay (9.6 see.) in the correlations. 3. WIND MACHINE RESPONSE
In this section we examine the dynamic response of the two-blade vertical-axis Darrieus wind machine at Christopher Point. Equation 2 and Fig. 1 given earlier are approximations to the power vs wind speed response function. They represent only the static characteristic expected for steady winds. Figures 3(a) and 3(e) show that, in practice, very rapid changes are often encountered. We now examine these fluctuations in more detail, but first note a point of practical importance. A wind energy machine may have a response time of a few seconds (as this machine does) and can therefore follow many wind fluctuations. The practical problem is whether the system can fully use the power so generated. If the machine is feeding a power grid the fluctuation can cause voltage fluctuation unless the dynamical response of the primary generators are fast enough to respond and correct. Diesel generators with electronic governors can respond in about 1 sec. H y d r o generators are much slower. Both systems may respond 'wastefully.' That is if the system is running at an optimum load level, and a wind energy machine gives a positive increment of power to the grid, the primary generator can reduce its contribution, but it does this by running in a less efficient m o d e - - a t least temporarily. The net cost saving to the system may be positive, but may not be simply the contribution of the wind energy. The fluctuation effects can be reduced by adding the outputs of several wind machines, but Fig. 4 suggests that they will have to be quite widely separated. We cannot pursue this topic here, but proceed with an analysis of the time response of the 50 k W Darrieus machine. Our aim is to give a simple analysis which brings out the major characteristics of importance. Kirchoff has carried out a transfer function analysis for
a 25 kW machine[13]. While our methods are different, it is worth noting that his system shows a decay time of about 5 sec for decay to 1/e (or 33 sec for decay to 0.2%) for a resistive load compared with our result of about 2 sec for the Darrieus machine connected to the power grid. Our analysis begins by assuming a simple, single time constant, relaxation form for the wind machine coupled to the power grid (the load). Figure 5 gives the electric circuit analog. The windspeed v(t) is modeled as a voltage which drives an R-C circuit. The output voltage across the capacitor C is proportional to the machine power P(t) delivered to the grid. The frequency dependent transfer function is
1 + itor"
(4)
The model is applicable only in wind regimes where P(t) is positive. Since the step response and impulse response of the circuit are well known, we search the time records, P3 and Va (Fig. 3), for instances of each. Measuring the decay after ten such fluctuations and averaging we find the time constantt = R C = 2.2 _ 0.8 sec.
(5)
Using a different method, we note that Fig. 4(b) shows a delay of 9.6 sec between P3 and v4 (measured 40 m upstream). Since the mean wind speed for these runs is 11.9 m/s, 3.4 sec is accounted for by the 40 m separation. Thus 9.6 - 3.4 = 6.2 sec could be attributed to the time constant of the wind machine system. This method is approximate, however, since some of the large fluctuations contained in the P3 data carry us out of the linear approximation implicit in the model. We regard this result merely as an order of magnitude confirmation of the above measurement. A theoretical calculation proceeds as follows.
R
IV(t)-Vo ,
C
P(t)
-1-
Fig. 5. Circuit model for the response of the 50-kW Darrieus wind machine. Only wind speeds greater than vo are considered. For such winds the instantaneous speed v(t) has vo subtracted and the result is applied as a voltage to the circuit as shown. The voltage across C is proportional to the wind machine power output and the frequency response function is (I + itor)-1 where to is the frequency in radians per second. t x is defined as the time for decay to within 1/e of the final value. Some other authors have used a different definition which results in their time constant being larger by a factor of 2~r.
21
Wind energy machines Analysis of the torque that a single Darrieus blade contributes to the generator gives the form shown in Fig. 6. The torque has a steady component Tar, as well as large contributions at to and at 2to, where to is the rotational frequency of the machine. This torque is converted directly to power. The following assumptions can now be made. First, the blade is well designed so that the lift forces are much larger than the drag forces when averaged over 1 revolution. Second, the bearing and gear losses are small. Using these assumptions (which our estimates show to be reasonable), if the machine is generating at a rotational speed to and a torque T~, = a + bto, we assume the wind speed abruptly drops to zero. The load on the machine is the power being generated so that the equation of motion is I~b = - a
- bto
(6)
The solution is simply a constant plus a decaying exponential with time constant -r = I/b, where I is the moment of inertia of the rotating system and b the slope of the characteristic in Fig. 6. Using 5% as an estimate of the no-load to full-load slip speed of the induction motor (i.e., of b), and I = 1300
m2kg we calculate = 1.2 sec (calculated). This result is in satisfactory agreement with our measured result above, confirming that the response is principally a single relaxation type (Fig. 5) with the coupling to the load providing the damping. Finally, the response model can be used to predict the effect of the number of blades on the Darrieus output. The present machine has two blades. Its torque is therefore the sum of two curves as shown in Fig. 6(a), but with the second displaced 180° from the first. The maxima and minima of the second blade, therefore, fall on the maxima and minima of the first blade giving an output which is approximately a sinusoid. Using this as an input to a simple relaxation response function as developed above with to = 8.4 sec -1 and r = 1.2 sec, we obtain a power output of
P(t)
=
(1 + 0.10 cos 2tot)
for the lowest dominant harmonic.
(a) (ToTue)
I 0
',v
180
I 360
e
(degrees)
(b)
rov
Tav = a÷bC0
LO Fig. 6. (a) Quantitative behaviour of the torque generated by one blade of the Darrieus machine at a wind speed of 15 m/s versus angular position of the blade with respect to the undisturbed windstream. Zero degrees corresponds to the blade being directly downwind from the center of rotation. Lift and drag functions averaged over the entire blade have been assumed. The average torque Tar is the driving function of interest here which generates the power output. (b) Schematic behaviour of Tar as a function of rotation speed of the blades. The slope of the graph determines the response time of the wind machine-generator combination when it is coupled to its load.
(7)
22
A . S . BARKERJR.
In a portion of the trace where P = 30 kW, the P4 data shows
P(t) = 30(1 + .14 cos 2tot)kW
(8)
These fast fluctuations have been smoothed out in drawing Fig. 3(e), but are always present in the data. The agreement (within 40%) between eqn 7 and eqn 8 is further confirmation of the response model. Using the characteristic of Fig. 6(a), it is easy to show that three blades considerably reduce and four blades almost eliminate these fast power fluctuations. Work at Sandia Laboratories has documented similar power fluctuations for two bladed 100-kW Darrieus machines. In addition, workers there have considered the filtering effect of drive trains using compliance and damping, and also the resonant modes of the structure and their excitation by the torque fluctuations[14]. 4. WIND ENERGY POTENTIAL
Figure 1 shows graphically the point emphasized in the introduction that no energy is gathered from low speed winds. F o r this reason we wish to analyse the data of Fig. 2 in a manner which recognizes this limitation yet gives a realistic, yet simply calculated, assessment of the yearly wind energy potential at a site for which histogram data has been recorded. We define an annual availability A as follows. A survey of available wind machines in the 40-2000 kW power range suggests that a threshold windspeed of 5 m/s provides a reasonable value of Vo for our purposes. At a wind speed vr somewhere in the range 1.4 Vo-3 vo, many machines have reached their full rated power. We choose Vr = 14 m/s for our analysis. As Fig. 1 indicates, many machines are shut down at high s p e e d s - - g e n e r a l l y at a value in the range 3-10 vo. This shutdown is not as fundamental a limitation as the low speed threshold. Since our histograms have little weight above 25 m/s we will ignore the shut-down and count all winds above yr. A is calculated by first multiplying the wind speed histogram by the wind machine power curve normalized to a height of I. The result is then integrated over all wind speeds and divided by 8765, the number of hours in one year. Note that our discussion of response time shows that for the 50-kW Dardeus machines (and other similar machines), wind speed histograms should be constructed using a recorder with an averaging time of approximately 2 sec. The data for Tofino-Uclulet and Christopher Point were taken in this manner so that the value A = .22 properly includes the energy available in fluctuating winds. The remaining data are taken using various m e t h o d s - - o f t e n involving one reading per hour. These latter data give therefore a less reliable estimate of the wind energy. Figure 2 and Table 1 list the results of our analysis. The highest energy location is at the weather ship station. The availability value A = 0.55 predicts
that a 50-kW machine at this location would generate Etot = 0.55 x 8765 x 50 kW = 241 MWhr
(9)
annually. This 55% availability contrasts sharply with A = . 11 at Seattle-Tacoma airport. The same machine located at the airport would yield only 48 MWhr annually. The somewhat surprising result is the value A = 0.14 recorded over a two-year period for the coast near Tofino-Uclulet. In spite of the exposed shore location, the value is significantly lower than the offshore and island locations. The quarterly data showed a somewhat higher value of A for the spring quarter with a correspondingly lower value for the summer. Such short term quarterly data is not felt to be reliable enough to present here. The data show that Christopher Point, which has a shore location similar to the Tofino-Uclulet location, benefits from the channeling effect of the Juan de Fuca Strait. This effect is well known to small boat sailors who ply these waters. Finally, the two island locations (Tatoosh and Cape St. James) show extremely high wind energy potential due to their off-shore location and much larger unobstructed approach field for winds from many directions. The factors discussed above can be expected to be general features which apply to any coastline where the prevailing weather pattern is in the onshore direction. 5. CONCLUSIONS
In this study we have examined the dynamic response of a wind energy machine and noted the inadequacy of quoting the mean wind speed at a site as an indicator of wind power potential. While a complete wind speed record can be obtained using modern highspeed data recorders with megabyte memory capacity, we suggest recording a six-point histogram as a viable low-cost compromise. We note also that six points is sufficient to fit Weibull or Rayleigh wind speed distributions to the data if this is desired for comparison with other data records. In addition we have shown that the anemometer and data conditioning must allow recording of all fluctuations down to times as short as 1-2 sec. In the analysis carried out above, the histogram is used in a very simple way to give A, the fraction of rated power that a wind machine will generate on the average (sometimes called the load factor or capacity factor). A is a much more meaningful parameter for describing a site than the mean wind speed. It does, however, depend on the assumed power curve (Fig. 1). Of course the more fundamental description of a site is its wind speed histogram. The measurements for various west coast locations show A ranging from .14 to .49 and an rms fluctuation ratio of about 0.1 when the sea is disturbed, which is always the case when the machine
23
Wind energy machines is generating near its rated power. The 50-kW Darrieus was found theoretically and experimentally to have a response characterized by a simple decay time of about 2 sec. The power, however, shows much longer fluctuations on the average, controlled by the wind turbulence. Our method of recording rather short time series for study of autocorrelations and cross-correlations appears to be quite useful as long as small variations in t~ beyond T = 30 seconds are ignored or smoothed. The importance of wind fluctuations for power generation has been emphasized. A true cost-benefit estimate would have to include a full systems analysis including the fluctuations and their relation to system stability, number of machines and their spacing, number of blades per machines (especially if of Darrieus type), and response and efficiency of other generators supplying the same system. An important extension of this study would be the measurement of crosscorrelations of wind speeds over horizontally separated distances of 20 to 200 meters, both in relation to single wind machine performance and to the fluctuations to be expected from multiple-machine wind farms. An analysis has been given of the response time of the 50-kW Darrieus wind machine, and reasonable agreement obtained with the measured response. For larger machines of the same type we expect both the moment of inertia I and the slope b to increase. The change in response time • may be rather small, however the answer will depend on the detailed design of the blades and generating system. Once these are known, ~ can be calculated by the methods given above.
Acknowledgements--The author acknowledges with grat-
itude partial support of this work by Energy, Mines and Resources Canada, and the use of the Christopher Point wind site equipment by B.C. Hydro. It is a pleasure to acknowledge also many helpful contributions and conversations with N. Vander Kwaak, and with colleagues at the Simon Fraser Energy Research Institute.
REFERENCES
1. A. Betz, Wind Energy and its Utilization by Windmills. Bandenhock and Ruprecht, Gottingen (1926). 2. Solar Energy Technology Handbook. Edited by W. C. Dickinson and P. N. CheremisinoffPart A, Chapter 22 Marcel Dekker, N.Y. (1980). 3. E. W. Golding, The Generation of Electricity by Wind Power. Halsted Press-John Wiley & Sons, New York (1976). 4. Proceedings of Conference and Workshop on Wind Energy Characteristics and Wind Energy Siting. U.S. Dept. of Energy (1979). 5. H. Schlicting, Boundary Layer Theory, 6th Ed. McGraw Hill, New York (1968). Also H. A. Panofsky and J. A. Dutton, Atmospheric Turbulence. John Wiley and Sons, New York (1984). 6. Wind Energy Resource Atlas, Vol. 1 (Northwest Region) available from U. S. Dept. of Commerce-NTIS document PNL-3195-WERA-1. Canadian Climate Normals, Vol. 5--Wind. Gov. of 7. Canada (1982). 8. Unpublished data from Atmospheric Environment Service, Vancouver, B.C., Canada. 9. Unpublished data from B.C. Hydro for Christopher 10. Point Wind Energy site. D. K. C. MacDonald, Noise and Fluctuations. Wiley & Sons, NY, (1962). 11. J. C. Kaimal in ref. 4. 12. J. A. Dutton and J. Hojstrup, ref. 4, page 59. 13. R. H. Kirchoff, ref. 4, page 131. R. O. Nellums, Field test report of the D.O.E. 10014. kW vertical axis wind turbine. Report SAND84-0941 obtainable from Sandia Laboratories, Albuquerque, NM.
0038-092X/86 $3.00 + .00 © 1986 Pergamon Journals Ltd.
Printed in the U.S.A.
W I N D P O W E R P O T E N T I A L A N D TIME R E S P O N S E OF WIND E N E R G Y M A C H I N E S A. S. BARKERJR. Trinity Western University, Langley, B.C. Canada and Energy Research Institute, Simon Fraser University, Burnaby, B.C. Canada V5A 1S6 (Received 20 November 1984; revision received 19 November 1985; accepted 17 December 1985)
AbstractnLow-cost digital wind speed histogram recorders were designed to survey the west coast of British Columbia. Results are presented for several shore and island locations in terms of an available power parameter. Additional short term measurements of autocorrelation and cross-correlation functions showed ten-second exponential correlation in velocity fluctuations and gave values for the root mean square fluctuation. A derivation is given of the response time ofa Darrieus wind energy converter, which has implications for the sampling time of any wind speed recorder, and for the power fluctuations to be expected from such a converter. appropriate, however, for use in predicting wind converter performance. For example, if the wind It is well known that the power available from a blows for one hour at 4 msec-1 and then one hour steady wind blowing with velocity ~ is proportional at 32 msec -J, the average speed is ~ = 18 msec -1 to the magnitude of this velocity cubed times the and (v3)1/3 = 25.4 m s e c - 1. Both of these means precrossectional area of wind captured[l, 2, 3]. dict full power output using eqn 2; however, the detailed wind information tells us that a wind enP~Area × v3 (1) ergy converter which obeys eqn 2 will give zero output for this two hour period. In practice, the power extracted from the wind and The above discussion shows that a useful wind converted to electric energy by the best wind enrecorder must measure and store more than just the ergy converters follows a far different law princimean speed. In the present study we have chosen pally because of limitation on blade design[2] at low to record a six 'bin' histogram of the wind speed v, and limitations on generator design at high v distribution. The recorder continuously measures Many wind energy devices in fact give a power outwindspeed and adds one count to the appropriate put P which is approximately constant over a range bin every 2 seconds. Once each quarter the data are of speedst; e.g. retrieved and scaled to convert to hours. The first bin gives the number of hours the wind had speed ~constant, for 14 < v < 30 meter sec - j between 0 and 4.5 m s e c - ~; the second bin between P = [ 0 , for v < 5 and v - 30 meter sec -1. 4.5 and 9.0 msec -1, etc. For the first five points (2) the histogram interval is 4.5 msec -~. For the sixth point the histogram interval is the number of hours Figure 1 shows a typical power curve of this type the wind blew between 22.5 and 40 msec -~. The and defines the cut-in speed Vo, and rated speed Z'r. transducer which measures v is a three-cup aneMost wind energy converters$ are shut down mometer which gives a signal proportional to wind above some windspeed vs to protect the blades. In speed, but gives no information on wind direction. our example this speed is vs = 30 m/s. Most wind energy converters will generate power Wind speed data is often quoted as a mean speed from a wind which comes from any direction or for a given location. Reduction of weather data to even from a wind which changes direction slowly the mean speed or even to the mean of v3 is not (i.e., over a period of several seconds). For this reason the extra complexity of recording wind dit Most of the measurements discussed in this paper rection has been avoided. are made by a device which measures wind speed v, not To accomplish a survey of a large area (e.g., the velocity ~, under conditions where the fluctuations in 3 (in west coast of Vancouver Island) many wind redirection and magnitude and the spatial derivatives of each) are small on a scale of 1 sec for time changes and cording sites are required. In our work we have de20 cm for spatial changes. This ignoring of the vector na- cided to design an instrument which has low cost ture is valid for all of the considerations in this paper, and (less than $200 parts cost) and low power drain so we use "speed" in most of our discussion. thatit will operate on dry-cell batteries. These conThis paper follows the modern practice of naming siderations led to the concept of separating the remachines which couple energy from the wind and convert it to mechanical or electrical energy, wind energy con- corder and the readout devices. The major circuits for generating the time base, for counting and for verters (rather than windmills). I. INTRODUCTION
15
16 POWER
A. S. BARKER JR.
proach is aimed at a simple analysis which will allow for practical assessments of wind power potential. Section 4 presents a 'one number' average power estimate for some wind sites and discusses the assumptions underlying such estimates. 2. WIND DATA--COASTOF B.C.
Wind recorders were developed as discussed above and mounted near high water mark at three sites along the west coast of Vancouver Island between Tofino and Uclulet (see Fig. 2). Data was gathered over various periods (usually 3 month integration periods) with the cup anemometers 30100 meters inland from the high water mark and 30Vo Vs 40 m above mean sea level (AMSL). Since low cost WINDSPEED was important for this study, a 4-m mast was atFig. 1. Wind machine power output as a function of wind tached to the top of the tallest available tree at each speed for a hypothetical, but typical, modem wind energy site, which generally put the anemometer 10-15 m converter. The cut-in speed, rated speed where the con- above all nearby obstructions and considerably verter reaches rated power Pr, and shutdown speed are higher above all obstructions to the seaward side. noted on the horizontal axis. The data showed close equality between the sites when identical time periods were analysed. The the storage registers have been designed on one cir- data from the three sites have been averaged tocuit board which can be mounted with a battery in gether and presented as a 6-point histogram in Fig. a waterproof case. This device is called the wind 2. Figure 2 also shows data collected from various recorder. A second circuit, the readout instrument, others sources converted to the same type of hiscontains a four digit liquid crystal display panel and togram[6, 7, 8, 9]. Table 1 lists the recorder sites, the logic for interrogating the storage registers in period of observation and other details. Note that the wind recorder. A facility for resetting all storage while the first bar in the histogram for Tofino-Ucluregisters to zero is also provided. For the present let is quite high, this region of low velocities would study five wind recorders and two readout instru- theoretically yield very little power because of v3 ments were constructed and tested. law (eqn 1). For a practical wind energy converter In addition to considerations of wind power in with the power curve shown in Fig. 1, there would steady winds, one must question the effect of a fast be no power generated in this region. A significant rising puff of wind on power output. Most small wind power potential must rely on a significant cup-type anemometers are capable of measuring a height for the second, third and fourth histogram wind velocity 'pulse' with risetime of 1 to 3 sec. bars. This is discussed more fully in Section 4. Hot-wire anemometers can extend the measuring Figure 3 shows some wind speed data taken over range down to 20 Ix sec. If the wind velocity v puffs a two-day period in April 1982 at the Christopher by A v and falls again within 1 second say, should Point site (Table 1). This site has a tower 30 m inthis (v + Av) be cubed in a wind power assessment? land from high water mark with 3-cup anemometers To answer, we obviously must know the dynamic re- at 20, 28, and 38 m AMSL. 40 m further inland is sponse time of the wind energy machine. We also a 50 kW Darrieus type wind energy machine with must know whether such puffs exist over the entire a 2-blade rotor 11.2 m wide by 16.8 m high. Labarea of the wind machine, i.e., what are the cor- oratory tests were conducted on the anemometers relations in wind fluctuations between points sep- to determine their time response. For step function arated in space. changes in wind speed the anemometer output In this paper the above topics are dealt with from changed first quickly, then more slowly towards the the point of view of practical recording of wind data value appropriate to the final wind speed. This dyfor wind power assessment. The theory of wind namic response is often characterized approxiflow and fluctuations over anything other than fiat mately by a distance "constant" Lo, i.e., the prodfeatureless terrain is in its infancy. For a recent re- uct of the time taken to change to within 37% of view of such theories, the interested reader is re- the final speed multiplied by the initial speed. For ferred to the proceedings of the Portland Confer- our 3-cup anemometer this distance constant is 6 ence on Wind Energy Characteristics[4] and texts 8 meters for decreasing speeds which start at values on atmospheric turbulence[5]. In Section 2 wind in the range 4-20 m/s with the larger value holding data gathered on the west coast of British Columbia at the higher speeds. Therefore, the time constant is presented in histogram form. Some data on fluc- varies from 0.4 to 1.5 seconds according to the retuations is also presented. In Section 3, the dynamic lation, T = Lo/v. This is sufficiently fast for the response of a wind machine is considered. The ap- purposes of the present study. Figure 3(a) shows a
17
Wind energy machines
/ Christopher Point
_k iCop s
55ON
A=.49 Weothership
~ I~
~A I.55 ~
Sea - Toc Airport
.i - -A. 1 1
"~'~-~~
50°N
1450
140°
135=
130°
125° W
Fig. 2. Annual wind speed distribution histograms for six sites on the west coast of British Columbia and Washington. The Tofino-Uclulet graph was constructed using a two second response time recorder which gives a more accurate assessment of wind potential as described in the text.
Table !. Wind data stations
of Ane~meter (ANSI.) of Record ( y e a r s )
Name
Location Lat(N)-Long(W)
To f Ino-Ue l u l e t ( t h i s study)
49°-125°40 '
Sea-Tat airport (Ref. 6)
#7°27 ' - 1 2 2 ° 1 9 '
137m 17Y
.11
Tatoosh Is.
4802 3 ' - 1 2 4 0 4 4 °
35m
.)4
(source of data)
(Ref. 6)
Height Time P e r i o d
30-40m 2Y
Annual Energy Avatlabitlty A
.14
17Y
Cape St. 3ames (Ref. 7)
51056 ' -131001 '
89m 25Y
.49
Weather Ship
50°-145 °
2~m
.55
(Ref. 8 ) Chrlstopher Point ( t h i s study & Ref. ? )
15Y 48°20'-121)°40 '
28m 1.5Y
.22
18
A. S. BARKERJR.
short portion of a wind speed recording designated v2. Samples like v2 were digitized using a 1 sec sampiing rate for runs of 500 seconds. To study the fluctuations in v2 we must calculate power spectra or autocorrelation functions[10]. F o r each run, the mean ~ is calculated and subtracted from every point. The autocorrelation function tO of the speed about the mean is then calculated as
+~(T)
= (v(t) -
~)(v(t + T) -
(3)
~).
F o r a statistically stationary variable, the result will be independent of t and symmetric in T[10]. Figure 3b shows this function for a portion of the v2 data. We find that to always has a fast decay from a finite value at T = 0. There are smaller peaks beginning near T --- 20, but these depend on where in the data
(m/s)
(a)
~V2V 2 (mZ/s z)
E.8
~
(b)
;~XX~--~ Urms= ~ - ~ o ) = 13
12 ,
5
I0
15
20
rrel. t i m e
1.34m/s
Ilsec.
0
25 t
(s)
(s)
V4 (m/s)
(C)
13
(d)
( m2/s z)
2.6
= 1.6 m/s
Urms = ~ o )
2.0
12 correl.time
9 sec.
1.0
II I0
I
I
5
I0
v
I
I
15
20
v
I 25
o 2O
t (s)
T
(e)
30
,./FN, I i 40 ~,~60 ~
I
j /T'~.IOOX~-
T(s)
(f)
50
,,/3
= 21 kw
I/~Prms
20 --
= 7.0 kw
25 I0-I 15
\
20
0
25
30
35
(st )
20
40
60 ~ 8 0
~I00/ T(s)
Fig. 3. Short portions of various recordings of wind speed and wind machine output at Christopher Point. Because of the short (500 second) data records the correlation functions yield most accurate information over the first 20-40 sec.
19
Wind energy machines we begin the correlation calculation (i.e., on t). If we ignore these weak peaks, the data show that is approximately exponential with a correlation time for decay to 1/e for the fluctuations of 11 sec. The value of ~ at T = 0 yields the root mean square deviation about the mean. This value is shown in the figure. Figure 3(c) shows number 4 run from another day. Here the mean velocity 3 4 is slightly less than 32, but the fluctuations are larger and faster as shown by the autocorrelation in Fig. 3(d). We have taken Fourier transforms of ~ to find the power spectra of the fluctuations[ 10]. These spectra show a falling offat frequencies near and above 0.01 Hz as would be expected for the correlation times noted in Fig. 3. The peaks in ~ cause, however, extensive peaks in the power spectrum which make the latter of little value for analysis. Kaimal has taken extremely large data sets from the high tower at Boulder Colorado[ 11 ]. Using Fourier transforms, he found an f-5/s fall off for frequencies f greater than about 0.01 Hz for the horizontal components (the Kolmogorov law). He has also correlated spectral features with height of anemometer and with the depth of the convective boundary layer. These results for very large data sets confirm our methods using rather restricted sets as long as we restrict our interest to fluctuations with frequencies below a b o u t . 1 Hz. In this paper we are concerned mostly with power generation, and not as much with rotor stress. As will be shown, high frequency variations in windspeed are smoothed out by the response of the wind machine. They are not important for power generation, but may be very important as driving forces for high frequency mechanical stresses which can lead to machine breakdown. We do not analyze such stresses here. Figure 3 shows that two parameters may readily be extracted from the data. One of them is the rms fluctuation in velocity. Since we employ a single vertical axis anemometer our measurement yields a value appropriate to the longitudinal plus transverse horizontal components. These are usually denoted by u and v. Several studies have related the velocity fluctuations to the roughness of the surface over which the wind approaches the measurement point. We are interested principally in winds which approach a site over the sea so that the surface roughness which is appropriate is the wave height. Runs v2 and U4 yield t/~ms/32
=
~vlv z
(m2/s2)
(o)
1.0
~
o/ / I
J (o)
= 0.95
e o r r e l . time = IO.Os
;o
-20
T(s) (b)
~Jv4p 3
I0.0
/~~"~
~"--'----'------~ s h i f t 9 . 6 s
(o)= 9.e kWmI,
5.0
\f..~..--correl.
t i m e = iO.
s
0.10
u]m~4 = 0.13. F o r these runs wave heights were roughly 0.5 m and there were whitecaps on some wave crests. The preceding afternoon, however, during the initial set up of the instruments, there were somewhat slower but very steady winds (35 = 9.2 msec-~). At that time the sea was quite smooth and we measured U~ms/~ 5 =
While a detailed study of the dependence of the above ratio on sea surface conditions has not been attempted, the two higher values are consistent with values obtained on the coast of Denmark[12]. Figures 3(e) and 3(13 show power output from the Darrieus wind machine with a 3-phase induction motor feeding the local power grid. The nameplate rating of the motor (which acts as generator at sufficiently high wind speeds) is 50 kW. Again, 500 second records were taken for analysis. Figure 3(e) shows a short portion of run P3 containing a large fluctuation. This run is concurrent with run v4 (Fig. 3(c)), and the cause of the power fluctuation is seen immediately as the velocity fluctuation recorded 10 seconds earlier at the anemometer (which is 40 m upstream). The autocorrelation function of the P3 data clearly shows an exponential decay characteristic with a decay time of 10.5 sec. As with the speed data, the small peaks and structure in t~ beyond T = 30 sec are not stable features. They change for subsequent portions of the run while the exponential decay characteristic remains constant. Finally in Fig. 4 the spatial characteristics of the
.02
P
T(s)
Fig. 4. (a) Cross correlation for two anemometers spaced vertically by 10 meters. Peaks beyond T = 20 sec should be ignored, however the value of the correlation time is precise to better than 5%. (b) Cross correlation between wind speed and power generated by the wind machine located 40 meters downstream from the wind speed anemometer.
20
A. S. BARKER JR.
wind fluctuations are studied by calculating the cross correlation function between data records vl and v2. vl is not shown in Fig. 3 but is recorded 10 m higher on the same instrument mast. We thus obtain a measure of the correlation over a distance of 10 m which is approximately the diameter of the wind machine rotor. Figure 4 shows that the fluctuations are strongly coupled over this distance. The correlation time is the same as for ~)2 and O(0) has dropped by only 50%. While these data apply to a vertical correlation, we can anticipate the horizontal correlations will be somewhat similar. We conclude therefore that under the conditions at this type of test site a substantial portion of the fluctuations in the low frequency range affect the entire wind machine rotor and drive related fluctuations in power output. This result is confirmed by Fig. 4(b) where the crosscorrelation of v4 and P3 is calculated. Again a strong peak is obtained with, however, a significant delay (9.6 see.) in the correlations. 3. WIND MACHINE RESPONSE
In this section we examine the dynamic response of the two-blade vertical-axis Darrieus wind machine at Christopher Point. Equation 2 and Fig. 1 given earlier are approximations to the power vs wind speed response function. They represent only the static characteristic expected for steady winds. Figures 3(a) and 3(e) show that, in practice, very rapid changes are often encountered. We now examine these fluctuations in more detail, but first note a point of practical importance. A wind energy machine may have a response time of a few seconds (as this machine does) and can therefore follow many wind fluctuations. The practical problem is whether the system can fully use the power so generated. If the machine is feeding a power grid the fluctuation can cause voltage fluctuation unless the dynamical response of the primary generators are fast enough to respond and correct. Diesel generators with electronic governors can respond in about 1 sec. H y d r o generators are much slower. Both systems may respond 'wastefully.' That is if the system is running at an optimum load level, and a wind energy machine gives a positive increment of power to the grid, the primary generator can reduce its contribution, but it does this by running in a less efficient m o d e - - a t least temporarily. The net cost saving to the system may be positive, but may not be simply the contribution of the wind energy. The fluctuation effects can be reduced by adding the outputs of several wind machines, but Fig. 4 suggests that they will have to be quite widely separated. We cannot pursue this topic here, but proceed with an analysis of the time response of the 50 k W Darrieus machine. Our aim is to give a simple analysis which brings out the major characteristics of importance. Kirchoff has carried out a transfer function analysis for
a 25 kW machine[13]. While our methods are different, it is worth noting that his system shows a decay time of about 5 sec for decay to 1/e (or 33 sec for decay to 0.2%) for a resistive load compared with our result of about 2 sec for the Darrieus machine connected to the power grid. Our analysis begins by assuming a simple, single time constant, relaxation form for the wind machine coupled to the power grid (the load). Figure 5 gives the electric circuit analog. The windspeed v(t) is modeled as a voltage which drives an R-C circuit. The output voltage across the capacitor C is proportional to the machine power P(t) delivered to the grid. The frequency dependent transfer function is
1 + itor"
(4)
The model is applicable only in wind regimes where P(t) is positive. Since the step response and impulse response of the circuit are well known, we search the time records, P3 and Va (Fig. 3), for instances of each. Measuring the decay after ten such fluctuations and averaging we find the time constantt = R C = 2.2 _ 0.8 sec.
(5)
Using a different method, we note that Fig. 4(b) shows a delay of 9.6 sec between P3 and v4 (measured 40 m upstream). Since the mean wind speed for these runs is 11.9 m/s, 3.4 sec is accounted for by the 40 m separation. Thus 9.6 - 3.4 = 6.2 sec could be attributed to the time constant of the wind machine system. This method is approximate, however, since some of the large fluctuations contained in the P3 data carry us out of the linear approximation implicit in the model. We regard this result merely as an order of magnitude confirmation of the above measurement. A theoretical calculation proceeds as follows.
R
IV(t)-Vo ,
C
P(t)
-1-
Fig. 5. Circuit model for the response of the 50-kW Darrieus wind machine. Only wind speeds greater than vo are considered. For such winds the instantaneous speed v(t) has vo subtracted and the result is applied as a voltage to the circuit as shown. The voltage across C is proportional to the wind machine power output and the frequency response function is (I + itor)-1 where to is the frequency in radians per second. t x is defined as the time for decay to within 1/e of the final value. Some other authors have used a different definition which results in their time constant being larger by a factor of 2~r.
21
Wind energy machines Analysis of the torque that a single Darrieus blade contributes to the generator gives the form shown in Fig. 6. The torque has a steady component Tar, as well as large contributions at to and at 2to, where to is the rotational frequency of the machine. This torque is converted directly to power. The following assumptions can now be made. First, the blade is well designed so that the lift forces are much larger than the drag forces when averaged over 1 revolution. Second, the bearing and gear losses are small. Using these assumptions (which our estimates show to be reasonable), if the machine is generating at a rotational speed to and a torque T~, = a + bto, we assume the wind speed abruptly drops to zero. The load on the machine is the power being generated so that the equation of motion is I~b = - a
- bto
(6)
The solution is simply a constant plus a decaying exponential with time constant -r = I/b, where I is the moment of inertia of the rotating system and b the slope of the characteristic in Fig. 6. Using 5% as an estimate of the no-load to full-load slip speed of the induction motor (i.e., of b), and I = 1300
m2kg we calculate = 1.2 sec (calculated). This result is in satisfactory agreement with our measured result above, confirming that the response is principally a single relaxation type (Fig. 5) with the coupling to the load providing the damping. Finally, the response model can be used to predict the effect of the number of blades on the Darrieus output. The present machine has two blades. Its torque is therefore the sum of two curves as shown in Fig. 6(a), but with the second displaced 180° from the first. The maxima and minima of the second blade, therefore, fall on the maxima and minima of the first blade giving an output which is approximately a sinusoid. Using this as an input to a simple relaxation response function as developed above with to = 8.4 sec -1 and r = 1.2 sec, we obtain a power output of
P(t)
=
(1 + 0.10 cos 2tot)
for the lowest dominant harmonic.
(a) (ToTue)
I 0
',v
180
I 360
e
(degrees)
(b)
rov
Tav = a÷bC0
LO Fig. 6. (a) Quantitative behaviour of the torque generated by one blade of the Darrieus machine at a wind speed of 15 m/s versus angular position of the blade with respect to the undisturbed windstream. Zero degrees corresponds to the blade being directly downwind from the center of rotation. Lift and drag functions averaged over the entire blade have been assumed. The average torque Tar is the driving function of interest here which generates the power output. (b) Schematic behaviour of Tar as a function of rotation speed of the blades. The slope of the graph determines the response time of the wind machine-generator combination when it is coupled to its load.
(7)
22
A . S . BARKERJR.
In a portion of the trace where P = 30 kW, the P4 data shows
P(t) = 30(1 + .14 cos 2tot)kW
(8)
These fast fluctuations have been smoothed out in drawing Fig. 3(e), but are always present in the data. The agreement (within 40%) between eqn 7 and eqn 8 is further confirmation of the response model. Using the characteristic of Fig. 6(a), it is easy to show that three blades considerably reduce and four blades almost eliminate these fast power fluctuations. Work at Sandia Laboratories has documented similar power fluctuations for two bladed 100-kW Darrieus machines. In addition, workers there have considered the filtering effect of drive trains using compliance and damping, and also the resonant modes of the structure and their excitation by the torque fluctuations[14]. 4. WIND ENERGY POTENTIAL
Figure 1 shows graphically the point emphasized in the introduction that no energy is gathered from low speed winds. F o r this reason we wish to analyse the data of Fig. 2 in a manner which recognizes this limitation yet gives a realistic, yet simply calculated, assessment of the yearly wind energy potential at a site for which histogram data has been recorded. We define an annual availability A as follows. A survey of available wind machines in the 40-2000 kW power range suggests that a threshold windspeed of 5 m/s provides a reasonable value of Vo for our purposes. At a wind speed vr somewhere in the range 1.4 Vo-3 vo, many machines have reached their full rated power. We choose Vr = 14 m/s for our analysis. As Fig. 1 indicates, many machines are shut down at high s p e e d s - - g e n e r a l l y at a value in the range 3-10 vo. This shutdown is not as fundamental a limitation as the low speed threshold. Since our histograms have little weight above 25 m/s we will ignore the shut-down and count all winds above yr. A is calculated by first multiplying the wind speed histogram by the wind machine power curve normalized to a height of I. The result is then integrated over all wind speeds and divided by 8765, the number of hours in one year. Note that our discussion of response time shows that for the 50-kW Dardeus machines (and other similar machines), wind speed histograms should be constructed using a recorder with an averaging time of approximately 2 sec. The data for Tofino-Uclulet and Christopher Point were taken in this manner so that the value A = .22 properly includes the energy available in fluctuating winds. The remaining data are taken using various m e t h o d s - - o f t e n involving one reading per hour. These latter data give therefore a less reliable estimate of the wind energy. Figure 2 and Table 1 list the results of our analysis. The highest energy location is at the weather ship station. The availability value A = 0.55 predicts
that a 50-kW machine at this location would generate Etot = 0.55 x 8765 x 50 kW = 241 MWhr
(9)
annually. This 55% availability contrasts sharply with A = . 11 at Seattle-Tacoma airport. The same machine located at the airport would yield only 48 MWhr annually. The somewhat surprising result is the value A = 0.14 recorded over a two-year period for the coast near Tofino-Uclulet. In spite of the exposed shore location, the value is significantly lower than the offshore and island locations. The quarterly data showed a somewhat higher value of A for the spring quarter with a correspondingly lower value for the summer. Such short term quarterly data is not felt to be reliable enough to present here. The data show that Christopher Point, which has a shore location similar to the Tofino-Uclulet location, benefits from the channeling effect of the Juan de Fuca Strait. This effect is well known to small boat sailors who ply these waters. Finally, the two island locations (Tatoosh and Cape St. James) show extremely high wind energy potential due to their off-shore location and much larger unobstructed approach field for winds from many directions. The factors discussed above can be expected to be general features which apply to any coastline where the prevailing weather pattern is in the onshore direction. 5. CONCLUSIONS
In this study we have examined the dynamic response of a wind energy machine and noted the inadequacy of quoting the mean wind speed at a site as an indicator of wind power potential. While a complete wind speed record can be obtained using modern highspeed data recorders with megabyte memory capacity, we suggest recording a six-point histogram as a viable low-cost compromise. We note also that six points is sufficient to fit Weibull or Rayleigh wind speed distributions to the data if this is desired for comparison with other data records. In addition we have shown that the anemometer and data conditioning must allow recording of all fluctuations down to times as short as 1-2 sec. In the analysis carried out above, the histogram is used in a very simple way to give A, the fraction of rated power that a wind machine will generate on the average (sometimes called the load factor or capacity factor). A is a much more meaningful parameter for describing a site than the mean wind speed. It does, however, depend on the assumed power curve (Fig. 1). Of course the more fundamental description of a site is its wind speed histogram. The measurements for various west coast locations show A ranging from .14 to .49 and an rms fluctuation ratio of about 0.1 when the sea is disturbed, which is always the case when the machine
23
Wind energy machines is generating near its rated power. The 50-kW Darrieus was found theoretically and experimentally to have a response characterized by a simple decay time of about 2 sec. The power, however, shows much longer fluctuations on the average, controlled by the wind turbulence. Our method of recording rather short time series for study of autocorrelations and cross-correlations appears to be quite useful as long as small variations in t~ beyond T = 30 seconds are ignored or smoothed. The importance of wind fluctuations for power generation has been emphasized. A true cost-benefit estimate would have to include a full systems analysis including the fluctuations and their relation to system stability, number of machines and their spacing, number of blades per machines (especially if of Darrieus type), and response and efficiency of other generators supplying the same system. An important extension of this study would be the measurement of crosscorrelations of wind speeds over horizontally separated distances of 20 to 200 meters, both in relation to single wind machine performance and to the fluctuations to be expected from multiple-machine wind farms. An analysis has been given of the response time of the 50-kW Darrieus wind machine, and reasonable agreement obtained with the measured response. For larger machines of the same type we expect both the moment of inertia I and the slope b to increase. The change in response time • may be rather small, however the answer will depend on the detailed design of the blades and generating system. Once these are known, ~ can be calculated by the methods given above.
Acknowledgements--The author acknowledges with grat-
itude partial support of this work by Energy, Mines and Resources Canada, and the use of the Christopher Point wind site equipment by B.C. Hydro. It is a pleasure to acknowledge also many helpful contributions and conversations with N. Vander Kwaak, and with colleagues at the Simon Fraser Energy Research Institute.
REFERENCES
1. A. Betz, Wind Energy and its Utilization by Windmills. Bandenhock and Ruprecht, Gottingen (1926). 2. Solar Energy Technology Handbook. Edited by W. C. Dickinson and P. N. CheremisinoffPart A, Chapter 22 Marcel Dekker, N.Y. (1980). 3. E. W. Golding, The Generation of Electricity by Wind Power. Halsted Press-John Wiley & Sons, New York (1976). 4. Proceedings of Conference and Workshop on Wind Energy Characteristics and Wind Energy Siting. U.S. Dept. of Energy (1979). 5. H. Schlicting, Boundary Layer Theory, 6th Ed. McGraw Hill, New York (1968). Also H. A. Panofsky and J. A. Dutton, Atmospheric Turbulence. John Wiley and Sons, New York (1984). 6. Wind Energy Resource Atlas, Vol. 1 (Northwest Region) available from U. S. Dept. of Commerce-NTIS document PNL-3195-WERA-1. Canadian Climate Normals, Vol. 5--Wind. Gov. of 7. Canada (1982). 8. Unpublished data from Atmospheric Environment Service, Vancouver, B.C., Canada. 9. Unpublished data from B.C. Hydro for Christopher 10. Point Wind Energy site. D. K. C. MacDonald, Noise and Fluctuations. Wiley & Sons, NY, (1962). 11. J. C. Kaimal in ref. 4. 12. J. A. Dutton and J. Hojstrup, ref. 4, page 59. 13. R. H. Kirchoff, ref. 4, page 131. R. O. Nellums, Field test report of the D.O.E. 10014. kW vertical axis wind turbine. Report SAND84-0941 obtainable from Sandia Laboratories, Albuquerque, NM.