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The directional variation of wind probability and Weibull speed parameters
~nnospkric Printed
Enrrimnnvnr
ia Great
Vol. 18. No.
10, pp. 2091-2047.
19%
Britain.
THE DIRECTIONAL VARIATION OF WIND PROBABILITY AND WEIBULL SPEED PARAMETERS J. C. DIXON and R. H. SWIFT Engineering Mechanics, Faculty of Technology, The Open University, Milton Keynes, Bucks MK7 6AA, U.K. (First received 9 November 1983 and received for publicorion 3 May 1984) Abstract-A three-parameterwind model is proposed. Two of these are the familiar Weibull characteristic speed and shape factor; the third is a measure of directionality. The model is essentially empirical, and is easily applied, giving the directional probability and the Weibull parameters to be used for any particular wind direction. The I&Williams wind model, in which B is a dependent variable, and the Weibull model. where directionality is neglected, are each cases of the new m&l. The McWiIIiams mode1 has previously been shown to be a fair predictor of ground-levei winds. The modets are here compared with geostrophic wind data. The directional variation of probability, characteristic speed and power density can be fitted well, although shape factor varies erratically. Thus, the suggested model should provide a broadly realistic representation for any altitude, with the advantage that overall characteristic speed, shape factor and directionality can be independently varied as required. Key word index: Wind, geostrophic, direction, directional, Weibuli, ?&Williams, probability.
NOMENCLATURE T II
LARR VCH.MC
VCH (0) BETAR BETA, fi {@j A:0 i c a
Vchar
turbulent intensity wind sped (m s- ‘) wind speed (ms-‘) required characteristic speed (m s- ‘) actual characteristic speed (m s- ‘) directional characteristic speed (m s-l) required shape factor actual shape factor directional shape factor directionality wind direction, from prevailing direction (deg) hill shape exponent vector mean speed (m s-‘ f speed deviation (m s- ‘).
1. WlTROiXJCTION Wind models are required in engineering
design for both extreme value prediction (e.g. for design limit wind speeds) and for common value prediction (e.g. for wind energy analysis); it is the estimation of common values with which we are concerned here, and in particular with the directional dependence of the wind probability and speed characteristics. For engineering application, it is desirable that the model should be accurate, at least as far as the data permits judgement, and easily evaluated n~eri~liy. PreferabIy, it should also have a ready physical interpretation, since this facilitates judgement by the user regarding applicability, and assists detection of gross errors. Where directional wind speed properties are required, existing models are inadequate. The Weibull model (a two parameter distribution) says nothing about directional probability density, or about the
directional variation of its parameters (characteristic speed and shape factor). We show later that simply adding a directional ~o~bility distribution to a Weibull model gives a poor model. The McWilliams model (presented as a two parameter distribution using vector mean velocity and speed variance) is a fair predictor of directional properties. However, its appliability is limited by the fact that the overall Weibull shape factor for the McWiliiams wind is a dependent variable which has a value of 2.0-2.08 for realistic values of the independent variables, creating difficulties in modelling some real winds, for which the data indicates shape factors mostly in the range 1.7-2.3. Consequently, a three parameter distribution appears necessary. We suggest here a simple empirical model combining certain elements of both the models mentioned above, which uses the two parameters of the Weibulf mode1 combined with a third, directionality, parameter. The Weibull wind model has been thoroughly described, e.g. Justus et al. (1976), Swift-Hook (1979), Stevens and Smulders (1979), Takle and Brown (1978). It is a widely used model of common wind behaviour representing the speed distribution according to p(u
< v) = e-(“/“&B,
(1)
where Vchar(the characteristic speed) is the 63.21 percentik /l is the shape factor, taking a typical value around 2.0. It is desirable to separate out the zero-wind data before analysis (Takle and Brown, 1978) a methodology that we have used throughout this paper. Data on the directional probability density of winds is often separately available (as wind roses), but this gives no information on the directional variation of the Weibull parameters, if any, and so any such variation
2042
J. C. DIXON
and R. H. SWIPI
has historically been neglected. It is, however, included in the new model to be proposed here. 2. THE MCWILLIAMS
WIND
MODEL
This model has been described as follows (McWilliams et al., 1979): “. . the component of wind speed along the favoured wind direction is normally distributed with non-zero mean and a given variance while the component of wind speed along a direction at right angles is independent and normally distributed with zero mean and the same variance.” This can be re-expressed by saying that there is a probability ‘hill’ on the velocity plane, and that the directional nature of the wind corresponds to an offsetting of the hill from the origin. Because the variances along the prevailing and cross-prevailing directions are equal, the hill is an axially symmetric Gaussian bell, and its shape is dependent only on the variance, i.e. on the standard deviation a. The displacement of the hill from the origin is the vector mean wind velocity, of magnitude p. Non-dimensional properties of this joint probablility distribution are a function of the ratio r/a, which has values between 0 and 1 for realistic winds. McWilliams used y to represent p/o. We shall use 6, for reasons discussed later.
01
1
60
0
120
Application of this model to determine the probability density of any particular wind velocity I> straightforward. Given wind speed V, provided that the parameters p and u are known, and the wmd angle 6 relative to the prevailing direction is known. simple geometry finds the distance 0 from the velocity vector to the hill axis. Hence the probability density function for wind velocity (V. 01is
OV’
emo51L:
d
(21
2.1. Directional variation of properties Figure 1 illustrates, for three values of directionality, the directional dependence of probability, equivalent Weibull characteristic speed and shape factor, and power density, obtained by numerical integration. These curves are symmetrical about the prevailing
I l6C
, I
60
_ ‘77l
(where U = U (V, 0)). If the joint probability density of V and 0 is integrated for all angles, the probability density distribution for wind speed is recovered. If this is plotted as a cumulative probability on Weibull paper, it is found to be highly linear in the region of common probabilities for realistic values of 6; it is therefore well represented by a Weibull equivalent.
8
0
1
P(V,H)=
120
Fig. 1. Directional properties of McWilliams wind.
The directional variation of wind probability and Weibull speed parameters direction (fl = 0) so only one half is shown. The Weibull shape parameter /I was calculated using the 30 and 70 percentiles (in fact the result is not sensitive to the percentiles used because the Weibull distribution is a good approximation to the McWilliams marginal distribution). In finding the directionality from wind data, a rapid estimate may be made by examining the ratio R of prevailing to anti-prevailing probability (or other parameter) of a smoothed curve through the data. In the case of the directional probability density, in the realistic range of 6 (i.e. 6 < 1)
6 = 0.89 log,, R.
(3)
For characteristic speed, the constant of proportionality is 3.3; for power density it is 0.55. 2.2, Multi-directional properties
where 1 + 3T2 is the energy pattern factor associated with a one-dimensional longitudinal Gaussian turbulence distribution. Because the turbulent influence on any particular hourly mean speed is symmetrical with regard to speed, the mean speed is not affected by adding turbulence. The characteristic speed is not affected significantly for realistic turbulence levels. Again, the /? value has been evaluated between the 30 and 70 percentiles. The consequence is a reduction of /I as turbulence increases. However, the change is quite small for normal turbulence values. This effect of turbulence may appear to be of interest as one way of introducing, in principle, the possibility of Weibull shape factors less than 2.0 into McWilliams type wind models, but the effect is not really useful.
3. APPLICAT’ION TO WIND DATA
Integration of the joint probability density function for both speed and direction results in the multidirectional properties as a function of 6. For given a, the characteristic speed and mean speed increase with 6. For realistic values of 6( < 1.0) fi is limited to a very narrow range above 2.0, a significant limitation of the McWilliams wind model in matching some real winds. The energy pattern factor (EPF; the mean power per square metre divided by 0.5 p V&J reduces from 1.91, ultimately approaching 1.0 asymptotically. If the mean power is nondimensionalised against 0.5 pa3 instead, then the result increases without limit from an initial value of 3.8. Nondimensionalising against 0.5 p VA, gives the energy density factor (EDF). This is of similar shape to EPF, because the ratio of mean speed to characteristic speed is almost constant. 2.3. Turbulence If, contrary to original use, the basic McWilliams model is used to represent the wind behaviour exclusive of turbulence, then the influence of additional turbulence must be considered (McWilliams and Sprevak, 1982). One example would be representation of hourly mean windspeeds, to which must then be added the turbulence component at frequencies above the ‘spectral valley’. If the extra turbulent velocities are of constant standard deviation then an analytic solution may be available. In reality however, turbulent velocity and deviation are a function of short term mean windspeed, and also other atmospheric variables such as stability. As a lirst approximation, the standard deviation is proportional to speed, i.e. the turbulent intensity (standard deviation/speed) is constant, and isotropic. We are not aware of an analytic solution for this case. Considering a basic Weibull wind’(nondirectional, with shape factor fl = 2.0), the addition of one-dimensional turbulence, in the velocity direction, decreases the shape-factor and energy pattern factor. In fact EPF = 1.91(1+ 3T2),
2043
(4)
The Weibull model, through its two parameters, seems to be capable of an adequate engineering representation of the speed distribution for essentially any real wind for a speed range covering common probabilities. Parameter values fitted for prediction of common speeds are, however, not necessarily appropriate for prediction of abnormal conditions, such as 50-year return survival speeds, but they are generally acceptable up to, say, three times the mean speed. McWilliams and Sprevak (1980) presents groundlevel data for 19 sites compared with curves derived from his model. At any given site, the directional wind properties actually depend upon the surface drag coefficient over long fetch distances (e.g. 20 km); clearly this is generally variable with direction, in an erratic manner, out of keeping with the nature of the model. Within the obvious limitations of the twoparameter distribution, the model was nevertheless shown to manage quite well as a predictor of the marginal probability density distributions for speed and direction. It would appear that a markedly more complex, and site specific, model would be required to improve the model performance. For practical applications it is usually necessary to consider other altitudes other than just ground-level (10 m). The Meteorological Office has gathered data on the geostrophic wind, obtained by balloon at the 900 mb level (typically 850 m altitude). Moore (private communication) has reduced the data from six sites to Weibull parameters at directional intervals of 30 degrees. Although the geostrophic wind is not sensitive to surface friction, these parameters show significant scatter and irregularity, especially in shape factor. This is possibly due to the rather limited quantity of data. Figure 2 shows the mean parameter values for all six sites plus model curves for b = 0.5, u = 8 m s-i with prevailing direction 245 deg. The corresponding overall Weibull parameters are I&, = 11.3 ms-’ and /? = 2. Evidently, the form of the data is in general accordance with the model. The erratic variation of /?
J. C. DIXON
and R. H. SWIFT
0
I
I
1
60
e
120
180
e
Fig. 2. Geostrophic wind data, plus theory.
is, fortunately, not too important, the wind properties being relatively insensitive to this parameter. It is apparent from the above that the McWilhams model and therefore also the new model to be described later, is quite successful in fitting the directional distributions of wind properties for the examples of ground and geostrophic winds considered, and probably, therefore, for intermediate heights too. Evidently, however, at low altitudes where the topography is important, or surface friction over fetch distances change with direction, the fit is likely to be relatively poor. If a very simple model combining a Weibull wind plus a directional probability distribution is used, then the characteristic speed does not vary with direction, and the power density is the same shape as the probability density. Such a simple model would be easy to use, but is of poor accuracy and has serious limitations.
4. EXTENSION
OF THE MCWILLIAMS
MODEL
Notwithstanding these examples, there is one definite problem in the correspondence between theory and data; Weibull models fitted to real winds have shape factors anywhere in the range 1.7-2.3, whereas the overall shape factors emerging from the McWilhams wind are restricted to a very narrow range above 2.0, and even then are not independently adjustable. The McWilliams wind was presented as a two parameter distribution, with a normal (Gaussian) distribution of speeds along and perpendicular to the prevailing direction. A possible extension to this model therefore suggests itself; namely, admitting the hill shape exponent I as a third parameter p( v, 0) = _ 1 e-“.s~u/uJ”, u J27l
(5)
By allowing 1 to differ from 2.0 (compare Eqn 2), the
2045
The directional variation of wind probability and Weibull speed parameters requkd overall shape factor might be achieved. In fact, this development proves unsatisfactory. Even for small values of 6, rather large changes of hill shape factor are necessary. This distorts the directional distributions in an unrealistic way. Worse still, for d values around 0.6, it is effectively impossible to realise an overall @value above about 2.1. This development of the model is therefore, at best, suitable only for a very limited range of /I, and for small 6.
a cosine term, plus an exponential (Weibull-like) term
5. PROPOSAL FOR A SIMPLE EMPIRICAL MODEL
p(e) = c,+c,cose+CJ(e-(B’BIY-C,),
Because of the limi~tions of the existing wind models described above, we have sought an empirical model capable of adequate representation of directional winds. Since wind speed properties are very widely modelled by Weihull factors, it seems highly desirable that a practical directional wind model should be based on the following three parameters: (1) Weibull characteristic speed (integrated direction); (2) Weibull shape factor (integrated direction) and (3) a ‘d~~tio~~ty’ parameter. This would have the further advantage that in the systematic investigation of directionality effects, the overall Weibull parameters could be easily held constant. This requires the definition of a suitable parameter of directionality and the determination of simple empirical expressions for the corresponding directional variation of the probability density, characteristic speed and shape factor, preferably whilst respecting the successes of the McWilliams model.
P(0) = Cl + C2 COST+ C3 e-@@l)‘,
where the coefficients C,, C2 and C, are generally functions of 6. For the best fit, with increasing 6, initially the cosine term grows, but later diminish~, giving way to the exponential term. It is, of course, essential to preserve a total probability of 1.0, and therefore most convenient to think of the last two terms as redistributions of probability, totalling zero
The ratio da is the effective measure of directionality for the McWilliams wind. Although this does not retain exactly the same statistical meaning when applied in our empirical model, the same general physical interpretation remains relevant; essentially, the displacement of a probability hill on the velocity plane. We have therefore adopted the variable which we have called directionality, symbol&d by 6 for mnemonic convenience, and to distinguish the slightly general&d meaning compared with the McWilliams y. 5.2. Distribution o~probubifity The McWill~ms dir~tional probability distribution is a fairly complex function of both 6 and 8. For 6 = 0 the distribution in 6 is flat; for small 6 ( < 0.2) it contains a significant cosine term. Higher order terms become rapidly more significant; even for 6 = 0.5 a second order harmonic is not really adequate. For 6 = 1.0, the shape is more or less approaching a Weibuil type curve symmetrical about 6 = 0. Attempting to represent this family of curves by Fourier type series or polynomials requires an inconveniently large number of terms. However, a tolerably accurate and simple expression is possible on the basis of the above qualitative description; i.e. a constant, plus
(7)
where C, must be the mean value (2.7778 per thousand degrees). The cosine term automatically integrates to zero, and the exponential one will do so if CS is correctly chosen. In order to best fit the shape of probability distribution corresponding to 6 = 1.0, using no cosine contribution, we find that 8, = 63 deg and p = 1.65, requiring C, = 0.3126. Representation of the entire family is then found to be reasonable by making c, =3.5006(1-a)
(8)
C3 = 6.990 S’.
(9)
Thus for small 6, only the cosine term is significant, and its amplitude is, correctly, proportional to 6. However, as 6 increases, the exponential term rapidly becomes dominant. This representation agrees very closely with the ~cWi1~ dis~bution. In suck P = 2.7778+3.5006(1 +
5.1. Directionality parameter
(6)
-@OSt?
6.9906’ (e -(e163)“LJ - 0.3126)
= 2.7778(1+1.26OS(l
(10)
-S)ws8
+ 2.1566’ (e-@‘63)“”- 0.3126)). 5.3. distribution of chor~teristic factor
(11)
speed and shape
From Fig. 1, the functional shape of characteristic speed against direction is somewhat similar to that of probability, in that it progresses from harmonic to exponential form. A similar model gives a satisfactory fit
v,(e) = vl (1.0+0.3506(1 -s)c0se +0.718~z(e-(~~5~” -0,418)x
(121
where Vi ischosen to result in the desired final overall characteristic speed. This final characteristic speed depends not only upon the value of VI, but also upon the functions selected for the distributions of probability and shape factor. Since, in practice, the user wishes to specify the overall characteristic speed, not V, , the means must be provided to choose in advance a suitable value for V,: this operation will be described later. The rather scattered shape-factor data of Fig. 2 does not 6rrnly support the McWilliams model relationship for shape factor, but neither does it suggest any better
2046
J. C. DIXONand R. H. SWET
alternative. In order to maintain compatibility, it is therefore sensible to fit the theoretical curves as well as possible. The constant plus cosine plus exponential expression can again be fitted accurately /I(@)= /3i(l+o.1476(1
same reason, it would be difficult to refute this assumption, which does have the merit of simplicity.
6. VALUES OF DIRECTIONALTIY
-d)cosf?
i-0.32962fe-(e@8~’ -0.353)),
(13)
where /?i must be chosen to return the required overall shape factor. It now remains to find functions for the Vi and /I, that will return the required Vch, and j%with sufficient accuracy over an adequate range of fi (from 1.7 to 2.3) and of 6 (from 0 to 1.0). The degree of accuracy required is quite high, since if, for example, the wind kinetic power is of interest, this is a cubic function of speed. The foilowing entirely empirical relationships have been found to return the required values to much better than 1% Vi = F&J1 -o.212~~~9+o.020~~ +0.0056z~6(~-2))
McWiIhams (1980) gives probability versus direction curves for 19 sites, from which we have estimated the directionalities shown in Fig. 3. Also shown are estimates of the directionalities of individual sites for the Meteorological Office geostrophic wind data. This figure suggests that, for the British Isles at least, the gradient wind ~approxi~te~y 500 m altitude and above) has a directionality around 0.6 with modest variance amongst sites, whilst at ground-level (10 m) directionality ranges more widely and is of generally iower value, averaging about 0.35. These directionalities were derived by finding the ratio R of probability densities for prevailing/antiprevailing directions, and using Equation 3.
04) 7. IMPLEMENTATION
/I, = /I(1 -0.035s3~‘-0.0056~ +0.0656’.‘5@-2)).
05)
5.4. Application summary The above five equations for P(f)), I$,(@, @@I),Vi and fii are readily computer coded and meet the requirement specification of a three parameter wind for which the basic overall Weibull parameters can be specified in advance, and the directionality separately varied. They are immediately computed with no inte~ation or iteration or need for supporting software, and suitable for parameter ranges of 0 < 6 < 1 and 1.7 < fi < 2.3. The overall shape parameter /I is freely variable throughout the range 1.7-2.3, with no ill effects on the directional distribution of parameters, and there is no objection in principle to values outside this range, which, however, requires further effort for determination of a suitable fli value, e.g. by iteration. One physical implication of the model is that a high fi wind has a proportionately higher @for all directions. In view of the scatter in experimental values for the directional #I(@ distribution it would be difficult to make a definitive test against other mod&. For the
OF THE MODEL
The suggested empirical mode is represented by five equations. For computing purposes these are readily implemented. The most convenient form in FORTRAN is to use three statement functions for the directional variation of probabiiity, characteristic speed and shape factor (/I), which are functions of both wind directionality and direction (D and A, respectively here). STATEMENT
FUNCTIONS:
PROBABILITY: PR(D, A) = 2.7778 * (I .O+ 1.26*D*(l. - L))*COSL)( A) +2Slfi*D**2*(EXP(--(A/63.)**1.65)-0.3126))
CHARACTERISTIC
SPEED:
VCH(D,A) = VI* (1.0+0.3PL)C(I.-D)+COS~A) +0,718*D+*2*(EXP( -(A/95.)**2.10)-0.418)) SHAPE FACTOR: SH(I),A) = BETAl*
(l.O+O.l47*I)*(L -D)*COSD(A)
~0.329*~*~2*(~XP(-(A~88.)**2)-0.353))
v)
g0) 4 'ij P
i
2
I / 08
0
Directionality 8 Fig.
3.
Site directionality.
The directional variation of wind probability and Weibutl speed parameters
Given values of the required characteristic speed (VCHARR), directionality (D) and overall fi (BETAR), the values of VI and BETA1 are derived when required by normal assignment statements: VI =VCHARR
* (l.O-0.212*D**l.9~~.02O*D**6
+ 0.005*D**2.60 * (BETAR - 2.0) BETA1 = BETAR * (1.0--0.035*D**3.7 -O.OOS*D**5 $0.065*D**1.75
* (BETAR -2.0)).
There is an important conceptual difference between the requested values (VCHARR and BETAR) and the actual values obtained (YCHAR and BETA), the numericai difference bein dependent upon the goodness of the equations for VI and BETA I for VCHA RR and BETAR in the range of application. The directional probability in an angle step STEPA (degrees) is PRUBA = 0.001 * PR(D, A} * STEPA whilst for that angle range, the probability of speed I/ over the range STEPV is PROBVA = EXP( - (( V- O.S*STEPV)j (VCH(D, A) )**SH(D, A) ) -EXP(-((V+O.fSTEPV); VCH(D, A) )**SH(D, A)) and the actual combined probability is the probability product. This is sufficient to implement the model. To simulate the MeWilliams model, it is necessary to select j3 according to j? = 2.0 + 0.080 ~3~.
(16)
8. CONCLUSIONS E&sting models are limited in their ability to represent directional winds. The two-pamm~er Weibull mode1 says nothing about directionality. Adding a directional pro~bility is unlikely to be sufficient for many purposes because the directional variation of Weibull parameters is sibilant, The two-parameter McWiBiams model predicts directional properties with some success; the Weibuli shape factor is, however, a dependent not fit some real winds.
variable and does
A new empirical model is therefore proposed. This has three parameters: two of these are the familiar Weibull parameters, and the third is a directionality
2047
parameter. This therefore provides a natural extension to the widely used Weibull model. Such a mode1 could be implemented in various ways; the details proposed here have been guided by actual wind data, and by the McWiIIiams model. The new model encompasses both McWilliams and Weibull models. The McWilliams model has previously been shown to be a fair predictor of ground-level winds. ~om~rison with geostrophic wind data gives encouraging agreement, suggesting that the new model should be applicable over a wide range of altitudes. The model is easily apphed, and is ideal for computing. Essentially, it provides the dir~tiona~ probability and appropriate Weibull parameters to be used for any particular wind direction. Thus a limitation of the Weibuh model is overcome at the cost of increased ~mp~exity, and a limi~tion of the McWilliams model is overcome at the cost of a degree of empiricism. Those winds that have an erratic directional dependence are, of course, not well modefled in a detailed way, although they may be in a general way. Such winds are most likely to occur at low level and where the surface friction varies drastically in different directions. The mode1 is especially suited the systematic investigation of the effects of wind dir~tionality, and, indeed, was developed for this purpose.
REFERENCES Justus C. G., Hargreaves W. R, and Yaicin A. 11976) Nationwide assessment of potential output from windpowered generators. J. nppl. Met. 15,673-678. MeWilliams B., Newmann M. M, and Sprevak D. (1979) The probability distribution of wind velocity and direction. Wind Engng 3.269-273. Mc~~~rns B. and Sprevak D. (1980) The estimation of the parameters of the distribution of wind speed and direction, Wind Engng 4,227-238. McWi~~~s B. and Sprevak D. (1982) A simulation study of theeffectsofshort term wind fluctuations on the estimation of available wind power. Proc. Int. Ass. Advancement oj ~~~~l~n~ and Simdution Techniques~Paris, July 1982. Stevens M. 3. M. and SmuIders P. T. (1979) The estimation of the parameters af the Weibull wind speed distribution for wind energy utilisation purposes. W&d Enyng 3, 132-145. Swift-Hook D. T, 119791Describing wind data. Wind Enanu I”. 3. i67-t86. Takle E. S. and Brown J. M. (1978) Note on the use of Weibull statistics to characterise wind-speed data. J. appt. Met. 17, 5.56559.
Enrrimnnvnr
ia Great
Vol. 18. No.
10, pp. 2091-2047.
19%
Britain.
THE DIRECTIONAL VARIATION OF WIND PROBABILITY AND WEIBULL SPEED PARAMETERS J. C. DIXON and R. H. SWIFT Engineering Mechanics, Faculty of Technology, The Open University, Milton Keynes, Bucks MK7 6AA, U.K. (First received 9 November 1983 and received for publicorion 3 May 1984) Abstract-A three-parameterwind model is proposed. Two of these are the familiar Weibull characteristic speed and shape factor; the third is a measure of directionality. The model is essentially empirical, and is easily applied, giving the directional probability and the Weibull parameters to be used for any particular wind direction. The I&Williams wind model, in which B is a dependent variable, and the Weibull model. where directionality is neglected, are each cases of the new m&l. The McWiIIiams mode1 has previously been shown to be a fair predictor of ground-levei winds. The modets are here compared with geostrophic wind data. The directional variation of probability, characteristic speed and power density can be fitted well, although shape factor varies erratically. Thus, the suggested model should provide a broadly realistic representation for any altitude, with the advantage that overall characteristic speed, shape factor and directionality can be independently varied as required. Key word index: Wind, geostrophic, direction, directional, Weibuli, ?&Williams, probability.
NOMENCLATURE T II
LARR VCH.MC
VCH (0) BETAR BETA, fi {@j A:0 i c a
Vchar
turbulent intensity wind sped (m s- ‘) wind speed (ms-‘) required characteristic speed (m s- ‘) actual characteristic speed (m s- ‘) directional characteristic speed (m s-l) required shape factor actual shape factor directional shape factor directionality wind direction, from prevailing direction (deg) hill shape exponent vector mean speed (m s-‘ f speed deviation (m s- ‘).
1. WlTROiXJCTION Wind models are required in engineering
design for both extreme value prediction (e.g. for design limit wind speeds) and for common value prediction (e.g. for wind energy analysis); it is the estimation of common values with which we are concerned here, and in particular with the directional dependence of the wind probability and speed characteristics. For engineering application, it is desirable that the model should be accurate, at least as far as the data permits judgement, and easily evaluated n~eri~liy. PreferabIy, it should also have a ready physical interpretation, since this facilitates judgement by the user regarding applicability, and assists detection of gross errors. Where directional wind speed properties are required, existing models are inadequate. The Weibull model (a two parameter distribution) says nothing about directional probability density, or about the
directional variation of its parameters (characteristic speed and shape factor). We show later that simply adding a directional ~o~bility distribution to a Weibull model gives a poor model. The McWilliams model (presented as a two parameter distribution using vector mean velocity and speed variance) is a fair predictor of directional properties. However, its appliability is limited by the fact that the overall Weibull shape factor for the McWiliiams wind is a dependent variable which has a value of 2.0-2.08 for realistic values of the independent variables, creating difficulties in modelling some real winds, for which the data indicates shape factors mostly in the range 1.7-2.3. Consequently, a three parameter distribution appears necessary. We suggest here a simple empirical model combining certain elements of both the models mentioned above, which uses the two parameters of the Weibulf mode1 combined with a third, directionality, parameter. The Weibull wind model has been thoroughly described, e.g. Justus et al. (1976), Swift-Hook (1979), Stevens and Smulders (1979), Takle and Brown (1978). It is a widely used model of common wind behaviour representing the speed distribution according to p(u
< v) = e-(“/“&B,
(1)
where Vchar(the characteristic speed) is the 63.21 percentik /l is the shape factor, taking a typical value around 2.0. It is desirable to separate out the zero-wind data before analysis (Takle and Brown, 1978) a methodology that we have used throughout this paper. Data on the directional probability density of winds is often separately available (as wind roses), but this gives no information on the directional variation of the Weibull parameters, if any, and so any such variation
2042
J. C. DIXON
and R. H. SWIPI
has historically been neglected. It is, however, included in the new model to be proposed here. 2. THE MCWILLIAMS
WIND
MODEL
This model has been described as follows (McWilliams et al., 1979): “. . the component of wind speed along the favoured wind direction is normally distributed with non-zero mean and a given variance while the component of wind speed along a direction at right angles is independent and normally distributed with zero mean and the same variance.” This can be re-expressed by saying that there is a probability ‘hill’ on the velocity plane, and that the directional nature of the wind corresponds to an offsetting of the hill from the origin. Because the variances along the prevailing and cross-prevailing directions are equal, the hill is an axially symmetric Gaussian bell, and its shape is dependent only on the variance, i.e. on the standard deviation a. The displacement of the hill from the origin is the vector mean wind velocity, of magnitude p. Non-dimensional properties of this joint probablility distribution are a function of the ratio r/a, which has values between 0 and 1 for realistic winds. McWilliams used y to represent p/o. We shall use 6, for reasons discussed later.
01
1
60
0
120
Application of this model to determine the probability density of any particular wind velocity I> straightforward. Given wind speed V, provided that the parameters p and u are known, and the wmd angle 6 relative to the prevailing direction is known. simple geometry finds the distance 0 from the velocity vector to the hill axis. Hence the probability density function for wind velocity (V. 01is
OV’
emo51L:
d
(21
2.1. Directional variation of properties Figure 1 illustrates, for three values of directionality, the directional dependence of probability, equivalent Weibull characteristic speed and shape factor, and power density, obtained by numerical integration. These curves are symmetrical about the prevailing
I l6C
, I
60
_ ‘77l
(where U = U (V, 0)). If the joint probability density of V and 0 is integrated for all angles, the probability density distribution for wind speed is recovered. If this is plotted as a cumulative probability on Weibull paper, it is found to be highly linear in the region of common probabilities for realistic values of 6; it is therefore well represented by a Weibull equivalent.
8
0
1
P(V,H)=
120
Fig. 1. Directional properties of McWilliams wind.
The directional variation of wind probability and Weibull speed parameters direction (fl = 0) so only one half is shown. The Weibull shape parameter /I was calculated using the 30 and 70 percentiles (in fact the result is not sensitive to the percentiles used because the Weibull distribution is a good approximation to the McWilliams marginal distribution). In finding the directionality from wind data, a rapid estimate may be made by examining the ratio R of prevailing to anti-prevailing probability (or other parameter) of a smoothed curve through the data. In the case of the directional probability density, in the realistic range of 6 (i.e. 6 < 1)
6 = 0.89 log,, R.
(3)
For characteristic speed, the constant of proportionality is 3.3; for power density it is 0.55. 2.2, Multi-directional properties
where 1 + 3T2 is the energy pattern factor associated with a one-dimensional longitudinal Gaussian turbulence distribution. Because the turbulent influence on any particular hourly mean speed is symmetrical with regard to speed, the mean speed is not affected by adding turbulence. The characteristic speed is not affected significantly for realistic turbulence levels. Again, the /? value has been evaluated between the 30 and 70 percentiles. The consequence is a reduction of /I as turbulence increases. However, the change is quite small for normal turbulence values. This effect of turbulence may appear to be of interest as one way of introducing, in principle, the possibility of Weibull shape factors less than 2.0 into McWilliams type wind models, but the effect is not really useful.
3. APPLICAT’ION TO WIND DATA
Integration of the joint probability density function for both speed and direction results in the multidirectional properties as a function of 6. For given a, the characteristic speed and mean speed increase with 6. For realistic values of 6( < 1.0) fi is limited to a very narrow range above 2.0, a significant limitation of the McWilliams wind model in matching some real winds. The energy pattern factor (EPF; the mean power per square metre divided by 0.5 p V&J reduces from 1.91, ultimately approaching 1.0 asymptotically. If the mean power is nondimensionalised against 0.5 pa3 instead, then the result increases without limit from an initial value of 3.8. Nondimensionalising against 0.5 p VA, gives the energy density factor (EDF). This is of similar shape to EPF, because the ratio of mean speed to characteristic speed is almost constant. 2.3. Turbulence If, contrary to original use, the basic McWilliams model is used to represent the wind behaviour exclusive of turbulence, then the influence of additional turbulence must be considered (McWilliams and Sprevak, 1982). One example would be representation of hourly mean windspeeds, to which must then be added the turbulence component at frequencies above the ‘spectral valley’. If the extra turbulent velocities are of constant standard deviation then an analytic solution may be available. In reality however, turbulent velocity and deviation are a function of short term mean windspeed, and also other atmospheric variables such as stability. As a lirst approximation, the standard deviation is proportional to speed, i.e. the turbulent intensity (standard deviation/speed) is constant, and isotropic. We are not aware of an analytic solution for this case. Considering a basic Weibull wind’(nondirectional, with shape factor fl = 2.0), the addition of one-dimensional turbulence, in the velocity direction, decreases the shape-factor and energy pattern factor. In fact EPF = 1.91(1+ 3T2),
2043
(4)
The Weibull model, through its two parameters, seems to be capable of an adequate engineering representation of the speed distribution for essentially any real wind for a speed range covering common probabilities. Parameter values fitted for prediction of common speeds are, however, not necessarily appropriate for prediction of abnormal conditions, such as 50-year return survival speeds, but they are generally acceptable up to, say, three times the mean speed. McWilliams and Sprevak (1980) presents groundlevel data for 19 sites compared with curves derived from his model. At any given site, the directional wind properties actually depend upon the surface drag coefficient over long fetch distances (e.g. 20 km); clearly this is generally variable with direction, in an erratic manner, out of keeping with the nature of the model. Within the obvious limitations of the twoparameter distribution, the model was nevertheless shown to manage quite well as a predictor of the marginal probability density distributions for speed and direction. It would appear that a markedly more complex, and site specific, model would be required to improve the model performance. For practical applications it is usually necessary to consider other altitudes other than just ground-level (10 m). The Meteorological Office has gathered data on the geostrophic wind, obtained by balloon at the 900 mb level (typically 850 m altitude). Moore (private communication) has reduced the data from six sites to Weibull parameters at directional intervals of 30 degrees. Although the geostrophic wind is not sensitive to surface friction, these parameters show significant scatter and irregularity, especially in shape factor. This is possibly due to the rather limited quantity of data. Figure 2 shows the mean parameter values for all six sites plus model curves for b = 0.5, u = 8 m s-i with prevailing direction 245 deg. The corresponding overall Weibull parameters are I&, = 11.3 ms-’ and /? = 2. Evidently, the form of the data is in general accordance with the model. The erratic variation of /?
J. C. DIXON
and R. H. SWIFT
0
I
I
1
60
e
120
180
e
Fig. 2. Geostrophic wind data, plus theory.
is, fortunately, not too important, the wind properties being relatively insensitive to this parameter. It is apparent from the above that the McWilhams model and therefore also the new model to be described later, is quite successful in fitting the directional distributions of wind properties for the examples of ground and geostrophic winds considered, and probably, therefore, for intermediate heights too. Evidently, however, at low altitudes where the topography is important, or surface friction over fetch distances change with direction, the fit is likely to be relatively poor. If a very simple model combining a Weibull wind plus a directional probability distribution is used, then the characteristic speed does not vary with direction, and the power density is the same shape as the probability density. Such a simple model would be easy to use, but is of poor accuracy and has serious limitations.
4. EXTENSION
OF THE MCWILLIAMS
MODEL
Notwithstanding these examples, there is one definite problem in the correspondence between theory and data; Weibull models fitted to real winds have shape factors anywhere in the range 1.7-2.3, whereas the overall shape factors emerging from the McWilhams wind are restricted to a very narrow range above 2.0, and even then are not independently adjustable. The McWilliams wind was presented as a two parameter distribution, with a normal (Gaussian) distribution of speeds along and perpendicular to the prevailing direction. A possible extension to this model therefore suggests itself; namely, admitting the hill shape exponent I as a third parameter p( v, 0) = _ 1 e-“.s~u/uJ”, u J27l
(5)
By allowing 1 to differ from 2.0 (compare Eqn 2), the
2045
The directional variation of wind probability and Weibull speed parameters requkd overall shape factor might be achieved. In fact, this development proves unsatisfactory. Even for small values of 6, rather large changes of hill shape factor are necessary. This distorts the directional distributions in an unrealistic way. Worse still, for d values around 0.6, it is effectively impossible to realise an overall @value above about 2.1. This development of the model is therefore, at best, suitable only for a very limited range of /I, and for small 6.
a cosine term, plus an exponential (Weibull-like) term
5. PROPOSAL FOR A SIMPLE EMPIRICAL MODEL
p(e) = c,+c,cose+CJ(e-(B’BIY-C,),
Because of the limi~tions of the existing wind models described above, we have sought an empirical model capable of adequate representation of directional winds. Since wind speed properties are very widely modelled by Weihull factors, it seems highly desirable that a practical directional wind model should be based on the following three parameters: (1) Weibull characteristic speed (integrated direction); (2) Weibull shape factor (integrated direction) and (3) a ‘d~~tio~~ty’ parameter. This would have the further advantage that in the systematic investigation of directionality effects, the overall Weibull parameters could be easily held constant. This requires the definition of a suitable parameter of directionality and the determination of simple empirical expressions for the corresponding directional variation of the probability density, characteristic speed and shape factor, preferably whilst respecting the successes of the McWilliams model.
P(0) = Cl + C2 COST+ C3 e-@@l)‘,
where the coefficients C,, C2 and C, are generally functions of 6. For the best fit, with increasing 6, initially the cosine term grows, but later diminish~, giving way to the exponential term. It is, of course, essential to preserve a total probability of 1.0, and therefore most convenient to think of the last two terms as redistributions of probability, totalling zero
The ratio da is the effective measure of directionality for the McWilliams wind. Although this does not retain exactly the same statistical meaning when applied in our empirical model, the same general physical interpretation remains relevant; essentially, the displacement of a probability hill on the velocity plane. We have therefore adopted the variable which we have called directionality, symbol&d by 6 for mnemonic convenience, and to distinguish the slightly general&d meaning compared with the McWilliams y. 5.2. Distribution o~probubifity The McWill~ms dir~tional probability distribution is a fairly complex function of both 6 and 8. For 6 = 0 the distribution in 6 is flat; for small 6 ( < 0.2) it contains a significant cosine term. Higher order terms become rapidly more significant; even for 6 = 0.5 a second order harmonic is not really adequate. For 6 = 1.0, the shape is more or less approaching a Weibuil type curve symmetrical about 6 = 0. Attempting to represent this family of curves by Fourier type series or polynomials requires an inconveniently large number of terms. However, a tolerably accurate and simple expression is possible on the basis of the above qualitative description; i.e. a constant, plus
(7)
where C, must be the mean value (2.7778 per thousand degrees). The cosine term automatically integrates to zero, and the exponential one will do so if CS is correctly chosen. In order to best fit the shape of probability distribution corresponding to 6 = 1.0, using no cosine contribution, we find that 8, = 63 deg and p = 1.65, requiring C, = 0.3126. Representation of the entire family is then found to be reasonable by making c, =3.5006(1-a)
(8)
C3 = 6.990 S’.
(9)
Thus for small 6, only the cosine term is significant, and its amplitude is, correctly, proportional to 6. However, as 6 increases, the exponential term rapidly becomes dominant. This representation agrees very closely with the ~cWi1~ dis~bution. In suck P = 2.7778+3.5006(1 +
5.1. Directionality parameter
(6)
-@OSt?
6.9906’ (e -(e163)“LJ - 0.3126)
= 2.7778(1+1.26OS(l
(10)
-S)ws8
+ 2.1566’ (e-@‘63)“”- 0.3126)). 5.3. distribution of chor~teristic factor
(11)
speed and shape
From Fig. 1, the functional shape of characteristic speed against direction is somewhat similar to that of probability, in that it progresses from harmonic to exponential form. A similar model gives a satisfactory fit
v,(e) = vl (1.0+0.3506(1 -s)c0se +0.718~z(e-(~~5~” -0,418)x
(121
where Vi ischosen to result in the desired final overall characteristic speed. This final characteristic speed depends not only upon the value of VI, but also upon the functions selected for the distributions of probability and shape factor. Since, in practice, the user wishes to specify the overall characteristic speed, not V, , the means must be provided to choose in advance a suitable value for V,: this operation will be described later. The rather scattered shape-factor data of Fig. 2 does not 6rrnly support the McWilliams model relationship for shape factor, but neither does it suggest any better
2046
J. C. DIXONand R. H. SWET
alternative. In order to maintain compatibility, it is therefore sensible to fit the theoretical curves as well as possible. The constant plus cosine plus exponential expression can again be fitted accurately /I(@)= /3i(l+o.1476(1
same reason, it would be difficult to refute this assumption, which does have the merit of simplicity.
6. VALUES OF DIRECTIONALTIY
-d)cosf?
i-0.32962fe-(e@8~’ -0.353)),
(13)
where /?i must be chosen to return the required overall shape factor. It now remains to find functions for the Vi and /I, that will return the required Vch, and j%with sufficient accuracy over an adequate range of fi (from 1.7 to 2.3) and of 6 (from 0 to 1.0). The degree of accuracy required is quite high, since if, for example, the wind kinetic power is of interest, this is a cubic function of speed. The foilowing entirely empirical relationships have been found to return the required values to much better than 1% Vi = F&J1 -o.212~~~9+o.020~~ +0.0056z~6(~-2))
McWiIhams (1980) gives probability versus direction curves for 19 sites, from which we have estimated the directionalities shown in Fig. 3. Also shown are estimates of the directionalities of individual sites for the Meteorological Office geostrophic wind data. This figure suggests that, for the British Isles at least, the gradient wind ~approxi~te~y 500 m altitude and above) has a directionality around 0.6 with modest variance amongst sites, whilst at ground-level (10 m) directionality ranges more widely and is of generally iower value, averaging about 0.35. These directionalities were derived by finding the ratio R of probability densities for prevailing/antiprevailing directions, and using Equation 3.
04) 7. IMPLEMENTATION
/I, = /I(1 -0.035s3~‘-0.0056~ +0.0656’.‘5@-2)).
05)
5.4. Application summary The above five equations for P(f)), I$,(@, @@I),Vi and fii are readily computer coded and meet the requirement specification of a three parameter wind for which the basic overall Weibull parameters can be specified in advance, and the directionality separately varied. They are immediately computed with no inte~ation or iteration or need for supporting software, and suitable for parameter ranges of 0 < 6 < 1 and 1.7 < fi < 2.3. The overall shape parameter /I is freely variable throughout the range 1.7-2.3, with no ill effects on the directional distribution of parameters, and there is no objection in principle to values outside this range, which, however, requires further effort for determination of a suitable fli value, e.g. by iteration. One physical implication of the model is that a high fi wind has a proportionately higher @for all directions. In view of the scatter in experimental values for the directional #I(@ distribution it would be difficult to make a definitive test against other mod&. For the
OF THE MODEL
The suggested empirical mode is represented by five equations. For computing purposes these are readily implemented. The most convenient form in FORTRAN is to use three statement functions for the directional variation of probabiiity, characteristic speed and shape factor (/I), which are functions of both wind directionality and direction (D and A, respectively here). STATEMENT
FUNCTIONS:
PROBABILITY: PR(D, A) = 2.7778 * (I .O+ 1.26*D*(l. - L))*COSL)( A) +2Slfi*D**2*(EXP(--(A/63.)**1.65)-0.3126))
CHARACTERISTIC
SPEED:
VCH(D,A) = VI* (1.0+0.3PL)C(I.-D)+COS~A) +0,718*D+*2*(EXP( -(A/95.)**2.10)-0.418)) SHAPE FACTOR: SH(I),A) = BETAl*
(l.O+O.l47*I)*(L -D)*COSD(A)
~0.329*~*~2*(~XP(-(A~88.)**2)-0.353))
v)
g0) 4 'ij P
i
2
I / 08
0
Directionality 8 Fig.
3.
Site directionality.
The directional variation of wind probability and Weibutl speed parameters
Given values of the required characteristic speed (VCHARR), directionality (D) and overall fi (BETAR), the values of VI and BETA1 are derived when required by normal assignment statements: VI =VCHARR
* (l.O-0.212*D**l.9~~.02O*D**6
+ 0.005*D**2.60 * (BETAR - 2.0) BETA1 = BETAR * (1.0--0.035*D**3.7 -O.OOS*D**5 $0.065*D**1.75
* (BETAR -2.0)).
There is an important conceptual difference between the requested values (VCHARR and BETAR) and the actual values obtained (YCHAR and BETA), the numericai difference bein dependent upon the goodness of the equations for VI and BETA I for VCHA RR and BETAR in the range of application. The directional probability in an angle step STEPA (degrees) is PRUBA = 0.001 * PR(D, A} * STEPA whilst for that angle range, the probability of speed I/ over the range STEPV is PROBVA = EXP( - (( V- O.S*STEPV)j (VCH(D, A) )**SH(D, A) ) -EXP(-((V+O.fSTEPV); VCH(D, A) )**SH(D, A)) and the actual combined probability is the probability product. This is sufficient to implement the model. To simulate the MeWilliams model, it is necessary to select j3 according to j? = 2.0 + 0.080 ~3~.
(16)
8. CONCLUSIONS E&sting models are limited in their ability to represent directional winds. The two-pamm~er Weibull mode1 says nothing about directionality. Adding a directional pro~bility is unlikely to be sufficient for many purposes because the directional variation of Weibull parameters is sibilant, The two-parameter McWiBiams model predicts directional properties with some success; the Weibuli shape factor is, however, a dependent not fit some real winds.
variable and does
A new empirical model is therefore proposed. This has three parameters: two of these are the familiar Weibull parameters, and the third is a directionality
2047
parameter. This therefore provides a natural extension to the widely used Weibull model. Such a mode1 could be implemented in various ways; the details proposed here have been guided by actual wind data, and by the McWiIIiams model. The new model encompasses both McWilliams and Weibull models. The McWilliams model has previously been shown to be a fair predictor of ground-level winds. ~om~rison with geostrophic wind data gives encouraging agreement, suggesting that the new model should be applicable over a wide range of altitudes. The model is easily apphed, and is ideal for computing. Essentially, it provides the dir~tiona~ probability and appropriate Weibull parameters to be used for any particular wind direction. Thus a limitation of the Weibuh model is overcome at the cost of increased ~mp~exity, and a limi~tion of the McWilliams model is overcome at the cost of a degree of empiricism. Those winds that have an erratic directional dependence are, of course, not well modefled in a detailed way, although they may be in a general way. Such winds are most likely to occur at low level and where the surface friction varies drastically in different directions. The mode1 is especially suited the systematic investigation of the effects of wind dir~tionality, and, indeed, was developed for this purpose.
REFERENCES Justus C. G., Hargreaves W. R, and Yaicin A. 11976) Nationwide assessment of potential output from windpowered generators. J. nppl. Met. 15,673-678. MeWilliams B., Newmann M. M, and Sprevak D. (1979) The probability distribution of wind velocity and direction. Wind Engng 3.269-273. Mc~~~rns B. and Sprevak D. (1980) The estimation of the parameters of the distribution of wind speed and direction, Wind Engng 4,227-238. McWi~~~s B. and Sprevak D. (1982) A simulation study of theeffectsofshort term wind fluctuations on the estimation of available wind power. Proc. Int. Ass. Advancement oj ~~~~l~n~ and Simdution Techniques~Paris, July 1982. Stevens M. 3. M. and SmuIders P. T. (1979) The estimation of the parameters af the Weibull wind speed distribution for wind energy utilisation purposes. W&d Enyng 3, 132-145. Swift-Hook D. T, 119791Describing wind data. Wind Enanu I”. 3. i67-t86. Takle E. S. and Brown J. M. (1978) Note on the use of Weibull statistics to characterise wind-speed data. J. appt. Met. 17, 5.56559.