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Wind Velocity Vertical Extrapolation by Extended Power Law

Wind Velocity Vertical Extrapolation by Extended Power Law Hindawi Publishing Corporation Advances in Meteorology Volume 2012, Article ID 178623, 6 pages doi:10.1155/2012/178623 Research Article Zekai S¸en, Abdusselam ¨ Altunkaynak, and Tarkan Erdik Hydraulics Division, Civil Engineering Faculty, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey Correspondence should be addressed to Tarkan Erdik, tarkanerdik@hotmail.com Received 19 August 2012; Accepted 7 October 2012 Academic Editor: Harry D. Kambezidis Copyright © 2012 Zekai S¸en et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Wind energy gains more attention day by day as one of the clean renewable energy resources. We predicted wind speed vertical extrapolation by using extended power law. In this study, an extended vertical wind velocity extrapolation formulation is derived on the basis of perturbation theory by considering power law and Weibull wind speed probability distribution function. In the proposed methodology not only the mean values of the wind speeds at different elevations but also their standard deviations and the cross-correlation coefficient between different elevations are taken into consideration. The application of the presented methodology is performed for wind speed measurements at Karaburun/Istanbul, Turkey. At this location, hourly wind speed measurements are available for three different heights above the earth surface. 1. Introduction coefficients for different types of topography and geography [3]. Wind energy, as one of the main renewable energy sources in According to the calculations of wind resource analysis the world, attracts attention in many countries as the efficient program (WRAP) report, in 39 different regions, out of turbine technology develops. Wind speed extrapolation 7082 different wind shear coefficients, 7.3% are distributed might be regarded as one of the most critical uncertainty between 0 and 0.14 and 91.9% above 0.14, while 0.8% are factor affecting the wind power assessment, when consider- calculated as negative [7], due to the measurements error. ing the increasing size of modern multi-MW wind turbines. Different methods have been developed to analyze wind If the wind speed measurements at heights relevant to wind speed profiles, such as power, logarithmic, and loglinear laws energy exploitation lacks, it is often necessary to extrapolate [8]. Besides, in the literature various studies are conducted observed wind speeds from the available heights to turbine in order to estimate wind shear coefficient in the power law hub height [1], whichcausessomecriticalerrorsbetween only if surface data is available at hand [9–11]. estimated and actual energy output, if the wind shear The wind speed undergoes repeated changes, as a result coefficient, n, cannot be determined correctly. The difference of which the roughness and friction coefficients also change between the predicted and observed wind energy production depending on landscape features, the time of the day, the might be up to 40%, due to turbulence effects, time interval temperature, height and wind direction. The uncertainty of wind data measurement, and the extrapolation of the data is inhereted in the wind speed data and its extrapolation from reference height to hub heights [2]. to the hub height should be considered carefully and In the literature, the wind shear coefficient is generally preciously [12]. Moreover, this uncertainty is exacerbated in approximated between 0.14 and 0.2. However, in real situa- the offshore environment by the inclusion of the dynamic tions, a wind shear coefficient is not constant and depends surface [13]. Therefore, the mean wind speed profile of on numerous factors, including atmospheric conditions, the logarithmic type is developed by applying a stability temperature, pressure, humidity, time of day, seasons of correction for offshore sites [14]. the year, the mean wind speed, direction, and nature of It is crucial point for energy investors to accurately terrain [3–6]. Table 1 demonstrates the various wind shear predict the average wind speed at different wind turbine 2 Advances in Meteorology Table 1: Wind shear coefficient of various terrains [3]. of the averages and perturbation terms about their averages that render (1)to Terrain type Lake, ocean, and smooth-hard ground 0.1 Z V + V 1 1 = ,(2) Foot-high grass on level ground 0.15 Z V + V 2 2 Tall crops, hedges, and shrubs 0.2 where V and V are the perturbation terms with averages 1 2 Wooded country with many trees 0.25 equal to zero, V = V = 0. This last expression can be 1 2 Small town with some trees and shrubs 0.3 rewritten simply as City area with tall buildings 0.4 − 1 Z V V V 1 1 1 2 (3) = 1+ 1+ . V V V 2 2 1 2 hub heights and make realistic feasibility projects for these heights. In this study, a simple but effective methodology on This expression is referred to as the extended power law the basis of the perturbation theory is presented in order to in this paper. The third bracket on the right-hand side derive an extended power law for the vertical wind speed corresponds to a geometric series, which can be expressed extrapolation and then the Weibull probability distribution by the Binomial expansion as function (pdf) parameters. It is observed that on the contrary to the classical approach not only the means of wind speeds Z V V V V 1 1 1 2 2 = 1+ 1 − + are at different elevations, but also the standard deviations V V V V 2 2 1 2 2 and the cross-correlation coefficient should be taken into 3 4 consideration, if the wind speeds at different elevations are V V 2 2 − + −··· . not independent from each other. V V 2 2 (4) 2. Power Law This expression can still be simplified after the expansion of This law is the simplest way for estimating the wind speed the second and third brackets on the right-hand side and at a wind generator hub elevation from measurements at a then by considering the second-order term approximately reference level. In general, the power law expression is given yields as, Z V V V V 1 1 2 2 1 = 1 − + + Z V 1 1 = , (1) Z V V V V 2 2 2 2 1 Z V 2 2 (5) V V V V where terms in the brackets are the velocity and elevation 1 2 1 2 − + . ratios, V >V and Z /Z , respectively. Furthermore, V > 2 1 2 1 2 V V V V 1 2 1 2 V and Z >Z ;and n is the exponent of the power 1 2 1 After taking the arithmetic averages of both sides and then law, which is a complex function of the local climatology, considering that the odd order power term averages are equal topography, surface roughness, environmental conditions, to zero, (5) yields to meteorological lapse rate, and weather stability. It is clear ⎡ ⎤ that the effects of all these factors are embedded in the Z V V V V wind velocity time records, and consequently, their total 1 1 1 2 2 ⎣ ⎦ = 1 − + . (6) reflections are also expected in the value of the exponent, V V V V 2 2 1 2 2 n. Therefore, one tends to think whether there is a way of obtaining the estimation of this exponent from the wind By definition V = V = 0, exactly. In fact, for symmet- 1 2 speed time series. Power laws are used almost exclusively rical (i.e., Gaussian) perturbation terms, the odd number without any generally accepted methodology. Most often, arithmetic averages such as V V arealso equaltozero 1 2 only the arithmetic averages of the wind speed at two approximately by definition. elevations are considered in the numerical calculation of the In (6), the common arithmetic average of the perturba- exponent. Logically, other than the mean values, standard tion multiplication, V V ,attwo different elevations is equal 1 2 deviations and cross-correlation coefficient should enter the to the covariance of the perturbations. This can be written calculations, because these additional parameters arise as a in terms of the standard deviations S and S and cross- V1 V2 result of instability, roughness, and so forth. Provided that correlation, r , multiplication as V V = r S S .The 12 12 V V 1 2 1 2 there are wind speed records at two or more elevations, the second-order perturbation term average, V ,isequivalent following approach provides an objective solution. to the variance of the perturbation term as V = S .The 2 V substitution of these last two expressions into (6)leads to 3. Extended Power Law Z V S S 1 1 V V V 1 2 1 In (1), the only random variable that represents the weather = 1 − r + , (7) V V V 2 2 1 2 situation is the wind speeds, which can be written in terms 2 Advances in Meteorology 3 30 30 V (Z = 10 m) V (Z = 10 m) 25 25 20 20 15 15 10 10 5 5 V average average 0 0 0 1000 2000 3000 4000 0 1000 2000 3000 4000 Time index (hour) Time index (hour) (a) (b) V (Z = 10 m) average 0 1000 2000 3000 4000 Time index (hour) (c) Figure 1: Karaburun wind speed records at different levels. where S and S are the standard deviations and r is the in the subsequent wind energy, E, calculations through the V V 12 1 2 cross-correlation coefficient between the wind speed time classical formulation series at two elevations. In practice, most often the cross- correlation term in (7) is overlooked by assuming that there are no random fluctuations around the mean speed values (9) E = ρV , which bring the implication that the standard deviations are 2 equal to zero. These assumptions are not valid because in an actual weather, there are always fluctuations in the wind where ρ is the standard atmosphere air density which is equal speed records as in Figure 1. By definition in statistics, the 3 ◦ to 1.226 gr/cm at 25 C, and V is the wind speed. Most often ratio of standard deviation to the arithmetic mean is the the wind speed at a meteorology station is measured along coefficient of variation, and hence (7)can be rewrittenin atower at different elevations, and it is desired to be able to parameterized form as, find the wind profile at this station for further wind loadings or energy calculations. Some researchers have employed the Z V 1 1 Weibull pdf for empirical wind speed relative frequency = 1 − C C r + C . (8) V V 12 1 2 V Z V distribution (histogram), and a set of formulas are derived for the extrapolation of the Weibull pdf parameters [15–18]. In general, two-parameter Weibull pdf of wind speed, P(V), Herein, C and C are the coefficients of variation for the V V 1 2 wind speed records, V and V ,attwo different elevations, is given as, 1 2 respectively. k−1 k k V V 4. Weibull Distribution P(V) = exp − , (10) c c c Parameter Extrapolation Extrapolation of wind speed data to standard elevations poses a rather subjective approach based on the mean wind where k is a dimensionless shape parameter, and c is a scale velocity only. Unreliability in such extrapolations is reflected parameter with the speed dimension. In many applications, Wind speed, V (m/s) Wind speed, V (m/s) Wind speed, V (m/s) 4 Advances in Meteorology Hence, once the Weibull pdf parameters, c and k,are determined the power exponent can be calculated provided that the cross-correlation coefficient, r is found from the available wind speed time series data. It must be noticed that this last expression reduces to the classical counterpart in (1) after the substitution of k = 0. Therefore, this expression can be written as Ln(c /c ) 1 2 n = . (19) ( ) Ln Z /Z 1 2 Figure 2: The location of the Karaburun wind station. 5. Application In this paper, wind speed data from the Karaburun wind the basic Weibull pdf statistical properties are the expecta- station in Istanbul, Turkey are used, and this station is located tion, E(V), variance, Var(V), and the mth order moment at latitude 41.338 N and longitude 28.677 E(Figure 2). At E(V ) around the origin which are explicitly available as [19] this location hourly, wind speed measurements are available at three different heights (10 m, 20 m, and 30 m) above ( ) E V = cΓ 1+ , (11) the earth surface. The average wind speed, the standard deviation, and the coefficientofvariation foreachheightare 2 1 given in Table 2. Wind speed data measurement empirical 2 2 Var(V) = c r 1+ − r 1+ , (12) relative frequency distribution functions (histograms) at k k 10 m, 20 m, and 30 m are given together with the theoretically m m E(V ) = c Γ 1+ , (13) fitted Weibull pdf’s in Figure 3 for each height. A good fit between the empirical and theoretical counterparts at different heights are obtained through the Kolmogorov- respectively. Smirnov test at significance level of 5%. Table 1 presents the It is the purpose of this paper to present detailed Weibull pdf parameters, c (scale) and k (shape). The scale extrapolation formulations for the Weibull pdf parameters parameters are 8.32, 8.77, and 9.54 for heights of 10 m, 20 m, on the basis of perturbation approach and power law of and 30 m, respectively. The shape parameters are determined vertical wind velocity variation. as 2, 2.11, and 2.06, respectively for the same heights. The two-parameter Weibull pdf has the average and It is clear from this table that as the height increases, standard deviation as in (11)and (12); and by definition their the mean speed and standard deviation increase as expected. ratio gives the coefficient of variation as Coefficient of variation varies with different heights as in Table 2. This shows also that the closer the height to the earth ( ( )) Γ 1+ 2/k (14) C = − 1. surface, the greater is the instability of the air. The power law Γ (1+ (1/k)) exponent, n, calculation between any two heights are found from (18) and classically from (19); and they are presented in The k value can best be estimated by using the approximate Table 3. relationship for (14) as given by Justus and Mikhail [17], that For both classical and Weibull pdf approaches the is: greatest value lies between 20 m and 30 m, whereas the lowest value is between 10 m and 20 m. For all levels, the average k = (15) 1.086 values are 0.1360 and 0.1238 for classical and extended power laws, respectively. and (11) yields the scale parameter as V 6. Conclusions c = . (16) Γ(1+ (1/k)) A simple methodology on the basis of the perturbation theory is presented in order to derive an extended power The substitution of these last two expressions for two elevations with labels 1 and 2 into (8)leads aftersomealgebra law for the vertical wind speed extrapolation and then the Weibull probability distribution parameters. It is observed to that on the contrary to the classical approach not only the Z c 1 1 −0.921 −1.841 means of wind speeds are at different elevations, but also = 1 − r (k k ) + k . (17) 12 1 2 Z c 2 2 the standard deviations and the cross-correlation coefficient should be taken into consideration, if the wind speeds at By taking the logarithms of both sides, give different elevations are not independent from each other. Otherwise, consideration of the classical power law in the −0.921 −1.841 Ln(c /c ) +Ln 1 − r (k k ) + k 1 2 12 1 2 2 (18) calculations embodies the assumption that there are no n = . Ln(Z /Z ) 1 2 fluctuations in the wind speed time series around their Advances in Meteorology 5 0.1 0.1 0.08 0.08 c (scale) = 8.32 c (scale) = 8.77 k (shape) = 2 k (shape) = 2.11 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 510 15 20 25 0 510 15 20 25 Wind speed data (m/s) Wind speed data (m/s) V (Z = 10 m) data V (Z = 20 m) data 1 2 Weibull-pdf Weibull-pdf (a) (b) 0.1 0.09 c (scale) = 9.54 0.08 k (shape) = 2.06 0.07 0.06 0.05 0.04 0.03 0.02 0.01 010 20 30 40 50 Wind speed data (m/s) V (Z = 30 m) data Weibull-pdf (c) Figure 3: Karaburun Weibull pdf’s at different levels. Table 2: Wind speed summary statistics of Karaburun for three different heights. Weibull pdf parameters Height Mean speed Standard deviation Coefficient of variation (m) (m/s) (m/s) ck 10 7.37 3.86 0.52 8.32 2.0 20 7.75 3.88 0.50 8.77 2.11 30 8.44 4.30 0.51 9.54 2.06 Table 3: Exponent calculations both classical and extended power extrapolations are presented in this paper. The application law. of the developed methodology is presented for Karaburun, Istanbul, near the Black Sea coast wind speed measurement Height Classical Extended station data at three different levels. (m) (19) (18) 10–20 0.0760 0.0935 20–30 0.2076 0.1608 10–30 0.1245 0.1172 References Average 0.1360 0.1238 [1] M. Motta, R. J. Barthelmie, and P. Vølund, “The influence of non-logarithmic wind speed profiles on potential power respective mean values. The necessary formulations for output at danish offshore sites,” Wind Energy,vol. 8, no.2,pp. the Weibull distribution function wind speed parameter 219–236, 2005. Density Density Density 6 Advances in Meteorology [2] A. Tindal, K. Harman, C. Johnson, A. Schwarz, A. Garrad, [19] K. Conradsen, L. B. Nielsen, and L. P. Prahm, “Review of and G. Hassan, “Validation of GH energy and uncertainty Weibull statistics for estimation of wind speed distributions,” predictions by comparison to actual production,” in Proceed- Journal of Climate & Applied Meteorology,vol. 23, no.8,pp. ings of the AWEA Wind Resource and Project Energy Assessment 1173–1183, 1984. Workshop, Portland, Ore, USA, September 2007. [3] M.R.Patel, Wind and Solar Power Systems, CRC Press, 1999. [4] M.R.Elkinton,A.L.Rogers, andJ.G.McGowan,“An investigation of wind-shear models and experimental data trends for different terrains,” Wind Engineering,vol. 30, no.4, pp. 341–350, 2006. [5] R. H. Kirchhoff and F. C. Kaminsky, “Wind shear mea- surements and synoptic weather categories for siting large wind turbines,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 15, no. 1–3, pp. 287–297, 1983. [6] B. Turner and R. Istchenko, “Extrapolation of wind profiles using indirect measures of stability,” Wind Engineering, vol. 32, no. 5, pp. 433–438, 2008. [7] Minnesota Department of Commerce, “Wind resource analy- sis program (WRAP),” Minnesota Department of Commerce, St. Paul, Minn, USA, October 2002. [8] G. Gualtieri and S. Secci, “Comparing methods to calculate atmospheric stability-dependent wind speed profiles: a case study on coastal location,” Renewable Energy, vol. 36, no. 8, pp. 2189–2204, 2011. [9] M. Hussain, “Dependence of power law index on surface wind speed,” Energy Conversion and Management,vol. 43, no.4,pp. 467–472, 2002. [10] D. A. Spera and T. R. Richards, “Modified power law equations for vertical wind profiles,” in Proceedings of the Conference and Workshop on Wind Energy Characteristics and Wind Energy Siting, Portland, Ore, USA, June 1979. [11] A. S. Smedman-Hog ¨ strom ¨ and U. Hog ¨ strom, ¨ “A practical method for determining wind frequency distributions for the lowest 200 m from routine meteorological data,” Journal of Applied Meteorology, vol. 17, no. 7, pp. 942–954, 1978. [12] F. Banuelos-R ˜ uedas, C. Angeles-Camacho, and S. Rios- Marcuello, “Analysis and validation of the methodology used in the extrapolation of wind speed data at different heights,” Renewable and Sustainable Energy Reviews,vol. 14, no.7,pp. 2383–2391, 2010. [13] R. J. Barthelmie, “Evaluating the impact of wind induced roughness change and tidal range on extrapolation of offshore vertical wind speed profiles,” Wind Energ, vol. 4, pp. 99–105, [14] M. Motta, R. J. Barthelmie, and P. Vølund, “The influence of non-logarithmic wind speed profiles on potential power output at danish offshore sites,” Wind Energy,vol. 8, no.2,pp. 219–236, 2005. [15] C. G. Justus, W. R. Hargraves, and A. Yalcin, “Nationwide assessment of potential output from wind powered genera- tors,” Journal of Applied Meteorology, vol. 15, no. 7, pp. 673– 678, 1976. [16] C. G. Justus, W. R. Hargraves, and A. Mikhail, “Reference wind speed distributions and height profiles for wind turbine design and performance evaluation applications,” ERDA ORO/5107- 76/4, 1976. [17] C. G. Justus and A. Mikhail, “Height variation of wind speed and wind distribution statistics,” Geophysical Research Letters, vol. 3, pp. 261–264, 1967. [18] A. Altunkaynak, T. Erdik, I. Dabanlı, and Z. Sen, “Theoretical derivation of wind power probability distribution function and applications,” Applied Energy, vol. 92, pp. 809–814, 2012. 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Wind Velocity Vertical Extrapolation by Extended Power Law

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Hindawi Publishing Corporation
Copyright
Copyright © 2012 Zekai Şen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1687-9309
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10.1155/2012/178623
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Abstract

Hindawi Publishing Corporation Advances in Meteorology Volume 2012, Article ID 178623, 6 pages doi:10.1155/2012/178623 Research Article Zekai S¸en, Abdusselam ¨ Altunkaynak, and Tarkan Erdik Hydraulics Division, Civil Engineering Faculty, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey Correspondence should be addressed to Tarkan Erdik, tarkanerdik@hotmail.com Received 19 August 2012; Accepted 7 October 2012 Academic Editor: Harry D. Kambezidis Copyright © 2012 Zekai S¸en et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Wind energy gains more attention day by day as one of the clean renewable energy resources. We predicted wind speed vertical extrapolation by using extended power law. In this study, an extended vertical wind velocity extrapolation formulation is derived on the basis of perturbation theory by considering power law and Weibull wind speed probability distribution function. In the proposed methodology not only the mean values of the wind speeds at different elevations but also their standard deviations and the cross-correlation coefficient between different elevations are taken into consideration. The application of the presented methodology is performed for wind speed measurements at Karaburun/Istanbul, Turkey. At this location, hourly wind speed measurements are available for three different heights above the earth surface. 1. Introduction coefficients for different types of topography and geography [3]. Wind energy, as one of the main renewable energy sources in According to the calculations of wind resource analysis the world, attracts attention in many countries as the efficient program (WRAP) report, in 39 different regions, out of turbine technology develops. Wind speed extrapolation 7082 different wind shear coefficients, 7.3% are distributed might be regarded as one of the most critical uncertainty between 0 and 0.14 and 91.9% above 0.14, while 0.8% are factor affecting the wind power assessment, when consider- calculated as negative [7], due to the measurements error. ing the increasing size of modern multi-MW wind turbines. Different methods have been developed to analyze wind If the wind speed measurements at heights relevant to wind speed profiles, such as power, logarithmic, and loglinear laws energy exploitation lacks, it is often necessary to extrapolate [8]. Besides, in the literature various studies are conducted observed wind speeds from the available heights to turbine in order to estimate wind shear coefficient in the power law hub height [1], whichcausessomecriticalerrorsbetween only if surface data is available at hand [9–11]. estimated and actual energy output, if the wind shear The wind speed undergoes repeated changes, as a result coefficient, n, cannot be determined correctly. The difference of which the roughness and friction coefficients also change between the predicted and observed wind energy production depending on landscape features, the time of the day, the might be up to 40%, due to turbulence effects, time interval temperature, height and wind direction. The uncertainty of wind data measurement, and the extrapolation of the data is inhereted in the wind speed data and its extrapolation from reference height to hub heights [2]. to the hub height should be considered carefully and In the literature, the wind shear coefficient is generally preciously [12]. Moreover, this uncertainty is exacerbated in approximated between 0.14 and 0.2. However, in real situa- the offshore environment by the inclusion of the dynamic tions, a wind shear coefficient is not constant and depends surface [13]. Therefore, the mean wind speed profile of on numerous factors, including atmospheric conditions, the logarithmic type is developed by applying a stability temperature, pressure, humidity, time of day, seasons of correction for offshore sites [14]. the year, the mean wind speed, direction, and nature of It is crucial point for energy investors to accurately terrain [3–6]. Table 1 demonstrates the various wind shear predict the average wind speed at different wind turbine 2 Advances in Meteorology Table 1: Wind shear coefficient of various terrains [3]. of the averages and perturbation terms about their averages that render (1)to Terrain type Lake, ocean, and smooth-hard ground 0.1 Z V + V 1 1 = ,(2) Foot-high grass on level ground 0.15 Z V + V 2 2 Tall crops, hedges, and shrubs 0.2 where V and V are the perturbation terms with averages 1 2 Wooded country with many trees 0.25 equal to zero, V = V = 0. This last expression can be 1 2 Small town with some trees and shrubs 0.3 rewritten simply as City area with tall buildings 0.4 − 1 Z V V V 1 1 1 2 (3) = 1+ 1+ . V V V 2 2 1 2 hub heights and make realistic feasibility projects for these heights. In this study, a simple but effective methodology on This expression is referred to as the extended power law the basis of the perturbation theory is presented in order to in this paper. The third bracket on the right-hand side derive an extended power law for the vertical wind speed corresponds to a geometric series, which can be expressed extrapolation and then the Weibull probability distribution by the Binomial expansion as function (pdf) parameters. It is observed that on the contrary to the classical approach not only the means of wind speeds Z V V V V 1 1 1 2 2 = 1+ 1 − + are at different elevations, but also the standard deviations V V V V 2 2 1 2 2 and the cross-correlation coefficient should be taken into 3 4 consideration, if the wind speeds at different elevations are V V 2 2 − + −··· . not independent from each other. V V 2 2 (4) 2. Power Law This expression can still be simplified after the expansion of This law is the simplest way for estimating the wind speed the second and third brackets on the right-hand side and at a wind generator hub elevation from measurements at a then by considering the second-order term approximately reference level. In general, the power law expression is given yields as, Z V V V V 1 1 2 2 1 = 1 − + + Z V 1 1 = , (1) Z V V V V 2 2 2 2 1 Z V 2 2 (5) V V V V where terms in the brackets are the velocity and elevation 1 2 1 2 − + . ratios, V >V and Z /Z , respectively. Furthermore, V > 2 1 2 1 2 V V V V 1 2 1 2 V and Z >Z ;and n is the exponent of the power 1 2 1 After taking the arithmetic averages of both sides and then law, which is a complex function of the local climatology, considering that the odd order power term averages are equal topography, surface roughness, environmental conditions, to zero, (5) yields to meteorological lapse rate, and weather stability. It is clear ⎡ ⎤ that the effects of all these factors are embedded in the Z V V V V wind velocity time records, and consequently, their total 1 1 1 2 2 ⎣ ⎦ = 1 − + . (6) reflections are also expected in the value of the exponent, V V V V 2 2 1 2 2 n. Therefore, one tends to think whether there is a way of obtaining the estimation of this exponent from the wind By definition V = V = 0, exactly. In fact, for symmet- 1 2 speed time series. Power laws are used almost exclusively rical (i.e., Gaussian) perturbation terms, the odd number without any generally accepted methodology. Most often, arithmetic averages such as V V arealso equaltozero 1 2 only the arithmetic averages of the wind speed at two approximately by definition. elevations are considered in the numerical calculation of the In (6), the common arithmetic average of the perturba- exponent. Logically, other than the mean values, standard tion multiplication, V V ,attwo different elevations is equal 1 2 deviations and cross-correlation coefficient should enter the to the covariance of the perturbations. This can be written calculations, because these additional parameters arise as a in terms of the standard deviations S and S and cross- V1 V2 result of instability, roughness, and so forth. Provided that correlation, r , multiplication as V V = r S S .The 12 12 V V 1 2 1 2 there are wind speed records at two or more elevations, the second-order perturbation term average, V ,isequivalent following approach provides an objective solution. to the variance of the perturbation term as V = S .The 2 V substitution of these last two expressions into (6)leads to 3. Extended Power Law Z V S S 1 1 V V V 1 2 1 In (1), the only random variable that represents the weather = 1 − r + , (7) V V V 2 2 1 2 situation is the wind speeds, which can be written in terms 2 Advances in Meteorology 3 30 30 V (Z = 10 m) V (Z = 10 m) 25 25 20 20 15 15 10 10 5 5 V average average 0 0 0 1000 2000 3000 4000 0 1000 2000 3000 4000 Time index (hour) Time index (hour) (a) (b) V (Z = 10 m) average 0 1000 2000 3000 4000 Time index (hour) (c) Figure 1: Karaburun wind speed records at different levels. where S and S are the standard deviations and r is the in the subsequent wind energy, E, calculations through the V V 12 1 2 cross-correlation coefficient between the wind speed time classical formulation series at two elevations. In practice, most often the cross- correlation term in (7) is overlooked by assuming that there are no random fluctuations around the mean speed values (9) E = ρV , which bring the implication that the standard deviations are 2 equal to zero. These assumptions are not valid because in an actual weather, there are always fluctuations in the wind where ρ is the standard atmosphere air density which is equal speed records as in Figure 1. By definition in statistics, the 3 ◦ to 1.226 gr/cm at 25 C, and V is the wind speed. Most often ratio of standard deviation to the arithmetic mean is the the wind speed at a meteorology station is measured along coefficient of variation, and hence (7)can be rewrittenin atower at different elevations, and it is desired to be able to parameterized form as, find the wind profile at this station for further wind loadings or energy calculations. Some researchers have employed the Z V 1 1 Weibull pdf for empirical wind speed relative frequency = 1 − C C r + C . (8) V V 12 1 2 V Z V distribution (histogram), and a set of formulas are derived for the extrapolation of the Weibull pdf parameters [15–18]. In general, two-parameter Weibull pdf of wind speed, P(V), Herein, C and C are the coefficients of variation for the V V 1 2 wind speed records, V and V ,attwo different elevations, is given as, 1 2 respectively. k−1 k k V V 4. Weibull Distribution P(V) = exp − , (10) c c c Parameter Extrapolation Extrapolation of wind speed data to standard elevations poses a rather subjective approach based on the mean wind where k is a dimensionless shape parameter, and c is a scale velocity only. Unreliability in such extrapolations is reflected parameter with the speed dimension. In many applications, Wind speed, V (m/s) Wind speed, V (m/s) Wind speed, V (m/s) 4 Advances in Meteorology Hence, once the Weibull pdf parameters, c and k,are determined the power exponent can be calculated provided that the cross-correlation coefficient, r is found from the available wind speed time series data. It must be noticed that this last expression reduces to the classical counterpart in (1) after the substitution of k = 0. Therefore, this expression can be written as Ln(c /c ) 1 2 n = . (19) ( ) Ln Z /Z 1 2 Figure 2: The location of the Karaburun wind station. 5. Application In this paper, wind speed data from the Karaburun wind the basic Weibull pdf statistical properties are the expecta- station in Istanbul, Turkey are used, and this station is located tion, E(V), variance, Var(V), and the mth order moment at latitude 41.338 N and longitude 28.677 E(Figure 2). At E(V ) around the origin which are explicitly available as [19] this location hourly, wind speed measurements are available at three different heights (10 m, 20 m, and 30 m) above ( ) E V = cΓ 1+ , (11) the earth surface. The average wind speed, the standard deviation, and the coefficientofvariation foreachheightare 2 1 given in Table 2. Wind speed data measurement empirical 2 2 Var(V) = c r 1+ − r 1+ , (12) relative frequency distribution functions (histograms) at k k 10 m, 20 m, and 30 m are given together with the theoretically m m E(V ) = c Γ 1+ , (13) fitted Weibull pdf’s in Figure 3 for each height. A good fit between the empirical and theoretical counterparts at different heights are obtained through the Kolmogorov- respectively. Smirnov test at significance level of 5%. Table 1 presents the It is the purpose of this paper to present detailed Weibull pdf parameters, c (scale) and k (shape). The scale extrapolation formulations for the Weibull pdf parameters parameters are 8.32, 8.77, and 9.54 for heights of 10 m, 20 m, on the basis of perturbation approach and power law of and 30 m, respectively. The shape parameters are determined vertical wind velocity variation. as 2, 2.11, and 2.06, respectively for the same heights. The two-parameter Weibull pdf has the average and It is clear from this table that as the height increases, standard deviation as in (11)and (12); and by definition their the mean speed and standard deviation increase as expected. ratio gives the coefficient of variation as Coefficient of variation varies with different heights as in Table 2. This shows also that the closer the height to the earth ( ( )) Γ 1+ 2/k (14) C = − 1. surface, the greater is the instability of the air. The power law Γ (1+ (1/k)) exponent, n, calculation between any two heights are found from (18) and classically from (19); and they are presented in The k value can best be estimated by using the approximate Table 3. relationship for (14) as given by Justus and Mikhail [17], that For both classical and Weibull pdf approaches the is: greatest value lies between 20 m and 30 m, whereas the lowest value is between 10 m and 20 m. For all levels, the average k = (15) 1.086 values are 0.1360 and 0.1238 for classical and extended power laws, respectively. and (11) yields the scale parameter as V 6. Conclusions c = . (16) Γ(1+ (1/k)) A simple methodology on the basis of the perturbation theory is presented in order to derive an extended power The substitution of these last two expressions for two elevations with labels 1 and 2 into (8)leads aftersomealgebra law for the vertical wind speed extrapolation and then the Weibull probability distribution parameters. It is observed to that on the contrary to the classical approach not only the Z c 1 1 −0.921 −1.841 means of wind speeds are at different elevations, but also = 1 − r (k k ) + k . (17) 12 1 2 Z c 2 2 the standard deviations and the cross-correlation coefficient should be taken into consideration, if the wind speeds at By taking the logarithms of both sides, give different elevations are not independent from each other. Otherwise, consideration of the classical power law in the −0.921 −1.841 Ln(c /c ) +Ln 1 − r (k k ) + k 1 2 12 1 2 2 (18) calculations embodies the assumption that there are no n = . Ln(Z /Z ) 1 2 fluctuations in the wind speed time series around their Advances in Meteorology 5 0.1 0.1 0.08 0.08 c (scale) = 8.32 c (scale) = 8.77 k (shape) = 2 k (shape) = 2.11 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 510 15 20 25 0 510 15 20 25 Wind speed data (m/s) Wind speed data (m/s) V (Z = 10 m) data V (Z = 20 m) data 1 2 Weibull-pdf Weibull-pdf (a) (b) 0.1 0.09 c (scale) = 9.54 0.08 k (shape) = 2.06 0.07 0.06 0.05 0.04 0.03 0.02 0.01 010 20 30 40 50 Wind speed data (m/s) V (Z = 30 m) data Weibull-pdf (c) Figure 3: Karaburun Weibull pdf’s at different levels. Table 2: Wind speed summary statistics of Karaburun for three different heights. Weibull pdf parameters Height Mean speed Standard deviation Coefficient of variation (m) (m/s) (m/s) ck 10 7.37 3.86 0.52 8.32 2.0 20 7.75 3.88 0.50 8.77 2.11 30 8.44 4.30 0.51 9.54 2.06 Table 3: Exponent calculations both classical and extended power extrapolations are presented in this paper. The application law. of the developed methodology is presented for Karaburun, Istanbul, near the Black Sea coast wind speed measurement Height Classical Extended station data at three different levels. (m) (19) (18) 10–20 0.0760 0.0935 20–30 0.2076 0.1608 10–30 0.1245 0.1172 References Average 0.1360 0.1238 [1] M. Motta, R. J. Barthelmie, and P. Vølund, “The influence of non-logarithmic wind speed profiles on potential power respective mean values. The necessary formulations for output at danish offshore sites,” Wind Energy,vol. 8, no.2,pp. the Weibull distribution function wind speed parameter 219–236, 2005. Density Density Density 6 Advances in Meteorology [2] A. Tindal, K. Harman, C. 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