1. Introduction

The wind power density (WPD) is required for the estimation of power potential from wind turbines. It is a nonlinear function of the wind speed probability density function (PDF). The wind speed PDF is usually estimated from data and then used as a functional in the WPD distribution function, which can be integrated to obtain the WPD (e.g., Çelik 2003a). The wind speed PDF has traditionally been estimated using a parametric model applied to wind speed data at turbine height. These models generally include the Weibull (Stevens and Smulders 1979), Rayleigh (Çelik 2003b), and lognormal functions (Zaharim et al. 2009). The two-parameter Weibull function is the most often used parameteric model for the wind speed PDF (Hennessey 1977; Justus et al. 1978 Pavia and O’Brien 1986; Ramirez and Carta 2005; Monahan 2006). However, it has also been shown that wind speed does not always have a Weibull-like distribution (Jaramillo and Borja 2004; Yilmaz and Çelik 2008). An extreme example of such a case is shown in Fig. 1 using 5-min 10-m height wind speed data from Boise City, Oklahoma, for the year 2000 taken from the Oklahoma Mesonetwork (Brock et al. 1995). The maximum likelihood estimation of the Weibull PDF parameters clearly results in an unacceptable fit to the wind speed histogram.

Given a small sample size, a good argument can usually be made for applying a parametric PDF model to wind data. However, when a robust, smooth histogram of the wind speed distribution can be determined from an abundance of available data like that shown in Fig. 1, nonparametric techniques (Izenman 1991; Silverman 1986) are usually used for PDF estimation because of their flexibility and the very real possibility that the underlying true density may not be adequately represented by a parametric model (Jaramillo and Borja 2004). A commonly used nonparametric method is the kernel method (Silverman 1986; Juban et al. 2007). Kernel techniques are quite flexible and usually provide an acceptable estimate of the wind speed PDF. There are, however, many instances (e.g., for engineering applications such as turbine design) where a functional, as opposed to graphical, representation of the wind power distribution is required. Such a representation of the PDF using the kernel method results in a number of terms equal to the number of unique data points used in the fitting process, resulting in a very unwieldy mathematical function. Thus, the kernel method is a suboptimal technique for providing a functional representation of the distribution of power with wind speed.

Because the WPD is a function of the product of the PDF, the air density and one-half the cubed wind speed, minor errors in the PDF estimation can lead to large errors in the WPD estimate (Çelik 2003a). This is especially true when those errors are made at medium and high wind speeds (e.g., using 5-min measurements), where the sample sizes are usually quite small, leading to an estimation error in the WPD that is likely to be quite large. In this paper, the process of estimating the wind speed PDF and then multiplying the resulting functional by the cubed wind speed times one-half the air density to obtain the WPD distribution is hereafter referred to as the “indirect” method. The present study develops a nonparametric, direct estimate of the WPD distribution function by incorporating the wind speed cubed in a Gauss–Hermite expansion. It will be shown that not only does this “direct” method provide a better estimate of the WPD distribution in terms of estimation bias and variance, but it also produces a tractable and computationally simple mathematical formulation. The paper begins with a description of the basic WPD formulation in section 2. A general description of Gauss–Hermite expansion is given in section 3. The development of the direct WPD estimator using the Gauss–Hermite expansion is described in section 4. The result of a comparison of the direct, indirect, and Weibull maximum likelihood estimators of the WPD using a resampling method is described in section 5. An experiment using wind speed from two sites in Oklahoma is described in section 6. Finally, the conclusions and summary are in section 7.

2. The wind power density formulation

The expression for the WPD (W m⁻²) as given by Li and Li (2005) is

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where υ is the wind speed, ρ is the air density, and f (υ) is the wind speed PDF. The WPD distribution as a function of wind speed is therefore

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If air density is assumed to be constant, then the WPD is simply a scaled version of the third raw moment of the wind speed. Because the PDF is essentially a weighting function, it is clear from Eq. (2) that the estimation error in e(υ) is

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where is the estimated wind speed PDF.

Thus, the error in e(υ) is a cubic function of the wind speed times the estimation error in the wind speed PDF. This can be an especially acute problem as the true wind speed PDF f (υ) is generally positively skewed for wind speed. Hence, there are less data available for estimating f (υ) at the higher wind speed values. This, together with the fact that the estimation error is a function of wind speed cubed, means that the estimation error in e(υ) will be greatest for the larger values of υ. Besides an insufficient number of data (i.e., sampling error), the estimation error is also a function of the quality of the fit of the selected function estimator. This paper will address both of the sources of error in the development of a new WPD distribution estimator.

A function estimator selected for a given task should have certain basic properties similar to that suggested by the data. For example, a Gaussian kernel method is commonly used to estimate probability densities because the weighted sum of these kernels is guaranteed to integrate to one and to be positive over the range of support of the data (Silverman 1986). The method is also flexible enough to mimic a wide range of PDF shapes. Another desired property of density estimators is that they are as parsimonious as possible for both computational reasons and further functional analysis. In this regard, the kernel method is not well suited for the reasons mentioned above. Thus, an optimal estimator of the WPD distribution would be one that has similar functional “shape” properties as the underlying distribution and produces a function that has reasonable mathematical tractability.

Perhaps the first step in designing a WPD distribution estimator is to experiment with functions that have the basic shape properties of the WPD distribution rather than the wind speed PDF. Interestingly, on the positive half–real line, the product of υ³/2 and a Weibull-like PDF results in a function with a near-Gaussian shape. This can be observed using a range of realistically shaped wind speed PDFs and plotting the resulting WPD distribution functions. The simple example shown below illustrates this. Assume that a PDF is formed from a mixed Weibull PDF, that is,

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where the shape α and scale β parameters largely represent the range of those found in the literature for wind speed. By varying parameter a the two different Weibull PDFs will have different weights. A plot of six mixed PDFs with a = 0, 0.2, 0.4, 0.6, 0.8, 1.0 is shown in Fig. 2. The corresponding WPD distribution functions (with ρ = 1.0 kg m⁻³) are created by multiplying each PDF by υ³/2 and are plotted in Fig. 3. The resulting functions are more or less quasi symmetric and certainly much more Gaussian-like than the original wind speed PDF.

Near-Gaussian functions have been shown to be quite well estimated using a Gauss–Hermite expansion (van der Marel and Franx 1993). This is because the expansion is a sum of weighted Hermite polynomials that are derived from the standard Gaussian function. In fact, only the first term in the expansion is required to reproduce a Gaussian-shaped function (see below). van der Marel and Franx (1993) demonstrated that additional terms in the expansion tend to account for any asymmetry in a quasi-Gaussian-like curve. They also showed that for near-Gaussian functions only a relatively small number of terms were required for an excellent approximation. Thus, it would appear reasonable that a Gauss–Hermite expansion fit directly to the WPD distribution function rather than the wind speed PDF may provide a better estimate.

3. The Gauss–Hermite orthogonal series estimator

This paper examines a new way to apply a specific class of nonparametric estimators of the WPD distribution, specifically, orthogonal polynomial expansions. Early work on these estimators was conducted by Cencov (1962), Schwartz (1967), Kronmal and Tarter (1968), Watson (1969), Walter (1977), and Liebscher (1990). All of these works involved designing a PDF estimator. In particular, Hall (1980) studied expansions on the positive half-line using Hermite and Laguerre polynomial series and found that Hermite polynomials were generally the superior estimator based on comparisons of the rate of convergence. Bryukhan and Diab (1993) applied a Laguerre polynomial expansion to South African wind speed data for the PDF estimation and found satisfactory results. Although well established in the mathematical literature, there has been little research on practical applications of orthogonal functions for the estimation of the WPD distribution function. In astronomy and astrophysics, however, different forms of these estimators have been successfully used to estimate the spectral line profiles of galaxies (van der Marel and Franx 1993; Blinnikov and Moessner 1998). Expansions of Hermite polynomials were chosen because the spectral profiles tended to be near Gaussian in shape and are modeled quite well by these expansions. Two common Hermite expansions, the Gram–Charlier Series A and the Gauss–Hermite expansions, have been closely studied for their utility in modeling these spectra (van der Marel and Franx 1993; Blinnikov and Moessner 1998). The study by Blinnikov and Moessner (1998) demonstrated quite convincingly the superior convergence properties of the Gauss–Hermite over the Gram–Charlier expansion. Therefore, this paper exclusively focuses on the Gauss–Hermite form of the expansion.

It will be assumed that wind speed measured at a location fixed in space is a random variable V with real-valued positive realizations υ in a temporal domain T. Hermite polynomials of order k, that is, H_k(υ), used in the expansion are orthogonal with respect to a weighting function for any real, continuous variable x, such that

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The Hermite polynomials defined in this manner can be generated using the recursion formula

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Using the wind speed mean μ and standard deviation σ, the PDF of V can be represented by the expansion

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where g(z) is the standard Gaussian

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and z is the standardized wind speed values. The orthogonality property of Hermite polynomials allows the coefficients ψ(k) of order k of the expansion to be represented by

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Thus, the kth expansion coefficient is a constant times the expected value of H_k(z)g(z). Because ψ(k) = π/2^k−1k!E[H_k(z)g(z)], the coefficients are in practice estimated using data (Schwartz 1967; Kronmal and Tarter 1968; Watson 1969; Walter 1977), that is,

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where n is the number of available data and i is a data index. Thus, the wind speed PDF can be estimated using

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The number of coefficients m that provide a reasonable fit for noncomplex PDFs (e.g., unimodal and moderately skewed) is usually between 10 and 15 (Blinnikov and Moessner 1998). Using the indirect method, Eq. (11) is substituted for f (υ) in (2) to obtain an estimate of the WPD distribution,

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given a set of n data values and m + 1 number of coefficients. The corresponding indirect method of estimating the WPD is therefore

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where the subscript I refers to indirect. Thus, the estimate in (12) is made up of the sum of m + 1 basis functions.

4. Estimating the wind power distribution directly

Any function that is square integrable (i.e., has a finite variance, E[ f ²(V)] < ∞) can be represented by an orthogonal expansion. Thus, instead of estimating f (υ) and then inserting this in (2) to obtain an estimate of e(υ), it is proposed to expand the wind power distribution function directly without first estimating f (υ). Because the WPD function appears to be near Gaussian, a Gauss–Hermite expansion should provide a good estimate of this function. For simplicity and to focus on the use of this approach for wind modeling, it will be assumed that air density is constant and equal to 1.0 kg m⁻³. This is not a restrictive assumption because it can be relaxed to incorporate variability in air density in the development.

The wind power distribution function can be represented by the expansion

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where z_d is the wind speed scaled to the wind power density-weighted (WPD-weighted) mean μ_d and standard deviation σ_d. The WPD-weighted wind values are defined by

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where the mean and standard deviation of z_d are defined as

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Estimates for these parameters can be made from data using

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where n is the number of data values available. By scaling υ to z_d the Hermite polynomials are now centered on the WPD function rather than on the wind speed PDF. By multiplying both sides of (14) by H_k(z_d)g(z_d), integrating from υ = 0 to ∞, and noting that dz_d = dυ/σ_d, the coefficients are now defined by

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which equals (again assuming that ρ equals 1.0 kg m⁻³)

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The coefficients are now estimated from the z_d values using

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This expression is a constant times an estimate of E[υ³g(z_d)H_k(z_d)]. By substituting these estimates for the coefficients in (20) into (14), the WPD distribution function can be estimated directly, that is,

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Note that the primary difference between the indirect and direct method is that ê_I(υ|n, m) is a function of an estimate of E[g(z_d)H_k(z_d)], whereas ê_D(υ|n, m) is a function of an estimate of E[υ³g(z_d)H_k(z_d)].

5. Experiment using resampling

Given two function estimators, ê_I(υ|n, m) and ê_D(υ|n, m), it is of interest to compare the estimation error of these estimates for a given number, that is, m + 1, of Gauss–Hermite terms. This can be accomplished by assuming the existence of a “true” PDF and using samples from this PDF to compute the mean integrated square error (MISE) for each estimate as a function of sample size. Although there are an infinite number of theoretical PDFs to choose from, an initial test was conducted using two Weibull functions with very different shape and scale parameters. A resampling scheme was designed by using the two estimators to produce WPD estimates from random values of a given sample size taken from each of the true PDFs and computing the MISE. Two resampling experiments were conducted (i.e., experiments 1 and 2) using the two Weibull PDFs shown in Fig. 4. Random samples of size n were extracted from the two PDFs to form eight sets of samples, with each set containing n number of random numbers for each experiment (j = 1, 2), that is,

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For experiment j and sample set χ*_j[n(i)], 200 estimates of and were constructed. The MISE for each experiment was formed using

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The MISE for the two functional estimates as a function of sample size n were computed by numerically summing over the t = 200 estimates (i.e., one WPD estimate for each of 200 samples for each sample size, n):

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The ê_I(υ|n, m) and ê_D(υ|n, m) functions were constructed using m = 15 (i.e., 16 basis functions) for each expansion (i.e., ψ̂(k), k = 0, 1, 2, … , 15). The results of the resampling experiment are shown in Figs. 5 and 6. Although the MISE differs between experiments, it is clear the direct method is more efficient than the indirect method across all sample sizes. The smaller difference between the two curves in Fig. 6 as compared with this difference in Fig. 5 simply indicates that the Gauss–Hermite expansion is more efficient for near-Gaussian shapes. Note that the wind speed PDF used for experiment 2 was very near Gaussian in shape (Fig. 4).

To assess how the direct and indirect orthogonal expansion methods compare with the indirect Weibull method, the resampling experiment was redone using 5-min 10-m wind speed measurements from Boise City, Oklahoma, taken for the year 2000 (i.e., 95 020 measurements). Thus, three estimators were formed: 1) an indirect Weibull (using the maximum likelihood fit of the Weibull function to estimate the wind speed PDF), 2) an indirect Hermite expansion [i.e., (12)], and 3) the direct Hermite expansion [i.e., (21)]. The indirect Weibull method involved constructing the Weibull PDF estimator [f̂_W(α, β)] from the given sample using the maximum likelihood method to estimate the two parameters α and β. The Weibull function estimator of the WPD distribution is thus

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where the MISE was computed using

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Given the large number of available records, the true WPD distribution curve [i.e., e(υ)] was assumed to be calculable from the wind speed data using a Gaussian kernel fit as

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where f̂_k(υ) was the wind speed PDF estimated from the kernel. As shown in Fig. 7, the kernel provided an excellent fit to the data histogram. Using e(υ) from (26) in (23) and (25), the MISE was then computed as a function of sample size via resampling for all the three estimators.

The results of the resampling experiment are shown in Fig. 8. In this situation, the direct method again outperformed the other two indirect methods across all sample sizes used. The Weibull function estimate simply was not flexible enough to capture the overall character of the WPD as a function of wind speed. The indirect Gauss–Hermite expansion method is clearly unacceptable as well.

The resulting three WPD distribution functions were compared with a bar chart of WPD distribution derived from the Boise City wind speed histogram in Fig. 9. The bar chart was constructed by calculating the relative frequency of wind speed in each bin and multiplying this by one-half the cube of the bin center value. Figure 9 illustrates the problems with indirect methods of calculating the WPD distribution. For Boise City the Weibull function provided an inadequate fit to the wind speed histogram (refer to Fig. 1), especially between the values of 3.5 and 9.0 m s⁻¹. Note that, although the Weibull estimator fit the right tail of the PDF reasonably well, the resulting WPD distribution estimates become increasingly poor after 10.0 m s⁻¹ because of the magnification of minor PDF errors by the υ³/2 factor in the WPD formulation. The indirect Gauss–Hermite expansion provided an adequate fit up to approximately 12 m s⁻¹. However, at values higher than this threshold the function oscillates, even becoming negative, indicating that error in the Gauss–Hermite expansion fit of the wind speed PDF produced moderate errors at the higher values, which were then significantly magnified by the υ³/2 factor in the WPD distribution formulation.

To check the consistency and generality of the previous results, the resampling experiments were rerun using 5-min 10-m wind speed data from 1994 through 2006 from a Mesonet site in Weatherford, Oklahoma, which represents 1 358 045 5-min measurements. The result of these experiments is shown in Figs. 10 and 11. As can be observed from these figures, these results also support those found using Boise City data in that the direct method is vastly superior to the indirect method in terms of being a much better estimator of the wind power density.

Obviously, the number of basis functions included in the expansion affects the quality of the fit. Because the objective of this study is to obtain the most mathematically parsimonious estimator, it is of interest to determine if the expansions converge with an increasing number of terms and what that convergence looks like.

6. An examination of the behavior of the orthogonal expansions

The results in the previous section depended upon the underlying shape of the function to be approximated and the number of terms used in the expansions. Thus, different results would most likely be obtained using different wind speed PDFs and a different number of terms. The “best” estimator is that which produces the smallest MISE for a given number of basis functions. To obtain insight into the behavior of the two expansions, the MISE was formed using the more skewed of the two Weibull functions f_W(υ|α, β) shown in Fig. 4, that is, the one with parameters α = 1.8 and β = 6.5 m s⁻¹.

Using f_W(υ|α, β), the direct and indirect functions were produced by numerically integrating

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for different values of m. The MISE for each estimator was formed from

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where

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represents the true WPD distribution. The results, shown in Fig. 12 for m = 0, 1, 2, 3, 4, 5, … , 15, indicate that given this particular fixed PDF the indirect method has very poor convergence properties, whereas the direct method converges monotonically with an increasing number of basis functions. Interestingly, the indirect method compared well with the direct method for m = 7 (8 terms) and m = 11 (12 terms), respectively. However, for any given number of terms the direct method had a consistently lower MISE than the indirect method throughout the range of included terms. The actual MISE values are given in Table 1. Given five basis functions (m = 4), the direct method has an MISE less than 0.5 W m⁻².

There are two postulated reasons for the better performance of the direct method. The first has to do with the shape of the function being estimated. As shown earlier, the WPD distribution curve appears to be near Gaussian in shape when given a range of Weibull wind speed PDFs. Plots of the individual terms for the indirect (Fig. 13) and direct (Fig. 14) expansion illustrate why the direct expansion fits the WPD distribution curve better. As the number of terms in the indirect expansion (Fig. 13) increases, the sum of k basis functions clearly have oddly varying shapes, notably in the right tail, which do not appear to have any consistent convergence to a given shape. On the other hand, the direct method appears to converge to the desired shape of the true WPD distribution curve (Fig. 14). To illustrate the relative simplicity of the mathematical formulation of the direct WPD expansion, the mathematical expression (27) for the exercise in this section, given five basis functions, reduces to the following expression

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The equation above indicates that the resulting expression for the wind power distribution function is a relatively simple product of an exponential with a fourth-order polynomial (the order, of course, depends upon the number of basis functions used).

7. Summary

An improved WPD distribution function estimator is required for wind resource assessment and turbine-engineering purposes. The aim of this project was to develop a new and improved wind resource modeling tool and examine its efficiency properties compared to commonly used methods. It was important that the estimator be mathematically tractable. Although the popular kernel method has been proven to be quite efficient at PDF estimation, its functional representation is highly unwieldy. This latter requirement eliminated the kernel method as a candidate estimator for this study’s objective. Truncated orthogonal Hermite polynomial series estimators seemed a likely candidate as they can provide a workable function [e.g., Eq. (32)]. An extensive literature review by the authors revealed a plethora of theoretical work on these estimators mostly prior to the year 2000. The theoretical work was mostly confined to the estimator’s convergence properties for PDFs. There is a lack of practical applications of orthogonal series estimators in the literature, especially for wind power estimation. The two most likely reasons for the seeming absence of applications are speculated to be the perception of translation invariance (Izenman 1991) on the abscissa of the Hermite system and the popularity of the kernel method for PDF estimation. An obvious correction to the translation problem is an appropriate data transformation given by van der Marel and Franx (1993) and shown in Eq. (15). Another potential drawback of using an orthogonal polynomial-based series is the possibility of negative values arising where the range of support is only on the positive real line. Although this is quite evident in Fig. 9 using the indirect orthogonal method, it did not appear to be a problem when applying the direct orthogonal estimator. Efromovich (1999) proposes several correction techniques in the case that negative values do arise.

Notwithstanding the lack of utilization of orthogonal series estimators, the results of the study indicated that the Gauss–Hermite expansion, used as an estimator of the WPD distribution function, does not appear to be efficient or convergent if first applied to estimate the wind speed PDF, and then subsequently using this estimate to construct the WPD distribution function (i.e., the indirect method). Alternatively, the use of the Gauss–Hermite expansion as a direct estimator of the WPD distribution results in an efficient, convergent, and mathematically tractable function. It is hypothesized that this results from the capability of Hermite polynomial expansions to naturally model near-Gaussian functions well (because Hermite polynomials are functions of the derivatives of the standard Gaussian), and the fact that estimation error in the wind speed PDF is not enhanced by the υ³/2 factor, which is characteristic of all indirect methods.