A New Approach to Estimate the Parameters of the Joint Distribution of the Wind Speed and the Wind Direction, Modelled with the Angular–Linear Model

Abstract

In order to assess the potential and suitability of a location to deploy a wind farm, it is essential to have a model of the joint probability density function of the wind speed and direction, f_V,Θ(v,θ). The angular–linear model is widely used to obtain the analytical expression of the joint density from the parametric estimation of the probability density functions of wind speed, f_V(v), and wind direction, f_Θ(θ). In previous studies, the parameters of the marginal distributions were obtained by fitting the wind measurements to the cumulative distribution function (CDF) using the least squares method and then calculating the probability density function (PDF). In this study, we propose to directly fit the probability density function and then calculate the cumulative distribution function. It is shown that it has both computational and goodness-of-fit advantages. In addition, previous studies have been expanded, analysing the effect of the number of intervals on which wind speed and direction ranges are divided. The new parameter fitting method is evaluated and compared with the original proposal in terms of goodness of fit, using the coefficient of determination R² as an estimator both in the probability density function (R²_pdf) and in the cumulative distribution function (R²_cdf). The computational times required to estimate the parameters using both methods will also be compared. The new approach is faster, and the goodness of the fitting is satisfactory for both estimators: it produces a better R²_pdf, without significantly affecting the R²_cdf, in contrast to the initial one where the R²_pdf is smaller.

1. Introduction

Studying the statistical behaviour of wind is a fundamental requirement when evaluating the location of a future wind farm [1,2,3]. Typically, this study has focused on modelling the probability distribution of wind speed, for which a multitude of models have been proposed [4]. From the probability distribution of the wind speed and the power curve of a wind turbine [5], it is possible to evaluate indicators such as wind energy density [6], sometimes also called wind power density [7], annual average energy, or capacity factor [8].

In addition to the wind speed, other wind variables and characteristics need to be studied. These include wind direction, which makes it possible to describe the variation in the direction of capture of a wind turbine, allowing the relative positions of the turbines within the wind farm to be selected, minimizing losses due to the wake effect [9,10] and thus maximizing the use of the wind resource. In addition, once the wind farm is in operation, knowledge of the wind direction allows it to be managed more efficiently in real time, reducing operating costs [11]. Through the joint assessment of wind speed and direction based on a joint probability distribution of both variables, it is possible to gain a more comprehensive understanding of the energy characteristics of wind, which is crucial for the effective design and planning of wind farms [12].

Many approaches and models have been proposed to estimate the joint distribution of wind speed and direction. They can be grouped according to the type of fitted statistical model: parametric or non-parametric. Some examples of the first type, which is the most classic one, are the isotropic Gaussian model [13,14], the anisotropic Gaussian model [15], the angular–linear model [12,16], and the Farlie–Gumbel–Morgenstern distribution [11], which consider the correlation between wind speed and direction. The availability of increased computational capacity and ease of recording empirical data have promoted the use of non-parametric statistical methods (data-driven). In this category, we can point out the kernel estimation of the density functions [17] and the multivariate kernel density estimation of wind speed and direction using the Bernstein empirical copula [18], which handles parametric marginal distributions, among many others.

This study will focus on the angular–linear model proposed in [12]. This model uses the linear probability density function of wind speed, f_V(v), and the angular probability density function of wind direction, f_Θ(θ), to construct the probability density function g(ζ), which models the relationship between wind speed and direction. All of them are used to obtain an analytical expression for the joint density function of wind speed and wind direction, f_V,_Θ(v,θ). The current proposal builds upon the methodology presented in [12] but introduces a key innovation: instead of estimating the parameters of the cumulative distribution functions F_V(v), F_Θ(θ), and G(ζ) and then deriving the probability density functions (PDFs), as performed in [12], we propose a direct estimation of the PDF parameters of f_V(v), f_Θ(θ), and g(ζ) using the least squares method.

This proposal is based on the fact that the CDFs of the distributions commonly used to model the statistical behaviour of wind speed and direction used in [12] have no analytical expression. Therefore, the parameter estimation problem formulated as an optimization nonlinear problem can be computationally expensive. However, the use of PDFs makes it possible to calculate an analytical expression, as will be shown in this article, resulting in shorter solving times. Moreover, from a statistical point of view, PDF-based estimation is often asymptotically efficient (it has the least variance among unbiased estimators). On the contrary, CDF-based estimation may be less efficient because it aggregates information into quantiles, potentially losing finer structural details present in the PDF. That is, PDF estimation directly models the local behaviour of the distribution, making it more sensitive to modes, skewness, and tail properties. In addition, in the case of estimating parametric distributions, PDF-based estimation is generally more statistically efficient and less biased when the model is correctly specified, because it uses all the local information in the data. It is difficult to analytically find the statistical efficiency and bias of the least squares estimator, so they are usually calculated by means of a simulation study in a specific family of distributions. These results confirm that it is better to adjust the PDF instead of the CDF; see, for example, [19]. The suitability of the proposed method is assessed in terms of computational time and goodness of fit, evaluated by the coefficient of determination R², which is the percentage of variance in the experimental data explained by each fitted model.

This article is structured as follows. Section 2 provides a more detailed explanation of the angular–linear model. Section 3 presents the formulation of the two alternatives proposed to obtain the parameters of the distributions. Section 4 outlines the metrics that will be used to compare the results of the two used methodologies. Section 5 shows the numerical results. Finally, Section 6 describes the conclusions.

2. Angular–Linear Model

The angular–linear model [12] employs the method proposed by Johnson and Wehrly in [16] to obtain any angular–linear distributions, Equation (1), and provides satisfactory results for modelling the joint density function of the wind speed and wind direction [11,12,18]. However, it has not always been the best among all the examined models [11,18].

$$f_{V,\Theta }\left(v,\theta \right)=2\pi g\left(\zeta \right)f_V\left(v\right)f_{\Theta }\left(\theta \right); 0\le \theta \le 2\pi ; 0\le v\le \infty$$

(1)

where f_V(v) is the PDF of the wind speed v, f_Θ(θ) is the PDF of the wind direction θ, and g(ζ) is the PDF of the circular variable ζ, which is defined according to Equation (2), where F_V(v) and F_Θ(θ) are the cumulative distribution functions of the wind speed and wind direction, respectively.

$$\zeta =\left\{\begin{array}{l}2\pi \left[F_V\left(v\right)-F_{\Theta }\left(\theta \right)\right], { F}_V\left(v\right)\ge F_{\Theta }\left(\theta \right)\\ 2\pi \left[F_V\left(v\right)-F_{\Theta }\left(\theta \right)\right]+2\pi ,{ F}_V\left(v\right)<F_{\Theta }\left(\theta \right)\end{array}\right.$$

(2)

The selection of the type of distribution used to model wind speed and direction, as well as the circular variable, affected the accuracy of the resulting model. The original proposal employed a mixture of a lower truncated normal distribution with a Weibull distribution (TNW) for the wind speed and a finite mixture of von Mises distributions (mvM) for the wind direction (which is multimodal) and for the circular variable ζ. This article maintains these choices in order to evaluate whether the proposed new method for obtaining the parameters of the distributions improves the results of the original proposal.

In Section 2.1, f_V(v) and F_V(v) will be defined and their analytical expressions will be derived. The same will be carried out for f_Θ(θ) and F_Θ(θ) in Section 2.2, and finally, in Section 2.3, it will be explained how g(ζ) has been modelled.

2.1. Probability Density Function of the Wind Speed f_V(v)

It is commonly accepted that wind speed behaviour can be modelled using Weibull or Rayleigh distributions [6]. However, these models have the drawback that they do not fit all wind regimes and cannot represent bimodal behaviours [20] or high probabilities of zero wind speed [21].

One way to overcome the above drawbacks is to use mixtures of distributions, being adequate a mixture of a lower truncated normal distribution with a Weibull distribution (TNW) [4,21].

The PDF of a TNW is defined in Equation (3), which is equivalent to that shown in [12,22].

$$f_V\left(v\right)={\omega }_0·{\displaystyle \frac{f_{VN}\left(v\right)}{F_{VN}\left(\infty \right)-F_{VN}\left(0\right)}}+\left(1-{\omega }_0\right)·f_{VW}\left(v\right) \forall 0\le \theta \le \infty$$

(3)

where ${\omega }_0$ is the weight of the mixture of the lower truncated normal distribution, defined in the interval [0, 1], f_VW(v) is the PDF of the Weibull distribution, f_VN(v) is the PDF of the normal distribution, and F_VN(v) is the CDF of the normal distribution given by Equations (4)–(6), respectively.

$$f_{VW}(v)={\displaystyle \frac{\alpha }{\beta }}{\left({\displaystyle \frac{v}{\beta }}\right)}^{\alpha -1}·e^{-{\left({\displaystyle \frac{v}{\beta }}\right)}^{\alpha }}$$

(4)

$$f_{VN}(v)={\displaystyle \frac{1}{{\varphi }_2·\sqrt{2\pi }}}·e^{\left[-{\displaystyle \frac{1}{2}}·{\left({\displaystyle \frac{v-{\varphi }_1}{{\varphi }_2}}\right)}^2\right]}$$

(5)

$$F_{VN}\left(v\right)={\int }_0^v{f}_{VN}\left(v\right)={\displaystyle \frac{1}{2}}\left[1+\mathrm{e}\mathrm{r}\mathrm{f}\left({\displaystyle \frac{v-{\varphi }_1}{{\varphi }_2\sqrt{2}}}\right)\right]$$

(6)

where β and α are the scale parameter and shape parameter of the Weibull distribution, ϕ₁ and ϕ₂ are the mean and standard deviation of the truncated normal distribution, respectively, and erf(x) is the error function, which is defined according to Equation (7).

$$\mathrm{erf}\left(x\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_0^x{e}^{-t^2}dt$$

(7)

Finally, Equation (8) shows the cumulative distribution function F_V(v) of a TNW which is obtained by integrating Equation (3).

$$F_V\left(v\right)={\omega }_0·{\displaystyle \frac{F_{VN}\left(v\right)}{F_{VN}\left(\infty \right)-F_{VN}\left(0\right)}}+\left(1-{\omega }_0\right)·F_{VW}(v) \forall 0\le \theta \le \infty$$

(8)

where F_VW(v) is given by Equation (9).

$$F_{VW}\left(v\right)=1-e^{-{\left({\displaystyle \frac{v}{\beta }}\right)}^{\alpha }}$$

(9)

2.2. Probability Density Function of the Wind Direction f_Θ(θ)

Several studies have shown that the use of a finite mixture of von Mises distributions provides a flexible model for representing and studying wind direction, which is multimodal, and its use has become widespread [22,23,24]. Therefore, this option is chosen to model f_Θ(θ) and F_Θ(θ), defined by Equation (10) and Equation (11), respectively.

$$f_{\Theta }\left(\theta \right)={\sum }_{j=1}^N{\omega }_j·{\displaystyle \frac{e^{k_j·\cos \left(\theta -{\mu }_j\right)}}{2\pi ·I_0\left(k_j\right)}} \forall 0\le \theta \le 2\pi$$

(10)

$$F_{\Theta }\left(\theta \right)={\sum }_{j=1}^N{\omega }_j·{\displaystyle \frac{{\int }_0^{\theta }{e}^{k_j·\cos \left(\theta -{\mu }_j\right)}d\theta }{2\pi ·I_0\left(k_j\right)}} \forall 0\le \theta \le 2\pi$$

(11)

where N is the number of components in the mixture, μ_j is the mean direction, k_j is the concentration parameter, ω_j is the weight of component j in the mixture, and I₀(k_j) is the modified Bessel function of the first kind and order zero defined by Equation (12).

$$I_0\left(k_j\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_0^{2\pi }{e}^{k_j·\cos \left(\theta \right)}d\theta ={\sum }_{k=0}^{\infty }{\displaystyle \frac{1}{{\left(k!\right)}^2}}·{\left({\displaystyle \frac{k_j}{2}}\right)}^{2k}$$

(12)

2.3. Probability Density Function of the Circular Variable g(ζ)

As ζ is a circular variable and also a multimodal variable that models the relationship between wind speed and wind direction, it is proposed to model g(ζ) as a finite mixture of von Mises distributions, in the same way as was carried out before for f_Θ(θ), in Equations (10)–(12), but now with the parameters μ_s, ω_s, and k_s. There are other alternatives for modelling g(ζ), such as those proposed in [25,26]. However, in this paper, it has been decided to keep the original proposal as stated above.

3. Parameter Estimation

It is common to estimate the parameters of a probability distribution using one of the following methods: the maximum likelihood, method of moments, or method of least squares [4]. However, as argued in [21], the method of least squares is the most appropriate method to estimate the parameters of a TNW. In [21], the least squares method is also chosen to estimate the parameters of the mvM.

In addition, in [11,12,21,27], the least squares method was applied to the CDF of the empirical data. Instead, in this paper, we propose to obtain the parameters of the distributions by applying the least squares method to the PDF of the empirical data. This proposal was motivated by the following reasons:

• In order to obtain the joint density distribution of wind speed and wind direction f_V,ϴ(v,θ), the PDFs of wind speed f_V(v), wind direction f_Θ(θ), and circular variable g(ζ) are needed. Therefore, we believe it may be more effective to obtain the parameters of the distributions by directly applying the method of least squares to these expressions. It is evident that to define the values of variable ζ, both F_V(v) and F_Θ(θ) are required.
• The cumulative probability distribution of the wind direction, F_Θ(θ), is modelled as a mixture of von Mises distributions, with several integral terms that do not have an analytical expression. Therefore, it is expected that the computational cost of obtaining the parameters using the least squares method will be lower if the probability density distribution f_Θ(θ) is used instead of the cumulative probability distribution F_Θ(θ). The same argument can be made for the circular variable.
• According to the previous point, if an optimization solver based on the gradient of the cost function and constraints, such as the SQP (sequential quadratic programming) method, is used to solve the fitting problems, the presence of integral functions in the constraints or in the cost function, for example in Equation (11), to model both the wind direction and the circular variable makes it difficult calculate the gradients and subsequently the number of iterations to find the optimum increase, thus augmenting the time to reach the solution.

For clarity, Section 3.1 presents the formulations for obtaining the parameters of the distributions by applying the least squares method to the PDF, and Section 3.2 presents the formulation for obtaining the parameters of the distributions by applying the least squares method to the CDF.

3.1. Initial Proposal: Fitting the Parameters of the Cumulative Distribution Function

The three optimization problems proposed to obtain the parameters that define the distributions of wind speed, wind direction, and the circular variable from their experimental frequencies are P.1.a, P.1.b, and P.1.c, respectively.

$$\mathbf{P}\mathbf{.}\mathbf{1}\mathbf{.}\mathbf{a}\mathbf{:}\underset{\{{\varphi }_1,{\varphi }_2,\alpha ,\beta \}}{\mathrm{m}\mathrm{i}\mathrm{n}} {\sum }_{i=1}^M{\left\{{VCDF}_i-F_V\left(v_i\right)\right\}}^2$$

(13)

$$\begin{array}{l}\mathrm{s}. \mathrm{to}:\\ {\varphi }_1\ge 0, {\varphi }_2\ge 0, \alpha \ge 0, \beta \ge 0\end{array}$$

(14)

where M is the number of intervals in which the wind speed data samples have been divided. This value is obtained by dividing the maximum speed of the data series by the desired interval size (I) and rounding the result up to the nearest integer. VCDF_i is the cumulative frequency until interval i obtained from the empirical data of wind speed, and F_V(v_i) is the value of the estimated analytical expression of the cumulative distribution for the same interval i.

$$\mathbf{P}\mathbf{.}\mathbf{1}\mathbf{.}\mathbf{b}\mathbf{:} \underset{\{{\mu }_j,k_j,{\omega }_j\}}{min} {\sum }_{k=1}^T{\left\{{\theta CDF}_k-F_{\Theta }\left({\theta }_k\right)\right\}}^2$$

(15)

$$\begin{array}{l}\mathrm{s}. \mathrm{to}:\\ k_j\ge 0, 0\le {\mu }_j\le 2\pi , 0\le {\omega }_j\le 1,\end{array}$$

(16)

$${\sum }_{j=1}^{N_1}{\omega }_j=1$$

(17)

where T is the number of intervals into which the wind direction data samples have been grouped, N₁ is the number of von Mises distributions used in the mixture (T ≥ N₁), θCDF_k is the cumulative frequency until interval k obtained from the empirical data of the wind direction, and F_Θ(θ_k) is the value of the cumulative probability until interval k obtained from the analytical expression of the cumulative distribution function.

$$\mathbf{P}\mathbf{.}\mathbf{1}\mathbf{.}\mathbf{c}\mathbf{:}\underset{\{{\mu }_s,k_s,{\omega }_s\}}{\mathrm{m}\mathrm{i}\mathrm{n}} {\sum }_{q=1}^T{\left\{{\zeta CDF}_q-G\left({\zeta }_q\right)\right\}}^2$$

(18)

$$\begin{array}{l}\mathrm{s}. \mathrm{to}:\\ k_s\ge 0, 0\le {\mu }_s\le 2\pi , 0\le {\omega }_s\le 1,\end{array}$$

(19)

$${\sum }_{s=1}^{N_2}{\omega }_s=1$$

(20)

where N₂ is the number of von Mises distributions used in the mixture (T ≥ N₂), ζCDF_q is the cumulative frequency of empirical data until interval q, and G(ζ_q) is the value of the analytical expression of the cumulative probability until interval q obtained from the cumulative distribution function. To generate ζCDF_q, it is first necessary to create e samples of variable ζ using Equation (2) and e pairs of empirical wind speed and wind direction data used to estimate F_V(v) and F_Θ(θ) (ζ_e = 2π[F_V(v_e) − F_Θ(θ_e)] if F_V(v_e) ≥ F_Θ(θ_e) or ζ_e = 2π[F_V(v_e) − F_Θ(θ_e)] + 2π if F_V (v_e) < F_Θ (θ_e)).

The cumulative frequencies (VCDF_i, θCDF_k, ζCDF_q) of the three variables from the data are obtained by dividing each dataset into a specified number of intervals (M and T). For each interval, the relative frequency is calculated and added to the relative frequencies of the preceding intervals. It is assumed that the cumulative frequency value for each interval corresponds to the upper bound of each interval.

Finally, we would like to emphasize that the cost function of optimization problems P.1.a, P.1.b, and P.1.c requires solving a numerical integration of, f_V(v), f_Θ(θ), or g(ζ) because there is no analytical expression for F_V(v), F_Θ(θ), or G(ζ). As will be discussed later, this fact generates convergence problems when gradient-based optimization solvers are used. However, our proposal does not have this drawback.

3.2. New Proposal: Fitting the Parameters of the Probability Density Function

The three optimization problems proposed to obtain the parameters that define the density functions of the wind speed, wind direction, and circular variable from their experimental frequencies are P.2.a, P.2.b, and P.2.c, respectively.

$$\begin{array}{l}\mathbf{P}\mathbf{.}\mathbf{2}\mathbf{.}\mathbf{a}\mathbf{:} \underset{\{{\varphi }_1,{\varphi }_2,\alpha ,\beta \}}{\mathrm{m}\mathrm{i}\mathrm{n}} {\sum }_{i=1}^M{\left\{{VPDF}_i-f_V\left(v_i\right)\right\}}^2\\ \mathrm{s}. \mathrm{to}:\\ \mathrm{Equation} (14)\end{array}$$

(21)

where VPDF_i is the empirical frequency for interval i and f_V(v_i) is the value of the analytical expression of the probability density function for the same interval i.

$$\begin{array}{l}\mathbf{P}\mathbf{.}\mathbf{2}\mathbf{.}\mathbf{b}\mathbf{:}\underset{\{{\mu }_j,k_j,{\omega }_j\}}{min} {\sum }_{k=1}^T{\left\{{\theta PDF}_k-f_{\Theta }\left({\theta }_k\right)\right\}}^2\\ \mathrm{s}. \mathrm{to}:\\ \mathrm{Equations} (16) \mathrm{and} (17)\end{array}$$

(22)

where θPDF_k is the empirical frequency for interval k and f_Θ(θ_k) is the value of the analytical expression of the probability density function for the same interval k.

$$\begin{array}{l}\mathbf{P}\mathbf{.}\mathbf{2}\mathbf{.}\mathbf{c}\mathbf{:} \underset{\{{\mu }_s,k_s,{\omega }_s\}}{\mathrm{m}\mathrm{i}\mathrm{n}} {\sum }_{q=1}^T{\left\{{\zeta PDF}_q-g\left({\zeta }_q\right)\right\}}^2\\ \mathrm{s}. \mathrm{to}:\\ \mathrm{Equations} (19) \mathrm{and} (20)\end{array}$$

(23)

where ζPDF_q is the empirical frequency for interval q obtained as previously described in Section 3.1 and g(ζ_q) is the value of the analytical expression of the probability density function for the same interval q.

The frequencies (VPDF_i, θPDF_k, ζPDF_q) of the three variables from the empirical data are obtained by dividing each dataset into a specified number of intervals (M and T). For each interval, its relative frequency is calculated and then divided by the interval width. It is assumed that the probability density value for each interval is assumed to correspond to the upper bound of each interval.

3.3. Parameter Initialization

When solving nonlinear programming problems, as those proposed before, it is recommended to provide a suitable starting point for the search algorithm.

The initial values of the parameters of the TNW function (P.1.a or P.2.a) are obtained with Equations (24)–(28) according to [21].

$${\omega }_0=0.5$$

(24)

$$\alpha ={\left({\displaystyle \frac{\sqrt{s^2}}{m}}\right)}^{-1.086}$$

(25)

$$\beta =m{\left[\Gamma \left(1+{\displaystyle \frac{1}{\alpha }}\right)\right]}^{-1}$$

(26)

$${\varphi }_1={\displaystyle \frac{2·m_2^{\prime }·m-m_3^{\prime }}{2·{\left(m\right)}^2-m_2^{\prime }}}$$

(27)

$${\left({\varphi }_2\right)}^2={\displaystyle \frac{m·m_3^{\prime }-{(m}_2^{\prime })}{2{·\left(m\right)}^2-m_2^{\prime }}}$$

(28)

where m, m′₂, and m′₃ are the first, second, and third moments, respectively, centred on the origin of the wind speed data sample for which we fit the distribution, s is the standard deviation of the data, and Γ(·) is the gamma function. On the other hand, to obtain the initial values for each component of the mvM (P.1.b, P.1.c, P.2.b, or P.2.c), one must first divide and group the sample data set, angles θ_j, into N sectors (N₁ or N₂) of size n_j. Then, using Equations (29)–(33) as per [22], the initial values of the parameters μ_j, ω_j, and k_j or μ_s, ω_s, and k_s can be calculated.

$${\omega }_j={\displaystyle \frac{1}{N}}$$

(29)

$${\mu }_j=\left\{\begin{array}{c}artan\left({\displaystyle \frac{\overline{s_j}}{\overline{c_j}}}\right) ; \overline{s_j}\ge 0, \overline{c_j}>0\\ {\displaystyle \frac{\pi }{2}} ;\overline{s_j}>0, \overline{c_j}=0 \\ \pi +artan\left({\displaystyle \frac{\overline{s_j}}{\overline{c_j}}}\right) ; \overline{c_j}<0 \\ \pi ; \overline{s_j}=0, \overline{c_j}=-1 \\ 2\pi +artan\left({\displaystyle \frac{\overline{s_j}}{\overline{c_j}}}\right) ; \overline{s_j}<0, \overline{c_j}>0 \\ {\displaystyle \frac{ 3\pi }{2}} ;\overline{s_j}=0, \overline{c_j}=-1 \end{array}\right.$$

(30)

$$\overline{s_j}={\displaystyle \frac{\sum _{i=1 }^{n_j}\sin {\theta }_i}{n_j}}$$

(31)

$$\overline{c_j}={\displaystyle \frac{\sum _{i=1 }^{n_j}\cos {\theta }_i}{n_j}}$$

(32)

$$A\left(k_j\right)={\displaystyle \frac{I_1\left(k_j\right)}{I_0\left(k_j\right)}}={\left(\overline{s_j}+\overline{c_j}\right)}^{{\displaystyle \frac{1}{2}}}$$

(33)

where n_j is the number of wind direction data points belonging to sector j, I₀(k_j) is the modified Bessel function of the first kind and order zero (12), and I₁(k_j) is the modified Bessel function of the first kind and order one. Since Equation (33) is implicit in k_j, a numerical solution method would likely need to be used, with the consequent computational cost. However, as only an initial value for k_j is being sought, an approximate equation proposed in [22] and shown in Equation (34) may alternatively be used.

$$k_j={\left\{23.29041409-16.8617370{\left({\overline{s}}_J^2+{\overline{c}}_J^2\right)}^{0.25}-17.4749884·e^{-\left({\overline{s}}_J^2+{\overline{c}}_J^2\right)}\right\}}^{-1}$$

(34)

4. Methodology

The results of both approaches will be compared in terms of calculation time and goodness of fit. The goodness of fit between the empirical data and estimated functions (cumulative distribution function or probability density function) is assessed with R²_cdf and R²_pdf, which are calculated according to Equations (35) and (36).

$$R_{cdf}^2=1-{\displaystyle \frac{\sum _{i=1 }^X{\left\{{YCDF}_i-F_Y\left(Y_i\right)\right\}}^2}{{\sum }_{i=1}^X{\left\{{YCDF}_i-\overline{YCDF}\right\}}^2}}$$

(35)

$$R_{pdf}^2=1-{\displaystyle \frac{{\sum }_{i=1}^X{\left\{{YPDF}_k-f_Y\left(Y_i\right)\right\}}^2}{{\sum }_{i=1}^X{\left\{{YPDF}_k-\overline{YPDF}\right\}}^2}}$$

(36)

where Y is the variable under study and can be the wind speed (V), wind direction (θ), both (V_θ), or the circular variable (ζ). X is the is the number of intervals (M for wind speed, T for wind direction or circular variable, and M × T for both wind speed and direction).

5. Results

The two proposals to obtain the parameters of f_V(v), f_Θ(θ), g(ζ), and their final effect on f_V,Θ(v,θ) will be tested and studied using simulated wind data from the New European Wind Atlas (NEWA) [28,29]. The database contains series of wind speed and direction at a specific location and at different heights from 2005 to 2018 in half-hourly time intervals, comprising 245,424 data points in total, of which 195 are missing. These data are available for download free of charge on the NEWA website [30].

This study is going to be carried out for two different locations, which correspond to the wind farms Páramo de Vega (WF1), located in the province of Burgos, Spain (Latitude 42° 30′ 59.6″, Longitude −3° 47′ 14.2″), and El Valle Valdenavarro (WF2), in the province of Navarra, Spain (Latitude 41° 55′ 18.9″, Longitude −1° 25′ 46.9″). For each location, the data series from 2005 to 2018 whose simulation height is closest to the height at which the wind turbine hub of the wind farms is located is chosen in the NEWA. For WF1, the height of 75 m is chosen, and for WF2, the height of 150 m is chosen, since the height at which the hubs are located are 78 and 150 metres, respectively [31,32].

The results are presented in the following outline: Section 5.1, wind speed modelling; Section 5.2, wind direction modelling; Section 5.3, ζ generation and circular variable modelling; and Section 5.4, joint wind speed and wind direction modelling.

The implementation and resolution of the problems have been carried out using the fmincon function from Matlab’s Optimization Toolbox [33,34] with the SQP algorithm as the optimization solver.

5.1. Wind Speed Modelling

In this section, we compare the results of modelling the wind speed when it is fitted to the parameters of the cumulative distribution function (P.1.a) or the parameters of the probability density function (P.2.a) for wind farms WF1 and WF2. For each wind farm, we have split the data with three different interval sizes. This decision was made because we want to test how the interval size affects the results, which would be fixed by the anemometer resolution in a study with real data.

Table 1. Goodness-of-fit for wind speed modelling in WF1.

WF1
I	R2pdf		R2cdf		Time (s)
	min P.2.a	min P.1.a	min P.2.a	min P.1.a	min P.2.a	min P.1.a
1	0.9973	0.9664	0.9954	0.9999	0.8225	1.0572
0.5	0.9969	0.9852	0.9991	0.9999	0.6155	1.34559
0.25	0.9964	0.9896	0.9996	0.9999	0.8042	1.8005

and

Table 2. Goodness-of-fit for wind speed modelling in WF2.

WF2
I	R2pdf		R2cdf		Time (s)
	min P.2.a	min P.1.a	min P.2.a	min P.1.a	min P.2.a	min P.1.a
1	0.9976	0.9713	0.9965	0.9990	0.5597	0.4209
0.5	0.9920	0.9654	0.9984	0.9991	0.5563	0.7309
0.25	0.9890	0.9602	0.9994	0.9991	0.6777	1.0486

show the value of the coefficients R²_pdf and R²_cdf and the time taken to solve the optimization problem for wind farms WF1 and WF2, respectively.

At first sight, in both cases, it can be seen that fitting directly the parameters of the probability density function (P.2.a) gives better results for R²_pdf than fitting the parameters using the cumulative distribution function (P.1.a). Conversely, the best results for R²_cdf are obtained when (P.1.a) is used instead of (P.2.a), except for WF2 and I = 0.25 m/s, marked in red in Table 2.

As the size of the intervals I decreases, the value of R²_pdf generally decreases, whether (P.2.a) or (P.1.a) is used. Conversely, R²_cdf improves for (P.2.a) and remains almost constant for (P.1.a).

Regarding the time to obtain the value of the parameters, it can be seen how estimating the parameters of the probability density function (P.2.a) requires slightly less time with a time reduction of about 20%, with the exception of WF2 and I = 1 m/s, marked in red in Table 2.

Figure 1 and Figure 2 show the fitting of f_V(v) and F_V(v) and the corresponding frequencies of the empirical wind speed data (VPDF and VCDF) for an interval size I = 1 m/s and I = 0.25 m/s, according to the results of Table 1 and Table 2. It is evident that there are differences in the shape of f_V(v) and F_V(v) depending on the method used to find the parameters and that these differences decrease with decreasing interval size I. The observed discontinuity in Figure 1 is due to the interval size used to discretize the distribution. A larger interval size causes slight irregularities, especially at the peak, as the resolution of the discretized data affects the smoothness of the plotted curve.

In general, estimating the parameters of the probability density function (P.2.a) allows us to obtain a satisfactory fit both for PDF and CDF functions, as R²_pdf and R²_cdf show. However, estimating the parameters of the cumulative distribution function (P.1.a) allows us to obtain the best fitting but only for the CDF function.

5.2. Wind Direction Modelling

In [22], it is shown how the number of von Mises distributions (N) affects the goodness of fit of the data, and it is found that from N = 6 there is no relevant change in the goodness of fit when it is evaluated by R².

In this paper, this comparison is repeated for different values of N (2, 3, 4, 5, and 6) for both P.1.b and P.2.b. Regarding the value of the number of intervals (T) into which the wind direction data samples are divided, it is important to choose a sufficiently large number. In this paper, it was decided to try dividing the data into 10, 36, 90, 180, or 360 intervals (36°, 10°, 4°, 2°, 1°). In the case of real data, the choice of this value would depend on the resolution of the measuring instrument.

Figure 3 and Figure 4 show as examples the settings obtained for f_Θ(θ) and F_Θ(θ) and their corresponding frequencies of empirical wind direction data (θPDF and θCDF) for the wind farm WF1 using T = 36 and N = 6 (Figure 3) and T = 360 and N = 6 (Figure 4) to solve P.2.b or P.1.b, whose numerical results are showed in

Table A5. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF1 and N = 6.

WF1
N = 6	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	1.0000	0.4691	0.9730	1.0000	0.5304	3.1694
T = 36	0.9997	0.9523	0.9985	1.0000	1.6397	76.2945
T = 90	0.9984	0.9895	0.9998	1.0000	2.3712	374.2246
T = 180	0.9977	0.9942	0.9999	1.0000	3.9549	894.5340
T = 360	0.9939	0.9927	1.0000	1.0000	4.9048	791.7207

.

From Figure 3, it is evident that fitting the parameters of the mixture of von Mises distributions by directly adjusting the cumulative probability distribution (P.1.b) yields worse results than directly fitting the parameters of the probability density function (P.2.b). This is evidenced by the fact that there is a significant difference between the two methods in the values of R²_pdf: R²_pdf = 0.9525 (P.1.b) and R²_pdf = 0.9997 (P.2.b). Meanwhile, for R²_cdf, there is hardly any difference: R²_cdf = 0.9985 (P.1.b) and R²_cdf = 1.0000 (P.2.b). From Figure 4, it can be seen that as the number of intervals into which the data sample (T) has been divided increases, the differences between fitting the parameters of the cumulative distribution function (P.1.b) or the probability density function (P.2.b) becomes insignificant for both R²_pdf and R²_cdf.

Figure 5, Figure 6 and Figure 7 show the graphical representation of the values of R²_pdf, R²_cdf and the time taken to fit the parameters as a function of the different values of N and T. In the case of R²_pdf, Figure 5 shows that a better result is obtained for this indicator when the parameters of the probability density function have been directly fitted (P.2.b). There are some exceptions, such as in WF2 (T = 360, N = 6, min P2.b R²_pdf = 0.960, min P.1.b R²_pdf = 0.9950), as can be seen in

Table A10. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF2 and N = 6.

WF2
N = 6	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	1.0000	0.5849	0.9331	1.0000	0.6029	3.6831
T = 36	0.9996	0.8480	0.9924	1.0000	1.4218	82.5337
T = 90	0.9982	0.9457	0.9987	0.9996	1.8090	260.5667
T = 180	0.9977	0.9875	0.9997	1.0000	3.1589	434.5785
T = 360	0.9604	0.9950	0.9911	1.0000	3.4800	1127.9038

. In the case of R²_cdf, a better result for this indicator is obtained in all cases when the parameters of the cumulative probability function are fitted (P.1.b), as shown in Figure 6.

For both R²_pdf and R²_cdf, the greatest differences between using P.2.b and P.1.b occur when the number of T intervals is reduced, with the difference being greater for R²_pdf and for the original problem P.1.b.

With regard to the time required to fit the parameters, Figure 7 shows how the calculation time increases with an increase in the number of components in the mixture (N) or in the number of intervals (T) into which the data wind direction data are divided. It can also be observed that in the majority of cases directly adjusting the parameters of the probability density function (P.2.b) requires less computational time when compared with the formulation P.1.b, where the parameters of the cumulative distribution function are fitted.

It is evident that P.2.b exhibits superior scaling properties in comparison to P.1.b. As the number of components in the mixture (N) or the number of intervals (T) into which the data are divided increase, the computational time of the proposed approach (P.2.b) increases linearly, unlike the original method, which increases exponentially.

For example, to guarantee R²_pdf and R²_cdf values above 0.99, the first method P.1.b requires T = 180 or 360 and N = 5 or 6 with a computational time of about 800 s, but our approach (P.2.b) needs only 4.9 s in the worst case, as can be seen in

Table A4. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF1 and N = 5.

WF1
N = 5	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	1.0000	0.5042	0.9762	1.0000	0.5736	3.0452
T = 36	0.9975	0.9386	0.9985	0.9999	1.0714	15.17452
T = 90	0.9960	0.9878	0.9998	1.0000	1.4712	205.0244
T = 180	0.9924	0.9749	0.9997	0.9999	1.8937	85.1508
T = 360	0.9938	0.9929	1.0000	1.0000	3.2127	812.8730

and Table A5 for WF1 and

Table A9. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF2 and N = 5.

WF2
N = 5	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	1.0000	0.4604	0.9436	1.0000	0.6374	5.4395
T = 36	0.9991	0.8553	0.9923	1.0000	0.9597	41.4387
T = 90	0.9971	0.9631	0.9982	1.0000	1.2076	115.3523
T = 180	0.9970	0.9878	0.9996	1.0000	1.7334	271.4960
T = 360	0.9946	0.9933	0.9994	1.0000	2.396	817.3426

and Table A10 for WF2, both in Appendix A.

The numerical values presented in Figure 5, Figure 6 and Figure 7 and the corresponding goodness of each fit can be found in

Table A1. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF1 and N = 2.

WF1
N = 2	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	0.9847	0.5055	0.9768	0.9995	0.5875	0.4373
T = 36	0.9575	0.9069	0.9974	0.9996	0.5666	0.8899
T = 90	0.9544	0.9390	0.9989	0.9996	0.5686	2.1501
T = 180	0.9533	0.9444	0.9991	0.9996	0.6168	3.7769
T = 360	0.9519	0.9454	0.9992	0.9996	0.6364	8.7883

,

Table A2. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF1 and N = 3.

WF1
N = 3	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	0.9953	0.4888	0.9781	0.9999	0.5994	2.3977
T = 36	0.9776	0.9069	0.9981	0.9996	0.6	2.1133
T = 90	0.9748	0.9650	0.9994	0.9998	0.6692	14.2521
T = 180	0.9739	0.9444	0.9996	0.9996	0.7683	12.0320
T = 360	0.9726	0.9709	0.9997	0.9998	0.9041	52.0740

,

Table A3. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF1 and N = 4.

WF1
N = 4	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	1.0000	0.5194	0.9766	1.0000	0.6354	3.0979
T = 36	0.9854	0.9385	0.9979	0.9999	0.6597	15.2572
T = 90	0.9849	0.9705	0.9993	0.9999	0.8836	36.0721
T = 180	0.9848	0.9802	0.9995	0.9999	1.0116	85.6814
T = 360	0.9838	0.9809	0.9996	0.9999	1.4091	179.1024

, Table A4, Table A5,

Table A6. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF2 and N = 2.

WF2
N = 2	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	0.9809	0.5358	0.7919	0.9978	0.5248	0.5052
T = 36	0.9170	0.8012	0.9818	0.9963	0.6105	1.2996
T = 90	0.9033	0.8587	0.9890	0.9962	0.5367	2.8707
T = 180	0.9004	0.8641	0.9899	0.9961	0.6113	5.5376
T = 360	0.8995	0.8647	0.9900	0.9961	0.5995	12.4109

,

Table A7. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF2 and N = 3.

WF2
N = 3	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	1.0000	0.5776	0.9385	1.0000	0.5769	2.0221
T = 36	0.9699	0.8666	0.9906	0.9995	0.5632	7.9804
T = 90	0.9580	0.9375	0.9962	0.9995	0.5796	16.6834
T = 180	0.9573	0.9412	0.9926	0.9995	0.7047	28.5370
T = 360	0.9566	0.9397	0.9808	0.9994	0.9006	41.7003

,

Table A8. Goodness-of-fit for wind direction modelling, f_Θ(θ) and F_Θ(θ), in WF2 and N = 4.

WF2
N = 4	R2pdf		R2cdf		Time (s)
	min P.2.b	min P.1.b	min P.2.b	min P.1.b	min P.2.b	min P.1.b
T = 10	1.0000	0.5671	0.9431	1.0000	0.5317	3.2113
T = 36	0.9974	0.8679	0.9918	0.9995	0.7737	16.6579
T = 90	0.9952	0.9375	0.9980	0.9995	0.8159	49.1095
T = 180	0.9574	0.9763	0.9942	0.9998	0.9415	101.3827
T = 360	0.9941	0.9397	0.9991	0.9994	1.4965	153.2017

, Table A9 and Table A10 in Appendix A.

5.3. ζ. Generation and Angular Variable Modelling

Samples of the circular variable ζ are generated from the historical wind speed and direction data and the fitted F_V(v) and F_Θ(θ). With these data samples, parameters must be fitted to the model g(ζ), using one of the two proposed methods (P.1.c or P.2.c). When using a mixture of von Mises functions to model g(ζ), as for f_Θ(θ), the differences found between fitting the parameters of the probability density function (P.2.c) or fitting the parameters of the cumulative distribution function (P.1.c) remain, as in the case of wind direction modelling (Section 5.2), and the exact same conclusion can be reached. The numerical values of the fits are shown in

Table A11. Goodness-of-fit for circular variable modelling, g_ζ(ζ) and G_ζ(ζ), in WF1 and N = 6.

WF1
N = 6	R2pdf		R2cdf		Time (s)
	min P.2.c	min P.1.c	min P.2.c	min P.1.c	min P.2.c	min P.1.c
T = 10	0.9985	−0.6080	1.0000	1.0000	0.1469	2.6798
T = 36	0.9878	0.9602	1.0000	1.0000	1.5429	122.6574
T = 90	0.9652	0.9592	1.0000	1.0000	14.7182	240.3376
T = 180	0.9383	0.9348	1.0000	1.0000	28.6691	717.9321
T = 360	0.8954	0.8862	1.0000	1.0000	63.4637	968.2772

and

Table A12. Goodness-of-fit for circular variable modelling, g_ζ(ζ) and G_ζ(ζ), in WF2 and N = 6.

WF2
N = 6	R2pdf		R2cdf		Time (s)
	min P.2.c	min P.1.c	min P.2.c	min P.1.c	min P.2.c	min P.1.c
T = 10	1.0000	0.8534	0.9901	1.0000	0.3657	5.5427
T = 36	0.9993	0.9813	0.9989	1.0000	3.2684	71.4667
T = 90	0.9970	0.9934	0.9998	1.0000	5.0959	157.6568
T = 180	0.9951	0.9940	0.9999	1.0000	8.5651	1341.1430
T = 360	0.9877	0.9894	1.0000	1.0000	46.3624	1680.6885

of Appendix A.

5.4. Joint Wind Speed and Wind Direction Modelling

Finally, by combining the results of Section 5.1, Section 5.2, and Section 5.3 in Equation (1), we obtain f_V,Θ(v,θ), and we can evaluate the estimator R²_pdf between the analytical expression obtained and the empirical data of wind speed and wind direction (VθPDF). As can be seen in

Table 3. Results for joint wind speed and wind direction modelling, f_V,Θ(v,θ) and F_V,Θ(v,θ), in WF1 with N = 6 and I = 0.25.

WF1
N = 6 I = 0.25	R2pdf		R2cdf		Time (s)
	min P.2	min P.1	min P.2	min P.1	min P.2	min P.1
T = 10	0.8693	0.6230	0.9852	0.9987	1.355	6.8978
T = 36	0.8578	0.8218	0.9799	0.9987	3.8603	200.0005
T = 90	0.8484	0.8310	0.9986	0.9987	17.7671	615.6108
T = 180	0.8379	0.8242	0.9987	0.9987	33.3017	1613.5147
T = 360	0.8185	0.8060	0.9986	0.9987	69.0462	1761.0465

and

Table 4. Results for joint wind speed and wind direction modelling, f_V,Θ(v,θ) and F_V,Θ(v,θ), in WF2 with N = 6 and I = 0.25.

WF2
N = 6 I = 0.25	R2pdf		R2cdf		Time (s)
	min P.2	min P.1	min P.2	min P.1	min P.2	min P.1
T = 10	0.8068	0.5494	0.9384	0.9957	1.6463	10.2744
T = 36	0.8361	0.7024	0.9857	0.9945	5.3679	155.049
T = 90	0.8244	0.7972	0.9922	0.9940	7.5826	419.2721
T = 180	0.8203	0.8147	0.9937	0.9942	12.4017	1776.7701
T = 360	0.7906	0.8143	0.9919	0.9940	50.5201	2809.6409

, the best fit for f_V,Θ(v,θ) (R²_pdf) is obtained when the parameters of the probability density functions f_V(v), f_Θ(θ), and g(ζ) have been directly estimated (P.2).

As for the cumulative distribution of wind speed and direction, F_V,Θ(v,θ), there is no analytical function, so a numerical method must be used to integrate it from f_V,Θ(v,θ). As can be seen in Table 3 and Table 4, the goodness of the fit is very similar for both methods P.1 and P.2 with an R²_cdf greater than 0.99.

However, in terms of computation time, calculated as the sum of the computation time of each fitting problem (P.1.a, P.1.b, and P.1.c or P.2.a, P.2.b, and P.2.c), solving P.1 to fit the parameters of the cumulative distribution functions is much longer than solving it using our approach (P.2), as shown in Table 3 and Table 4. For example, the best fit for the probability density function is R²_pdf = 0.831 (T = 90) for solving P.1 in WF1 and takes 615 s, but under the same conditions, solving P.2 gives an R²_pdf = 0.848 with a computation time of only 17 s.

As an example, Figure 8 and Figure 9 show the function f_V,Θ(v,θ) generated for I = 0.25, N = 6, and T = 90 together with the histogram of the empirical data of wind speed and wind direction (VθPDF).

Finally, Figure 10 and Figure 11 show, as an example, the F_V,Θ(v,θ) generated for I = 0.25, N = 6 and T = 90, together with the histogram of the empirical data of wind speed and wind direction (VθCDF).

6. Conclusions

One of the most commonly used models for modelling the joint probability density function of wind speed and wind direction f_V,Θ(v,θ) is the angular–linear model, which is based on using the individual probability density functions of wind speed f_V(v) and wind direction f_Θ(θ). In this paper, it is proposed to directly estimate the parameters of the probability density functions (f_V(v) and f_Θ(θ)) instead of estimating the parameters of the cumulative distribution functions (F_V(v) and F_Θ(θ)) as an alternative, since the CDF used to model the wind speed and direction has no analytical expression.

In the case of wind speed, it has been shown that the fit of f_V(v) fitting the probability density function (P.2.a) is slightly better than fitting the cumulative distribution function (P.1.a), regardless of the size of the interval (I) into which the data are divided. The computation time required to fit the parameters is slightly lower for the proposed approach; however, the absolute value remains low even for large problems. Thus, while the increase is notable in relative terms, it is not substantial. For the wind direction, the advantages of fitting using the probability density function (P.2.b) are that regardless of the number of intervals (T) into which the experimental data are divided, the parameters obtained always give similar or better fits than those obtained with the cumulative distribution function (P.1.b). Furthermore, as with wind speed, solving P.2.b is, in general, always much faster than solving P.1.b. Moreover, in this case, the differences in absolute computation time increase exponentially as the fitted model becomes more complex, in particular, as T and N increase.

Finally, once f_V(v), f_Θ(θ), and g(ζ), have been defined, it is possible to construct the joint function f_V,Θ(v,θ), which provides a better fit and much less time consumption if the parameters of all probability density functions are fitted, that is, our proposal (P.2).

In summary, this paper has proposed the utilization of the least squares method with probability density functions as a novel approach to estimate the parameters of the marginal distributions that compose the joint distribution of wind speed and direction. This novel approach offers a slight improvement in goodness of fit. The most significant benefit of this method is that it enables the assessment of complex models, characterized by small interval sizes and a high number of von Mises mixtures (N), in a considerably shorter time than the conventional approach.