# Multivariate Weibull Distribution for Wind Speed and Wind Power Behavior Assessment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multivariate Weibull Distribution

- (1)
- The key point of the procedure is to define a change of variable from a Standard Normal to a Weibull distributed one. This transformation must be differentiable and the inverse function has to exist. The Standard Normal distributed variable is created in order to use its known features and transfer them to the Weibull one;
- (2)
- Establish as many changes as the number of variables, n, considering the different Weibull parameters for each case;
- (3)
- Obtain the multivariate Weibull PDF from the multivariate Standard Normal PDF applying the change of variable.

#### 2.1. Normal to Weibull Change of Variables

_{u}(u) is the CDF of u.

_{x}(x) is the CDF of x.

_{u}= F

_{u}(u) and y

_{x}= F

_{x}(x) are uniform distributed variables.

_{u}and y

_{x}can be matched in order to establish a relationship between a Weibull and a Normal distributed variable. The whole process can be understood in two steps: First, the conversion of a Normal distributed variable into a Uniform one and then the conversion of this one into a Weibull one. Therefore, if both CDFs are matched, then the Weibull distributed variable, u, can be expressed as a function of the Normal one x, such as in Equation (3):

_{i}, the notation is expressed in Equation (4):

_{i}and k

_{i}are the parameters corresponding to the ith variable.

^{−1}(u;C,k) where u is the Weibull variable. For a single variable, u

_{i}, that function is shown in Equation (5):

^{−1}( ) is the inverse of the error function [18]. The function ntw

^{−1}(u;C,k) is shown in Figure 1 for various values of the parameter C.

**Figure 1.**Standardized Normal variable as a function of a Weibull one, with various values of the scale parameter C.

#### 2.2. Multivariate Normal to Weibull Change

**x**is a vector formed by several Standard Normal variables; n is the number of them; det( ) means the determinant of a matrix,

**Σ**is the covariance matrix;

**Σ**= (Var

_{ij}) i,j = 1,…,n; Var

_{ii}= 1; and Var

_{ij}= ρ

_{ij}(i ≠ j); ρ

_{ij}is the correlation coefficient between variables x

_{i}and x

_{j};

**Σ**

^{−1}means inversion of matrix

**Σ**; and

**x**

^{t}means transposition of vector

**x**.

**u**is the vector formed by several Weibull variables; J is the Jacobian matrix; and ǀǀ means absolute value. The elements of the Jacobian matrix are shown in Equation (10):

_{i}and k

_{i}) and one for every pair of them (ρ

_{ij}). However, the correlation coefficient ρ, between pairs of variables is established according to Standard Normal distributed variables, so the relationship between both parameters has to be obtained. By making a numerical approach based on the Cholesky decomposition (see Appendix), an interval can be obtained for each parameter where the correlation coefficients in Equation (13) can be considered according to Weibull variables, i.e., where the elements of the covariance matrix,

**Σ**, in Equation (13) correspond to the correlation coefficients between pairs of variables u

_{i}and u

_{j}. The intervals are shown in Equation (14):

#### 2.3. Multivariate Wind Speed Distribution

_{i}, can be described by a Weibull distribution with parameters C

_{i}and k

_{i}, and the relationship between every two distributions can be described by its correlation coefficient [19,20,21,22]. So, the multivariate wind speed PDF, or multilocation wind speed PDF, for a group of n variables (v

_{1},…,v

_{n}), with Weibull parameters C

_{i}and k

_{i}and correlation matrix R

_{v}referred to the pairs of Normally distributed variables, is shown in Equation (15):

^{−1}( ) and ntwʹ( ) are defined in Equations (5) and (6) respectively.

_{v}can also represent the correlation matrix of the v

_{i}variables.

#### 2.4. Multivariate Wind Power Distribution

_{i}is the power contained in an airstream that is flowing through a surface of area A

_{i}and d

_{i}is the air density at location i.

_{i}can be described by a Weibull distribution of parameters C

_{i}ʹ and k

_{i}ʹ [14], expressed in Equation (17):

_{i}and k

_{i}are the Weibull parameters of the wind speed at location i. Therefore, as defined in Equation (13) the multivariate wind power PDF is expressed in Equation (18) as a function of the parameters C

_{i}ʹ and k

_{i}ʹ.

_{P}contains the correlation coefficients corresponding to the pairs of normally distributed variables. In most cases the parameters of the Weibull distributions defined in Equation (17) lie outside the intervals expressed in Equation (14), so, as it is explained in Section 2.1, R

_{P}does not represent the correlation matrix of the P

_{i}variables.

## 3. Bivariate Weibull Distribution

#### 3.1. Bivariate Weibull Distribution Applied to Wind Speed

_{1}, k

_{1}, C

_{2}, k

_{2}and ρ, which stands for the correlation coefficient between x

_{1}and x

_{2}but, as has been stated above, can be considered as the correlation coefficient between v

_{1}and v

_{2}.

_{1}and x

_{2}, their relationships with v

_{1}and v

_{2}are shown in Equation (23):

_{1}= 8, k

_{1}= 2, C

_{2}= 8, k

_{2}= 2).

#### 3.2. Correlation Coefficient Inference

_{1}, u

_{2}), the sample correlation coefficient has to be tested with a hypothesis [23,24,25]. The sample value, r, is obtained using Equation (24), once the Weibull variables are changed to Normal ones (y

_{1}, y

_{2}).

_{ij}is the jth sample value of the variable y

_{i}and y

_{i}is the sample mean of the variable y

_{i}.

_{0}:

H

_{1}:

_{0}is a known value that corresponds to a bivariate Normal distribution, which needs to be tested. In order to do so, a new variable z is obtained from the sample correlation coefficient r, as expressed in Equation (25):

_{z}( ) is the CDF of the variable Z, according to the parameters of Equation (26).

_{0}can be accepted with significance level α, and if not, it cannot be accepted.

_{0}is accepted, the correlation coefficient corresponding to the bivariate Weibull distribution has to be obtained.

_{0}is accepted, it can be said that ρ

_{0}can be taken as an estimation of the correlation coefficient in a bivariate Weibull distribution with significance level of α.

#### 3.3. Bivariate Weibull Distribution Applied to Wind Power

_{1}and x

_{2}are shown in Equation (29):

_{1}= 8, k

_{1}= 2, A

_{1}= 7853 m

^{2}, d

_{1}= 1.225 kg/m

^{3}, C

_{2}= 8, k

_{2}= 2, A

_{2}= 7853 m

^{2}, d

_{2}= 1.225 kg/m

^{3}).

## 4. Case Study

_{ij}is the distance between location i and j; R

_{E}is the Earth’s radius; lat

_{i}and lon

_{i}are the latitude and longitude coordinates of the location I; sin( ) and cos( ) are the sine and cosine functions; and arccos( ) is the inverse of the cosine function.

^{−1}and b = 0.6589.

_{0}= 0.6012 is obtained.

_{min}< z < z

_{max}, ρ

_{0}= 0.6012 can be accepted as the correlation coefficient of the bivariate Weibull distributions corresponding to the wind speed data of those locations. The results are shown in Table 1.

Variable | Obtained value | Minimum value | Maximum value |
---|---|---|---|

r | 0.6202 | 0.5601 | 0.6394 |

z | 0.7253 | 0.6330 | 0.7571 |

## 5. Conclusions

## Appendix

^{*}means the conjugate transpose of the matrix L.

_{li}is distributed according to a N(µ

_{l},σ

_{l}) distribution, then y

_{ji}will follow a . Moreover, if µ

_{1}= µ

_{2}=…= µ and σ

_{1}= σ

_{2}=…= σ, then y

_{ji}will follow a distribution. And, in the Standard case, if µ = 0 and σ = 1, then y

_{ji}will be a distribution.

_{1}= µ

_{2}=…= 0, σ

_{1}= σ

_{2}=…= 1, y

_{ji}will be distributed by a N(0,1) distribution.

## Conflicts of Interest

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**MDPI and ACS Style**

Villanueva, D.; Feijóo, A.; Pazos, J.L. Multivariate Weibull Distribution for Wind Speed and Wind Power Behavior Assessment. *Resources* **2013**, *2*, 370-384.
https://doi.org/10.3390/resources2030370

**AMA Style**

Villanueva D, Feijóo A, Pazos JL. Multivariate Weibull Distribution for Wind Speed and Wind Power Behavior Assessment. *Resources*. 2013; 2(3):370-384.
https://doi.org/10.3390/resources2030370

**Chicago/Turabian Style**

Villanueva, Daniel, Andrés Feijóo, and José L. Pazos. 2013. "Multivariate Weibull Distribution for Wind Speed and Wind Power Behavior Assessment" *Resources* 2, no. 3: 370-384.
https://doi.org/10.3390/resources2030370