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In most studies related to wind energy, the quantity of the air density is considered constant, but actually, we know that it is variable and depending on others natural factors. We present a new proced ure to estimate the wind density power energy by simulating the components of the air density. The procedure uses the copula theory and demonstrates that the estimated power energy is higher if the air density is not constant.

Understanding the relationship between the components of wind power is fundamental to exploits wind energy, as well as the identification of suitable natural sites to perform the energy-efficient design and giving the economic rentability [

P = 1 2 ρ ( T , P ) S V 3 (1)

where ρ ( T , P ) is the air density. This term is dependent generally on the temperature T, and the pressure P, both of which vary with height, see for example [^{3}. In this case, the relationship between the power and wind speed is reduced to a linear function.

In this paper, we use the copula theory to handle the structure dependence beyond the linear correlation between the naturals variables: The wind speed, temperature, and pressure. The air density is considered not constant and the power should be affected by the dependence which is the idea of this research.

The term of copula was introduced by [

[

The remainder of this article is organized as: Section 2 describes the parametric and non-parametric distribution function to estimate the marginals probability distribution. The selected copulas used in the study are presented in Section 3. In Section 4 we explain a procedure to simulate wind speed density energy power. An application to real data is analyzed in the Section 5. Finally, we conclude.

Given X 1 , X 2 , ⋯ , X n a n independent and identically distributed sample with a unknown distri-bution function (cdf) F and probability density function (pdf) f. A natural way to estimates the distribution function F, is to consider the empirical distribution function

F e m p = 1 n ∑ i = 1 n I ( X i ≤ x ) (2)

where I ( X i ≤ x ) is the indicator function of a set I ( X i ≤ x ) . The correction of this approximation

F ˜ ( x ) = n n + 1 F e m p ( x ) (3)

is considered in the empirical estimation because using directly (2) to estimate the copula function can cause the boundary problems. That is the copula distribution function implanted to identify the dependence structure may not integrate one, for a discussion one can consult [

F ^ ( x ) = 1 n ∑ i = 1 n K ( x − X i h ) (4)

where K ( x ) = ∫ − ∞ x k ( t ) d t is some known kernel distribution function [

In this paper, we use the specific value, h = 3.572 σ n 1 / 3 , where σ = min ( s d , R / 1.34 ) and sd is the standard deviation of the sample and R is the sample interquartile range [

Finally, to fit wind speed distribution function we add a Weibull distribution function. Used generally in reliability field and failure times in physical systems, this distribution function is recommended by the International Standard IEC 61400-12 and is widely used in the context of eolian analysis of a wind speed, [

The CDF of Weibull distribution is: F k , λ = 1 − e − ( x λ ) k .

The pdf density function

f k , λ = { ( k λ ) ( x λ ) k − 1 e − ( k λ ) k s i x ≥ 0 0 s i x < 0

The mean and variance are, respectively:

E ( X ) = λ Γ ( 1 + 1 k ) and V a r ( X ) = λ 2 [ Γ ( 1 + 2 k ) + Γ 2 ( 1 + 1 k ) ]

such that, Γ ( x ) = ∫ 0 + ∞ t x − 1 e − t d t is the Euler’s Gamma function. To estimate the parameters k , λ , exists various methods like the mean squared method or the moment method [

In this paper we use the maximum likelihood method to estimate the parameters [

L V ( θ ) = ∏ i = 1 n f ( X i , θ ) (5)

where consist the set parameters to be estimated

To reveals the structure dependence between the pairs variable, (Wind speed, Temperature) and (Wind speed, Pressure), three parametric copula have been considered. A copula is a distribution function defined in the cubic interval [ 0 , 1 ] 2 with uniform marginal distribution functions U [ 0 , 1 ] . If F_{X}, and F_{Y} are marginals distribution of variables bivariate ( X , Y ) , then from the Sklar theorem [

The first copula considered is the Gaussian copula which pertains to the implicit copulas. Copula associated with elliptical distribution and represent a symmetrical dependency. In addition, they become important whether we are analyzing the right or left tail of the distribution function.

If we denote by ρ be the linear correlation coefficient between two random variables X and Y, the Gaussian copula with parameter ρ is expressed:

C ρ ( u , v ) = Φ ρ ( Φ − 1 ( u ) , Φ − 1 ( v ) ) = ∫ − ∞ Φ − 1 ( u ) ∫ − ∞ Φ − 1 ( v ) 1 2 π 1 − ρ 2 exp ( 2 ρ s t − s 2 − t 2 2 ( 1 − ρ 2 ) ) d s d t

where Φ ρ is the two-dimensional standard Normal distribution function with correlation coefficient equal to ρ , and Φ is the standard Normal one-di- mensional distribution.

The second copula family considered is the Sarmanov copula. The range of the subfamilies is infinite due to its way of constructing a copula. One can find a special sub-copula associated to each marginal distribution function. Let ( X , Y ) be a bivariate random vector with marginal probability distribution functions (pdfs) f X and f Y . Also, let ψ 1 and ψ 2 two bounded non-constant function such that:

∫ ∞ + ∞ f X ( t ) ψ 1 ( t ) d t = 0 , ∫ ∞ + ∞ f Y ( t ) ψ 2 ( t ) d t = 0

The joint bivariate pdf introduced by [

h ( x , y ) = f X ( x ) f Y ( y ) ( 1 + η ψ 1 ( x ) ψ 2 (y))

and the associated copula distribution function is:

C ( u , v ) = u v + η ∫ 0 u ∫ 0 v ψ 1 ( F X − 1 ( t ) ) ψ 2 ( F Y − 1 ( s ) ) d t d s (6)

The density is:

c ( u , v ) = 1 + η ψ 1 ( F X − 1 ( u ) ) ψ 2 ( F Y − 1 ( v ) ) (7)

where F X and F Y are the cumulative distribution functions (cdf’s) of X and Y, respectively. Parameter η is a real number that satisfies the condition for all x and y. 1 + η ψ 1 ( x ) ψ 2 ( y ) ≥ 0 for all x and y.

Note that when = 0, X and Y are independent. This parameter is related to the correlation between X and Y (if it exists), [

C o r r ( X , Y ) = η v 1 v 2 σ 1 σ 2 (8)

where ψ 1 ( x ) = x − μ X and ψ 2 ( x ) = x − μ X and μ X = E ( X ) and

μ Y = E ( Y ) (9)

To give a range of the parameter η, we use the result giving by [

max ( − ( b − a ) ( d − c ) ( μ X − a ) ( μ Y − b ) , − ( b − a ) ( d − c ) ( b − μ X ) ( d − μ Y ) ) ≤ ( b − a ) ( d − c ) η ≤ min ( ( b − a ) ( d − c ) ( μ X − a ) ( d − μ Y ) , ( b − a ) ( d − c ) ( b − μ X ) ( μ Y − c ) ) (10)

The ultimate copula is the Frank copula. This copula belongs to the so-called Archimedean copula, a family of dependence function used for here nice analytical proprieties [

The Frank copula is defined as:

C θ ( u , v ) = − 1 θ ln ( 1 − ( 1 − e − θ u ) ( 1 − e − θ v ) 1 − e − θ ) .

In this section, we describe a procedure to simulate the statistical behavior of the wind power density using a Monte Carlo method [

P w = 1 / 2 ρ ( T , P ) V 3 (11)

This term can represent the kinetic energy per unit area related to the wind. Now employing the ideal gas law, one expressed the air density [

ρ ( T , p ) = 1.225 [ 288.15 T ] [ P 1013.3 ] (12)

The wind power density (11) can be calculated in two way. The first way is considering the air density ρ ( T , p ) constant. In this case the mean power produced until an observation z is:

A P w ( z ) = ρ ( T , p ) 2 ∫ 0 z f ( v ) v 3 d v (13)

where f is the (pdf) of the wind speed. When the value z → ∞ we have the average of the power energy. The second case if the air density ρ ( T , P ) , is not constant. For n registration of the data ρ ( T , p ) = ( ρ ( T 1 , P 1 ) , ρ ( T 2 , P 2 ) , … , ρ ( T n , P n ) ) , the mean wind power energy can be calculated until the observation z_{k}.

M P w ( z k ) = 1 2 ∑ i = 1 n ρ ( T k , P k ) z k 3 (14)

Now to simulate the wind power energy density we start by simulating the wind speed variable coupled with the temperature and the pressure.

The same procedure describes in [

P ( V ≤ U | U = u ) = C u (v)

where C u ( v ) = lim Δ u → 0 + C ( u + Δ u , v ) − C ( u , v ) Δ u = ∂ C ( u , v ) ∂ u .

The following algorithm simulate the wind density power energy:

1) Start by fixing a copula C, of wind speed and temperature.

2) Generate two independent random variables u_{1} and z from a Uniform distribution U ( 0 , 1 )

3) Set u 2 = C u 1 [ − 1 ] ( z ) , where C u 1 [ − 1 ] denotes aquasi-inverse of C u 1 . The quasi-inverse is:

C u 1 [ − 1 ] ( z ) = { inf { x | C u 1 ≤ z } if z = 0 C u 1 − 1 if z ∈ ( 0 , 1 ) inf { x | C u 1 ≤ t } if z = 1

4) The desired first pair is ( u 1 , u 2 ) where u_{2} is a uniform variable related to the temperature variable.

5) Fixing the random variable u_{1}, and considering now the copula of wind speed and pressure, we repeated the procedure to give a pair ( u 1 , u 3 ) , where u_{3} is a uniform variable related to pressure variable.

6) Taking the inverse F − 1 ( u i ) , i = 1 , 2 , 3 , of the marginal distribution function used in (3), (for the inverse method one can consult [

7) Replacing this terms in (11).

The data represent the registration of wind speed, temperature, and pressure collected in the region Hrarza, situated in the north Morocco kingdom. Near on the straits of Gibraltar and surrounded by two seas, the Mediterranean and the Atlantic, this region suffers a gusty wind. The annual mean wind speed exceeds 6 m/s. For comparison with another region of the kingdom, we can consult [

The first lecture of the graphical behavior of this variable

As we noted in Section 2 the marginal adjustment is resumed in

Average | Std.Dev. | Kurtosis | Skewness | Min | Max | JB Test | |
---|---|---|---|---|---|---|---|

Wind Speed | 6.263 | 2.233 | 1.463 | 1.117 | 2.730 | 14.870 | 2.200 × 10^{−}^{16} |

Temperature | 25.236 | 7.258 | −1.255 | 0.100 | 12.870 | 40.410 | 5.522 × 10^{−}^{16} |

Pressure | 99.291 | 0.504 | 3.160 | −0.227 | 96.468 | 100.578 | 2.200 × 10^{−}^{16} |

and a scatter plot,

tool to identify the dependency structure between two variables confirm this relation, i.e. the point plotted is under the diagonal line wish indicate a negative dependency.

To visualize the intensity of the dependence in the tail and in the center the nonparametric estimation of copula density,

c ^ ( u , v ) = 1 n ∑ i = 1 n k ( U i − u h * ) k ( V i − v h * ) , (15)

where k is the kernel pdf function and h * is one of the smoothed parameters for copula estimation [

The package kdecopula [

The (IFM) method has an advantage that they avoid the excess time optimization. Using a global estimation of the log-likelihood function considering both the marginals and a copula doesn’t guarantee the existence of the minimum, [

Copula | Marginals (Wind vs Temp) | Estimate Dependence Parameter | CIC |
---|---|---|---|

Gauss | Emp/Emp CKE/CKE Weibull/Emp Weibull/CKE | −0.400 −0.438 −0.4258 −0.4479 | 32.251 33.884 34.427 35.621 |

Frank | Emp/Emp CKE/CKE Weibull/Emp Weibull/CKE | −2.531 −2.597 −2.689 −2.711 | 30.645 30.691 30.410 30.188 |

Sarmanov | Emp/Emp CKE/CKE Weibull/Emp Weibull/CKE | −0.007 −0.007 −0.007 −0.007 | 15.380 17.588 15.268 16.411 |

Copula | Marginals (Wind vs Press) | Estimate Dependence Parameter | CIC |

Gauss | Emp/Emp CKE/CKE Weibull/Emp Weibull/CKE | −0.178 −0.193 −0.207 −0.212 | 7.565 8.228 8.180 8.433 |

Frank | Emp/Emp CKE/CKE Weibull/Emp Weibull/CKE | −0.780 −0.818 −0.920 −0.939 | 4.043 4.155 4.600 4.652 |

Sarmanov | Emp/Emp CKE/CKE Weibull/Emp Weibull/CKE | −0.090 −0.090 −0.090 −0.090 | 7.957 7.823 7.647 6.713 |

The CIC criteria give the best selection of copula among another copula. Like the Akaike information criterion [

A simulation of daily energy is illustrated in

In this work, we have analyzed the statistical behavior of the wind energy in

north Morocco, from the point the view of the copula. Our objective is to give a most accurate simulation method for a density power energy in the eolian park. Fitting the marginal probability component of the wind density variable, we have noted that the nonparametric approach can give the best fit. To capture the negative dependence between the variables we have used two type of copula. The Archimedean and the elliptical copula. We have incorporated Sarmanov copula wish has never been used in this type of data and giving a suitable model. The procedure introduced to simulate the wind density power energy can generalize to the other field related to renewable energies since the density of the air is always present.

The authors declare that there is no conflict of interest regarding the publication of this paper and there has been no significant financial support for this work that could have influenced its outcome.

Bahraoui, Z., Bahraoui, F. and Bahraoui, M.A. (2018) Modeling Wind Energy Using Copula. Open Access Library Journal, 5: e4984. https://doi.org/10.4236/oalib.1104984