Wind speed distribution modeling for wind power estimation: Case of Agadir in Morocco

Abstract

To estimate a wind turbine output, optimize its dimensioning, and predict the economic profitability and risks of a wind energy project, wind speed distribution modeling is crucial. Many researchers use directly Weibull distribution basing on a priori acceptance. However, Weibull does not fit some wind speed regimes. The goal of this work is to model the wind speed distribution at Agadir. For that, we compare the accuracy of four distributions (Weibull, Rayleigh, Gamma, and lognormal) which have given good results in this yield. The goodness-of-fit tests are applied to select the effective distribution. The obtained results explain that Weibull distribution is fitting the histogram of observations better than the other distributions. The analysis deals with comparing the error in estimating the annual wind power density using the examined distributions. It was found that Weibull distribution presents minimum error. Thus, wind energy assessors in Agadir can use directly Weibull distribution basing on a scientific decision made via statistical tests. Moreover, assessors worldwide can use the followed methodology to model their wind speed measurements.

Keywords

Wind speed modeling probability distribution function Weibull distribution Rayleigh distribution Gamma distribution lognormal distribution wind power density goodness-of-fit tests

Introduction

Morocco is the only country in North Africa which has no reserves in fossil fuel. The demographic and economic growth of the kingdom will increase the energy demand up to 70% till 2025. Therefore, Morocco imports 97% of its energy needs. That makes this developing country sensitive to fluctuations of energy prices in global market. Since 2008, the government has planned a new strategy for development and investment in renewable energy projects. Its aim is to increase the share of renewable energy to 42% by 2020, where 14% is wind energy (Ettaik, 2016). For that reason, an accurate wind power assessment is extremely necessary.

Jourdier (2015) has stated that in recent years, industrialists in North America have observed an under-production (approximately 10%) of existing wind farms and that a negative bias in estimating the expected wind energy on more than 100 wind farms is noted in the United States and Canada. For Morocco, there are no official statistics. In fact, if the installed wind turbines produce less than expected, the profitability of the wind power project is in risk, especially if these differences continue during the project lifetime (20 years in general). In addition, the overestimating of potential increases the loan interest. Indeed, to assess the economic profitability for a wind energy farm, all economic indicators (net present value, payback period, income, annual saving, etc.) are based on the estimated cost of energy (COE), which is defined as the average cost per kWh of the produced wind electricity. It is obtained by dividing the annualized cost (AC) by the annual electricity generation $E_{T}$ (in kWh/year). It is given by

COE = \frac{AC}{E_{T}}

(1)

The previous equation shows that COE depends strongly on the annual electricity generation. Therefore, $E_{T}$ should be estimated with maximum precision in order to reduce the uncertainty on the economic profitability (investigation on AC impact is the subject of a separate work). In general, $E_{T}$ is estimated using the statistical technique. For a wind turbine and a specific site, the average wind turbine energy output $E_{T}$ is given by

E_{T} = T \int_{0}^{\infty} f (v) \cdot P (v) d (v)

(2)

where T is the time in hours of a full year, f(v) is the wind probability density function (PDF), P(v) is the machine power curve, and v is the wind speed (m/s).

As one can see in equation above, to avoid the overestimating of potential, it is crucial to choose the PDF that models the wind speed distribution accurately. Moreover, that helps to analyze and interpret the wind speed variation, manage the wind farms, and optimize its dimensioning. The power curve has also impact on the accuracy of wind power assessment. But, its uncertainty is less than estimation of the PDF (Morgan et al., 2011) (investigation on impact of modeling the power curve will be the subject of a future study).

To model the wind speed, many researchers have used directly the Weibull distribution. However, they are based on a priori acceptance, because it does not always display some wind speed characteristics well, and it may not fit all wind speed regimes encountered in nature. For that, several distributions (such as Weibull, lognormal, Rayleigh, Inverse Gaussian, Gamma, generalized Normal, etc.) are examined by many researches.

In this wise, the goal of this study is to model the wind speed at Agadir in Morocco in order to reduce the financial risk in investing in a wind energy project and give investors the confidence in its economic feasibility. For that, after presenting a literature overview of many studies in wind speed modeling, we compare four distributions (Weibull, Rayleigh, Gamma, and lognormal distributions) which are commonly used by researches (Alavi et al., 2016; Aries et al., 2018; Sohoni et al., 2016; Wang et al., 2016) and which have shown good results in fitting the unimodal wind speed distribution according to the literature. To carry out those distribution performances regarding the modelization of wind speed regime at Agadir, we conduct a statistical study using four tests of goodness of fit, namely, the coefficient of determination, the root mean square error (RMSE), the Chi-square test, and the mean bias absolute error (MBAE). For more accuracy in modeling the wind speed at Agadir, we deal with calculating the error between the estimated wind power density via the studied theoretical distributions and the calculated wind power density using measurements.

Literature review

Many researches compared several PDFs in order to select the most efficient in fitting their measurement. Morgan et al. (2011) have compared the non-parametric kernel density distribution with six parametric distributions (Weibull, Rayleigh, lognormal, Gamma, Normal, and loglogistic) in four sites in China. The achieved results show that the non-parametric estimation outperforms the parametric models. Pobočíková et al. (2017) have evaluated the suitability of four analytical distributions (two-parameter Weibull; three-parameter Weibull, Gamma, and lognormal) in fitting the wind speed distribution in Slovakia. It was found that the three-parameter Weibull performs as the best distribution. Yilmaz and Heçeli (2008) have compared 10 PDFs, namely, Erlang, Beta, exponential, loglogistic, lognormal, Gamma, Pearson V, Pearson VI, Weibull, and Uniform. Weibull was found to be the best one to fit the wind speed distribution at Aegean region in Turkey. Morgan et al. (2011) have modeled the wind speed at 178 stations in United States. They have concluded that the bimodal Weibull gives the best results. For higher wind speeds, authors have found that the Wakeby and Kappa distributions are the most suited, those distributions estimate the average wind turbine power output very with more accuracy. For extreme wind speeds, the two-parameter lognormal distribution performs better. Sohoni et al. (2016) have compared Gamma, Weibull, Rayleigh, lognormal, Inverse Gaussian PDFs in India; they have found that Weibull distribution is efficient in fitting the wind speed data with less skewness, and Gamma distribution fitted better to wind speed data which presented highly skewed histograms. Ben Amar and Elamouri (2011) have developed a new model using maximum entropy. They have compared this model with Weibull distribution. As a result, the proposed model adjusts better the measured data.

To reduce the uncertainty in wind potential evaluation in four sites in Algeria, Aries et al. (2018) have assessed the wind speed distribution models. They have compared eight distributions, namely, Weibull, Gamma, Inverse Gaussian, lognormal, Gumbel, Generalized Extreme Value, Nakagami, and Generalized Logistic. As a result, the Generalized Extreme Value and Gamma are the most suited. Furthermore, authors deal with comparing power density error. Gamma, Inverse Gaussian, and lognormal were found to be the most appropriate. Parajuli (2016) has fitted the measured data using Weibull and Rayleigh PDFs; the author has concluded that Weibull is accurate in estimation of wind power at Jumla in Nepal. Akgül et al. (2016) have used the Inverse Weibull distribution to model the wind speed variation at two sites in Turkey. This PDF has been compared with Weibull distribution and it was concluded that Inverse Weibull performances better than Weibull. In order to estimate the wind speed distribution at five stations in Iran, Alavi et al. (2016) have evaluated a various PDFs using the actual and the truncated data. They have compared the performance of four typically used distribution functions (Weibull, Gamma, Rayleigh, and lognormal functions). Results indicate that for actual wind speed data, the effective PDF is lognormal while Weibull PDF performs better for the truncated data. Ouarda et al. (2015) have analyzed the wind speed distribution at nine stations in United Arab Emirates. The authors have carried out the performance of non-parametric models, parametric models, and mixture models. Results show that Generalized Gamma and Kappa distributions give the best fit to the observations. The two-parameter Weibull shows also a good fit comparing with Generalized Extreme Value and three-parameter lognormal. For bimodal regimes, the mixture distributions provide a very good result than non-parametric distribution

Methodology

Case study and wind speed data

The wind speed data in hourly time series format were recorded at a height of 10 m at the International Airport of Agadir Al Massira over the year 2016 (geographical coordinates: 30.33N, 9.40O). Figure 1 represents the monthly averages of the wind speed. The maximum monthly average is 3.71 m/s, while the minimum is 3.05 m/s. The wind speed at Agadir appeared to be uniform during the year. The annual standard deviation was found to be equal to 1.91. It is weak enough, which means that observations revolve around the average value. The monthly mean wind speed and the annual standard deviation are calculated using the following equations, respectively

\bar{v} = \frac{1}{N} \sum_{i = 1}^{N} v_{i}

(3)

σ = {[\frac{1}{N - 1} \sum_{i = 1}^{N} {(v_{i -} \bar{v})}^{2}]}^{\frac{1}{2}}

(4)

where N is the number of measurement.

Figure 1.

Monthly averages of wind speed at height of 10 m.

The available data are summarized in Table 1. For each wind speed class with an equal width of 1 m/s, the number of occurrences is presented (second column), the measured wind speed frequency $f_{i}$ , and its cumulative probability are calculated and given in the third and fourth column, respectively.

Table 1.

Wind speed number of occurrences, its frequency, and its cumulative probability.

Wind speed, $v_{i}$	Number of occurrences	Measured frequency, $f_{i}$	Cumulative probability
0	74	0.0087	0.0087
1	1052	0.1243	0.1330
2	2390	0.2823	0.4154
3	1730	0.2044	0.6197
4	1065	0.1258	0.7455
5	953	0.1126	0.8581
6	642	0.0758	0.9340
7	289	0.0341	0.9681
8	147	0.0174	0.9855
9	67	0.0079	0.9934
10	33	0.0039	0.9973
11	14	0.0017	0.9989
12	8	0.0009	0.9999
13	1	0.0001	1.0000

Wind speed distribution modeling

In order to choose the PDF that fits better the wind speed at Agadir region, we compare in this study four distributions namely, Weibull, Rayleigh, Gamma, and lognormal. Those distributions were selected because they have been used commonly by researches and they have shown good performances regarding the unimodal wind speed distribution modeling. Those distributions definitions including their PDF and cumulative distribution function (CDF) are described below. Their other statistical properties are reported in Table 2, in which Γ is the Gamma function. It is expressed as

Γ (x) = \int_{0}^{\infty} \exp (- t) t^{x - 1} d t

(5)

Table 2.

Statistical properties of the studied distributions.

Distribution	Statistical properties	Expression
Weibull	Mean	$\bar{v} = c Γ (1 + \frac{1}{k})$
	Variance	$σ^{2} = c^{2} [Γ (1 + \frac{2}{K}) - Γ^{2} (1 + \frac{1}{K})]$
	Third moment	$\bar{v^{3}} = c^{3} Γ (1 + \frac{3}{K})$
Rayleigh	Mean	$\bar{v} = c \sqrt{\frac{π}{2}}$
	Variance	$σ^{2} = \frac{4 - π}{2} c^{2}$
	Third moment	$\bar{v^{3}} \approx 3.76 c^{3}$
Gamma	Mean	$\bar{v} = k θ$
	Variance	$σ^{2} = θ^{2} k (k + 1)$
	Third moment	$\bar{v^{3}} = θ^{3} k (k + 2) (k + 1)$
Lognormal	Mean	$\bar{v} = e^{μ + \frac{σ^{2}}{2}}$
	Variance	$σ^{2} = (e^{σ^{2}} - 1) e^{2 μ + σ^{2}}$
	Third moment	$\bar{v^{3}} = e^{3 μ + \frac{9 σ^{2}}{2}}$

Weibull distribution function

The Weibull distribution is a two-parameter function, namely, scale parameter c (m/s) and shape parameter k (dimensionless). It is the most used distribution in wind energy field, because it presents the majority of wind distribution in worldwide. Furthermore, it is flexible and simple to use (Tizgui et al., 2016). The Weibull PDF (f) and CDF (F) are given by equations (6) and (7), respectively

f (v) = (\frac{k}{c}) {(\frac{v}{c})}^{k - 1} \exp (- {(\frac{v}{c})}^{k})

(6)

F (v) = 1 - \exp (- {(\frac{v}{c})}^{k})

(7)

Rayleigh distribution function

The Rayleigh PDF has just one parameter that makes it the simplest distribution and the most used by researchers to model the wind speed. It is a particular case of Weibull PDF in which the shape parameter k = 2. The Rayleigh PDF and CDF are expressed mathematically as

f (v) = \frac{2 v}{c^{2}} \exp (- {(\frac{v}{c})}^{2})

(8)

F (v) = 1 - \exp (- {(\frac{v}{c})}^{2})

(9)

where c is the scale parameter.

Gamma distribution function

The gamma distribution is also applied to wind speed by many researches. It indicates the sum of the exponentially distributed random variables. The expression of the gamma distribution PDF and CDF are presented by the following equations, respectively (Pobočíková et al., 2017)

f (v) = \frac{v^{k - 1}}{Γ (k) θ^{k}} \exp (- \frac{v}{θ})

(10)

F (v) = \frac{γ (k, \frac{v}{θ})}{Γ (k)}

(11)

where k is the shape parameter, θ is the scale parameter, and $γ$ is the incomplete Gamma function, and it is given by

γ (p, x) = \int_{0}^{x} e^{- t} t^{p - 1} d t

(12)

Lognormal distribution function

The lognormal distribution is one of the versatile distributions. Since it provides a good result regarding the wind speed distributions modeling, it has been used by many researches. If the logarithm of the wind speed is distributed in a normal manner, the wind speed can be modeled by lognormal PDF. The formula of the lognormal distribution is (Aries et al., 2018)

f (v) = \frac{1}{σ \sqrt{2 π}} \exp (- \frac{{(\ln v - μ)}^{2}}{2 σ^{2}})

(13)

F (v) = \emptyset (\frac{\ln v - μ}{σ})

(14)

where $μ$ µ is the location parameter, σ is the scale parameter, and $\emptyset$ is the CDF of the standard normal distribution.

Estimation the distributions parameters

Many methods can be used to estimate the studied distributions parameters (Tizgui et al., 2017). In this study, the maximum likelihood method (MLM) is applied. This method gives the values of the parameters which maximize the probability of obtaining the measured data. For each parameter $α$ , MLM consists in estimating its value which maximizes the likelihood function (L) by solving the following equation

\frac{d \log L}{d α} = 0

(15)

The likelihood function (L) for a random sample $v_{1}$ , $v_{2}$ , …, $v_{n}$ and theoretical PDF (f) is represented by equation

L = \prod_{i}^{n} f (v_{i}, α)

(16)

Table 3 lists the equations to estimate the parameters of the used PDFs using the MLM.

Table 3.

Parameters estimation using MLM.

Distribution	Estimator MLM
Weibull	$k = {[\frac{\sum_{i = 1}^{N} v_{i}^{k} \ln v_{i}}{\sum_{i = 1}^{N} v_{i}^{k}} - \frac{1}{N} \sum_{i = 1}^{N} \ln v_{i}]}^{- 1}$ and $c = {[\frac{1}{N} \sum_{i = 1}^{N} v_{i}^{k}]}^{\frac{1}{k}}$
Rayleigh	$c = \sqrt{\frac{1}{2 n} \sum_{i = 1}^{n} v_{i}^{2}}$
Gamma	$θ = \frac{\bar{v}}{k}$ and $φ (k) = \ln (k) - \ln (\bar{v}) + \frac{1}{n} \sum_{i = 1}^{n} v_{i}$ with $φ (p) = \frac{\partial 3 (p)}{\partial p}$ is the digamma function
Lognormal	$μ = \frac{1}{n} \sum_{i = 1}^{n} v_{i}$ and $σ^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(\ln v_{i} - \frac{1}{n} \sum_{i = 1}^{n} \ln v_{i})}^{2}$

MLM: maximum likelihood method

Goodness-of-fit tests

In order to determinate the distribution that fit better the wind speed, we conduct a statistical study, which is based on calculating the differences between the predicted values from the models and the experimental values. The used statistical tests are given below.

For all equations (17)–(20), n is the number of bins, $y_{i} = f_{i}$ is the value of frequency observations at time stage i, $x_{i} = f (v_{i})$ the calculated value from the theoretical PDF for the same stage, and $\bar{y}$ is the mean of $y_{i}$ .

The coefficient of determination

The coefficient of determination $(R^{2})$ evaluates the linear correlation between the predicted values from the theoretical distribution and reel observations. It ranges from 0 to 1. A greater value of $R^{2}$ means a good fit of the considered PDF to the measured data. It is expressed as (Akgül et al., 2016)

R^{2} = 1 - \frac{\sum_{i = 1}^{n} {(y_{i} - x_{i})}^{2}}{\sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}}

(17)

RMSE

RMSE calculates the residuals of frequency of examined PDF and the measured data (Pobočíková et al., 2017). A model trend to be precise, if the RMSE is close to zero. The RMSE formula is given by

R M S E = {[\frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - x_{i})}^{2}]}^{0, 5}

(18)

Chi-square test

Chi-square test $(χ^{2})$ is commonly used to examine the performance of the considered PDFs for modeling the wind speed distribution. The authors have stated that $χ^{2}$ is recommended to analyze only big samples (more than 20 individuals). This statistic criterion is independent of the population variance and mean. The examined PDF models the data distribution if $χ^{2}$ is close to zero. It is determined by

χ^{2} = \sum_{i = 1}^{n} \frac{{(y_{i} - x_{i})}^{2}}{y_{i}}

(19)

MBAE

MBAE can be also used to assess the quality of each used distributions. It measures the average of total absolute bias errors between the theoretical PDF and the frequency obtained by actual observations. MBAE is defined by Jung and Schindler (2017) as

MBAE = \frac{1}{n} \sum_{i = 1}^{n} | y_{i} - x_{i} |

(20)

Available wind power evaluation

One distribution may be better in fitting the histogram of observations but not on estimating the power density. For example, Weibull and Rayleigh models were compared by Ahmed and Mahammed (2012); they found that for the whole year, Weibull distribution fit the measured data better than the Rayleigh, but the Rayleigh distribution provides more accuracy in estimating the power density in 9 months. For that, we deal with comparing the estimated wind power using the four distributions and observed data. For a wind device located perpendicular to the direction of the wind speed, the average of available wind power density is

P = \frac{1}{2} ρ v^{3}

(21)

$ρ$ is the air density in kg/m³. It can be calculated using the following relationship

ρ = ρ_{0} - 1.194 \times 10^{- 4} \times H_{m}

(22)

where $H_{m}$ is the site elevation (m) and $ρ_{0}$ is the air density at sea level, $ρ_{0} = 1.225 k g / m^{3}$ . In this study, the site elevation is 74 m, so $ρ = 1.216 k g / m^{3}$ . For the observed data, the measured wind power density $(P_{M})$ can be written as follows

P_{M} = \frac{1}{2} ρ \frac{1}{N} \sum_{i = 1}^{N} v_{i}^{3}

(23)

For a theoretical distribution, the wind power density estimated using the theoretical PDF $(P_{T h})$ can be expressed as

P_{T h} = \frac{1}{2} ρ \sum_{i = 1}^{i = n} f (v_{i}) v_{i}^{3}

(24)

$f (v_{i})$ is the theoretical PDF value at stage i, n is number of bins, and N is the number of hours.

Absolute error in estimating the wind power density

In order to select the PDF that estimates the wind power density with more accuracy, we calculate the absolute error (AE) between the wind power density estimated using the measured data and that obtained using the four theoretical distributions. This error can be expressed as (Ben Amar and Elamouri, 2011)

AE = 100 * | \frac{P_{T h} - P_{M}}{P_{M}} |

(25)

Results and discussion

Measured data adjustment

As we mentioned, in this study, we highlight the effectiveness of four distributions which are widely applied to fit wind speed, namely, Weibull, Rayleigh, Gamma, and lognormal. Figures 2 and 3 depict the histogram of the wind speed data adjusted by the four PDFs and its CDFs, respectively. Graphically, it can be noticed that Weibull, Rayleigh, and Gamma PDFs may be acceptable, but lognormal PDF is poor in fitting the data distribution and its cumulative function. Table 4 presents the parameters of the considered PDFs obtained using the MLM under MATLAB software.

Figure 2.

Data distribution fitted by the four theoretical PDFs.

Figure 3.

Data cumulative frequency and the four theoretical cumulative functions.

Table 4.

Estimated parameters for the studied distributions.

Distribution	Parameters values
Weibull	c = 3.72 and k = 1.73
Rayleigh	c = 2.72
Gamma	k = 2.24 and θ = 1.49
Lognormal	µ = 0.97 and σ = 1.12

Goodness of fits

To address the sensibility of the studied PDFs in term of fitting with more accuracy the wind speed distribution at Agadir region, we applied the most used statistical indicators, namely $R^{2}$ , RMSE, $χ^{2}$ and MBAE. Table 5 presents the statistical indicator values and rank obtained for the studied distributions. Those indicators are presented graphically in Figure 4.

Table 5.

Comparison of Weibull, Rayleigh, Gamma, and lognormal PDFs.

Criterion	Weibull	Rank	Rayleigh	Rank	Gamma	Rank	Lognormal	Rank
$R^{2}$	0.9260	1	0.9047	3	0.9197	2	0.6159	4
RMSE	0.0481	1	0.0633	2	0.0936	3	1.5185	4
$χ^{2}$	0.0235	1	0.0263	3	0.0244	2	0.0537	4
MBAE	0.0124	1	0.0134	2	0.0153	3	0.0372	4

PDF: probability density function; RMSE: root mean square error; MBAE: mean bias absolute error.

Figure 4.

Goodness-of-fit tests: (a) $R^{2}$ , (b) RMSE, (c) $χ^{2}$ , and (d) MBAE.

The values of $R^{2}$ are 0.9260, 0.9047, and 0.9197 for the Weibull, Rayleigh, and Gamma distributions, respectively. Lognormal presents the lowest value in term of $R^{2}$ . That indicates that Weibull distributions fit data set very well followed by Gamma distribution. Regarding the RMSE test, the Weibull PDF shows again a good performance followed by Rayleigh PDF with values of 0.0481 and 0.0633, respectively. The Gamma PDF is ranked third with an RMSE value equal to 0.0936. While lognormal presents the lowest value in term of RMSE too. In term of $χ^{2}$ test, Weibull and Gamma distributions perform better than others with values of 0.0235 and 0.0244, respectively. Rayleigh distribution is ranked third. Lognormal present always the poorest fitting. For the MBAE, Weibull shows the minimum value followed Rayleigh then Gamma. Lognormal is ranked fourth.

In general, it can be seen that the Weibull PDF gives the maximum value of coefficient of determination, minimum values of RMSE, Chi-square, and MBAE, followed by Rayleigh and Gamma distributions. While the lognormal PDF is the worst distribution in modeling the measurements, it is ranked fourth for all indicators.

Error in estimating the power density

In addition to goodness-of-fit indicators ( $R^{2}$ , RMSE, $χ^{2}$ , and MBAE), the reliability of each model is assessed in terms of estimating wind power density using the AE between the measured wind power density and that expected from the tested models.

Table 6 presents the annual wind power density calculated using measured data and the annual wind power density estimated via the four studied distributions. The error on estimating the power density using the considered PDFs is also given in the table. The average error was found to be 0.88% for Weibull PDF, 6.12%, 18.49%, and 83.14% for Rayleigh, Gamma, and lognormal, respectively. We concluded that Weibull PDF presents a minimum error in estimating the annual wind power density. So, it is the effective and most accurate distribution to model the wind speed at Agadir.

Table 6.

Errors in estimating wind power density using the four studied distributions.

	Measured	Weibull	Rayleigh	Gamma	Lognormal
Available wind power density (W/m²)	49.07	49.51	46.07	58.15	89.88
AE (%)	–	0.88	6.12	18.49	83.14

AE: absolute error.

Conclusion

The overestimating of potential influences negatively the profitability of a wind energy project. It may be caused by the uncertainty in wind speed distribution modeling. In fact, it is common to fit the wind speed distribution by Weibull PDF. However, Weibull PDF may not always model the wind speed. Therefore, in this study, the wind speed data in hourly time series format over the year 2016 at Agadir are statistically analyzed and fitted by four candidate PDFs (Weibull, Rayleigh, Gamma, and lognormal) were fitted to a wind speed sample at Agadir. Their parameters were estimated using MLM. Graphically, the PDFs and cumulative functions curves indicate that the lognormal gives the worst adjustment to the measurements and the other PDFs are acceptable. In order to make a scientific decision, four tests of goodness of fit (coefficient of determination, RMSE, Chi-square test, and MBAE) are used to select the effective distribution that fit better the histogram of observations. As a result, lognormal is ranked fourth for all statistical indicators. The Weibull PDF was found to be the best one. It is ranked first and fits well to wind speed data with maximum value of coefficient of determination and minimum values of RMSE, Chi-square, and MBAE. Gamma distribution is ranked second according to $R^{2}$ and $χ^{2}$ tests and third using RMSE and MBAE. Rayleigh distribution is ranked second for RMSE and MBAE. It is ranked third for $R^{2}$ and $χ^{2}$ tests.

Furthermore, we deal with calculating the error between the annual estimated wind power density via the studied theoretical distributions and the calculated wind power density using measurements. The error was found to be 0.88%, 6.12%, 18.49% and 83.14% for Weibull, Rayleigh, Gamma, and lognormal, respectively. We note that Weibull PDF presents minimum error in estimating the annual wind power density. Therefore, we concluded that Weibull PDF is the most suitable distribution that models wind speed at Agadir. Thus, wind energy assessors in the studied region can use directly Weibull distribution basing on a scientific decision made via statistical tests. Moreover, assessors worldwide can use the followed methodology to model their wind speed measurements with more accuracy.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This study was supported by the Moroccan National Center for Scientific and Technical Research (CNRST).

ORCID iD

Hassane Bouzahir

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Yilmaz

Heçeli

(2008) A statistical approach to estimate the wind speed distribution: The case of Gelibolu region. Doğuş Üniversitesi Dergisi 9(1): 122–132.