Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4, 1937-1955 2024 Publisher: Learning Gate DOI: 10.55214/25768484.v8i4.1568 © 2024 by the authors; licensee Learning Gate © 2024 by the authors; licensee Learning Gate * Correspondence: guenoukpatib@gmail.com Optimal power density estimation in Sub-Saharan Africa regions based on Weibull distribution parameters Agbassou Guenoukpati1,2,3*, Akuété Pierre Agbessi1,2, Komi Amoussou1,2,3, Adekunlé Akim Salami1,2, Seydou Ouedraogo4,3 1Department of Electrical Engineering, Ecole Polytechnique de Lomé (EPL), University of Lome, Lome-Togo; guenoukpatib@gmail.com (A.G.). 2Centre d’Excellence Régional pour la Maîtrise de l’Electricité (CERME), University of Lome, Lome-Togo. 3Laboratoire de Recherche en Sciences de l’Ingénieur (LARSI), University of Lome. 4Polytechnic University of bobo-Dioulasso, Burkina-Faso. Abstract: The production of wind energy requires knowledge of certain wind speeds and directions. Several tools are used for this purpose to characterize the wind power, including the Weibull distribution function with two parameters, shape factor k, and the scale factor c. In this paper six methods Graphical Method (GP), Empirical Method of Justus (EMJ), Empirical Method of Lysen (EML), Energy Pattern Factor (EPF), Maximum Likelihood (ML), and Moroccan Method (MMa) are used to estimate Weibull distribution parameters to evaluate the wind potential an its power density. Twelve sites in West African sub-region such as Abuja, Accra Kotoka, Bamako Senou, Conakry Gbessia, Cotonou Cadjehoun, Kano Mallam Aminu, Lagos Ikeja, Lome Tokoin, Niamey, Niamtougou, Ouagadougou, Tambacounda were selected as case studies. For each selected site, hourly wind speed data collected at 10 m height for the twelve years from January 2011 to December 2023 are used. The evaluations of each method were carried out every month and statistical criteria to provide a more complete analysis. The results show that the EPF, EMJ, EML, and ML provide highly desirable better performance while the GP and MMa showed poor performance for all stations. For all sites, the EPF was recognized as the most appropriate method except for the Lagos site where the EMJ ranks first on the others. The methods that ranked second after EPF varied among the sites. Keywords: Numerical estimation methods, Power density, Weibull distribution, Wind speed. 1. Introduction The problem of electrification in West Africa has been growing in recent years. Additionally, the depletion of fossil fuels and the environmental impacts have led to a shift towards other sources of electrical energy production. In fact, West Africa has significant hydroelectric potential, strong wind potential for the deployment of wind energy, high solar radiation, particularly in desert areas, and considerable hydrocarbon resources, accounting for about half of the continent's reserves [1]. In terms of wind power, only 43 megawatts of installed capacity were deployed, with another 230 megawatts being installed in the region by 2011. The only wind farm operational on a commercial scale is Cabeolica in Cape Verde, which has the largest installed capacity at over 28 MW. Consequently, West Africa lags behind with few projects in the wind sector. For example, a project to build and operate a 25.2 MW wind power plant in Lome is planned but has not yet been implemented. However, the West Africa region is well positioned especially in its coastal areas to take advantage of its wind energy potential. 1938 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Other aspects to consider in a wind farm project include comprehensive assessments of the wind resource, as well as the power and energy densities required to verify the financial viability of this potential. The power density of a wind turbine is an important factor to consider when assessing wind resources and implementing wind farm projects, as it helps in selecting the optimum turbines for a particular area [2]–[4]. Power density is calculated from the amount of energy available, which can be converted into electricity using wind turbines. There are two approaches to calculating the power density of a wind turbine. In the first approach, wind power density is calculated empirically from measured wind speed data. The wind energy at a given location depends on the cube of the wind speed. Thus, the power density for the time series of actual wind speed data can be calculated using the parameters of the Weibull distribution [5]. Several methods have been proposed in the literature to estimate the parameters of the Weibull distribution with two parameters k and c, respectively the shape and scale factors [2], [6]–[9]. The Graphical method, Justus' empirical method, Lysen's empirical method, the Energy pattern factor method, the Moroccan method, the Maximum likelihood method, the Modified maximum likelihood method, the Method of moments are commonly used [10], [11]. It can be concluded that the relevance of the methods may vary depending on the sample size of the data, the distribution of the sample data, the sample data format and the precision of the fit tests. In this paper, the GP, EMJ, EML, EPF, ML, and MMa are used to compute of the Weibull parameters of twelve sites in West Africa. The aim is to determine the most accurate for wind power density estimation in this region. The rest of the study is organized as follows: in section 2 the study background is presented, section 3 depicts the study area and data description, Sections 4 and 5 exhibits respectively the wind speed and wind potential based on Weibull distribution and evaluation metrics. The case study results and discussions of twelve West African sites are presented in Section 6 concludes with section 5. 2. Study Background Two parameters of the Weibull distribution are known as form (k) and scale (c) parameters. As mentionné, l’estimation de ces paramètres de la distribution de la vitesse du vent est faite grâce à plusieurs méthodes proposées dans la littérature. For example, [11] compared the performance of five methods for calculating the shape and scale parameters of the Weibull function to characterize the wind speed distribution. The results indicate that the maximum likelihood method outperforms the other methods in terms of representing the wind speed distribution. In [12], the authors evaluated the performance of four parameter estimation methods of the Weibull function for the monthly wind speed distribution modeling in Halabja, Pakistan. [13] compared the performance of six different methods to compute shape and scale parameters for estimating the wind speed distribution. According to the results, the maximum likelihood method followed by the modified maximum likelihood method showed the highest performance, while the graphical methods had the lowest performance. The study in [14] evaluated the Weibull parameters to represent the distribution of wind speed in Garoua, Nigeria. Their results showed that the use of the energy pattern factor has more aptitude than the other examined methods. [2] has evaluated the performance of six numerical methods as GP, EMJ, EML, EPF, ML, and MML to determine the k and c parameters of the Weibull distribution function to evaluate wind energy density at four stations distributed in the province of Canada, Alberta. À cet égard, la densité d'énergie éolienne estimée à l'aide de la fonction de Weibull est comparée à celle obtenue avec les données de vent mesurées. The daily and monthly results indicated that the accuracy changed with numerical methods compared to empirical results. It was found that the EMJ, EML, EPF, and ML methods provided the best performance than the GP method which ranks last for all the considered stations. Another observation was that the EMJ and EML methods are very close in terms of efficiency. A study conducted in [15], analyzed wind characteristics using wind speed data collected from five meteorological sites in Lebanon. The authors found that the power density method gives the best estimate of the measured distribution for all sites except Quaraoun where Justus' empirical method gives the best estimate. In the Northeast region of Brazil [13], the EPF and GP are efficient in fitting 1939 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Weibull distribution for wind speed data from the coastal area of Ceará, based on data collected from the cities of Camocim and Paracuru and analyzed using statistical tests. The authors in [16], have determined for the Lome site, the best model that corresponds to la forte fréquence des vents calmes observee to accurately estimate the amounts of recoverable wind energy. They propose using the Weibull hybrid distribution approach. Two other traditional approaches are often used in this context to evaluate the la pertinence de cette approach. They found that if the frequencies of calm winds are relatively high, the Weibull distribution is inadequate. In this case, the hybrid Weibull distribution function is the best solution. The same authors in [17] also presented the characterization and evaluation of the wind potential at annual and monthly scales of the Lome site and specified the characteristics of the wind turbines to be installed on this site. The wind speed data collected over two years at a height of 10 meters above the ground show that the mean annual speed is 2.9 m/s. These studies also showed that February, March, April, July, August, and September have a monthly average speed close to 4 m/s. Il est ressorti que that wind turbines with low nominal speeds of the order of 6 m/s to 8 m/s will be the most suitable for optimal exploitation of electrical energy from wind energy on the Lome site from a height of 25 meters above ground. Other approaches as, Rayleigh distribution [18], mixture hybrid Weibull distribution [19], le machine learning [20] are also used in the literature. The methods for assessing wind resources make them suitable for installing commercial wind turbines. However, understanding the wind characteristics and potential in West Africa has gaps due to a lack of specifically documented information to assess the wind potential of study sites and select appropriate commercial turbines. Additionally, no studies have directly addressed the wind regimes and unique atmospheric conditions prevailing in West Africa over various months. The relevance of power density in evaluating wind resources and selecting suitable turbines is crucial. Further research is needed to obtain specific information for wind project feasibility studies. The study used the Weibull distribution method, with the best numerical fit, to analyze monthly wind patterns and assess energy potential 3. Study Area and Data Analyses For this study, twelve sites Abuja, Accra Kotoka, Bamako Senou, Conakry Gbessia, Cotonou Cadjehoun, Kano Mallam Aminu, Lagos Ikeja, Lome Tokoin, Niamey, Niamtougou, Ouagadougou, Tambacounda were selected as case studies. Figure 1 illustrates the location of the selected sites on a map of Africa. Table 1 shows the geographic location of the twelve selected sites. For each selected site, hourly wind speed data collected at 10 m height for the twelve years from January 2011 to December 2023 are used. Table 2 shows the statistics; mean speed, standard deviation, kurtosis, skewness, and power density, calculated from the measured wind speed data for the selected sites. 1940 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Figure 1. Geographical location of the selected sites. Table 1. Graphical location of selected sites. Sites OACI Code Latitude (°N) Longitude (°E/°W) Altitude (m) Abuja. Nigeria DNAA 9.25N 7.00 °E 344 Accra Kotoka. Ghana DGAA 5.60N 0.17 °W 69 Bamako Senou. Mali GABS 12.53N 7.95 °W 381 Conakry Gbessia. Guinée GUCY 9.34N 13.36 °W 22 Cotonou Cadjehoun. Benin DBBB 6.35N 2.38 °E 9 Kano Mallam Aminu. Nigeria DNKN 12.05N 8.53 °E 481 Lagos Ikeja. Nigeria DNMM 6.58N 3.33 °E 38 Lome Tokoin. Togo DXXX 6.17N 1.25 °E 25 Niamey. Niger DRRN 13.48N 2.17 °E 227 Niamtougou. Togo DXNG 9.77N 1.10 °E 343 Ouagadougou. Burkina Faso DFFD 12.35N 1.52 °W 306 Tambacounda. Sénégal GOTT 13.77N 13.68 °W 50 1941 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Table 2. Descriptive statistics of the wind data used from the selected sites. Sites Mean (m/s) Std (m/s) Skewness Kurtosis Abuja 2.43045 1.27036 1.25746 11.03356 Accra 4.16032 2.21591 0.08801 2.76699 Bamako 2.79606 1.72889 0.82169 5.09067 Conakry 3.35295 1.65166 0.54587 6.07934 Cotonou 4.01159 1.81438 -0.12249 2.49218 Kano 4.78558 2.32290 0.25828 4.31923 Lagos 2.71353 2.24166 1.23657 5.64726 Lome 3.52870 2.02964 0.26247 2.33358 Niamey 3.31053 1.84692 1.08897 5.66665 Niamtougou 2.61775 1.79118 0.41532 3.62772 Ouagadougou 2.99562 1.66267 0.78947 4.59290 Tambacounda 2.95754 1.64106 1.36276 8.59716 We note that Kano, Accra and Cotonou have the highest average wind speeds at 4.78558m/s, 4.16032m/s and 4.01159m/s respectively. Lome follows with an average speed of 3.52870m/s. On the other hand, the lowest wind speed is observed for Abuja. In addition, for the stations of Abuja and Tambacounda the kurtosis coefficient is significantly higher than the stations of Lome, Accra and Cotonou. It is observed that for all stations the values of the skewness coefficients are positive except for Cotonou, indicating that all distributions are skewed to the right except for Cotonou which is skewed to the left. The standard deviation for all stations is between 1 and 2.5. It should be noted, however, that Accra has the largest standard deviation, which is 2.21591. 4. Weibull Distribution The wind energy at a given location depends on the wind speed cube. Thus, the power density for time series of actual wind speed data can be calculated using Equation (1). where  denotes the air density, a parameter that varies with latitude and temperature, but is generally considered to be constant and averages about 1.25 kg/m3 which depends on altitude, air pressure and temperature; and v is the wind speed in m/s. S is the swept area by the wind turbine in the previous expression shows that the available power varies with the average cubic speed of the observed wind. 31 2 P Sv= (1) The latter method is based on a statistical treatment of the raw wind data and the calculation of frequencies at a given threshold of speed. The two-parameters Weibull probability density function is given by Equation (2) [21]. 1 ( ) exp k k k v v f v c c c −       = −            (2) where k is the shape parameter that indicates the wind distribution of any region, c is the scale parameter in m/s indicates how windy the location is. The cumulative function can be obtained by calculating the integral of the probability density function. The cumulative distribution function is expressed by Equation (3). ( ) ( ) 0 1 exp k V v F v f V dv c    = = − −       (3) 1942 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate The wind density energy calculated from the density of the Weibull probability density function is estimated using the following Equation (4) [1]. The wind density energy calculated from the density of the Weibull probability density function is estimated using the following Equation (4). where () denotes gamma function. 3 2 1 3 1 2 W P c k m      =  +        (4) The estimation of the parameters of the wind speed distribution is important in terms of selecting the wind turbine to be implemented in order to obtain a good return on wind energy production and also the economic viability of the project. There are several methods in the literature to compute the parameters k and c of the Weibull distribution function. In this study, six methods such as the graphical method (GP), the empirical method of Justus (EMJ), the empirical method of Lysen (EML), the energy pattern factor method (EPF), the maximum likelihood method (ML), and the Moroccan method (MMa) are used to calculate these parameters. The graphical method is achieved using the cumulative distribution function. In this method, the wind speed data are interpolated using least-squares regression. The observed wind speeds are divided into v1,...,vn intervals. The discrete probability of these wind speeds is respectively given by Equation (5). ( )   1 exp k i i i v F v F v v c   =  = −     (5) By taking twice the logarithm of the equation (5) [2], [11], [22], we obtained as the Equation (6). ( )  ( ) ( )ln ln 1 ln lniF v k v k c− − = −   (6) Plotting ln(v) as the axis x compared to the first member of (6) as the axis of the y presents a straight line in which k is the slope of the line and the ordinate at the origin is -kln(c) [7], [17], [23]. This determines the linear regression line of yi according to xi in the form given by Equation (7). i iy a x b=  + (7) With a=k, and b= - kln(c). The Weibull parameters are then calculated according to the Equation (8). exp exp k a b b c a k  =       = − = −        (8) Based on the empirical method introduced by Justus, the parameters k and c are calculated respectively by Equations (9) and (10) [2], [24], [25]. 1,086 k v  −   =     (9) ( )1 1 v c k =  + (10) In the empirical method proposed by Lysen, k is calculated by Equation (9) as in the Justus method. The only difference is the calculation of c given by Equation (11) [2], [26]. 1943 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate 1 0,433 0,586 k c v k −   = +    (11) To calculate the parameters k and c by this process, the Energy pattern factor as a parameter used for the aerodynamic design of the turbines must be defined first. The energy pattern factor is obtained by using Equation (12) [2], [3]. ( ) ( ) 3 3 1 3 3 1 1 1 3 1 1 1 n ii pf n ii v kvnE v k v n = =  + = = =  +   (12) where 𝑣3̅̅ ̅ is the average wind speed cube, �̅�3 is the cube of the average speed. Then, the parameter k can be calculated by Equation (13). ( ) 2 3,69 1 pf k E    = +     (13) The parameter c is also calculated in the same way using Justus empirical method given by the Equation (10) with the obtained parameter. The maximum likelihood method is a mathematical expression recognized as a likelihood function of wind speed data in time series format. In this method, extended numerical iterations are required to determine the k and c parameters of the Weibull distribution. Using the maximum likelihood method, these parameters are respectively calculated by Equations (14) and (15) [3], [23], [27], [28]. where vi is the wind speed at time i in m/s and n is the number of non-zero wind speed data points. ( ) ( ) 1 1 1 1 ln ln n nk i i ii i n k ii v v v k nv − = = =    = −        (14) 1 1 n kk ii v c n =    =      (15) This method was used in the evaluation of the wind potential in Morocco [15]. The parameters k and c are determined respectively by the Equation (16) and (10) using the obtained value of the parameter k. ( )( ) 0,51 1 0,483 2k v= +  − (16) 5. Evaluation Metrics To evaluate the performance of the six methods for wind energy density estimates, different statistical approaches, including seven reliable statistical indicators are used in this study. Several statistical indicators including mean absolute percentage error (MAPE), mean absolute error (MABE), root mean square error (RMSE). In their formula, Pi,w et Pi,M are respectively the i-order wind power densities calculated by the Weibull function and the i-order wind power densities calculated by the measured data. In addition Pw,avg et PM,avg are the averages of the values of Pi,w et Pi,M and n is the total number of wind speed. The MAPE shows the average absolute percentage difference between the wind powers calculated using the Weibull function and those reached by the measured values. The MAPE is calculated by Equation (17). 1944 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate , , 1 , 1 100 n i W i M i i M P P MAPE n P= − =  (17) The MABE represents the mean total absolute amount of polarization errors between the wind powers calculated by the Weibull function and those obtained by the measured values. The MABE is defined by Equation (18). , ,1 1 n i W i Mi MABE P P n =   = −     (18) The RMSE identifies the accuracy of the model by comparing the difference between the values obtained by the Weibull function and those of the measured data. The RMSE always has a positive value and is calculated by Equation (21) [22], [29], [30]: The root mean square error is very useful for comparing several estimators. It measures the performance of the estimators based on the mean of the squares of the errors. In our study we also use it to determine the method that gives a Weibull function that better follows the frequency histograms of the different classes of measured wind speeds. The fit is better when the RMSE is low (very close to 0). ( ) 2 , ,1 1 n i W i Mi RMSE P P n = = − (19) 6. Results and Discussion Figures 2 to 11 show some monthly curves of k and c parameters estimated with each method and the Weibull distribution functions plotted with the obtained values. For all sites, the shape parameters follow practically the same variations for all methods. But it should be noted that for some sites there are some curves that are slightly out of phase: the MMa method for the Abuja site, the GP and MMa methods for Conakry, the MMa method for Cotonou and Kano and the ML method for Niamey. The peak for k values is reached for Cotonou with k equals to 4 for the ML method. But generally, for the k methods varies between 1 and 3. As for the variations in c curves, they are mostly in the same range for all sites. As for the variations in c curves, they are mostly in the same range for all sites. The Weibull distributions vary according to the values of the parameters k and c calculated for each method. It can be seen that for all stations the values of the parameter c for all methods are quite close to each other and only minor differences are found for the GP method and sometimes also for the ML method. Nevertheless, the values of the parameter k for the EMJ, EML and EPF methods are in the same range for all months while for the ML, GP and MMa methods the values of k are sometimes higher or lower than other methods These differences of k and c values for each method highlight the difference in the calculated values of wind power densities with the measured data. Although the summarizes provide significant insights especially with respect to the distribution of wind power density but they cannot be used solely to determine the level of precision of the monthly power density calculation methods. Therefore, the statistical indicators are used in order to identify the level of precision of each method. The statistical indicators introduced in this section are used to assess the performance of the six estimation methods. Tables 3 to 14 provide the results of descriptive statistics and power density estimation, and Figure 12 the evaluation of the performance of the six selected methods on a monthly basis in terms of MAPE, MABE, RMSE, RMSE, respectively for Abuja, Accra, Bamako, Conakry, Cotonou, Kano, Lagos, Lome, Niamey, Niamtougou, Ouagadougou, and Tambacounda. 1945 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Figure 2. Monthly variation of k and c values, and Weibull distribution for Abuja. Figure 3. Monthly variation of k and c values, and Weibull distribution for Accra. 1946 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Figure 4. Monthly variation of k and c values, and Weibull distribution for Bamako. Figure 5. Monthly variation of k and c values, and Weibull distribution for Conakry. 1947 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Figure 6. Monthly variation of k and c values, and Weibull distribution for Cotonou Figure 7. Monthly variation of k and c values, and Weibull distribution for Kano 1948 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Figure 8. Monthly variation of k and c values, and Weibull distribution for Lagos Figure 9. Monthly variation of k and c values, and Weibull distribution for Niamey 1949 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Figure 10. Monthly variation of k and c values, and Weibull distribution for Niamtougou Figure 11. Monthly variation of k and c values, and Weibull distribution for Ouagadougou 1950 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Table 1. Descriptive statistics and power density for Abuja site. Results Real values GP EPF EMJ EML ML MMa Vit.moy. 2.43776 2.0605 2.4378 2.4378 2.439 2.5368 2.4378 Ec.typ. 1.24492 1.367 1.2832 1.233 1.2337 1.1775 1.7101 Dens.puiss. 17.4943 15.229 17.462 16.612 16.638 17.177 25.541 Kurtosis 8.91043 6.9212 7.4299 9.2685 9.2434 10.451 3.1199 Skewness 1.04677 1.6215 0.9168 1.0794 1.0747 0.9273 0.4831 Table 2. Descriptive statistics and power density for Accra site. Results Real values GP EPF EMJ EML ML MMa Vit.moy. 4.15723 3.55301 4.15723 4.15723 4.15930 4.43428 4.15723 Ec.typ. 2.14720 2.01581 2.06641 2.12624 2.12736 1.92788 2.15189 Dens.puiss 82.12856 60.10977 82.17615 84.62223 84.74185 87.43053 86.26740 Kurtosis 2.92631 4.65209 3.39907 3.04094 3.03396 4.57299 3.03982 Skewness 0.08977 1.21591 0.10664 0.09400 0.09078 -0.41216 0.12566 Table 3. Descriptive statistics and power density for Bamako site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 2.79765 2.37492 2.79765 2.79765 2.79934 3.01251 2.79765 Ec.typ. 1.62835 1.65579 1.63012 1.60993 1.61095 1.49114 1.83500 Dens.puiss 31.39636 25.65127 31.29463 30.79267 30.84730 31.90702 36.68506 Kurtosis 6.00753 6.71940 5.81369 6.30352 6.28229 8.14727 3.92648 Skewness 0.91573 1.66482 0.89034 0.95025 0.94505 0.66773 0.66526 Table 4. Descriptive statistics and power density for Conakry site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 3.33148 3.00024 3.33148 3.33148 3.33280 3.46414 3.33148 Ec.typ. 1.60201 1.81798 1.59259 1.58902 1.58965 1.50113 1.91915 Dens.puiss 40.69449 40.50231 40.56237 40.46758 40.51499 41.68727 49.60185 Kurtosis 6.44260 4.09952 6.33274 6.65489 6.64252 8.07227 3.04739 Skewness 0.48592 0.88131 0.47693 0.49775 0.49461 0.25320 0.27832 Table 5. Descriptive statistics and power density for Cotonou site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 4.02330 3.33855 4.02330 4.02330 4.02384 4.10702 4.02330 Ec.typ. 1.69862 1.60903 1.68343 1.69004 1.69037 1.59787 2.11385 Dens.puiss 63.79816 42.86103 65.06774 65.18791 65.20482 65.20710 80.72832 Kurtosis 2.69366 4.49615 2.78383 2.74248 2.74035 3.46772 1.21240 Skewness -0.14677 1.41403 -0.13552 -0.14623 -0.14725 -0.35331 -0.05093 1951 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Table 6. Descriptive statistics and power density for Kano site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 4.71277 4.36121 4.71277 4.71277 4.71446 5.10566 4.71277 Ec.typ. 2.25113 2.27993 2.21547 2.23338 2.23425 1.96098 2.30906 Dens.puiss 112.7769 101.1049 112.9734 113.7094 113.8243 120.7997 118.0493 Kurtosis 4.51041 4.58891 4.78343 4.65114 4.64290 8.46210 4.44586 Skewness 0.28197 0.77250 0.28125 0.28838 0.28574 -0.54907 0.26923 Table 7. Descriptive statistics and power density for Lagos site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 2.70026 2.28902 2.70026 2.70026 2.70148 3.18704 2.70026 Ec.typ. 2.16501 2.00671 2.09101 2.14185 2.14275 1.92858 1.78841 Dens.puiss 45.03188 33.96930 43.48667 45.64691 45.71185 47.88559 32.88635 Kurtosis 6.37692 9.48775 7.22131 6.64091 6.62908 8.87452 13.44755 Skewness 1.27921 2.26594 1.40586 1.31888 1.31577 0.81327 2.23905 Table 8. Descriptive statistics and power density for Lome site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 3.53826 2.91092 3.53826 3.53826 3.54031 3.70109 3.53826 Ec.typ. 1.94333 1.72320 1.87766 1.92214 1.92333 1.82855 1.97795 Dens.puiss 55.29799 35.38807 55.31737 56.78762 56.88031 58.08273 59.19682 Kurtosis 2.40566 5.58399 2.75858 2.51307 2.50547 2.99925 2.37624 Skewness 0.26244 1.84950 0.29571 0.27278 0.26891 0.01376 0.28701 Table 9. Descriptive statistics and power density for Niamey site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 3.32004 2.99810 3.32004 3.32004 3.32192 3.40313 3.32004 Ec.typ. 1.77855 1.92446 1.81939 1.75988 1.76090 1.72087 1.91881 Dens.puiss 47.52339 44.26524 47.32130 45.62783 45.70436 46.48500 50.15246 Kurtosis 6.39370 5.23938 5.58891 6.65319 6.63403 6.92268 4.28204 Skewness 1.12474 1.43759 1.03090 1.15983 1.15461 1.06955 0.86939 Table 10. Descriptive statistics and power density for Niamtougou site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 2.60855 2.27461 2.60855 2.60855 2.61052 3.17567 2.60855 Ec.typ. 1.73066 1.55230 1.63536 1.70829 1.70956 1.39119 1.80945 Dens.puiss 27.76409 21.10048 27.68213 29.44720 29.51493 32.85143 31.83122 Kurtosis 3.77115 6.61094 4.70233 3.96792 3.95486 9.74914 3.23382 Skewness 0.35281 1.35784 0.41839 0.36637 0.36213 -1.38592 0.30581 1952 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Table 11. Descriptive statistics and power density for Ouagadougou site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 2.99765 2.58985 2.99765 2.99765 2.99936 3.12567 2.99765 Ec.typ. 1.60551 1.64402 1.61133 1.58862 1.58954 1.51770 1.83538 Dens.puiss 34.01567 27.77934 33.88799 33.38351 33.43866 34.33200 39.77047 Kurtosis 4.98264 5.57020 4.77967 5.19495 5.17930 5.92498 2.82876 Skewness 0.80675 1.55254 0.78654 0.83299 0.82820 0.66273 0.53418 Table 12. Descriptive statistics and power density for Tambacounda site. Results Real values GP EPF EMJ EML ML MMa Vit.moy 2.97820 2.69982 2.97820 2.97820 2.97987 3.05704 2.97820 Ec.typ. 1.57577 1.83930 1.64332 1.55950 1.56039 1.51973 1.83429 Dens.puiss 34.72223 35.61662 34.56636 32.65425 32.70756 33.35933 39.48413 Kurtosis 10.16318 5.60248 8.01953 10.61158 10.57942 11.59459 5.24357 Skewness 1.41329 1.28658 1.20300 1.45946 1.45355 1.41555 0.87873 It is important to note that each statistical parameter offers different perspectives that are useful for comparing methods. Thus, the combination of all these statistical indicators offers a possibility to compare the differences between the wind power calculated by the measured data and that of the Weibull distribution function with much more reliability. The results show that the accuracy of the computed wind power density values changes with estimates methods. It is clear that for all stations when the four methods EPF, EMJ, EML and ML are used to compute the Weibull parameters, the values of wind power density computed by the Weibull distribution function are in favorable agreement with the value of wind power density computed by measured data. This conclusion is drawn by the low values of the MAPE, MABE, RMSE. On the other hand, it can be seen that the lowest agreement indices are reached when the MMa and GP methods are applied for the calculation of k and c parameters. Abuja, Accra, Bamako, Conakry, Cotonou, Kano, Lagos, Lome, Niamey, Niamtougou and Tambacounda present the best results in terms of wind energy density calculation when the EPF method is used to calculate the k and c parameters. After the EPF method, for the stations Accra, Conakry, Cotonou, Kano, Lagos, Lome, Niamtougou, the most accurate results are obtained using the EMJ method. For Abuja, Niamey, Ouagadougou and Tambacounda stations, the ML method is the most accurate after the EPF method. As for the Bamako station, the EML method. However, the EPF method gives the best accuracy for all sites. The reason why the most appropriate methods come after the EPF method are different between sites with the wind characteristics variation. It is also important to note that the performances of the EMJ and EML methods are very close to each other based on all statistical indicators with a slight difference but the highest precision is often obtained by the EMJ process. In each table, the most accurate method for each station is indicated in bold. With respect to the weakest methods, the GP and MMa methods show relatively high differences from the other selected methods. In the rankings they occupy the last places so that their use leads to much higher errors than the other methods. 1953 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate Figure 12. Metrics for methods evaluation. 7. Conclusion Knowledge of wind power density is therefore of vital importance in assessing the potential of wind energy and in determining the suitability of the site for wind energy development. The two parameters Weibull distribution function has been widely used in various wind energy applications because of its simplicity, adaptability and accuracy. In this work, the performance of six numerical methods namely GP, EPF, EMJ, EML, ML and MMa were evaluated to determine the k and c parameters of the Weibull distribution function for the calculation of wind energy density at twelve stations distributed in the 1954 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 4: 1937-1955, 2024 DOI: 10.55214/25768484.v8i4.1568 © 2024 by the author; licensee Learning Gate West African sub-region. To achieve this goal, the wind power density derived from the Weibull function is compared to the power wind density calculated using measured wind data. Evaluations were performed on monthly basis in order to provide more complete analysis. The results indicated that by using different estimation methods to determine the k and c parameters, the accuracy of calculating wind power density values using the Weibull function changes. Based on the analysis of the results of, it is proven that the EPF, EMJ, EML and ML methods provide highly desirable better performance while the GP and MMa methods showed poor performance for all stations. Furthermore, the analysis of the results shows that the most appropriate parameter estimation methods are not the same for all stations examined due to the wind characteristics. At all sites, the EPF method was recognized as the most appropriate of the methods except for the Lagos site where some indicators put the EMJ method in first place. The methods that occupy second place after EPF vary according to the sites: the EMJ method for Accra, Conakry, Cotonou, Kano, Lome, Niamtougou; EML for Bamako, ML for Abuja, Niamey, Ouagadougou and Tambacounda. The parameters estimated can be used with excellent performance to represent the monthly wind speed distribution and determine the different statistical properties of the power density. Nevertheless, it should be mentioned that since each station benefits from specific wind power characteristics, the results obtained in this study with respect to the efficiency of the estimation methods of the Weibull distribution function parameters can only be extended to regions with identical wind power characteristics. Funding: This work was supported by the World Bank, through the Regional Center of Excellence for Electricity Management (CERME). Author contributions: Conceptualization, Agbassou Guenoukpati; methodology, Agbassou Guenoukpati, Pierre Akuété Agbessi and Komi Amoussou; validation, Adekunlé Akim Salami and Agbassou Guenoukpati; formal analysis, Adekunlé Akim Salami and Agbassou Guenoukpati; writing-original draft preparation, Agbassou Guenoukpati; writing-review and editing, Pierre Akuété Agbessi and Komi Amoussou; supervision, Adekunlé Akim Salami. All authors have read and agreed to the published version of the manuscript. Copyright: © 2024 by the authors. 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