Seasonal Wind Energy Characterization in the Gulf of Mexico

: In line with Mexico’s interest in determining its wind resources, in this paper, 141 locations along the states of the Gulf of Mexico have been analyzed by calculating the main wind characteristics, such as the Weibull shape ( c ) and scale ( k ) parameters, and wind power density (WPD), by using re-analysis MERRA-2 (Modern-Era Retrospective Analysis for Research and Applications version 2) data with hourly records from 1980–2017 at a 50-m height. The analysis has been carried out using the R free software, whose its principal function is for statistical computing and graphics, to characterize the wind speed and determine its annual and seasonal (spring, summer, autumn, and winter) behavior for each state. As a result, the analysis determined two di ﬀ erent wind seasons along the Gulf of Mexico;, it was found that in the states of Tamaulipas, Veracruz, and Tabasco wind season took place during autumn, winter, and spring, while for the states of Campeche and Yucatan, the only two states that shared its coast with the Caribbean Sea and the Gulf of Mexico, the wind season occurred only in winter and spring. In addition, it was found that by considering a seasonal analysis, more accurate information on wind characteristics could be generated; thus, by applying the Weibull distribution function, optimal zones for determining wind as a resource of energy can be established. Furthermore, a k -means algorithm was applied to the wind data, obtaining three clusters that can be seen by month; these results and using the Weibull parameter c allow for selecting the optimum wind turbine based on its power coe ﬃ cient or e ﬃ ciency.


Introduction
Global electricity demand grew 4% in 2018, almost twice than that for 2010, where renewables and nuclear power met most of the growth in demand [1]. According to the International Renewable Energy Association (IRENA) [2], 171 GW were added in 2018 worldwide after a strong growth in the last decade in renewable energy capacity. The total use of all renewables increased by 7.9%, where wind and solar energy contributed 84% of this total, and it is expected that in 2023, this increase for wind and solar energy will be 12.4% and 24%, respectively, in 2030, where the solar photovoltaic and wind power will be key energy sources since they are the energies with the highest growth, with the latter being one of the most profitable sources of energy in the world [3]. One of the most studied atmospheric parameters for decades is the direction of the wind [4]; nowadays this parameter is essential for the installation of a wind farm, where it is important that the wind turbines are not reducing their energy capacity due to poor design [5].
Weibull parameters, and found out that variation of power density with time was significant; therefore, they divided the year in two periods, period I (spring and summer) and period II (autumn and winter).
The ideas of Faghani et al. [31] are considered for this study and are extended for seasonal analysis (spring, summer, autumn, and winter) to determine its Weibull parameters and the characteristics of wind speed. To achieve this, a statistical analysis was conducted, and with this information, wind can be utilized effectively as a resource for electric generation.
The main objective of this study was to identify seasonal wind characteristics to assess them and determine their potential using different types of wind turbines based on their power curves and power coefficients. We considered a very important step in the process of wind turbine selection, which is the assessment or characterization of the wind speed. In this study, we also include a proposal to relate the wind turbine efficiency through its power coefficient and the conditions of the wind at each specific site.

Sites and Data
Mexico has five states along the Gulf of Mexico: Tamaulipas, Veracruz, Tabasco, Campeche, and Yucatan, as seen in Figure 1.
In order to carry out this study, data for 141 different sites along the Mexican Gulf were collected from the Modern-Era Retrospective Analysis for Research and Applications version 2 (MERRA-2) from the National Aeronautics and Space Administration (NASA); these are re-analysis data that are long-term, model-based analyses of multiple datasets using a fixed assimilation system [32]. MERRA-2 has presented data every hour from 1980 until 2017. According to Shang [33] the network measures derived from the empirical observations are often poor estimators of the true structure of system as it is impossible to observe all components and all interactions in many real-world complex systems. This problem occurs when there is missing data; in this study, MERRA-2 has no missing data; therefore, this problem is avoided and can be considered to be unbiased data.
In Figure 2, we give the geographic positions where MERRA-2 data were taken from.

The Weibull Distribution and Seasonal Wind
There are some statistical distribution functions for analyzing wind data, such as the lognormal, normal, Rayleigh, and Weibull probability distributions [34,35]. The Weibull function is the most-used function to assess wind energy potential because shows variables as shape and scale parameters [36], where these parameters are obtained using estimation methods, such as the maximum likelihood method, and the goodness of the resulting fits are evaluated using several indicators, e.g., the coefficient of determination (R 2 ) [24,[37][38][39]. The R 2 is a statistical measure that represents the proportion of the variance for a dependent variable that is explained by an independent variable; R 2 is generally interpreted as the percentage of a value's movements that can be explained by movements of another variable [40].

Wind Model
The Weibull distribution and cumulative distribution functions are expressed in Equations (1) and (2), respectively [41]: where k is the shape parameter (dimensionless), which is considered a Weibull form parameter because it specifies the shape of the distribution taking place within values between 1 and 3. A small value for k signifies very variable winds, while constant winds are characterized by a larger k [42]; when the shape parameter is 2, it is considered to represent Rayleigh distribution. The scale parameter c has the same units as wind speed (m/s) and is proportional to the mean wind speed (v m ), where v (m/s) is the wind speed registered in the site.
If the mean wind speed in Equation (3), and the standard deviation (σ) are known, k and c can be calculated using Equations (4) and (5) as follows [43]: The gamma function, Γ, can be calculated using Equation (6):

Wind Power Analysis
The observed wind power density (WPD O ) can be obtained using Equation (7): where ρ is the air density and is calculated using the ideal gas law (see Equation (8)), in which T is the absolute temperature (K), p is the absolute pressure (Pa), and R is the specific gas constant (J/kg·K): As an approximation, R = 0.286 is used for dry air.

Wind Variables Analysis
Calculations for wind variables were performed by implementing a self-written code on the R free software environment and language [44]. This environment, besides being free, provides powerful tools for statistical computing and graphical display via eight packages, and when required, they can be extended with additional packages available through the comprehensive R archive network (CRAN) family available on the Internet. In fact, in the former study, six base packages were used (tools, stats, graphics, grDevices, utils, and base) [44], which were complemented with the extra packages (lubridate [45], RColorBrewer [46], ggplot2 [47], and gridExtra [48]).

Data Processing
An iterative process was run for all 141 studied sites, which were distributed in a grid-shaped arrangement along the coast of the Gulf of Mexico (see Figure 2) using the previously mentioned MERRA-2 files as input data. These files contained complete time series data (no missing values) of the surface pressure (kPa), air temperature ( • C) at 2 m and 10 m above ground level, wind speed (m/s) at 50 m above ground level, and wind direction ( • ) at 60 m above ground level; all these variables have hourly records from 1980 to 2016.
An annual and a seasonal analysis was performed for every site, where for the sake of simplicity, spring was considered to include the months of March, April, and May; summer included June, July, and August; autumn included September, October, and November; and Winter included December, January, and February.
The structure of the program consisted of an initial pre-processing of the MERRA-2 data, which generated an output file that was used for obtaining the geolocated values of the wind variables, as well as customized plots via two scripts named Weibull Analysis and Directional Analysis, as can be observed on Figure 3, and is explained as follows.

Pre-Processing
In the pre-processing step, a time adjustment was performed on the MERRA-2 data in order to assign the corresponding UTM zone (for Mexico −6 h) and make summertime corrections. Data was classified for years, months, days, and hours for further manipulation, and the air density and observed WPD were obtained via Equations (9) and (10), respectively.
Once the pre-processing was finished, an output was generated for feeding the following two independent and complementary scripts.

Weibull Analysis
The Weibull probability distribution function (PDF) and cumulative distribution function (CDF) were obtained and plotted according to Equations (1) and (2), respectively, for the annual and seasonal series.
Tables were generated with Weibull shapes and scale factors (k, c), mean wind speed (v m ), its corresponding standard deviation (σ), Weibull most-probable wind speed (v mp ), mean air density (ρ m ), and the observed wind power density (WPD O ).

Directional Analysis
In order to complement the Weibull distribution, a directional analysis was performed. For this, data was grouped according to the wind direction into 12 bins, corresponding to sectors as 30 • segments. The cumulative WPD O was obtained for every sector by considering the one with the highest value as the prevailing direction, which was obtained for complete years, as well as for every season.

Wind Clustering
The wind performance was grouped into several clusters using a k-means clustering model to identify its monthly behavior, which was analyzed to diagnose the time where the wind speed could be used to generated power. The k-means algorithm is widely used in data mining for the partitioning of n measured quantities into k clusters [49]; according to Sugar and James [50], the classification of observations into groups requires computing the distance between the observations [51][52][53][54]. The k-means algorithm is one of the simplest unsupervised machine learning algorithms, where unsupervised algorithms make inferences from datasets using only an input vector without referring to known, or labelled, outcomes [53]. We can define a cluster as a data set with similar characteristics. The k-means algorithm identifies k centroids and then allocates every data point to the nearest cluster [53]. This algorithm has been used in a study done by Wang et al. [55], where the k-means clustering algorithm was used to find the largest historical samples that had the greatest influence on forecasting accuracy to improve the efficiency of the proposed model.
A method for clustering was proposed by Deng et al. [52], which used a Weibull distribution to establish that an unclustered dataset P can be represented using Equation (9): where p i denotes the ith element and N i is the number of observations. The set of clusters C can be given as: and the set of observations within a cluster, C j , is represented by: C jk and N jk are the kth observation with the jth cluster and the number of observations, respectively, in cluster set C j . The set of centroids associated with the clusters W is given as: where W j is the jth centroid. For wind clustering, each observation P is assigned to a cluster C j and its centroid will be represented by its mean wind speed.

Wind Turbine Selection
The energy available for conversion mainly depends on the wind speed and the swept area of the wind turbine. Using Newton's Law F = ma: where E is the kinetic energy, m is the mass, a is the constant acceleration, and s is the distance. From kinematics, the acceleration can be expressed using Equation (14): Because the initial velocity of the object is 0, then u = 0, such that: Substituting Equation (15) in Equation (13) gives: The rate of change of energy of the power in the wind is given by: The mass flow rate is expressed as: and the distance's rate of change is: Substituting Equation (18) into Equation (19) gives: Hence, from Equation (17), the wind power output generated by a wind turbine can be expressed using Equation (21): where P WT is the rated power, ρ is the air density, A is the rotor area, and U is the wind speed approaching the turbine. According to Grillo et al. [56], the power coefficient is a function of the tip speed ratio, known as lambda (λ), and the blade pitch angle (β) ( • ). λ can be calculated Equation (22): where ω is the rotational speed of the rotor (rad/s) and R is the rotor radius (m).
To characterize the wind speed in this study, a proposal for wind turbine selection is given, where this proposal considers the air density, rotor area, wind turbine power curve, and power coefficient (C p ).
Song et al. [57] defined the power coefficient C p as the ratio of the power extracted from the wind turbine to the available power. C p can be calculated as follows: The C p of a wind turbine is a measurement of how efficiently the wind turbine converts the energy in the wind into electricity.
Wind speed is one of the most important parameters in determining the electric power, and the general equation is related to the density of air, wind speed, and swept area, as in Equation (21), and represents the total energy obtained from the wind resource; however, in terms of generating electricity, only a certain proportion of energy can be converted and is expressed by Equation (24), as follows: P e = η e η m C p P wT , where P e is the amount of electric power generated, η e is the electrical conversion efficiency of the wind turbine, and η m is the mechanical efficiency [58].
There is an optimization proposal that uses the wind turbine efficiency as a fundamental variable to determine the power output generated by a wind power farm [59]. In this proposal, as in this study to determine it, a Weibull distribution was used. The parameter c from the Weibull distribution was used to select a wind turbine according to its C p ; in this case, the maximum efficiency of a wind turbine was compared to the parameter c calculated from the wind speed site studied.

Results and Discussion
The annual Weibull parameters c and k, WPD F , WPD O , ρ, v m , v mp , and v maxE were obtained for the 141 MERRA-2 data locations. Table 1 only presents the places with the highest and lowest mean wind speeds for each state (colored blue in Figure 2); these values were used as references for the seasonal analysis. Tamaulipas was the state with more zones with high values of wind than the other states, where its maximum v m = 7.34 m/s, scale and shape parameters were between 4.17-8.26 m/s and 1.97-2.91, respectively. In this case, by comparing v m , c, and k, it can be established that in the time domain, most of the wind resource had values higher than its mean wind speed. In Veracruz, the scale and shape parameters showed values between 3.86-6.88 m/s and 1.72-2.41, respectively. Tabasco presented more wind speed and wind resource than Veracruz with Weibull parameters ranging between 4.02-7.48 m/s for c, and 2.51-3.07 for k. Campeche and Yucatan showed a higher k's than the others states, with 2.71 and 2.95, respectively, which can be interpreted as a long period of time with winds higher than their means; c was 7.47 m/s and 7.88 m/s for Campeche and Yucatan, respectively.
According to Katinas et al. [29], the Weibull parameter c has the same behavior as the WPD, where in all cases, when the scale factor grows, the WPD grows as well. It is important to mention that the lowest shape parameter corresponds to the lowest wind speed, which means that the frequency of data below the mean was greater than that above it.

Seasonal Weibull Parameters and WPD
Seasonal wind speed characterization was carried out by dividing the data according to the four annual seasons: spring (March, April, and May); summer (June, July, and August); autumn (September, October and November); and winter (December, January and February). For the sake of analysis, two sites for each state were chosen by considering its highest and lowest mean wind speed (see .
In Figure 4, it can be observed that the points Tam34 and Tam48 had the highest and the lowest mean wind speed-8.26 m/s and 4.18 m/s, respectively-in the state of Tamaulipas. At the annual level, Tam34 had k = 2.71, which represents a very good value due its higher magnitudes of wind speed; at the seasonal level, the Weibull distribution shows that spring and winter are the periods of time during an average year where there was more available wind resource.
In the state of Veracruz, 31 locations were studied, with the highest scale parameter being found for Ver29, where this was located is in the Isthmus of Tehuantepec, one of the windiest zones in the world. During an average year, its highest mean wind speed was 5.72 m/s, its c was 6.46 m/s, and k was 2.40, and had a windy season during spring, autumn, and winter (see Figure 5).
In Tabasco (Figure 6), Tab2 had c = 6.90 m/s, k = 2.84, and a mean wind speed of 6.15 m/s; these values represent a good wind resource because analyzing Weibull parameters showed that c was higher than its mean, and k showed that most of the time, the wind speed data was above its mean. In contrast, the seasonal Weibull of Tab9, which was the point with the lowest values in this state (c = 4.02 m/s, k = 2.76, v m = 3.58 m/s), allowed to determine a period of wind during an average year that had constant magnitudes of wind values, namely winter, spring, and summer, as shown.
The analysis of Campeche (Figure 7) showed its windiest location was Cam14, which had a mean wind speed of 5.82 m/s, and c and k values equal to 6.51 m/s and 3.08, respectively. The lowest valued location, Cam20, had c = 2.90 m/s and v m = 2.64 m/s, where both values are similar with a difference of 0.26 m/s, which can be considered as a location that did not have variations during an average year. Its k = 3.04 could be considered a very good frequency of wind speed; however, due its low wind speed, it did not represent an impactful wind resource, which can be seen in the seasonal analysis because its variation was minimal.
In Yucatan, as seen in Figure 8, its seasonal wind variation was divided in two periods, highest one between winter and spring and the lowest one between summer and autumn. The location with the highest wind speed was Yuc2 (v m = 4.56 m/s, c = 5.10 m/s, and k = 3.07), and the lowest location, Yuc17, was inside a rainforest, which reduced the magnitude of the wind speed (v m = 2.67 m/s, c = 2.99, and k = 3.10).   These results are in agreement with the study developed by Herrero-Novoa [30], where it was established that dividing an average year into two periods (spring-summer and fall-winter) provides more accuracy regarding wind characterization; in a similar fashion, for the current study, an average year was divided into four periods of time (seasons) where even more accurate wind periods were established. This allowed us to determine the wind characteristics, specifically its WPD, in the most relevant period. By observing Figure 8 and considering the values of the parameters obtained in Table 1, we established that in Yuc2, the parameter c = 5.10 m/s was higher than its average of 4.56 m/s, and its k = 3.07; therefore, at this site, the wind speed was over 4.56 m/s most of the time. Furthermore, according to Vazquez et al. [42], this value of k means that the site had constant winds.
In addition to the Weibull analysis, in Figures 9 and 10, the prevailing wind directions and the magnitude of its WPD O can be observed for the annual and seasonal behavior, respectively. The highest WPD O points were along Tamaulipas and Veracruz, although Tamaulipas had more locations than Veracruz, while Veracruz has the highest resource next to the sea, Tamaulipas had it at the border with the United States of America. As can be observed in Figure 10, the seasons with greatest WPD O were autumn and winter, and the states with greatest resource were Tamaulipas and Veracruz. Regarding wind orientation, this figure shows that Tamaulipas had two periods with different prevailing directions, with the first one during spring and summer and the second one in autumn and winter. The WPD in southern Veracruz presented the same direction most of the time (southeast), meanwhile Tabasco, which is the southernmost state along the Gulf of Mexico, had different prevailing orientations throughout an average year. It can also be seen that Campeche had a prevailing direction in winter and spring, and for Yucatan, autumn, winter, and spring had northeast as the predominant direction, while it was southeast in summer.

Wind Speed and Wind Power Clustering
Using k-means clustering, the wind speed could be clustered into months, and in these clusters, the power output could be calculated. Figure 11 shows the clustering done for wind speed from 1980-2017. Wind speed was divided into three clusters-C 1 , C 2 , and C 3 -as shown in Figure 11 where the blocks represent the amount of wind speed each month.
Taking these results and knowing the windy seasons throughout an average year, the wind power output could be calculated. A critical point in this stage was to select the wind turbine, where in Table 2, we present the C p of different wind turbines as a first step to know the efficiency of each one of them. The wind turbines were selected according to parameter c of the points analyzed. Six wind turbines were found such that their C p fit with the parameter c calculated from the wind speed data. Table 3 presents the wind turbine selected for each point along the Gulf of Mexico.  A total of 58 points were fitted with an appropriate wind turbine C p , where the wind speed of the site was related to the wind speed at the maximum wind turbine efficiency; with this information, the best option for the wind turbine to be used could be determined based on its C p . The probable energy production was calculated for each wind turbine selected using Equation (16) based on the electrical and mechanical efficiency given by the manufacturer.

Conclusions
The seasonal characterization of wind speeds along the Mexican states of the Gulf of Mexico was done for 141 locations with MERRA-2 data, with records between 1980-2017. An average year of wind data for each location was obtained and these were divided according to the seasons of the year (spring, summer, autumn, and winter). This distinction allowed us to describe the wind characteristics more accurately, especially the WPD, which allowed for establishing wind seasons properly, as well as the description of the prevailing wind direction. It is was found that there were different wind seasons along the states of the Gulf of Mexico and they were established for each state. The wind season in Tamaulipas occurred in winter and spring; Veracruz and Tabasco had a wind season in autumn, winter, and spring; and in Campeche and Yucatan, the wind season was in winter and spring, where these states shared their coast with both the Gulf of Mexico and the Caribbean Sea. This variation allowed us to determine that the wind season could depend upon other factors regarding the climatology of each state.
The state with the greatest wind resource was Tamaulipas, as seen in Figure 9; it had plenty of locations with good values for wind, scale, and shape parameters, where its highest v m , c, and k was 7.34 m/s, 8.26 m/s, and 2.91 respectively. North and south Veracruz had the highest values of wind speed, where its southern zone corresponded to the Isthmus of Tehuantepec, one of the zones with the greatest wind energy resource in the world; its highest parameters were v m = 6.10 m/s, c = 6.88 m/s, and k = 2.41. Tabasco presented the following parameters: v m = 6.67 m/s, c = 7.48 m/s, and k = 3.07. These three states (Tamaulipas, Veracruz, and Tabasco) presented the same wind season.
The highest parameters for Campeche were v m = 6.57 m/s, c = 7.39 m/s, and k = 2.71, and the Yucatan parameters were v m = 7.88 m/s, c = 7.04 m/s, and k = 3.07; these two states presented the same wind season.
Three clusters were found, which were useful for determining the wind speed in monthly groups and to visualize the wind stations in the study area of the Gulf of Mexico.
Using C p , optimal wind turbines were determined by comparing it with the parameter c of Weibull distribution. In this study, 58 sites were fitted with a wind turbine C p , and by analyzing these data, three wind turbines were selected: Acciona AW70/1500 Class III with a C p = 0.4188 at 6.5 m/s, Acciona AW70/1500 Class II with a C p = 0.4457 at 7.5 m/s, and Vestas V90 1.8 MW with a C p = 0.4381 at 8 m/s.
At the end of this study, the probable energy production was calculated for each wind turbine applied at each site with its own respective conditions.
Regarding future research lines, we propose following the proposal to select the most efficient wind turbine by adding more constraints, such as wind direction, roughness, and orography.