Joint Modeling of Wind Speed and Power via a Nonparametric Approach

Abstract

Power output from wind turbines is influenced by wind speed, but the traditional theoretical power curve approach introduces uncertainty into wind energy forecasting models. This is because it assumes a consistent power output for a given wind speed. To address this issue, a new nonparametric method has been proposed. It uses K-means clustering to estimate wind speed intervals, applies kernel density estimation (KDE) to establish the probability density function (PDF) for each interval and employs Monte Carlo simulation to predict power output based on the PDF. The method was tested using data from the MERRA-2 database, covering five wind farms in Brazil. The results showed that the new method outperformed the conventional estimation technique, improving estimates by an average of 47 to 49%. This study contributes by (i) proposing a new nonparametric method for modeling the relationship between wind speed and power; (ii) emphasizing the superiority of probabilistic modeling in capturing the natural variability in wind generation; (iii) demonstrating the benefits of temporally segregating data; (iv) highlighting how different wind farms within the same region can have distinct generation profiles due to environmental and technical factors; and (v) underscoring the significance and reliability of the data provided by the MERRA-2 database.

1. Introduction

The growing demand for energy from renewable sources has led to the construction of many wind farms in Brazil and other countries. However, two main factors can hamper the development of the wind power industry: the probabilistic nature of wind speed and the uncertainty involving its production [1,2]. These factors require more robust and reliable models that better replicate the behavior of local wind generation to support the planning, monitoring and operational management of the power grid and increase the reliability of the electrical network [3,4,5,6].

For the correct dimensioning of power generation, it is necessary to understand the functioning of wind turbine generators [1]. It is well established that the power from a wind turbine varies in function with the wind speed, air density and turbine blade parameters (size, design, tip speed and pitch angle ratio) [7]. The manufacturers usually supply a power curve for their models, which is typically generic and related only to the wind speed for the power generated [8]. Nevertheless, this information serves as a parameter for initial monitoring of the performance of a wind farm’s turbines [9].

For managers of wind farms and power systems, it is not advisable to apply the manufacturer’s power curve directly since the real working conditions of a turbine can be very different. The power curve made available by the manufacturer is projected for the functioning of a single wind turbine operating under ideal conditions. It does not consider the various types of interference they generate with each other within a wind farm. Hence, a need exists to develop models to estimate wind power in specific generation locations [9].

Researchers have described various deterministic or probabilistic strategies to create reliable estimation models. These power curve models are also subdivided into parametric and nonparametric techniques [7,10,11]. The parametric methods are based on mathematical formulations, such as the linearized segmented model; polynomial power curves [12,13]; exponential [14], cubic [7] and maximum principle methods; the least squares method [10]; cubic spline interpolation; dynamic power curves; and models based on probabilistic distributions [6,10,15], among others. In general, they are methods that are easy to apply and calculate the parameters, need little historical data, and have precise adjustments. Their drawbacks are poor accuracy, the need for comprehensive data from the manufacturer and the local conditions, and failure to consider the variability of these data [11].

The nonparametric models include copulas [16,17], kernel density estimation, artificial neural networks [18], fuzzy logic systems, support vector machines, response surface methodology and data mining algorithms (such as random forest and clustering) [18], among many others [9,19,20,21,22]. These do not impose any model specified in advance, are more precise, contemplate the variance of the data and estimate the power curve as closely as possible with the available data subject to smoothing of the fit. However, these methods require long historical data series and are complex to implement [11,23].

In [18], the authors used machine learning, artificial neural networks (ANNs), decision trees (DTs) and random forests (RFs) to infer the best alternative to represent the outputs of a wind turbine. The same evaluation is described in [23] via three dynamic Bayesian models considering wind speed and power. In turn, refs. [16,17] propose the use of copula functions to model this relationship through the measured data, while [16] also uses a probabilistic method to exclude outliers. The article by [24] evaluates the performance of multivariate models based on the Gaussian mixture copula model (GMCM), artificial neural networks and Bayesian artificial neural networks (BANNs) to characterize the power curve and estimated power.

Many other studies have also used wind speed distribution as a basis to model the relationship between wind speed and power output. Most of these have recommended using a two-parameter Weibull distribution [25]. More recent studies have suggested modifications of the Weibull distribution [26] and the use of a four-parameter Kappa probability distribution or five-parameter Wakeby probability distribution to obtain better results [27].

In [7], the authors describe the fitting of polynomial, exponential and cubic equations to represent the power curves of turbines. The authors found the polynomial and cubic approaches to be most precise. Still, the polynomial approach required complex equations, making it hard to find a general expression, leading to the recommendation of the cubic, which only depends on the parameters provided by the manufacturer. Ref. [11] warns that these mathematical models should only be used for initial evaluation of the power output, because they do not precisely consider the inflection point of the power curve and can result in large forecasting errors. They serve to evaluate new undertakings, dimensioning and the optimization of costs. Ref. [28] suggests using sigmoid, logistic and Hill functions, while [10] proposes the spline regression method to obtain better performance and overcome the problems of determining the curve’s inflection point. However, although these mathematical expressions are widely used, there is little evidence that these curves fit the data pertaining to real wind power turbines [12,13,14].

For this reason, this study aims to describe a model that more accurately represents this relationship between wind speed and power, requiring only historical data for both variables. Although the method uses only these two data points, it can indirectly incorporate the complexity inherent in the variability of the different sources of meteorological data and those specific to the turbine designer, which can also influence the prospection of the resulting wind energy.

In this context, a method is presented to segment the wind speed ranges through clustering data by K-means, mapping possible wind power outputs associated with each range and constructing the probability density function of the power data using the kernel density estimation method. Finally, the Monte Carlo simulation is applied to make the model flexible regarding wind power generation.

To improve the modeling of the relationship between wind speed and wind energy generation, this paper combines these methods to create a more reliable and effective methodology. This will help us make more accurate projections that align with real-life conditions in wind turbines. Once implemented, this model will enable us to use wind speed projections to estimate wind power effectively.

This article is divided into five sections, including this introduction. The second section presents the method used to model the relationship between wind speed and power; the third section presents the data used in the tests; the fourth section describes and discusses the results; and the fifth section presents the conclusions and some recommendations for future research.

2. Proposed Methods

The purpose of this article is to describe a method that can model the nonlinear relationship between the speed of the wind that passes through wind turbines and the power generated by them, thus enabling the exploration of the peculiarities of the source and place and the depiction of the probability distribution of generation for a given wind speed value. The steps used in this study to attain this goal are presented in Figure 1.

Section 2.1 presents the locations used to test the proposed method. Section 2.2 summarizes the steps employed in the predecessor contribution [29], which details steps 2.2.1 to 2.2.4. It discusses how to obtain and treat the wind speed reanalysis data from MERRA-2 for points in Brazilian territory to have data representing local reality and overcome the shortage of measured data points. We describe the main contributions of this article in Section 2.3.1, Section 2.3.2 and Section 2.3.3 regarding the wind power modeling and forecasting method. Section 2.4 presents a simple and widely adopted method to compare our proposal here against other methods used to relate wind speed and wind power generation. Finally, Section 2.5 offers the metrics adopted to evaluate the method.

2.1. Wind Farm Data

To test the method presented in this article, information from five wind farms located in the same state in Brazil’s Northeast Region is used; three farms are along the coast and the other two are in the interior of the state. The information the wind farm managers provided included the geographic locations, installed capacities and generation time series, which cannot be disclosed due to a secrecy agreement.

The other information necessary to apply the method, such as the technical characteristics of the turbines used by the wind farms, was obtained from the Electric Sector Georeferenced Information System of Brazil’s National Electric Energy Agency (SIGEL/ANEEL) [30] and The Wind Power database [31].

These wind farms were chosen due to the convenience of measuring data availability and their location in the Northeast Region, where there is significant wind potential, the number of wind plants has grown and methodologies are needed to help their work [23,29,32,33].

2.2. Wind Speed Data

The wind farms chosen for testing do not have any wind speed history, only historical generation data. However, the power generated by these turbines is directly related to the wind speed at the height of the corresponding rotor, and modeling this relationship is the purpose of this article.

To overcome this limitation, MERRA-2 reanalysis data were used [34]. They provide a free historical series on climate variables and atmospheric data since 1980, available at grid points covering the Earth’s entire land surface. The MERRA-2 databases used most often are “MERRA-2 tavg1_2d_slv_Nx” and “MERRA-2 inst1_2d_asm_Nx” to obtain historical wind speed data. The main difference is that the wind speed data are hourly averages in the former, and in the latter, the data are collected instantaneously. The first database was adopted for this study because it presented the best results in preliminary tests and the wind power data were also hourly averages [34].

Interpolation, extrapolation and bias correction were used to adjust the wind speed time series of MERRA-2 to the natural conditions of the turbines in question [29]. The interpolation adopted was based on the nearest neighbor technique, which consists of obtaining the data from the point on the MERRA-2 grid closest to the geographic location of the wind farm to represent the speeds that occurred at that point. In turn, the power law is used as the extrapolation technique to adjust the historical wind speed data of MERRA-2 to the values that occurred at the hub height [32]. Finally, bias correction involves applying an average hourly correction factor using data from the nearest National Meteorology Institute (INMET) station. This technique consists of calculating the hourly wind speed averages of the MERRA-2 and INMET datasets at the same height (in this case, 10 m) and calculating the ratio between them each hour to generate an hourly correction factor to be applied to the extrapolated wind speed time series to adjust it to the real winds measured in Brazilian territory and contemplate the local particularities [13,20]. This method is suggested by [29] to obtain data from MERRA-2 for Brazil.

2.3. Modeling the Speed Versus Power Relationship

A particular wind speed value can generate different power values from the same turbine at distinct moments. A combination of factors, such as variations in wind direction, relative air humidity, atmospheric pressure and other climatic and atmospheric variables, can explain this. However, mapping all these wind generation variations in the function of such variables would make the model highly complex and still would not assure the identification of all the factors that can impact the generation [7].

Thus, this proposed methodology seeks to simplify the modeling of wind generation to enable its projection using a single variable, the wind speed, striving to create a model that explains all the randomness arising from the various data sources solely according to wind speed. To materialize the idea, the wind speeds that can occur in a wind farm are segmented, and each range’s generation possibilities are dimensioned based on the speeds that are believed to occur. The techniques used to materialize this method are presented in Section 2.3.1 Section 2.3.2 and Section 2.3.3.

2.3.1. Clustering of Wind Speeds

The K-means algorithm assigns each input data point to a group, also known as a cluster, and seeks to minimize the sum of squares within each group [21,35]. This article uses this technique to identify the wind speed ranges. It groups the wind speed data into k clusters (speed range groups), where each data point is assigned to a cluster with more similar data. This approach is an alternative to using multivariate probability density functions and reducing the problem’s dimensionality using coupling speeds that can be considered similar behavior.

The technique’s algorithm starts by randomly defining $k$ centroids (centers of clusters). Then, the distance is calculated between each data point and the initial $k$ centroids, and the value is attributed to the nearest cluster. Next, new centroids are defined, each being the average of all the data composing the respective cluster. This process is repeated until the centroids remain fixed after multiple iterations [22].

The K-means method has a premise consisting of the number of clusters, $k$, to be adopted. Hence, to find the most suitable number of clusters, the elbow method is used, which is applied by gradually increasing the number of clusters, and with each addition verifying whether, on average, the standard deviation of the clusters is reduced, such that the accuracy of the centroid reflecting the historic data is increased. Thus, the number of clusters is defined when the variations of the standard deviation become negligible when including another cluster [36].

2.3.2. Density Curve Estimation

Since power generation is a continuous variable, and it was desired that the frequency of its values for each cluster of velocities was understood, the probability density function (PDF) was estimated for the generation of each cluster. This association is possible because each wind speed value is associated in time with a historic occurrence of wind farm generation.

The technique used to estimate the probability density functions is kernel density estimation (KDE), a nonparametric estimation approach. Its main advantage is that it is not necessary to assume the distribution of the sample data in advance, thus avoiding the introduction of subjective previous information. This allows for a generalization of the proposal independent of the data used [37,38].

Since the KDE involves fitting the density function concerning the data, this process is more precise and robust compared to parametric approaches [37]. If $x$ denotes the sample dataset, $f\left(x\right)$ can be expressed as in Equation (1):

$$f\left(x\right)=F^{\prime }\left(x\right)={\displaystyle \frac{F\left(x+h\right)-F\left(x-h\right)}{2h}},$$

(1)

where $h$ is a non-negative constant, called the bandwidth, and $F\left(x\right)$ represents the empirical distribution of the power data that compose the sample.

To assure a better fit of the probability distribution, the sample size is allowed to tend to infinity ($n\to \infty$) and the bandwidth to zero ($h\to 0$) [37,38]. Hence, the expression for the estimator of $f\left(x\right)$ can be defined as in Equation (2):

$$\widehat{f}\left(x\right)={\displaystyle \frac{1}{nh}}{\displaystyle \sum _{i=1}^n}K\left({\displaystyle \frac{x-x_i}{h}}\right),$$

(2)

where $K(.)$ is the kernel function. Based on Equation (2), note that the determination of the estimates, $\widehat{f}\left(x\right)$, mainly depends on $h$ and $K(.)$ [35].

2.3.3. Simulation

Monte Carlo simulation is a statistical method that uses a large number of random samples to obtain numerical results that would previously be deterministic [39]. It was used to guarantee the variability in wind energy generation as a function of wind speed.

According to the previous steps, each speed range from a cluster has an associated probability density function, for which the cumulative density function is calculated. This randomly generates a value from a uniform [0, 1] distribution, which is used in the cumulative function to find the power value that produces the corresponding cumulative probability [6,10,39,40].

In order to calculate the power values, one needs to determine which cluster the next wind speed value belongs to and find the corresponding cumulative function. Then, a new random probability value is generated for the cumulative function. This process is repeated until the sequence of wind power generation estimates is completed [38].

2.4. Power Curve

The conventional deterministic approach to obtain the available power of the wind passing through the rotor-swept area is expressed in Equation (3) [7,41]:

$$P_t\left(v\right)={\displaystyle \frac{1}{2}}A\rho v^3$$

(3)

where $P_t\left(v\right)$ is the theoretical power generated in watts when the wind speed $v$, in m/s, passes through a turbine with rotor area $A$, in m², and $\rho$ is the air density.

However, the real power generated, $P_r\left(v\right)$, is lower than $P_t\left(v\right)$ due to mechanical and electrical losses and the aerodynamic factors of the blades. The ratio between the powers generates the power coefficient, $C_p$, which is typically available from the turbine manufacturer. The theoretical maximum value of the power coefficient is 0.593, the Betz limit, but in practice this value is not attained in the turbines, with the maximum value being 0.5 [6,7].

Equation (4) represents this mathematically:

$$P\left(v\right)=\left\{\begin{array}{cc}0,& v<V_{Ci}\\ P*\left(v\right),& V_{Ci}\le v<V_r\\ P_r,& V_r\le v\le V_{co}\\ 0,& v>V_{co}\end{array}\right.$$

(4)

where $P\left(v\right)$ is the electric power generated, $V_{Ci}$ is the initial cutoff wind speed, $V_r$ is the nominal wind speed, $V_{co}$ is the final cutoff wind speed, $P_r$ is the nominal power and $P*\left(v\right)$ is the power related nonlinearly to wind speed.

The format of the nonlinear part of Equation (4) is related to the strategy of controlling the maximum power extraction from the wind, which is normally moderated by a cubic power curve, as expressed in Equation (5):

$$P*\left(v\right)={\displaystyle \frac{1}{2}}\pi r^2{C}_p\rho v^3$$

(5)

where $r$ is the rotor radius [7].

The regions identified in Equation (4) are not clearly demarcated in the operation of the wind turbines. The corresponding wind speeds that delimit the regions are found from averages of repeated measurements. This is another limitation of this approach [7,23].

2.5. Evaluation

The evaluation metrics adopted in this study were root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE) and coefficient of determination (R²). These are the most commonly used and recommended in the literature [3,7,18,27,36,37,39,42,43,44]. They measure to what extent the real (observed) power values differ from those predicted by the techniques used, reproducing the degree of dispersion between the two metrics. The RMSE and MAE are expressed per unit (p.u.), while the MAPE and R² are expressed in percentages.

In the first three metrics, lower values are associated with better performance, while for R², higher values mean a better result.

3. Data

This section presents the data used in this study and their descriptive analyses.

3.1. Wind Power Data

The historical time series from the wind farms are the hours of active power and did not need treatment. There were no missing data, power levels higher than installed capacity, or negative power values. There were differences in the historical dimensions of the wind farms due to their different startup dates. The farms located along the coast are the oldest and have been in operation for 9 years and 2 months (1 January 2013 to 28 February 2022), while those located inland have only been operating for 4 years and 4 months (1 November 2017 to 28 December 2022). The other data necessary to apply the method, such as the technical characteristics of the turbines, were obtained from the SIGEL/ANEEL platform [30] and are presented in

Table 1. Information about the wind farms.

Wind Farm	Location in the State	Installed Capacity (KW)	Number of Turbines	Total Wind Turbine Height (m)	Rotor Diameter (m)	Operating Period
1	Interior	31,500	15	194.00	114.00	1 November 2017 to 28 February 2022
2	Interior	63,000	30	194.00	114.00
3	Coast	10,200	13	98.85	48.00	1 January 2013 to 28 February 2022
4	Coast	48,000	60	99.60	48.00
5	Coast	4500	3	123.50	77.00

.

Figure 2 and Figure 3 depict the characteristics of the historical data on the active power of the wind farms. Power data are used p.u. (per unit), which consists of dividing the registered powers by the maximum power. This is carried out to make a fairer comparison between wind farms and the generation of different orders of magnitude. From Figure 2a, it can be noted that the probability density functions of each region are similar. The density curves of the wind farms on the coast are unimodal, with a top value near 10% of the installed capacity. In contrast, the density curves of the interior wind farms are bimodal, with peaks at the extremes. The power data from interior wind farms are more variable than those on the coast, and higher power values are also rarer due to the presence of outliers in their boxplots, as illustrated in Figure 2b.

Figure 3 presents boxplots that indicate the generation variability during the different hours and months of operation. The wind farms in the interior have the lowest generation during the afternoon, while those located along the coast have highest generation during that period. Nevertheless, from a monthly standpoint, all the wind farms had higher generation during the year’s second half, with peaks in August and September.

3.2. Wind Turbines

Although the theoretical power curve has apparently well-defined regions, the limiting speeds of those regions vary compared with the manufacturer’s specifications. The turbine models used by the wind farms examined in this study are identified in

Table 2. Turbine models on the wind farms.

Wind Farm	Manufacturer	Model	Rotor Diameter (m)	${\mathit{v}}_{\mathit{c}\mathit{i}}$ (m/s)	${\mathit{v}}_{\mathit{r}}$ (m/s)	${\mathit{v}}_{\mathit{c}\mathit{o}}$ (m/s)	${\mathit{C}}_{\mathit{p},\mathit{m}\mathit{a}\mathit{x}}$
1	Gamesa (Spain)	G114/2100	114	3.5	10	25	0.45
2	Gamesa (Spain)	G114/2100	114	3.5	10	25	0.45
3	Enercon (Brazil)	E48/800	48	3	14	25	0.50
4	Enercon (Brazil)	E48/800	48	3	14	25	0.50
5	IMPSAr (Argentina)	IV-77-1500	77	3	13	22	0.42

. This information was obtained from a previous study [7] and The Wind Power database [31].

4. Results and Discussion

This section will present the results of the wind power estimates achieved by the traditional approach and the different application strategies of the proposed method. The work presented here was developed using R software, version 4.3.0 [45].

Table 3. Data from adjusting the wind speed data from MERRA-2.

Wind Farm	Hub Height (m)	Distance
		MERRA-2—Wind Farm (km)	INMET—Wind Farm (km)
1	137.00	18.64	52.97
2	137.00	14.95	54.65
3	74.85	3.41	19.86
4	75.60	9.01	17.92
5	85.00	7.61	13.49

presents some of the data used and calculated in adjusting the wind speed time series from MERRA-2 to each wind farm’s conditions. Gruber et al. [32] recommend applying the bias correction step to MERRA-2 data only when the INMET station is 40 km from the wind farm. Otherwise, bias correction is not advisable due to the considerable distance between the station and wind farm, such that the INMET data may not adequately represent the real local conditions of the wind farm.

In this study, certain wind farms were situated over 40 km away from an INMET station. To assess the influence of utilizing these data in bias correction, tests were conducted on the wind speed time series both with and without bias correction.

Other information used to generate the results of this work, such as the geographic coordinates of the points on the MERRA-2 network closest to the wind farms and INMET stations, is not shown here due to the confidentiality agreement signed with the wind farm operators.

With the wind speed data, the wind power estimations were made using different techniques and approaches. The theoretical power curve method used the information from the manufacturer presented in Table 2 to delineate the regions of zero, constant and variable generation in terms of the wind speed, as depicted in Equation (4).

The modeling of the nonlinear region was performed with the cubic power curve presented in Equation (5), with the parameter $\rho$ set to 1.16 kg/m³, as recommended by [46]. The $C_p$ values were adjusted empirically to enable the maximum power to be found at the nominal wind speed; the other parameters of the curve were assumed to be known and accurate to adjust [46]. Comparing the power outputs generated by the curves based on the wind speed time series with the historical power time series through the evaluation metrics, the results will be listed.

The results achieved with the traditional cubic power curve approach were used to develop the steps of the proposed method, which was tested in four approaches: single-period, monthly, hourly, and monthly–hourly.

In the first step of the proposed method, K-means clustering was performed only once with all data in the single-period approach. In the monthly approach, the clustering was carried out 12 times, each with the wind speed data for the respective month, while this was 24 times in the hourly approach and 288 times in the monthly–hourly case. In all cases, defining the number of clusters with the elbow method was first necessary. In the single-period approach, the number of clusters varied from 20 to 22, in the monthly case it ranged from 17 to 22, in the hourly case it ranged from 12 to 20, and in the monthly–hourly case it ranged from 3 to 14.

Table 4. Number of clusters obtained using each approach.

Wind Farm	Speed Treatment	Approaches of the Proposed Method
		Singular	Monthly	Hourly	Monthly–Hourly
1	EXT	23	18–22	12–20	3–12
	INMET	20	17–22	17–20	3–11
2	EXT	20	17–21	15–20	5–12
	INMET	20	18–22	16–20	5–12
3	EXT	21	18–22	14–20	6–12
	INMET	22	18–21	15–20	5–10
4	EXT	22	18–22	13–20	6–12
	INMET	21	17–22	13–19	5–12
5	EXT	22	18–22	12–19	5–12
	INMET	22	18–22	12–20	6–12

details the number of clusters used in each test. Irrespective of the wind farm and the steps adopted to construct the wind speed time series, the number of clusters defined with the elbow method for applying clustering was practically the same, only diminishing when the data segregation was increased to apply the new proposed method.

Figure 4 presents the result of applying the elbow method to Wind Farm 5 using a single period with extrapolated speed data (single-EXT) and a dispersion graph of the speed and power data. In the second graph, the different blue shades indicate the different wind speed clusters, permitting visualization of the variability in the power levels generated in each speed range. Corresponding graphs for the other farms and approaches are contained in the Supplementary Materials.

Then, KDE was applied to estimate the probability density function of the power values for each cluster resulting from the previous step. Figure 5 contains the PDF resulting from this step for each wind speed interval on Wind Farm 5 by the single-EXT approach. Note that as the wind speed values of the intervals increase, the peak of the density curve shifts toward the end of the interval. This behavior was seen for all the wind farms and approaches and was expected since faster wind speeds tend to generate more energy. Figures with the probability distributions of all the speed ranges for all the wind farms and approaches are shown in the Supplementary Materials.

It is important to mention that the KDE technique adopted in this work, based on [37,38], generates probability distributions that can infer negative power values. Other methods were tested, such as KDE Beta, but the results were worse, as observed by visualizing the graphs and the evaluation metrics. This highlights the great advantage of kernel density estimation, which adjusts according to the data and allows the proposal to be generalized regardless of the data used.

To emulate the variability in the behavior of a turbine in operation, Monte Carlo simulation was performed, which requires the wind speed time series and the density functions of the clusters. In applying the simulation for a single period, only the PDF utilized varied based on the generating wind speed cluster. At the same time, in the approaches with temporal segmentation, the choice of the PDF changed due to the wind speed and instant of time for estimating the wind power.

To measure the quality of the results, the metrics RMSE, MAE, MAPE and R² were used, compared to the active, historical and estimated power time series. To construct the estimates using the method proposed here, 100 power time series simulations were generated, and each time point’s average power was calculated. The number of scenarios adopted was found experimentally by ceasing to increase the number of scenarios when no further gains were obtained based on the evaluation metrics (only increases in computational cost).

Table 5. Results reached by the methods according to the evaluation metrics.

Metrics	Wind Farm	Speed Treatment	Methods
			Cubic Power Curve	Proposals
				Singular	Monthly	Hourly	Monthly–Hourly
RMSE	1	EXT	0.6012	0.2315	0.2210	0.2029	0.1772
		INMET	0.4489	0.2428	0.2310	0.2032	0.1771
	2	EXT	0.5971	0.2009	0.1892	0.1775	0.1538
		INMET	0.3224	0.2103	0.1942	0.1778	0.1538
	3	EXT	0.1624	0.1597	0.1546	0.1559	0.1454
		INMET	0.3439	0.1713	0.1640	0.1559	0.1455
	4	EXT	0.1590	0.1444	0.1385	0.1372	0.1249
		INMET	0.2804	0.1482	0.1416	0.1373	0.1248
	5	EXT	0.2590	0.1580	0.1537	0.1330	0.1245
		INMET	0.2767	0.1536	0.1494	0.1330	0.1247
MAE	1	EXT	0.4529	0.1883	0.1784	0.1578	0.1349
		INMET	0.3697	0.1998	0.1884	0.1581	0.1347
	2	EXT	0.4449	0.1615	0.1509	0.1382	0.1179
		INMET	0.2554	0.1707	0.1557	0.1385	0.1180
	3	EXT	0.1245	0.1245	0.1201	0.1206	0.1112
		INMET	0.2826	0.1343	0.1276	0.1206	0.1113
	4	EXT	0.1209	0.1118	0.1060	0.1050	0.0930
		INMET	0.2226	0.1142	0.1074	0.1051	0.0930
	5	EXT	0.2064	0.1230	0.1186	0.1018	0.0949
		INMET	0.2209	0.1177	0.1140	0.1019	0.0950
MAPE	1	EXT	0.4648	0.1932	0.1831	0.1620	0.1384
		INMET	0.3794	0.2050	0.1934	0.1623	0.1383
	2	EXT	0.4669	0.1695	0.1584	0.1451	0.1238
		INMET	0.2681	0.1792	0.1634	0.1453	0.1239
	3	EXT	0.1249	0.1249	0.1205	0.1210	0.1115
		INMET	0.2834	0.1347	0.1280	0.1210	0.1117
	4	EXT	0.1279	0.1183	0.1122	0.1112	0.0984
		INMET	0.2356	0.1209	0.1137	0.1113	0.0984
	5	EXT	0.2077	0.1238	0.1194	0.1025	0.0955
		INMET	0.2223	0.1184	0.1148	0.1025	0.0956
R2	1	EXT	−2.8184	0.4338	0.4840	0.5650	0.6682
		INMET	−1.1294	0.3769	0.4364	0.5637	0.6687
	2	EXT	−3.5560	0.4842	0.5426	0.5972	0.6979
		INMET	−0.3280	0.4348	0.5183	0.5961	0.6978
	3	EXT	0.4201	0.4387	0.4744	0.4657	0.5351
		INMET	−1.6019	0.3542	0.4083	0.4651	0.5343
	4	EXT	0.2889	0.4139	0.4603	0.4704	0.5616
		INMET	−1.2102	0.3825	0.4365	0.4696	0.5619
	5	EXT	−0.8726	0.3029	0.3405	0.5063	0.5673
		INMET	−1.1378	0.3416	0.3772	0.5061	0.5660

depicts the results obtained by the different estimation strategies, with the best result of each test highlighted in boldface. Note that irrespective of the evaluation metric, wind farm analysis or steps adopted to construct the wind speed time series employed in the test, the new proposed method always performed better than the cubic power curve, and the monthly–hourly strategy was the best in all cases.

Regarding the new proposed method, as the segmentation of the data for applying the three techniques increased, the simulations based on the metrics indicated in Table 5 improved, with only one exception. In the test of Wind Farm 3 using speed EXT data, the monthly approach performed better than the hourly approach, as identified by the four evaluation metrics.

Table 6. RMSEs attained by the methods per month.

Wind Farm	Methods	Speed Treatment	Months
			Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
1	Theoretical Curve	EXT	0.5298	0.4716	0.4596	0.4721	0.5240	0.5509	0.5497	0.6939	0.7611	0.7447	0.7119	0.6194
		INMET	0.3777	0.3204	0.3133	0.3549	0.4359	0.5188	0.5472	0.5663	0.5303	0.4645	0.4522	0.4353
	Singular	EXT	0.2376	0.2327	0.2439	0.2345	0.2206	0.2288	0.2212	0.2098	0.2087	0.2328	0.2518	0.2426
		INMET	0.2477	0.2405	0.2540	0.2391	0.2280	0.2369	0.2326	0.2261	0.2231	0.2476	0.2655	0.2586
	Monthly	EXT	0.2305	0.2145	0.2238	0.2238	0.2177	0.2165	0.1975	0.1842	0.1999	0.2320	0.2498	0.2398
		INMET	0.2399	0.2211	0.2283	0.2272	0.2246	0.2259	0.2091	0.1974	0.2102	0.2467	0.2623	0.2548
	Hourly	EXT	0.2152	0.2208	0.2360	0.2266	0.2104	0.2084	0.2026	0.1745	0.1582	0.1775	0.1917	0.1993
		INMET	0.2146	0.2215	0.2369	0.2270	0.2104	0.2081	0.2029	0.1755	0.1590	0.1775	0.1923	0.1998
	Monthly–Hourly	EXT	0.2027	0.1996	0.2028	0.2020	0.1938	0.1807	0.1608	0.1361	0.1382	0.1478	0.1602	0.1779
		INMET	0.2034	0.1985	0.2025	0.2027	0.1931	0.1809	0.1600	0.1361	0.1384	0.1463	0.1598	0.1785
2	Theoretical Curve	EXT	0.5111	0.4643	0.4494	0.4743	0.5428	0.5540	0.5620	0.7154	0.7735	0.7322	0.6850	0.5895
		INMET	0.2857	0.2441	0.2382	0.2716	0.3149	0.3645	0.3792	0.3812	0.3561	0.3374	0.3368	0.3241
	Singular	EXT	0.2023	0.2007	0.2150	0.2075	0.1971	0.1972	0.1950	0.2018	0.1882	0.1931	0.2084	0.2019
		INMET	0.2092	0.2182	0.2367	0.2283	0.2203	0.2133	0.2070	0.2076	0.1913	0.1924	0.2008	0.1998
	Monthly	EXT	0.1972	0.1810	0.1877	0.1869	0.1900	0.1891	0.1764	0.1763	0.1775	0.1921	0.2059	0.1998
		INMET	0.2007	0.1881	0.1946	0.2002	0.2107	0.2064	0.1899	0.1731	0.1744	0.1886	0.1978	0.1992
	Hourly	EXT	0.1817	0.1867	0.2045	0.1966	0.1897	0.1831	0.1818	0.1731	0.1488	0.1493	0.1606	0.1705
		INMET	0.1815	0.1880	0.2055	0.1974	0.1908	0.1824	0.1821	0.1726	0.1488	0.1501	0.1606	0.1699
	Monthly–Hourly	EXT	0.1742	0.1644	0.1725	0.1650	0.1676	0.1606	0.1486	0.1376	0.1290	0.1227	0.1366	0.1536
		INMET	0.1739	0.1648	0.1722	0.1654	0.1685	0.1608	0.1488	0.1375	0.1299	0.1233	0.1349	0.1533
3	Theoretical Curve	EXT	0.1368	0.1344	0.1285	0.1438	0.1588	0.1933	0.2139	0.1981	0.1851	0.1568	0.1390	0.1327
		INMET	0.2974	0.2875	0.2444	0.2534	0.2588	0.3278	0.3736	0.4292	0.4522	0.4296	0.3635	0.3277
	Singular	EXT	0.1352	0.1344	0.1287	0.1433	0.1595	0.1895	0.1987	0.1948	0.1834	0.1572	0.1385	0.1320
		INMET	0.1434	0.1430	0.1350	0.1464	0.1584	0.1884	0.1985	0.1975	0.2081	0.1912	0.1718	0.1560
	Monthly	EXT	0.1335	0.1321	0.1284	0.1420	0.1504	0.1796	0.1901	0.1926	0.1803	0.1472	0.1299	0.1284
		INMET	0.1417	0.1411	0.1298	0.1406	0.1441	0.1816	0.1957	0.1900	0.1951	0.1777	0.1633	0.1506
	Hourly	EXT	0.1296	0.1282	0.1198	0.1365	0.1527	0.1856	0.1950	0.1870	0.1808	0.1583	0.1421	0.1329
		INMET	0.1302	0.1278	0.1195	0.1361	0.1526	0.1857	0.1953	0.1868	0.1812	0.1588	0.1418	0.1334
	Monthly–Hourly	EXT	0.1254	0.1246	0.1179	0.1324	0.1369	0.1701	0.1819	0.1783	0.1754	0.1410	0.1186	0.1199
		INMET	0.1250	0.1247	0.1181	0.1336	0.1373	0.1695	0.1828	0.1787	0.1747	0.1413	0.1186	0.1199
4	Theoretical Curve	EXT	0.1206	0.1165	0.1170	0.1387	0.1640	0.2044	0.2350	0.2087	0.1767	0.1393	0.1139	0.1115
		INMET	0.2468	0.2352	0.1985	0.2008	0.2040	0.2573	0.2966	0.3551	0.3755	0.3557	0.2941	0.2683
	Singular	EXT	0.1224	0.1183	0.1123	0.1269	0.1411	0.1686	0.1852	0.1834	0.1692	0.1439	0.1186	0.1155
		INMET	0.1247	0.1219	0.1126	0.1218	0.1321	0.1583	0.1724	0.1750	0.1838	0.1711	0.1477	0.1360
	Monthly	EXT	0.1192	0.1149	0.1119	0.1255	0.1327	0.1573	0.1751	0.1809	0.1653	0.1337	0.1099	0.1101
		INMET	0.1233	0.1209	0.1101	0.1169	0.1198	0.1498	0.1673	0.1696	0.1744	0.1586	0.1397	0.1290
	Hourly	EXT	0.1178	0.1121	0.1024	0.1157	0.1280	0.1561	0.1677	0.1625	0.1646	0.1492	0.1299	0.1213
		INMET	0.1177	0.1120	0.1023	0.1159	0.1284	0.1562	0.1687	0.1622	0.1645	0.1494	0.1297	0.1212
	Monthly–Hourly	EXT	0.1114	0.1081	0.1013	0.1116	0.1114	0.1362	0.1520	0.1525	0.1582	0.1298	0.1023	0.1045
		INMET	0.1116	0.1084	0.1012	0.1116	0.1113	0.1363	0.1513	0.1528	0.1577	0.1297	0.1027	0.1044
5	Theoretical Curve	EXT	0.1956	0.1843	0.1849	0.2150	0.2515	0.3200	0.3568	0.3349	0.3237	0.2609	0.1957	0.1915
		INMET	0.2531	0.2460	0.2060	0.1990	0.2106	0.2541	0.2920	0.3539	0.3400	0.3365	0.2965	0.2707
	Singular	EXT	0.1470	0.1425	0.1320	0.1358	0.1459	0.1650	0.1838	0.1959	0.1867	0.1663	0.1408	0.1359
		INMET	0.1361	0.1326	0.1239	0.1277	0.1384	0.1562	0.1724	0.1878	0.1813	0.1726	0.1536	0.1420
	Monthly	EXT	0.1454	0.1397	0.1299	0.1324	0.1384	0.1576	0.1782	0.1928	0.1861	0.1609	0.1310	0.1316
		INMET	0.1352	0.1321	0.1210	0.1214	0.1286	0.1509	0.1702	0.1818	0.1791	0.1664	0.1474	0.1390
	Hourly	EXT	0.1267	0.1190	0.1111	0.1136	0.1243	0.1451	0.1521	0.1477	0.1474	0.1410	0.1325	0.1261
		INMET	0.1272	0.1193	0.1113	0.1138	0.1241	0.1446	0.1521	0.1482	0.1473	0.1406	0.1322	0.1261
	Monthly–Hourly	EXT	0.1224	0.1159	0.1088	0.1077	0.1122	0.1316	0.1448	0.1388	0.1428	0.1300	0.1139	0.1160
		INMET	0.1229	0.1158	0.1087	0.1089	0.1125	0.1312	0.1445	0.1391	0.1433	0.1297	0.1142	0.1164

The best result for each wind farm is shown in bold.

and

Table 7. RMSEs attained by the methods per hour.

Wind Farm	Methods	Speed Treatment	Hours
			1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
1	Theoretical Curve	EXT	0.8585	0.7680	0.6671	0.5637	0.4918	0.4500	0.4154	0.3856	0.3657	0.5635	0.7824	0.7139	0.5953	0.4851	0.3998	0.3363	0.3079	0.3244	0.3817	0.4410	0.7188	0.8698	0.8972	0.8661
		INMET	0.4374	0.4393	0.4426	0.4511	0.4559	0.4554	0.4679	0.5026	0.4989	0.4709	0.3822	0.2862	0.2212	0.2001	0.2161	0.2578	0.3207	0.4025	0.5509	0.6563	0.6324	0.5978	0.5377	0.4630
	Singular	EXT	0.2526	0.2448	0.2342	0.2249	0.2216	0.2178	0.2123	0.2201	0.2151	0.2073	0.2269	0.2340	0.2130	0.1861	0.1640	0.1522	0.1699	0.2133	0.3110	0.3477	0.2546	0.2450	0.2467	0.2501
		INMET	0.2622	0.2565	0.2440	0.2296	0.2242	0.2181	0.2098	0.2197	0.2167	0.2307	0.1703	0.1951	0.2354	0.2357	0.2132	0.1825	0.1708	0.2090	0.3194	0.3965	0.3024	0.2669	0.2483	0.2516
	Monthly	EXT	0.2290	0.2216	0.2138	0.2089	0.2098	0.2059	0.2022	0.2065	0.2016	0.1933	0.2181	0.2347	0.2196	0.1981	0.1769	0.1623	0.1684	0.2055	0.2966	0.3355	0.2504	0.2254	0.2224	0.2246
		INMET	0.2300	0.2287	0.2212	0.2128	0.2127	0.2069	0.1998	0.2043	0.2012	0.2106	0.1682	0.2017	0.2406	0.2431	0.2260	0.1937	0.1782	0.2010	0.2981	0.3729	0.2909	0.2510	0.2291	0.2240
	Hourly	EXT	0.2501	0.2435	0.2297	0.2218	0.2158	0.2128	0.2032	0.2021	0.1957	0.2044	0.1549	0.1319	0.1203	0.1167	0.1259	0.1401	0.1669	0.1958	0.2237	0.2371	0.2292	0.2398	0.2463	0.2471
		INMET	0.2495	0.2415	0.2302	0.2213	0.2170	0.2124	0.2034	0.2031	0.1968	0.2062	0.1553	0.1332	0.1188	0.1184	0.1265	0.1409	0.1676	0.1957	0.2252	0.2367	0.2312	0.2389	0.2461	0.2477
	Monthly–Hourly	EXT	0.2075	0.1999	0.1979	0.1948	0.1980	0.1960	0.1905	0.1836	0.1806	0.1635	0.1429	0.1270	0.1168	0.1135	0.1244	0.1369	0.1571	0.1783	0.1886	0.1808	0.1811	0.2020	0.2127	0.2130
		INMET	0.2064	0.1985	0.1975	0.1948	0.1984	0.1965	0.1904	0.1842	0.1787	0.1650	0.1411	0.1262	0.1166	0.1126	0.1227	0.1370	0.1593	0.1788	0.1896	0.1817	0.1826	0.2037	0.2114	0.2099
2	Theoretical Curve	EXT	0.8431	0.7459	0.6414	0.5348	0.4559	0.4094	0.3861	0.3728	0.3606	0.5546	0.7666	0.6952	0.5766	0.4657	0.3856	0.3273	0.2976	0.3145	0.3642	0.4365	0.7875	0.9266	0.9239	0.8679
		INMET	0.2562	0.2419	0.2759	0.3181	0.3559	0.4008	0.4365	0.4770	0.4874	0.4540	0.3340	0.2397	0.1788	0.1509	0.1545	0.1973	0.2459	0.3041	0.3744	0.3911	0.2730	0.2798	0.2423	0.3135
	Singular	EXT	0.1993	0.1923	0.1829	0.1781	0.1795	0.1839	0.1901	0.2091	0.2130	0.1986	0.2091	0.2208	0.2011	0.1742	0.1536	0.1499	0.1643	0.2034	0.2590	0.2690	0.2227	0.2143	0.2066	0.2006
		INMET	0.2187	0.2111	0.2001	0.1907	0.1875	0.1967	0.2152	0.2491	0.2651	0.2639	0.1755	0.1754	0.1906	0.1986	0.1906	0.1754	0.1734	0.1876	0.2233	0.2367	0.2234	0.2261	0.2131	0.2184
	Monthly	EXT	0.1813	0.1765	0.1715	0.1670	0.1690	0.1724	0.1769	0.1956	0.2004	0.1973	0.2091	0.2224	0.2078	0.1843	0.1642	0.1524	0.1562	0.1848	0.2398	0.2521	0.1963	0.1779	0.1736	0.1738
		INMET	0.1899	0.1893	0.1824	0.1764	0.1716	0.1795	0.1969	0.2288	0.2452	0.2391	0.1720	0.1805	0.1989	0.2063	0.2002	0.1808	0.1699	0.1792	0.2115	0.2190	0.1816	0.1779	0.1746	0.1804
	Hourly	EXT	0.1954	0.1869	0.1768	0.1711	0.1675	0.1715	0.1747	0.1863	0.1869	0.1967	0.1608	0.1378	0.1223	0.1175	0.1214	0.1434	0.1634	0.1852	0.2062	0.2143	0.2144	0.2087	0.2001	0.1939
		INMET	0.1942	0.1878	0.1776	0.1716	0.1677	0.1710	0.1754	0.1869	0.1882	0.1972	0.1605	0.1386	0.1220	0.1169	0.1211	0.1425	0.1628	0.1851	0.2074	0.2159	0.2141	0.2100	0.1996	0.1938
	Monthly–Hourly	EXT	0.1643	0.1578	0.1548	0.1541	0.1534	0.1587	0.1597	0.1710	0.1744	0.1684	0.1503	0.1334	0.1211	0.1148	0.1160	0.1336	0.1453	0.1587	0.1653	0.1573	0.1587	0.1666	0.1646	0.1650
		INMET	0.1627	0.1567	0.1557	0.1545	0.1519	0.1576	0.1610	0.1709	0.1739	0.1678	0.1500	0.1321	0.1218	0.1144	0.1172	0.1326	0.1467	0.1585	0.1663	0.1588	0.1586	0.1670	0.1664	0.1650
3	Theoretical Curve	EXT	0.1558	0.1580	0.1623	0.1659	0.1677	0.1715	0.1757	0.1831	0.1702	0.1411	0.1300	0.1377	0.1474	0.1539	0.1624	0.1662	0.1687	0.1701	0.1686	0.1698	0.1679	0.1676	0.1638	0.1606
		INMET	0.3368	0.3288	0.3262	0.3269	0.3258	0.3268	0.3156	0.2998	0.3060	0.3193	0.3316	0.3375	0.3397	0.3434	0.3512	0.3568	0.3620	0.3678	0.3688	0.3788	0.3847	0.3820	0.3672	0.3502
	Singular	EXT	0.1576	0.1597	0.1642	0.1666	0.1687	0.1723	0.1778	0.1854	0.1729	0.1438	0.1309	0.1341	0.1397	0.1458	0.1537	0.1580	0.1592	0.1605	0.1570	0.1622	0.1636	0.1654	0.1620	0.1605
		INMET	0.1657	0.1653	0.1697	0.1746	0.1755	0.1780	0.1756	0.1776	0.1716	0.1625	0.1553	0.1622	0.1652	0.1725	0.1790	0.1807	0.1790	0.1740	0.1658	0.1651	0.1731	0.1770	0.1744	0.1701
	Monthly	EXT	0.1457	0.1508	0.1556	0.1596	0.1605	0.1649	0.1673	0.1719	0.1602	0.1403	0.1353	0.1381	0.1453	0.1497	0.1573	0.1606	0.1601	0.1604	0.1553	0.1567	0.1556	0.1543	0.1503	0.1482
		INMET	0.1539	0.1551	0.1602	0.1654	0.1658	0.1676	0.1651	0.1631	0.1570	0.1547	0.1532	0.1645	0.1676	0.1706	0.1727	0.1739	0.1712	0.1665	0.1592	0.1640	0.1736	0.1683	0.1621	0.1580
	Hourly	EXT	0.1558	0.1582	0.1616	0.1653	0.1667	0.1704	0.1714	0.1712	0.1627	0.1390	0.1290	0.1332	0.1391	0.1455	0.1531	0.1557	0.1557	0.1548	0.1508	0.1535	0.1570	0.1596	0.1612	0.1605
		INMET	0.1562	0.1581	0.1622	0.1647	0.1666	0.1701	0.1714	0.1723	0.1638	0.1391	0.1289	0.1326	0.1401	0.1453	0.1525	0.1550	0.1549	0.1545	0.1508	0.1543	0.1575	0.1605	0.1614	0.1600
	Monthly–Hourly	EXT	0.1422	0.1447	0.1498	0.1556	0.1545	0.1596	0.1566	0.1513	0.1423	0.1329	0.1260	0.1302	0.1355	0.1413	0.1473	0.1508	0.1510	0.1494	0.1437	0.1441	0.1449	0.1439	0.1425	0.1436
		INMET	0.1436	0.1448	0.1503	0.1561	0.1552	0.1582	0.1576	0.1499	0.1429	0.1329	0.1262	0.1303	0.1370	0.1410	0.1476	0.1501	0.1503	0.1495	0.1441	0.1453	0.1444	0.1439	0.1417	0.1438
4	Theoretical Curve	EXT	0.1582	0.1613	0.1655	0.1686	0.1727	0.1771	0.1879	0.1928	0.1638	0.1270	0.1235	0.1324	0.1432	0.1470	0.1535	0.1529	0.1571	0.1612	0.1645	0.1643	0.1570	0.1559	0.1547	0.1557
		INMET	0.2518	0.2406	0.2374	0.2348	0.2316	0.2310	0.2298	0.2342	0.2546	0.2856	0.3045	0.3102	0.3116	0.3111	0.3127	0.3139	0.3121	0.3081	0.2988	0.3017	0.3058	0.3035	0.2871	0.2678
	Singular	EXT	0.1405	0.1410	0.1437	0.1453	0.1472	0.1503	0.1588	0.1632	0.1426	0.1274	0.1254	0.1270	0.1365	0.1403	0.1474	0.1495	0.1504	0.1494	0.1445	0.1477	0.1457	0.1483	0.1450	0.1419
		INMET	0.1380	0.1360	0.1384	0.1389	0.1383	0.1400	0.1456	0.1518	0.1446	0.1583	0.1559	0.1496	0.1478	0.1500	0.1552	0.1581	0.1587	0.1568	0.1499	0.1477	0.1476	0.1527	0.1495	0.1428
	Monthly	EXT	0.1251	0.1271	0.1319	0.1349	0.1357	0.1381	0.1462	0.1490	0.1287	0.1275	0.1312	0.1339	0.1414	0.1458	0.1527	0.1538	0.1527	0.1494	0.1414	0.1422	0.1367	0.1360	0.1301	0.1268
		INMET	0.1257	0.1239	0.1258	0.1272	0.1267	0.1276	0.1318	0.1373	0.1301	0.1544	0.1557	0.1519	0.1516	0.1500	0.1529	0.1525	0.1543	0.1511	0.1470	0.1485	0.1476	0.1454	0.1371	0.1300
	Hourly	EXT	0.1360	0.1347	0.1369	0.1372	0.1376	0.1381	0.1441	0.1482	0.1367	0.1216	0.1195	0.1229	0.1311	0.1355	0.1407	0.1417	0.1422	0.1420	0.1378	0.1396	0.1400	0.1440	0.1422	0.1393
		INMET	0.1359	0.1345	0.1369	0.1376	0.1374	0.1383	0.1433	0.1478	0.1375	0.1223	0.1194	0.1234	0.1308	0.1349	0.1412	0.1406	0.1423	0.1422	0.1379	0.1401	0.1401	0.1444	0.1432	0.1397
	Monthly–Hourly	EXT	0.1169	0.1165	0.1194	0.1227	0.1205	0.1213	0.1255	0.1255	0.1187	0.1178	0.1162	0.1203	0.1273	0.1302	0.1351	0.1363	0.1371	0.1357	0.1284	0.1289	0.1257	0.1256	0.1216	0.1193
		INMET	0.1170	0.1159	0.1205	0.1224	0.1200	0.1216	0.1252	0.1265	0.1183	0.1168	0.1154	0.1207	0.1271	0.1299	0.1357	0.1359	0.1375	0.1362	0.1288	0.1290	0.1253	0.1244	0.1218	0.1193
5	Theoretical Curve	EXT	0.2785	0.2799	0.2823	0.2850	0.2934	0.3019	0.3155	0.3085	0.2480	0.1740	0.1718	0.1869	0.1966	0.2023	0.2093	0.2177	0.2391	0.2772	0.2939	0.2866	0.2813	0.2689	0.2677	0.2706
		INMET	0.2036	0.1963	0.1951	0.1934	0.1902	0.1887	0.1941	0.2277	0.2922	0.3560	0.3822	0.3876	0.3869	0.3790	0.3668	0.3440	0.2932	0.2360	0.2177	0.2279	0.2408	0.2440	0.2301	0.2151
	Singular	EXT	0.1297	0.1320	0.1357	0.1360	0.1398	0.1425	0.1541	0.1636	0.1560	0.1761	0.1864	0.1956	0.2055	0.2115	0.2098	0.1959	0.1622	0.1356	0.1296	0.1271	0.1253	0.1234	0.1220	0.1245
		INMET	0.1112	0.1123	0.1113	0.1094	0.1090	0.1114	0.1244	0.1455	0.1667	0.2182	0.2197	0.2017	0.1907	0.1837	0.1789	0.1712	0.1615	0.1682	0.1646	0.1469	0.1248	0.1173	0.1132	0.1122
	Monthly	EXT	0.1229	0.1256	0.1310	0.1336	0.1356	0.1388	0.1475	0.1505	0.1434	0.1703	0.1844	0.1931	0.2024	0.2092	0.2065	0.1951	0.1587	0.1330	0.1280	0.1263	0.1231	0.1190	0.1157	0.1168
		INMET	0.1068	0.1087	0.1086	0.1071	0.1073	0.1093	0.1183	0.1329	0.1527	0.2105	0.2134	0.1972	0.1886	0.1823	0.1760	0.1673	0.1559	0.1619	0.1608	0.1483	0.1287	0.1161	0.1091	0.1074
	Hourly	EXT	0.1077	0.1065	0.1084	0.1073	0.1066	0.1071	0.1190	0.1435	0.1551	0.1506	0.1467	0.1529	0.1588	0.1648	0.1674	0.1657	0.1508	0.1317	0.1201	0.1190	0.1170	0.1160	0.1118	0.1086
		INMET	0.1074	0.1067	0.1083	0.1070	0.1066	0.1071	0.1188	0.1438	0.1553	0.1499	0.1468	0.1535	0.1584	0.1649	0.1686	0.1652	0.1508	0.1321	0.1197	0.1187	0.1170	0.1164	0.1121	0.1084
	Monthly–Hourly	EXT	0.0999	0.1000	0.1028	0.1018	0.1007	0.1017	0.1065	0.1189	0.1296	0.1408	0.1419	0.1470	0.1523	0.1586	0.1601	0.1583	0.1427	0.1249	0.1139	0.1134	0.1100	0.1090	0.1040	0.1001
		INMET	0.1005	0.1001	0.1026	0.1024	0.1002	0.1014	0.1066	0.1192	0.1298	0.1410	0.1417	0.1475	0.1542	0.1587	0.1590	0.1590	0.1436	0.1248	0.1138	0.1127	0.1104	0.1087	0.1042	0.1011

The best result for each wind farm is shown in bold.

detail the results of Table 5, where it is possible to see the behavior of the RMSE metric by month and hour in each test, respectively, and verify whether a trend exists other than that observed in Table 5. Again, the best performance was achieved by the monthly–hourly approach in all the wind farms in all months and hours. Nevertheless, in a few isolated months and hours, the theoretical power curve method performed better than the proposed technique, with the single-period and monthly approaches being better than the hourly approach. All these analyses were carried out using the metrics MAE, MAPE and R²; the respective tables are in the Supplementary Materials. The conclusions were the same as those with the RMSE.

The type of treatment applied to wind speed did not influence the performance of the modeling strategies presented here for the evaluation metrics. Mainly, it did not impair the identification of the best approach to estimate the wind power. Thus, as pointed out in a previous study [29], the wind speed time series data that only underwent interpolation and extrapolation (EXT) also achieved a good representation of the series measured on the farms; in some cases, this was better than the series that underwent the third step, bias correction (INMET-H). This might have happened because the quality of the correction is directly related to the quality and quantity of measurements supplied by the INMET station and its distance from the wind farm. It is not always possible to satisfy these factors. For example, the EXT series produced better performance at Wind Farms 1 and 2, where the maximum distance requirement between the farm and INMET station was not satisfied. At the other wind farms, the best performance between the two treatment options in the monthly case was alternated.

Figure 6 and Figure 7 compare the behavior of the wind farms’ historical data with the estimates resulting from each approach of the proposed method. The blue dots and lines in the graphs refer to historical data, while the black ones denote the estimated values. Figure 6 presents the dispersion graphs, evidencing the behavior of the power variable using the wind speed variable. From these graphs, it is possible to infer that the strategies described in this study managed to better replicate the inherent generation variability for the same speed values, with a highlight on the monthly–hourly proposal, which best captured the existing stochasticity. Figure 7 contains the frequency polygon graphs that estimate the density curve based on the frequencies of the power value intervals. Once again, it can be noted that the suggested strategies (single-period, monthly, hourly and monthly–hourly) replicate the power frequencies obtained from the historical data better than the cubic power curve.

5. Conclusions

Wind speed significantly influences wind power generation, although it is not the sole determining factor. The power curve traditionally represents the relationship between wind speed and power output, often employing the cubic approach. However, this method fails to fully capture the variability in this relationship and necessitates data, such as the air density, turbine height and efficiency factor, which may not always be readily available. Moreover, the theoretical power curve does not entirely reflect the dynamic regions of wind power generation.

To address this limitation, this paper developed a method that replicates wind generation variability based solely on wind speed, without the need for additional data. The approach leverages clustering, probability density function estimation and simulation, making it a probabilistic and nonparametric technique. A key advantage of this proposal is its reliance solely on historical wind power and speed data; however, one downside is its requirement for measured wind generation data, which may be unavailable or limited for new or recently completed wind farms.

This paper utilized historical active wind power data and MERRA-2 reanalysis wind speed data in a study involving five wind farms in Northeast Brazil. The monthly–hourly strategy outperformed the traditional cubic power curve approach.

This new power estimation technique accurately represents power generation variability concerning wind speed, facilitating enhanced project planning and understanding of wind power utilization. It improves data segregation and performance across evaluation criteria by effectively reflecting the probability distribution of historical power production and capturing wind power data characteristics and stochastic behavior.

When accurate wind speed data were challenging to acquire, the proposal turned to the MERRA-2 dataset as a dependable alternative. Two types of wind speed series from MERRA-2 were employed: one with interpolation and extrapolation (EXT) and the other with these steps and bias correction (INMET). Both types yielded favorable results in the four approaches considered. When the performances were compared, they alternated in producing the best results and sometimes even yielded the same results for each evaluation metric. This pattern persisted across all tests, irrespective of the evaluation metric, wind farms and analysis details.

Notably, wind speed data from the INMET station consistently indicated low values, potentially resulting in underestimation after the bias correction step. This impact was observed across all wind farms, including the recommended bias correction sites (Farms 3, 4, and 5). Conversely, the power curve method did not consistently demonstrate complete formation, often leaving the maximum turbine generation unspecified. However, the new method proposed in this article remains unaffected, as all power possibilities derive from occurring wind speeds, effectively addressing the underestimation.

Future endeavors include evaluating this method with other renewable energy sources to ascertain its efficacy, exploring additional clustering and PDF estimation approaches, and considering improved methods for segmenting the data to apply the new method. Furthermore, leveraging a multivariate modeling approach for evaluating other variables from MERRA-2 and wind speed to estimate wind generation warrants consideration.