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The Weibull probability density function (PDF) has mostly been used to fit wind speed distributions for wind energy applications. The goodness of fit of the results depends on the estimation method that was used and the wind type of the analyzed area. In this paper, a study on a particular area (Galicia) was performed to test the performance of several fitting methods. The goodness of fit was evaluated by well-known indicators that use the wind speed or the available wind power density. However, energy production must be a critical parameter in wind energy applications. Hence, a fitting method that accounts for the power density distribution is proposed. To highlight the usefulness of this method, indicators that use energy production values are also presented.

In recent years, the use of wind energy has been continuously growing, even at double-digit rates in several countries. Spain has the fourth largest wind capacity in the world rankings, with 21,674 MW installed in 2012 [^{2} in 2012, which is greater than the mean value of Spain (4.3 MW/100 km^{2}) and also greater than that in Denmark (9.2 MW/100 km^{2}), Germany (8.8 MW/100 km^{2}) and The Netherlands (5.5 MW/100 km^{2}).

In this context, characterizing the wind speed at a specific location or area is extremely important. This task is complex due to the random nature of the wind, which does not exactly follow any known statistical distribution [

Difficulty arises in choosing the best PDF that fits the wind speed distribution [

Weibull parameters are typically obtained using well-known estimation methods, e.g., maximum likelihood, and the goodness of the resulting fits are evaluated by several indicators, e.g., ^{2} [

To evaluate the proposed fitting method and the proposed fitness indicators, data from several weather stations distributed around Galicia (northwest Spain) were used in this study. As background work related with this paper, certain studies that have had similar objectives must be emphasized. Carta

The analyzed region in this paper is Galicia, a region in northwest Spain, which is located in one of the windiest areas of Europe [

Wind speed is typically measured at weather stations at a height of 10 m every 10 m. The resulting wind speed series can be represented as:

The basic representation of wind speed data is the histogram. The most common form of the histogram is obtained by splitting the range of data into equally sized bins, called classes. Each class is represented by the middle value of the bin. Therefore, each bin _{j}_{j}_{j}_{j}

Finally, the chosen bin width Δ

The most widely used PDF to fit wind data is the Weibull distribution, which is defined as [

The available power of the wind that crosses the rotor of a WTG is [_{w}(^{3} [_{p}

When evaluating the available energy at a wind site, the following function is used:

This function is called the wind power density distribution and represents the distribution of wind energy at different wind speeds per unit of time and rotor area (W/m^{2}). For a specific wind site, it can be obtained from

Therefore, the total wind power density _{w} is:

When a Weibull PDF is considered, the following equation can be used [_{w} values will be discussed in Section 5.2.

The energy production of a WTG can be obtained by means of its power curve, where the relationship between the wind speed and the delivered power is established, and can be expressed by the following (see _{ci} is the cut-in wind speed; _{co} is the cut-out wind speed; _{r} is the rated wind speed; _{r} is the rated power; and

Several expressions can be used to represent the non-linear part of the power curve _{p}_{,eq} is a constant equivalent to the power coefficient. Using the equation above, the relationship between the rated power and rated wind speed is

To evaluate the energy production, the distribution of the energy generated by a WTG at different wind speeds per unit of time and rotor area is considered, which is called the power density distribution ^{2} and is defined as:

Taking into account

The total power density

As a particular case, the power density _{w}, shown in _{p}_{,eq}, this complexity is primarily related with the power curve behavior at the rated wind speed _{r}.

The impact of Weibull parameters, _{w} and power density _{w} with respect to the Weibull parameters, _{w} values were obtained as

In conclusion, the power density uncertainty cannot be associated with any variation interval of

To characterize wind power curves, a database with WTG parameters [

The rated wind speed has a greater effect than the cut-in wind speed or cut-out wind speed from the point of view of energy production [

There are several ways to estimate a Weibull PDF to fit a wind speed distribution. The most widely used methods to calculate the Weibull parameters are the following (see

least square method (LSQM) or the graphical method [

maximum likelihood estimation (MLE) and modified maximum likelihood estimation (MMLE) [

moment method (MM) [

density power method (DPM) [

In all the methods used in this paper, the lower wind speeds (calms) were treated separately [

To determine if a theoretical PDF is suitable to describe the wind speed data, several indicators can be used. For each wind site and fitting method, the following indicators were considered:

The relative mean wind speed error (_{ν}_{m}) and the mean wind speed data (_{m}) are compared with the resulting mean wind speed (

The relative error of the available power density (_{E}_{w}):
_{w} is calculated from the measured data; and

The coefficient of determination of the wind speed distribution (^{2}):
_{rj}^{2} is applied to the distribution of the wind power density, the following indicator is obtained:
_{w} is the mean of the _{w}_{j})

Root mean square error (

The goodness of fit is better when the

Goodness of fit parameters related to hypothesis testing methods, such as the Chi-square or Anderson-Darling methods, were not used in this work because their values strongly depend on the number of data points, which makes it difficult to compare results from different wind sites.

The energy produced by WTGs should be taken into account when Weibull PDFs are used in wind energy applications. For this reason, a set of indicators are proposed in the following paragraphs to include the energy output of WTGs in indicator calculations, which is done to increase the independence of the results from those obtained supposing a particular WTG or a reduced set [

For the aforementioned purpose, a set of power curves is considered, which is defined by selecting different rated wind speeds using _{r} is defined between 10 m/s and 17 m/s, according to the values shown in Section 5.4 (see _{r} is the number of rated wind speeds used in the histogram. Therefore, at each wind site, a set of wind energy distributions is calculated using the different power curves defined by the rated wind speeds in

Relative error of power density (_{mE}_{l}_{r}_{l}

Coefficient of determination of the power density distribution
^{2} values calculated with the power density distribution at different rated wind speeds using the wind speed data and fitted PDF:
_{l}_{r}_{l}_{l}_{l}_{j}

Regarding the methods to estimate the Weibull parameters, despite the fact that the estimation of energy produced by WTGs is one of the primary objectives in evaluating wind sites, the only method that partially takes it into account is the power density method (PDM), which uses the available energy associated with a wind speed distribution. In this context, the method, named the PDEM, which considers the typical behavior of power curves, is proposed in this paper, which is accomplished using a method that minimizes the following index:
_{s}

The selection of _{s} in _{mE}_{s} values. As can be seen, at the selected _{s}, the mean error is at its minimum value. Therefore, a value of _{s} equal to 12 m/s was selected.

After choosing all the estimation methods (MLE, MMLE, LSQM, MM, PDM and PDEM), the methods were used to estimate the Weibull PDF parameters for each wind site. An example of the estimated PDF and distribution of the wind power density is plotted in

The Weibull parameters obtained for each wind site using the different fitting methods are shown in

Once the parameters of the Weibull distribution are obtained, the goodness of fit indicators is calculated. The results for each wind site are displayed in

To evaluate the fitting methods, all the fitness indicators should be taken into account because, for example, higher ^{2} values, when fitting wind, wind energy or energy, do not imply lower wind, wind energy or energy errors [

The following are the main conclusions regarding the fitting methods:

The proposed PDEM method exhibited the best behavior when the ^{2} values for wind power density and power density distributions were considered. In addition, this method has an acceptable behavior when the relative error of the wind power density and power density were taken into account. The overall behavior of the proposed method is extremely satisfactory.

MM exhibited the best behavior in wind distribution fitting according to the ^{2} and RMSE values [

PDM's behavior is satisfactory with respect to estimating the mean wind speed; however, the method failed when the wind power production was considered [

MLE and MMLE exhibited similar behavior [

LSQM, in terms of relative error, strongly depended on the wind site data, as can be seen in the high values shown by the STD of its errors (_{νm}_{Ew}_{mE}

In conclusion, the proposed PDEM is the most suitable method when the focus of the analysis is on the energy produced by WTGs. Nevertheless, other methods, particularly MM, exhibited satisfactory results in terms of energy fitness.

To evaluate the robustness of the proposed PDEM, the behaviour of the different fitting methods with an estimated wind speed at different hub heights is shown in

This paper presents an analysis of wind speed data based on using fitting curve methods to obtain the parameters of the Weibull PDF. The most widely used methods were selected for this analysis. Furthermore, a method, called PDEM, which takes into account the power density distribution and the typical performance of WTGs, is presented.

The results of the fitting methods in obtaining the Weibull parameters were evaluated by a set of indicators defined from wind speeds and wind power density distributions. Additionally, the indicators, _{mE}

As the primary conclusion, the proposed PDEM method exhibited the best results when the energy produced by the WTGs is considered. Furthermore, its result when all indicators are taken into account, are extremely satisfactory. Nevertheless, other methods, particularly MM, exhibited satisfactory results in terms of energy fitness. In this paper, wind speed data from weather stations from a specific region, northwest Spain (Galicia), were used.

The authors would like to thank the personnel at Sotavento Experimental Wind Park for their contribution and help with field experience and for their accessibility to the measurement data. This work was supported in part by the Consellería of Innovación e Industria xunta de Galicia, Spain) under contract 07REM008V19PR and the Ministry of Science and Innovation (Spain; under contracts ENE 2007-67473 and ENE 2009-13074.

The authors declare no conflict of interest.

In the following paragraphs, the most common methods to obtain the scale and form parameters for a Weibull PDF are described.

The Weibull parameters are those that maximize their joint probability of occurrence and can be obtained by solving the following:
_{i}

Using data from a histogram, the modified maximum likelihood method (MMLM) results [_{j}

The LSQM, also known as the Weibull plot, is based on logarithmic transformations applied to the Weibull cumulative distribution function

The Weibull parameters can be obtained from:

Finally, the line parameters are obtained from:
_{m}_{m}_{i}_{i}

The LSQM is extremely popular due to its simplicity; however, the logarithmic transformations used during the calculation tend to cause some inaccuracy [

The MM is based on obtaining the Weibull parameters using certain statistical moments calculated using wind speed data. When the mean wind speed (_{m}

The shape parameter

The comparison between the approximate solution using

The PDM uses the Energy Pattern Factor [

This factor relates to the shape parameter

Using the NR method to solve this equation, the following expressions are obtained:

The following approximated expression can be used, which assumes that _{pf} is typically between 1.45 and 4.4:

Both results from using

In order to evaluate the robustness of the proposed PDEM method, it has been evaluated using wind speeds at different hub heights. For this purpose, wind speeds have been estimated using a logaritmic wind profile with a roughness length of 0.05 m, which is a common value in wind farms [_{νm}_{Ew}_{mE}^{2}_{ew}^{2}_{me}^{2}) for heights between 10 m and 150 m can be seen in

Meteogalicia weather stations, where the size of each circle is proportional to the measured mean wind speed, and the number inside each circle is the site number. The circles at the top left portion of the figure indicate the scale.

Wind power curve.

Power density _{r} of 11 m/s and 15 m/s) and wind power density _{w} for different Weibull parameters.

Distribution of the cut-in, rated and cut-out wind speeds for wind turbine generators (WTGs).

Candlestick chart with the power density at the considered wind sites for different values of rated, cut-in and cut-out wind speeds.

Evolution of the mean and standard deviation of the relative error _{mE}_{s}.

Histogram derived from the estimated Weibull probability density function (PDF) compared with the histogram of the wind speed data (bar diagram) at wind site n° 21 (O Cebreiro). MLE: maximum likelihood method; MMLE: modified maximum likelihood estimation; LSQM: least square method; MM: moment method; PDM: power density method; and PDEM: part density energy method.

Distribution of the wind power density derived from the estimated Weibull PDF compared with that derived from the wind speed data (bar diagram) at wind site n° 21 (O Cebreiro).

Representation of the mean wind speed calculated from data, scale and shape parameters for all wind sites using the different estimation methods.

Representation of the _{νm}

Representation of the _{Ew}

Representation of ^{2} at the different wind sites.

Representation of

Representation of root mean square error (

Representation of each component of _{mE}_{mE}

Representation of each component of

Shape parameter estimated with MM and density power method (DPM).

Mean values of the indicators of the fitness (_{νm}_{Ew}_{mE}^{2}_{ew}^{2}_{me}^{2}) at different hub heights.

Meteorological stations (No.: wind site number; MWS: mean wind speed in m/s; and ND: amount of data in years).

1 | P.E. Sotavento 20 m | 5.5 | 6.6 | 16 | LU Fragavella | 4.7 | 4.1 |

2 | P.E. Sotavento 40 m | 6.1 | 6.6 | 17 | LU Guitiriz | 4.1 | 9.7 |

3 | CO A Gandara | 6.8 | 1.1 | 18 | LU O Cebreiro | 4.4 | 1.1 |

4 | CO Aldea Nova | 3.8 | 1.3 | 19 | OU Alto de Rodicio | 4.5 | 5.6 |

5 | CO Corrubedo | 4.1 | 9.1 | 20 | OU Cabeza de Manzaneda | 6.4 | 3.2 |

6 | CO Corunha Dique | 4.9 | 2.8 | 21 | OU Lardeira | 5.4 | 3.0 |

7 | CO Lira | 5.8 | 0.4 | 22 | OU Serra do Eixe | 4.3 | 2.4 |

8 | CO Malpica | 6.5 | 4.6 | 23 | OU Xares | 4.4 | 2.9 |

9 | CO Marco da Curra | 5.3 | 6.4 | 24 | PO Castro Vicaludo | 6.1 | 6.3 |

10 | CO Muralla | 6.9 | 3.8 | 25 | PO Coron | 4.8 | 8.1 |

11 | CO Punta Candieira | 8.0 | 4.4 | 26 | PO Fornelos de Montes | 4.7 | 7.1 |

12 | CO Rio do Sol | 6.1 | 1.5 | 27 | PO O Viso | 3.7 | 2.3 |

13 | CO Salvora | 5.7 | 4.0 | 28 | PO Ons | 5.5 | 5.1 |

14 | LU Ancares | 4.9 | 8.4 | 29 | PO Serra do Faro | 6.2 | 3.9 |

15 | LU Burela | 5.5 | 3.6 | - | - | - | - |

Weibull PDF parameters: scale parameter

| ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 6.3 | 2.1 | 6.3 | 2.1 | 5.5 | 1.7 | 6.3 | 2.0 | 6.3 | 2.0 | 5.9 | 1.7 |

2 | 7.0 | 2.1 | 7.0 | 2.1 | 6.3 | 1.8 | 7.0 | 2.1 | 7.0 | 2.0 | 6.6 | 1.8 |

3 | 7.7 | 2.1 | 7.7 | 2.1 | 7.2 | 1.7 | 7.6 | 2.0 | 7.6 | 1.9 | 7.5 | 1.9 |

4 | 4.4 | 1.8 | 4.4 | 1.8 | 3.4 | 1.4 | 4.4 | 1.7 | 4.4 | 1.7 | 4.5 | 1.9 |

5 | 4.8 | 1.6 | 4.8 | 1.6 | 3.8 | 1.2 | 4.8 | 1.5 | 4.8 | 1.5 | 4.5 | 1.4 |

6 | 5.7 | 1.9 | 5.7 | 1.9 | 4.8 | 1.6 | 5.7 | 1.8 | 5.7 | 1.8 | 5.6 | 1.7 |

7 | 6.5 | 1.6 | 6.5 | 1.6 | 5.4 | 1.3 | 6.5 | 1.6 | 6.5 | 1.6 | 6.5 | 1.6 |

8 | 7.4 | 1.9 | 7.4 | 1.9 | 6.4 | 1.7 | 7.4 | 1.9 | 7.4 | 1.9 | 7.2 | 1.8 |

9 | 6.1 | 2.0 | 6.1 | 2.0 | 5.3 | 1.6 | 6.1 | 1.9 | 6.1 | 1.9 | 5.7 | 1.6 |

10 | 7.9 | 1.9 | 7.9 | 1.9 | 6.8 | 1.7 | 7.8 | 1.9 | 7.8 | 1.9 | 7.7 | 1.8 |

11 | 9.2 | 1.7 | 9.2 | 1.7 | 7.8 | 1.5 | 9.2 | 1.7 | 9.2 | 1.8 | 8.9 | 1.7 |

12 | 6.9 | 2.3 | 6.9 | 2.3 | 6.4 | 2.0 | 6.9 | 2.3 | 6.9 | 2.2 | 6.7 | 2.0 |

13 | 6.6 | 1.9 | 6.6 | 1.9 | 5.7 | 1.6 | 6.5 | 1.9 | 6.5 | 1.9 | 6.3 | 1.7 |

14 | 5.8 | 1.6 | 5.8 | 1.6 | 4.7 | 1.3 | 5.7 | 1.5 | 5.7 | 1.5 | 5.3 | 1.4 |

15 | 6.3 | 1.9 | 6.3 | 1.9 | 5.6 | 1.5 | 6.3 | 1.8 | 6.3 | 1.7 | 5.7 | 1.5 |

16 | 5.6 | 1.6 | 5.6 | 1.6 | 4.6 | 1.3 | 5.5 | 1.5 | 5.5 | 1.5 | 5.0 | 1.4 |

17 | 4.9 | 1.9 | 4.8 | 1.9 | 3.9 | 1.6 | 4.8 | 1.9 | 4.8 | 1.9 | 4.6 | 1.7 |

18 | 5.2 | 1.9 | 5.2 | 1.9 | 4.3 | 1.6 | 5.2 | 1.9 | 5.2 | 1.9 | 4.8 | 1.6 |

19 | 5.4 | 1.8 | 5.4 | 1.8 | 4.5 | 1.5 | 5.4 | 1.8 | 5.4 | 1.7 | 5.0 | 1.6 |

20 | 7.2 | 1.7 | 7.2 | 1.7 | 6.4 | 1.5 | 7.2 | 1.6 | 7.2 | 1.6 | 6.8 | 1.5 |

21 | 6.1 | 1.5 | 6.1 | 1.5 | 5.1 | 1.2 | 6.0 | 1.3 | 6.0 | 1.3 | 5.6 | 1.2 |

22 | 5.0 | 1.8 | 5.0 | 1.8 | 4.1 | 1.3 | 4.9 | 1.7 | 4.9 | 1.6 | 4.3 | 1.3 |

23 | 6.4 | 1.7 | 6.4 | 1.7 | 5.5 | 1.4 | 6.3 | 1.6 | 6.3 | 1.5 | 5.0 | 1.3 |

24 | 7.0 | 1.6 | 7.0 | 1.6 | 6.0 | 1.3 | 7.0 | 1.5 | 7.0 | 1.5 | 6.7 | 1.4 |

25 | 5.7 | 1.7 | 5.7 | 1.7 | 4.5 | 1.4 | 5.7 | 1.7 | 5.7 | 1.7 | 5.9 | 1.9 |

26 | 5.5 | 1.7 | 5.5 | 1.7 | 4.5 | 1.3 | 5.5 | 1.6 | 5.5 | 1.6 | 5.2 | 1.5 |

27 | 4.2 | 1.6 | 4.2 | 1.6 | 3.2 | 1.1 | 4.2 | 1.5 | 4.2 | 1.4 | 3.7 | 1.2 |

28 | 6.6 | 2.0 | 6.6 | 2.0 | 5.7 | 1.7 | 6.6 | 2.0 | 6.6 | 1.9 | 6.3 | 1.8 |

29 | 7.1 | 2.1 | 7.1 | 2.1 | 6.2 | 1.8 | 7.1 | 2.1 | 7.1 | 2.0 | 6.9 | 1.9 |

Summary of the results (best values of the mean and standard deviation (STD) for each fitness indicator are displayed in bold).

_{νm} |
Mean | 2.0% | 2.0% | −9.7% | 1.9% | 2.1% | |

STD | 1.2% | 2.5% | 1.3% | 1.4% | 3.0% | ||

| |||||||

_{Ew} |
Mean | −2.9% | −2.9% | −17.4% | 2.3% | −0.6% | |

STD | 3.8% | 3.8% | 5.7% | 2.5% | 4.2% | ||

| |||||||

_{mE} |
Mean | 2.7% | 2.7% | −16.2% | 3.7% | 5.4% | |

STD | 2.5% | 7.7% | 2.9% | 4.1% | 4.9% | ||

| |||||||

^{2} |
Mean | 0.976 | 0.976 | 0.931 | 0.976 | 0.957 | |

STD | 0.012 | 0.013 | 0.019 | 0.014 | 0.038 | ||

| |||||||

^{2}_{ew} |
Mean | 0.922 | 0.922 | 0.911 | 0.938 | 0.944 | |

STD | 0.074 | 0.074 | 0.055 | 0.059 | 0.057 | ||

| |||||||

^{2}_{me} |
Mean | 0.952 | 0.952 | 0.930 | 0.959 | 0.959 | |

STD | 0.039 | 0.039 | 0.053 | 0.035 | 0.038 | ||

| |||||||

Mean | 0.3% | 0.3% | 0.6% | 0.3% | 0.4% | ||

STD | 0.1% | 0.1% | 0.1% | 0.1% | 0.2% |