Abstract
Accurate wind speed modeling is critical in estimating wind energy potential for harnessing wind power effectively. The quality of wind speed assessment depends on the capability of chosen probability density function (PDF) to describe the measured wind speed frequency distribution. The objective of this study is to describe (model) wind speed characteristics using three mixture probability density functions Weibullextreme value distribution (GEV), Weibulllognormal, and GEVlognormal which were not tried before. Statistical parameters such as maximum error in the KolmogorovSmirnov test, root mean square error, Chisquare error, coefficient of determination, and power density error are considered as judgment criteria to assess the fitness of the probability density functions. Results indicate that WeibullGEV PDF is able to describe unimodal as well as bimodal wind distributions accurately whereas GEVlognormal PDF is able to describe familiar bellshaped unimodal distribution well. Results show that mixture probability functions are better alternatives to conventional Weibull, twocomponent mixture Weibull, gamma, and lognormal PDFs to describe wind speed characteristics.
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Background
Growing global population along with fast depleting reserves of fossil fuels is influencing researchers to search for clean and pollutionfree sources of energy such as solar, wind, and bioenergies. Wind energy is a never ending natural resource which has shown its great potential in combating climate change while ensuring clean and efficient energy. Further, rapid advances in wind turbine technology led to significant growth of wind power generation across the world. However, wind energy is more sensitive to variations with topography and wind patterns compared to solar energy. Wind energy can be harvested economically if the turbines are installed in a windy area and suitable turbine is properly selected. Wind speed forecasting is a critical factor in assessing wind energy potential and performance of wind energy conversion systems. Several probability density functions (PDFs) have been used in literature to describe wind speed characteristics which include Weibull, Rayleigh, bimodal Weibull, lognormal, gamma, etc.
Islam et al. [1] used twoparameter Weibull distribution function for wind speed forecasting and assessed wind energy potentiality at Kudat and Labuan, Malaysia. Celik [2] used Weibullrepresentative wind data instead of the measured data in timeseries format for estimating the wind energy and had shown that estimated wind energy is highly accurately. Celik [3] made statistical analysis of wind data at southern region of Turkey and summarized that Weibull model was better than Rayleigh model in fitting the measured data distributions. Akdag et al. [4] discussed the suitability of twoparameter Weibull wind speed distribution and the twocomponent mixture Weibull distribution (WWPDF) to estimate wind speed characteristics. Carta et al. [5] used WWPDF because it is able to represent heterogeneous wind regimes in which there was evidence of bimodality or bitangentiality or, simply, unimodality. Maximum likelihood and leastsquare methods were used to estimate WWPDF parameters. In [6], wind speed distributions were shown to be satisfactorily described with a lognormal function, and in [7], Weibull and lognormal distribution functions were used to fit wind speed distributions. Kiss and Imre [8] used Rayleigh, Weibull, and gamma distributions to model wind speeds both over land and sea. They found that generalized gamma distribution, which has independent shape parameters for both tails, provides an adequate and unified description almost everywhere. Generalized extreme value (GEV) distribution that combines the Gumbel, Frechet, and Weibull extreme value distributions were used to model extreme wind speeds [9–12]. In recent past, mixture distributions were used to estimate wind energy potential that are quite accurate in describing wind speed characteristics. Jaramillo and Borja [13] used mixture Weibull distribution (WW) to model bimodal wind speed frequency distribution. Akpinar et al. [14] used mixture normal and Weibull distribution (NW), which is a mixture of truncated normal distribution, and conventional Weibull distribution to model wind speeds. Tian Pau [15] employed mixture gamma and Weibull distribution (GW) which is a combination of gamma and Weibull distributions, and also mixture normal distribution (NN) which is a mixture function of twocomponent truncated normal distribution for wind speed modeling.
The objective of this study is to propose three new mixture distributions, viz., Weibulllognormal (WL), GEVlognormal (GEVL), and WeibullGEV (WGEV) for wind speed forecasting. Comparison of the proposed mixture distributions with existing distribution functions is done to demonstrate their suitability in describing wind speed characteristics.
The rest of this paper is organized as follows: wind distribution models and goodness of fit tests used in this paper are presented in the section ‘Methods’. Results derived from this study are discussed in ‘Results and discussions’ section. Details about the data used for the analysis are given in this section. Conclusions are presented in the ‘Conclusions’ section.
Methods
Significance
The most suitable wind turbine model which needs to be installed in a wind farm is selected by careful wind energy resource evaluation. Accurate evaluation could be done using best fit distribution model. Using inappropriate distribution models results in inaccurate estimation of wind turbine capacity factor and annual energy production which in turn leads to improper estimation of levelized production cost [13]. Hence, it is important to choose an accurate distribution model which closely mimics the wind speed distribution at a particular site.
Wind distribution models
Wind distribution modeling requires analysis of wind data over a number of years. To reduce the expenses and time required to process longterm wind speed data, it is desirable to use statistical distribution functions for describing the wind speed variations. The primary tools to describe wind speed characteristics are probability density functions. The parameters of probability distribution functions which describe windspeed frequency distribution are estimated using statistical data of a few years. Many PDFs have been proposed in recent past, but in present study Weibull, Lognormal, gamma, GEV, WWPDF, mixture gamma and Weibull distribution, mixture normal distribution, mixture normal and Weibull distribution, and three new mixture distributions, viz., Weibulllognormal, GEVlognormal, and WeibullGEV are used to describe wind speed characteristics. Parameters defining each distribution function are calculated using maximum likelihood method.
Weibull distribution
The Weibull function is commonly used for fitting measured wind speed probability distribution. Weibull distribution with two parameters is given by [1]:
Weibull PDF
Weibull cumulative distribution function (CDF):
Weibull shape and scale parameters are calculated using the maximum likelihood method [16] which is given by:
where v_{ i } is the wind speed in time step i and n is the number of data points. To evaluate (3) an iterative technique is used. Scale parameter is obtained by
Generalized extreme value distribution
GEV distribution is a flexible model that combines the Gumbel, Frechet and Weibull maximum extreme value distributions [11, 17]. GEV PDF is given by
GEV CDF [17] is given by
GEV parameters are calculated using the maximum likelihood method which maximizes the Logarithm of likelihood function given by
Lognormal distribution
Lognormal distribution is probability distribution of a random variable whose logarithm is normally distributed.
Lognormal PDF is given by [18, 19]
Lognormal CDF is written as [18]
where
Lognormal parameters λ and Φ estimated using maximum likelihood method which do not need an iterative procedure are given by [20]
Gamma distribution
The probability density function of gamma distribution is expressed using the below function [15]
The cumulative Gamma distribution function is given by [20]
Gamma distribution parameters are estimated using maximum likelihood method that maximizes the logarithm of likelihood function which is given by:
Twocomponent mixture Weibull distribution
The probability density function, which depends on five parameters (v; k_{1}, c_{1}, k_{2}, c_{2}, w) is given by [5]
The cumulative distribution function is given by [5]
Relevant likelihood function is
Mixture gamma and Weibull distribution
The probability density function and cumulative distribution function of the mixture gamma and Weibull distribution are given by [15]
Relevant likelihood function is
Mixture normal distribution
The probability density function of singly truncated normal distribution is given by [15]
where I(μ,σ) is the normalization factor that leads the integration of the truncated distribution to one is expressed as
The cumulative truncated normal distribution is given by
The mixture function of two component truncated normal distribution from the above can be written as [15]
The cumulative distribution function is given by
Relevant likelihood function to estimate the five parameters is
Mixture normal and Weibull distribution
The probability density function of the mixture distribution comprising of truncated normal and conventional Weibull is written as [15]
Its cumulative distribution function is given as
Relevant likelihood function to estimate the five parameters is
Mixture Weibull and GEV distribution
The probability density function of the mixture distribution comprising Weibull and GEV functions which is applied for the first time to model wind speed distribution is written as
Its cumulative distribution function is given as
Relevant likelihood function to estimate the six parameters is
Mixture Weibull and lognormal distribution
The probability density function of the mixture distribution comprising Weilbull and lognormal of functions which is applied for the first time to model wind speed distribution is written as
Its cumulative distribution function is given as
Relevant likelihood function to estimate the five parameters is
Mixture GEV and lognormal distribution
The probability density function of the mixture distribution comprising GEV and lognormal functions which is applied for the first time to model wind speed distribution is written as
Its cumulative distribution function is given as
Relevant likelihood function to estimate the five parameters is
Goodnessoffit tests
Goodnessoffit tests are used to measure the deviation between the predicted data using theoretical probability function and the observed data. In this paper five statistical errors are considered as judgment criteria to evaluate the fitness of PDFs.
KolmogorovSmirnov test
The first one is the KolmogorovSmirnov test (KS), which is defined as the maximum error in cumulative distribution functions [21].
where C(v) and O(v) are the cumulative distribution functions for wind speed not exceeding v calculated by distribution function and observed wind speed data respectively. Lesser KS value indicates better fitness of the PDF.
R^{2} test
R^{2} test is used widely for goodnessoffit comparisons and hypothesis testing because it quantifies the correlation between the observed cumulative probabilities and the predicted cumulative probabilities of a wind speed distribution. A larger value of R^{2} indicates a better fit of the model cumulative probabilities \widehat{F} to the observed cumulative probabilities F. R^{2} is defined as [22]:
where \overline{F}=\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}{\widehat{F}}_{i}} The estimated cumulative probabilities \widehat{F} are obtained from cumulative distribution functions (CDFs).
Chisquare error
Chisquare error is used to assess whether the observed probability differs from the predicted probability. Chisquare error is given by
Root mean squared error
Root mean squared error (RMSE) provides a termbyterm comparison of the actual deviation between observed probabilities anead predicted probabilities. A lower value of RMSE indicates a better distribution function model.
Power density error (PDE)
The relative error between the wind power density calculated from actual timeseries data and that from theoretical probability function is expressed as [23]
where PD_{ ts } is the wind power density calculated from actual timeseries data which is given by
where PD_{ tp } is the wind power density based on a theoretical probability density function fn(v) which is given by
Results and discussion
Wind speed data from four wind stations were used in evaluating different PDFs to assess their suitability. Wind speed data provided by National Data Buoy Center [24] at five stations 42056 (Yucatan Basin), 46012 (Half Moon Bay, 24NM South Southwest of San Francisco, CA), 46014 (PT Arena, 19NM North of Point Arena, CA), and 46054 (Santa Barbara W 38 NM West of Santa Barbara, CA) are used for wind speed analysis. Tenminute mean wind speed data recorded at 5 m above the sea level are used for present studies.
Wind data of over the period 2008 to 2010 is used for wind station 42056.
For station 46012, wind data over a period of 10 years (2001 to 2010) is used for statistical analysis.
Wind data of station 46014 over a period of three years (2008 to 2010) is analyzed for wind distribution modeling.
For wind station 46054 data over the period (1999 to 2000) is used for statistical analysis.
In the present study, suitability of the PDFs is assessed using goodnessoffit tests. All computational procedures are carried out in MATLAB software package. Computed parameter values of different PDFs used for all the four stations are presented in Table 1.
The mean and standard deviation of observed wind speed for Station 42056 are 6.91888 m/s and 2.56768 m/s, respectively. Wind frequency histogram resembles familiar bellshaped curve; hence, Weibull PDF fits the observed distribution well. The statistical parameters for fitness evaluation of PDFs currently analyzed are presented in Table 2. All the PDFs except lognormal, GEV, and gamma are able to describe the wind speed characteristics well which is evident from their small power density errors shown in Table 2. Considering KS error, χ^{2} error, RMSE and PDE, the distribution functions lognormal, GEV and Gamma have large errors indicating their inadequacy in modeling wind speeds. Results presented in Table 2 show clearly that proposed mixture GEVL PDF provided the best fit of observed wind speed distribution. From Figure 1, it is evident that GEVL distribution provides a close fit throughout the entire wind speed spectrum when compared to other distributions. The higher value of R^{2} and the lower values of KS error, RMSE and chisquare error indicate that proposed GEVL distribution is more accurate than other PDFs in modeling wind speeds of Station 42056.
Station 46012 has a mean and standard deviation of 5.59185 m/s and 3.13391 m/s, respectively, for the observed wind speed. Statistical errors, KS, R^{2}, χ^{2}, and RMSE given in Table 3 indicate that proposed mixture WGEV distribution provides best fit for the observed wind frequency distribution, which is closely followed by GW, WL, GEVL, and WW mixture distributions. Convention PDFs such as lognormal and gamma, over predicted wind speeds which are in the range of 2 to 5 m/s and 13 to 24 m/s, respectively. These PDFs have under predicted speeds between 5 to 11 m/s. Apart from WGEV and WL, other mixture PDFs and conventional PDFs over predicted wind speeds in the range of 3 to 5 m/s which are reflected by the overestimated predicted probabilities as depicted in Figure 2. Results indicate that mixture PDFs perform better compared to conventional single PDFs.
Mean and standard deviation of wind speed for Station 46014 are 6.19907 m/s and 3.71888 m/s, respectively. From Figure 3, it is seen that WGEV, WG, WW, and WN distributions are able to model wind speed characteristics better than other PDFs. All other distributions have either over or underpredicted wind speeds apart from these three. Results presented in Table 4 clearly show that, considering KS error, χ^{2}, and RMSE, GW PDF has the smallest error followed by WGEV, WW, and NW. If R^{2} error is considered, GW has a value very close to 1.0, confirming its superiority in performance followed by WGEV, WW, and NW distributions.
Wind regime of Station 46054 has bimodal distribution with mean and standard deviation of 8.1261 m/s and 4.2390 m/s, respectively. Compared to conventional single PDFs, mixture PDFs have performed well in modeling the wind speeds. Lognormal, gamma, Weibull, and GEV fared poorly in describing the wind characteristics compared to other mixture PDFs. As seen from Figure 4 and statistical parameters from Table 5, two component mixture Weibull distribution (WW) provided the best fit for the observed wind data, closely followed by proposed mixture function WGEV.
Figures 1 to 4 show that mixture PDFs fit much better than the conventional Weibull, lognormal, and gamma distributions. Proposed mixture distributions GEVL for Station 42056 and WGEV for Station 46012 have outperformed other existing mixture and conventional single distributions. For Station 46054, WW distribution provided a better fit than others, while WGEV being the close second best fit.
Conclusions
In the present article, a comparison of distribution models has been undertaken for describing wind regimes of four wind stations. Common conventional PDFs and mixture PDFs along with three proposed new mixture PDFs, viz., WGEV, GEVL, and WL are used for wind speed modeling. It is shown that conventional PDFs, such as Weibull, lognormal, and gamma, are inadequate; hence, mixture functions are used to model the observed wind speed distributions better. Though the superiority of proposed mixture functions in Station 46014 is not very significant, the proposed mixture distributions GEVL for Station 42056 and WGEV for Station 46012 have provided better fit of the empirical data than other existing mixture distributions. For Station 46054, both WW and WGEV are found more suitable for describing wind speed distributions than other distributions. The performance difference between WW and WGEV distributions is not significant for this station. Results show clearly that proposed mixture PDFs, WGEV and GEVL, provide viable alternative to other mixture PDFs in describing wind regimes. Mixture PDFs which include GEV are able to provide close fit, particularly for high speed ranges of the wind spectrums. This is critical for wind speed applications as wind power is proportional to the cube of wind speed. Hence, mixture combinations of GEV with other conventional distributions need to be tried out for further analysis.
Authors' information
RK is an assistant professor in the Electrical Engineering Department, Jawaharlal Nehru Technological University, Kakinada, India. His areas of interest include distribution system planning and distributed generation. SRR is an associate professor of the same university. His areas of interest include electric power distribution systems and power systems operation and control. SVLN is a professor of Computer Science and Engineering in the School of IT, Jawaharlal Nehru Technological University, Hyderabad, India. His areas of interests include real time power system operation and control, IT applications in power utility companies, web technologies, and egovernance. KMP is a postgraduate student of the Department of Electrical and Electronics Engineering at JNTU Kakinada, India. His areas of interest include probabilistic DG planning and energy systems.
Abbreviations
 α :

shape parameter of gamma distribution (dimensionless)
 β :

scale parameter of gamma distribution (m/s)
 χ ^{2} :

chisquare error
 δ:

scaleparameter of GEV distribution (m/s)
 Γ :

gamma function
 λ :

mean of natural logarithm of wind speed (m/s)
 μ :

mean of wind speed (m/s)
 Φ :

standard deviation of natural logarithm of wind speed (m/s)
 ρ :

air density (kg/m^{3})
 σ :

standard deviation of wind speed (m/s)
 ζ:

shapeparameter of GEV distribution (dimensionless)
 c :

scale parameter of Weibull (m/s)
 CDF:

cumulative distribution function
 e :

GEV PDF
 E :

GEV CDF
 erf :

error function
 \widehat{F} :

predicted cumulative probability at time stage i
 {\overline{F}}_{i} :

observed cumulative probability at time stage i
 f :

Weibull PDF
 F :

Weibull CDF
 ff :

twocomponent mixture Weibull PDF
 FF :

twocomponent mixture Weibull CDF
 fn :

theoretical PDF
 g :

gamma PDF
 G :

gamma CDF
 GEV:

generalized extreme value distribution
 GEVL:

mixture GEV and lognormal distribution
 GW:

mixture gamma and Weibull distribution
 h :

gammaWeibull PDF
 H :

gammaWeibull CDF
 k :

shape parameter of Weibull (dimensionless)
 KS:

KolmogorovSmirnov
 l :

location parameter
 ln :

lognormal PDF
 LN :

lognormal CDF
 LL :

logarithm of likelihood function
 MLM:

maximum likelihood method
 n :

number of data points
 NN:

mixture normal distribution
 NW:

mixture normal and Weibull distribution
 PDF:

probability density function
 q :

singly truncated normal PDF
 Q :

singly truncated normal CDF
 r :

normalnormal PDF
 R :

normalnormal CDF
 R ^{2} :

coefficient of determination
 RMSE:

root mean square error
 s :

normalWeibull PDF
 S :

normalWeibull CDF
 t :

WeibullGEV PDF
 T :

WeibullGEV CDF
 u :

Weibulllognormal PDF
 U :

Weibulllognormal CDF
 v _{i} :

wind speed in time stage i
 \overline{{v}^{3}} :

mean of wind speed cubes
 v :

GEVlognormal PDF
 V :

GEVlognormal CDF
 w :

weight of mix parameter (dimensionless)
 W:

Weibull distribution
 WW:

twocomponent Weibull distribution
 WL:

mixture Weibull and lognormal distribution
 WGEV:

mixture Weibull and GEV distribution.
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Acknowledgments
The authors are very much thankful to Sri GT Rao, professor in Electronics and Communication Engineering Department, G.V.P. College of Engineering, Visakhapatnam for performing the language corrections of the manuscript. RK would like to acknowledge Dr M Ramalinga Raju, professor, JNTU Kakinada and Sri KVSR Murthy, associate professor, G.V.P. College of Engineering for their encouragement and support.
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Authors' contributions
RK conceived the concept, performed the modeling, carried out the software simulation, and drafted the manuscript. KMP provided assistance for the software simulation and modeling. SRR checked the modeling analysis, verified the results, and helped in drafting the manuscript. SVLN reviewed the entire manuscript and offered technical help. All the authors read and approved the final manuscript.
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Kollu, R., Rayapudi, S.R., Narasimham, S. et al. Mixture probability distribution functions to model wind speed distributions. Int J Energy Environ Eng 3, 27 (2012). https://doi.org/10.1186/22516832327
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DOI: https://doi.org/10.1186/22516832327