Inverted Kumarswamy distribution for modeling the wind speed data: Lake Van, Turkey

Highlights

• A review of the literature is carried out, on studies employed probability distribution functions in modeling wind speeds.

• The Inverted Kumaraswamy (IKum) distribution is used for the first time in modeling wind speeds.

• Least squares and Maximum product of spacing methods are utilized for the first time in estimating parameters of the IKum.

• A Monte Carlo simulation study is conducted for comparing efficiencies of methods.

• For the first time, a comprehensive study on the modeling of wind speeds around Lake Van has been conducted.

Abstract

In calculating the wind energy potential of a region, some important points such as determining the distribution used to model wind speeds and estimating the parameters of the distribution accurately should be considered. Many different distributions have been proposed in wind energy literature over the years. In this paper, some of these studies are reviewed. Then, Inverted Kumaraswamy (IKum) distribution is used for the first time to model wind speed data as an alternative to the well-accepted Weibull distribution. Maximum Likelihood, Least Squares, and Maximum Product of Spacing methodologies are employed in estimating the parameters of the IKum distribution. A Monte Carlo simulation study is conducted for comparing the efficiencies of these methods. The wind speed data sets considered in this study include wind speeds from 6 stations located around Lake Van in Turkey. Modeling performances of the Weibull and IKum distributions are evaluated with the well-known goodness-of-fit criteria and power density error values. Results show that the IKum distribution can be considered as an alternative to the well-accepted Weibull distribution.

Introduction

The world's energy consumption is increasing day by day. Moreover, limited and unsustainable fossil fuels mostly supply energy needs. It is a known fact that the use of fossil fuels has adverse effects on the environment. Also, the use of fossil fuels is not sustainable, both environmentally and economically. Consequently, switching to alternative renewable energy sources can be considered as a necessity.

Renewable energy sources play an important role among all energy sources as they provide sustainable economic development by taking environmental factors into account [1]. Wind energy is a renewable energy source that contributes not only to satisfy the increasing energy demand but also to reduce the effects of climate change.

Wind power generation has snowballed over the past two decades. It has been used as an essential renewable energy source in electricity production in many developed countries [2]. According to the 2017 report of Global Wind Energy Council (GWEC), global wind power market remained above 50 GW (GW) in that year. Nine states have more than 10,000 MW (MW) of installed capacity, including China, the United States of America (USA), Germany, India, Spain, the United Kingdom (UK), France, Brazil, and Canada. The wind power installations continue to spread in the rest of the world. Turkey has also been increasing emphasis on renewable energy sources see; Fig. 1. In Turkey, there are currently more than 30 projects amounting to 800 MW, under construction. The Ministry of Energy and Natural Resources also announced the national target for wind power is set at 20 GW by 2023 [3]. However, many regions of Turkey have not been examined yet.

According to the 2017 report of the GWEC, the data concerning global installed wind power capacities for the top 10 countries are illustrated in Fig. 1.

Some crucial points such as characteristics of the wind speed regimes and wind capacity of the candidate region should be considered in planning the installation of wind power plants and using wind turbines effectively [4]. Accurate determination of wind energy potential requires detailed information about the wind characteristics of the candidate region [5]. A small error in wind speed modeling may result in a more significant error in calculating the energy output [6].

In wind energy studies, the Weibull distribution introduced by Walodi Weibull in 1951 [7] having a probability density function (pdf) with a scale parameter σ and a shape parameter α

$$f_{Weibull}\left(x;\sigma ,\alpha \right)=\alpha \sigma x^{\alpha -1}\exp \left[-\sigma x^{\alpha }\right];x>0,\sigma >0,\alpha >0,$$

is the most widely used distribution.

As it is known, skewness and kurtosis measures are functions of the shape parameter(s). If a distribution has only one shape parameter, skewness and kurtosis measures take values in a limited range for different values of the shape parameter. This result delimits the modeling capability of a distribution; see Fig. 4. In this context, the Weibull distribution has a deficiency in terms of the skewness and kurtosis measures, since it has only one shape parameter; see Rouse [8] for skewness and kurtosis measures of the Weibull distribution. Hence, some papers in the literature mentioned that the Weibull distribution might not adequately model all kinds of wind regimes. For example, Wais [4], Akgül et al. [6], Arslan et al. [9], and references therein stated that the Weibull distribution performs inadequate modeling for wind speed data having more skewness and/or kurtosis. Also, Takle and Brown [10] showed that problems may occur when using the Weibull distribution to model data including calm or very low wind speeds. Therefore, Takle and Brown [10] proposed to use hybrid density function in such a situation. Based on these conclusions, different distributions should be considered to model wind speed more efficiently and minimize errors in the wind power estimation.

The originality of this work comes from the fact that the three-parameter Inverted Kumaraswamy (IKum) distribution introduced by Al-Fattah et al. [11] is utilized to model wind speed data as an alternative to the Weibull distribution. The IKum distribution is obtained by transformation $X=\left(1/T\right)-1$to the random variable T that follows the Kumaraswamy distribution. The pdf and cdf of the IKum distribution are

$$f_{IKum}\left(x;\sigma ,\alpha ,\beta \right)=\sigma \alpha \beta {\left(1+\sigma x\right)}^{-\left(\alpha +1\right)}{[1-{\left(1+\sigma x\right)}^{-\alpha }]}^{\beta -1}$$

and

$$F_{IKum}\left(x;\sigma ,\alpha ,\beta \right)={[1-{\left(1+\sigma x\right)}^{-\alpha }]}^{\beta };x>0,\sigma ,\alpha ,\beta >0,$$

respectively. Here σ is the scale parameter, and α and β are the shape parameters.

The IKum distribution exhibits a longer right tail than several widely-used distributions, and this feature positively affects the distribution's ability to fit the rare events occurring in the right tail, i.e., outlying observations at the right tail. Besides, it is a flexible distribution with two shape parameters along with a scale parameter. Therefore, values of skewness and kurtosis measures of the IKum distribution take values on a surface rather than a line as the Weibull distribution; see Fig. 4. The skewness and kurtosis measures of the IKum distribution have a more extensive range than the Weibull distribution. This feature makes it attractive in modeling the data having different skewness and kurtosis values. It should also be noted that the Lomax (Pareto type-II), Beta type-II (inverted Beta), and Log-logistic (Fisk) distributions are sub-models of the IKum distribution. Several distributions such as the exponentiated Weibull, exponentiated Burr type-XII, Kumaraswamy Dagum, Kumaraswamy inverse Weibull distributions, and sub-models of them can be obtained by applying appropriate variable transformation on the IKum random variable. Details of the IKum distribution are not provided here for brevity; see Al-Fattah et al. [11] for further information. The Weibull distribution is also taken into consideration due to its common usage to make the study complete.

This study has the following contributions to the related literature. (i) The suggested distribution, the IKum, is being used for the first time in modeling the wind speed data. (ii) The parameters of the IKum distribution are estimated by the maximum likelihood (ML), least squares (LS), and maximum product of spacing (MPS) methods. The ML, LS, and MPS methods' efficiencies are compared for the IKum distribution via a Monte-Carlo (MC) simulation study. To the best of authors’ knowledge, there is no study concerning the other estimation methods except the ML and Bayesian in the related literature. (iii) Since there have been plans for establishing a wind power plant to Van, this is the first comprehensive study conducted in the considered region. This region is located in the highlands of eastern Anatolia, Turkey, at $N{38}^{\circ }38\text{'}$, $E{42}^{\circ }49\text{'}$. It includes the largest soda lake on Earth.

The wind speed data used in this study are taken from the Turkish State Meteorological Service and contains wind speeds obtained from 6 meteorological stations Van, Gevas, Tatvan, Ahlat, Ercis, and Muradiye around Lake Van in Turkey. Gevas, Ercis, and Muradiye are the districts of Van province. Tatvan and Ahlat are the districts of Bitlis province. Bitlis has tourism potential with its natural beauties, e.g., Mount Nemrut stratovolcano with Nemrut Crater Lake and Mount Suphan stratovolcano. Also, Van has the largest population with over one million in eastern Anatolia, Turkey. It has recently been announced that 50 textile factories will be established in Van to improve its industrialization.

Renewable energy sources play an essential role in serving natural beauties and sustainable industrialization. The accurate estimation of the wind energy potential in this region will be essential for future investments to be efficient. In this context, the outcomes of this study will guide the decision-makers during the installation of wind power plants in Bitlis and Van.