Pearson System Distribution Approximation in Wind Energy Potential Analysis

Transcription

1 International Conference on Future Environment and Energy IPCBEE vol.8() ()ICSIT Press, Singapoore Pearson System Distriution pproximation in Wind Energy Potential nalysis E.. zrulhisham, K.P. Zakaria,. Samizee and M.B.M. Juhari Malaysia France Institute, Universiti Kuala Lumpur, Malaysia stract. Wind energy is considered as one of the solution to the worldwide depletion of fossil fuel resources as well as the economic alternatives in protecting the atmosphere from the adverse consequences of gloal warming. Introducing of wind machine at a particular site however requires the knowledge of the distriution of the wind speed. The work presented in this paper considers the possiility of using Pearson system in approximation of wind speed distriution using the northern Borneo island wind speed data. Considering the stochastic properties of the wind speed, the density function approximation was otained y considering the first to fourth statistical moments. The wind energy potential was evaluated ased on the Pearson system frequency curve. pplication of Pearson system has advantages in capaility to take on variety shapes of distriution which makes it particularly applicale to the analysis of wind energy potential characterized y random wind speed data. Keywords: Wind energy, Wind speed distriution, Pearson system, Density function approximation. Introduction The threats of climate change caused y urning fossil fuels as well as escalating fossil fuel prices make the further rapid development of renewale energy sources a gloal imperative. Energy provided y the winds is considered as one of the renewale energy alternatives that meet the needs of modern societies in reducing the dependence on fossil fuels whilst at the same time delivering sustantial reduction in greenhouse gas emissions. However introduction of winds as an alternative source of energy is often criticised for the reason that the power provided from the wind is intermittent and virtually uncontrollale. Considering the stochastic properties of wind speed, the assessment of power potential and the economic feasiility elonging to a particular site of wind machine are largely relies on knowledge of wind speed distriution []. Conventional statistical distriutions used in fitting the measured wind speed data are generally ased on the mean-variance analysis ut it has een well suggested y numers of research that the oserved performance distriutions are not always fully captured y the first two moments of the distriution []. The motivation for modelling skewness and kurtosis in the wind power production system has followed from attempts to understand the ehaviour of a wide range of wind speed data ased on various statistical moments. Cheng et al. [], among others, have proposed that production costing and reliaility of a wind power system is related to dynamic changes of skewness and kurtosis coefficients of the wind speed distriution. Given the needs in handling variety shapes of wind speed distriution, it has ecome interesting to propose frameworks that are flexile enough to accommodate distriutions with road range of properties. This paper is aimed to provide further insight into the wind energy potential analysis ased on the wind speed distriution selected from the Pearson family. The Pearson system is a parametric family of continuous proaility distriutions capale to model a wide scale of distriutions with various skewness and kurtosis []. Firstly introduced y Karl Pearson in 895 as an effort to model visily skewed distriution, the Pearson family of distriution is amendale to oth theoretical and empirical prolems where density function must e explicitly expressed. 6

2 In this study, descriptive statistics of northern Borneo wind speed data was otained from the meteorological data provided y Malaysian Meteorological Department (MMD) []. Data measured y utomatic Weather Stations located in Kota Kinaalu Saah was selected in view of the low and unpredictale wind speed characterized y various monsoon of equatorial region. The density function approximation of the wind speed data was otained using the Pearson system employing the first to fourth statistical moments. The wind power produced y the proailistic mean wind speed was then evaluated considering the gearless lade tip wind turine system.. Proailistic Wind Energy Model.. The Pearson System The Pearson system is a parametric family of distriutions used to model a road scale of distriutions with various third and fourth moments. This method of moments is a statistical technique to estimate proaility distriutions y equating their theoretical moments with the moments of empirical distriutions. Pearson distriutions are defined y a separale first order differential equation of f '( P( = = x( Q( x a + x + x () where f is a density function and a,, and are the parameters of the distriution. n important characteristic of the Pearson system is the direct correspondence etween the parameters and the first four moments (µ,..., µ ) of the distriution [5]. The expressions for the moments are otained y solving Equation using integration y part. The expressions for the parameters a,, and in terms of the first four moments can e otained from the set of Equation (). μ = a / ( μ + μ ) μ β ( β + ) ( μ μ μ 6μ ) ( β β 6) μ ( μ μ μ ) μ ( β β ) () where the two moment ratios β = μ / μ and / μ β = μ respectively denote skewness and kurtosis. The scaling parameters and are otained from Classification and selection of the distriution are achieved y finding the roots of the denominator of quadratic Equation and the Pearson s coefficient given y β = μμ 8μ μ = β 8 K = () Depending on the values of these parameters, different types of Pearson curves can e otained such that.if K <, roots are real and of opposite signs. This corresponds to eta distriution or type-i distriution in the Pearson system. 6

3 .If K >, roots are real and have the same sign. This corresponds to eta distriution of the second kind, or type-vi distriution..if < K <, roots are complex. This corresponds to type IV-distriution. Pearson proposed further class distinctions y taking into account certain distriutions and oundaries etween classes and classified the solutions into types numered to []... Descriptive Statistics of Wind Speed Data Located on the west coast of Saah on the northern portion of the island of Borneo, the data sets of this study consist of daily wind speed measured in meter per second. Measured y the MMD at the unmanned weather oservation station situated in Kota Kinaalu, the data consists of monthly wind speed for the year of 9. The descriptive statistics for the data set is shown in Tale. Jan Fe Mar pr May June July ug Sept Oct Nov Dec Mean Std. Dev Skewness Kurtosis Sample Size Tale : Monthly Descriptive Statistics Kota Kinaalu features a tropical rainforest climate which characterized y the Northeast and the Southwest Monsoon [6]. The Northeast Monsoon occurs etween Novemer and March while the Southwest Monsoon occurs etween May and Septemer. s shown in Tale the mean wind speed recorded during the Southwest Monsoon is higher than mean wind speed recorded during the Northeast Monsoon. The descriptive statistics also indicated that the highest mean wind speed was recorded in Septemer which is the transition period from the Southwest to Northeast Monsoon... Density Function pproximation Potential indices of a particular site for a wind energy project such as the mean wind power are typically expressed as single numerical values. These numerical indices provide the expected values of the distriutions that fully descrie the particular indicator eing expressed. The expected value of a distriution is one of its several parameters, which provide a complete mathematical description of the distriution function. This mean value alone does not, however, provide any information on the variaility of the index eing descried. Therefore, a potential wind energy index of a particular site is est represented as a proaility distriution, or proaility density function, []. In this study, proaility distriution of the wind speed was estimated using the Pearson system. Based on the first through fourth statistical moments, the type of the Pearson system was determined y Equation and the quadratic roots which represent the Pearson s criterion for fixing the distriution family as shown in Tale. Jan Fe Mar pr May June July ug Sept Oct Nov Dec Coeff. K a a Tale : Pearson s Criterion for Fixing the Distriution Family Taking into account that the Pearson s coefficient K < and the roots are real and opposite sign the distriution is classified as eta distriution or Pearson s type. The frequency curve for the type Pearson s family of distriution for the wind speed is represented y: 6

4 f ( = k + x a m where k is the normalization constant. Integrating the frequency curve the values of percentage points in the distriution can represented y the cumulative density function,. Figure shows and of the wind speed approximated using the Pearson system. Referring to cumulative density function, the proailistic mean of the wind speed is represented in Tale. x a m ().5..5 Jan Fe Mar pr Jan Fe Mar pr May Jun July.9 ug May Jun July ug Sept Oct Nov Dec Sept Oct Nov Dec 5 6 Figure : and of the wind speed Considering the gearless lade tip wind turine with the lade diameter, D of.8 meter and the proailistic mean wind speed, v the availale power in the wind y was evaluated y the equation of D P Wind = ρ π v (5) where ρ represents the density of the air approximated at.kg/m. Figure shows the mean of availale power in the wind for the months periods. 6

5 Jan Fe Mar pr May June July ug Sept Oct Nov Dec Power Tale : Proailistic mean of the wind speed.5 Wind Power, (watt) Months, 9 Figure : vailale power in the wind. Conclusion Conventional statistical distriutions used in fitting the measured wind speed are largely relies on the mean-variance analysis assuming a symmetric distriution. However it has een well suggested y numers of research that the oserved performance distriutions are not always fully captured y the first two moments of the distriution. The use of the Pearson curve in approximating the proaility distriution of wind speed has the advantages in terms of capaility to take on variety shapes of distriution which makes it particularly applicale to the analysis of wind energy potential characterized y random wind speed data.. cknowledgements The author would like to thanks Universiti Kuala Lumpur for providing funds and necessary support for this research. 5. References [] H. Cheng, Y. Hou and F. Wu. Proailistic wind power generation model: Derivation and applications. International Journal of Energy. 5, 5(): 7-6. []. ndreev,. Kanto and P. Malo. Computational examples of a new method for distriution selection in the Pearson system. Journal of pplied Statistics. 7, (): [] E.. zrulhisham, K.P. Zakaria and M. Muhammad.Pearson curves density function approximation for power production reliaility evaluation of GDC turine. Proc. of Int. Conf. on Production, Energy & Reliaility. Kuala Lumpur: Univ. Teknologi Petronas.. []. lani, M.Z. Irahim and K.H. Yong. Investigation on wind energy potential at Saah State of Malaysia. Proc. of Univ. Malaysia Terengganu Int. nnual Symposium. Terengganu: Univ. Malaysia Terengganu., pp [5]. Stuart and K. Ord. Kendall's dvanced Theory of Statistics, Volume : Distriution Theory. 6th Edition. Edward rnold, 99. [6] N.. dullah, S.H. Shuhaimi, T.Y. Ying,.H. Shapee and M. Mohamad. The study of seasonal variation of PM concentration in Peninsular, Saah, and Sarawak. JMM Research Pulication. Vol