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Evaluating wind power density models and their statistical properties
Energy xxx (2015) 1e9
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Evaluating wind power density models and their statistical properties Nurulkamal Masseran a, b, * a b
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia Centre for Modeling and Data Analysis (DELTA), Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 September 2014 Received in revised form 29 December 2014 Accepted 5 March 2015 Available online xxx
Information about the wind power df(density function ) is very important when measuring the wind energy potential for a specific area. Usually, the wind power df provides knowledge about the mean power, which is an indicator of the energy potential. However, the mean power does not describe well the characteristics of power density. Thus, by knowing information about other statistical properties, such as standard deviation, skewness and kurtosis, better insight about the characteristics and properties of power density can be obtained. This study proposes a method to derive a wind power density model and its statistical properties particularly from well-known dfs, namely, the Weibull, Gamma and Inverse Gamma dfs. Applying the method of transformation and Monte Carlo integration has been discussed to address the difficulty of finding the different statistical properties of power density. In addition, an application of the proposed method is demonstrated by a case study that involves wind speed data from several stations in Malaysia. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Statistical modeling Wind energy Wind power density Wind speed probability distribution
1. Introduction Currently, wind energy has become an important alternative source of energy because it is cost-effective and readily available. In addition to its low cost and its availability everywhere in the world, wind energy also provides advantages with its low, local and manageable impact on the environment. These advantages exist specifically because the process involved in generating the electricity from wind turbines does not release carbon dioxide. These advantages have allowed wind energy to be among the potential alternatives for renewable clean energy; thus, wind energy could substitute for fossil-fuel-based energy sources, which contaminate the lower layers of the troposphere [1]. In addition, wind energy does not pose a transportation problem, and its utilization does not require advanced technology [2]. In fact, investment into wind energy will provide benefits in terms of employment, research, economic activity and energy independence in the electricity sector [3]. In developed countries such as Germany, USA and Spain, the use of wind energy has gained recognition, and currently, it is
* School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia. Tel.: þ60 3 8921 3424; fax: þ60 3 8925 4519. E-mail address: [email protected].
gaining more and more advanced implementation. In addition, Asia, China and India have also actively enhanced and expanded the use of wind energy applications [4]. Thus, to take advantage of this freely available energy source, a large amount of research has been conducted toward the development of an accurate and reliable wind energy assessment model using many different approaches [5]. An effective utilization of wind energy requires detailed knowledge of wind characteristics in a specific area. The characteristics of wind speed can be explained by using a wind speed density function. The wind speed density function is important in determining the selection of suitable sites for a wind generator, designing the wind farm, designing the power generator, determining the dominant direction of the wind and evaluating the management operations of the wind power conversion system [6,7]. Thus, it can be concluded that the information regarding the wind speed density function is very important for assessing the capacity and the potential performance of wind energy in a specific area. A wind speed density function is usually described by its pdf (probability density function). Then, the energy density E in term of W/m2 for a specific wind site and a wind turbine can be obtained by using the power curve and the probability density function of wind speed [8]. Morgan et al. [9] and Carta et al. [10] stated that engineering practice has identified the mean of the wind energy that
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can be produced by a wind turbine associated with the probability distribution function of wind speed which can be calculated as follows:
Z∞ Pw ¼
Pw ðxÞf ðxÞdx
(1)
0
where P w is the mean power that is produced, x is the value for the wind speed random variable X, f(x) is a pdf for the wind speed, andPw(x) is a turbine power curve. The uncertainties in the wind power estimation when using Equation (1) depend on the pdf for wind speed, f(x). This relationship occurs because the value of the turbine power curve can usually be determined accurately. Thus, the selection of the ‘good’ wind speed pdf can provide a more precise estimation and a better result for the analysis of wind energy potential. In fact, the mean of the wind power P w, described by Equation (1) can also be known as a first-moment of the wind speed density function corresponding to the turbine power curvePw(x). The turbine power curve mentioned in Equation (1) defined wind power as a proportion of the cube of wind speed X. Instead of the wind speed, the power curve of a wind turbine also depends on the value of constant air density, rk and the area of the airstream that has been measured at a perpendicular plane to wind speed direction, A [11,12]. Thus, the wind power equation, based on the criteria of the power curve, can be written as follows:
Pw ðXÞ ¼
1 Ar X 3 2 k
(2)
Equation (2) provides the power of the wind that flows at the speed of X through a blade sweep area A. This equation has been used to compute the availability of the wind power in a specific area. To be more precise, we must consider Betz's law in the power coefficientCp, which is used in windmills after considering several factors in the process of generating power [11]. Thus, the complete wind power equation is:
PCp ðXÞ ¼
1 Ar X 3 Cp ðl; bÞ 2 k
(3)
where Cp(l,b) is the power coefficient value (which complies with Betz’ law and the type of wind turbine that is used). However, the calculation involved in Equation (3) must be performed by involving several engineering aspects, such as the parameter of the rotational speed of a wind turbine, the turbine blade's angle of attack, the pitch angle etc. [13]. Thus, in this study, I attempt to provide a theoretical approach to the estimation of wind power using Equations (1) and (2). The researchers who are involved with the real application of wind power generators (using Equation (3)) can easily make modifications to our approach in order to meet the requirements of their study. Based on the wind power equations discussed above, it can be concluded that the probability density function of the wind speed is very important in determining and evaluating wind energy potential. In fact, the Weibull pdf is among the most popular statistical distributions in the field of wind energy applications. For example, the Weibull pdf has been used widely in modeling and the assessment of wind energy potential for particular areas for examples,see [14e25]. The information from the Weibull pdf can be used to estimate the wind power corresponding to the wind turbine capacity factor see [26e30]. In addition to this, the Weibull pdf has also been used as an estimation model to evaluate the wind power performance system [31,32] and the failure model for the wind turbine [33]. However, not all of the wind regimes can be
modeled using the Weibull distribution. For example, Jaramillo & Borja [34] have shown that the mixture in the Weibull pdf can provide a better result in modeling the wind regime than using a bimodal distribution. Brano et al. [35] have found that the Burr distribution is among the best pdf models for wind regime in the urban area of Palermo, Italy. Safari [36] has found that the Gamma distribution can also be used to model the wind-regime data in Rwanda. Morgan et al. [9] compare 14 different wind speed pdfs to determine the best model for offshore wind speed calculations in North America. The results of their study conclude that the Lognormal, Kappa, Bimodal Weibull and Wakeby pdfs can perform better for modeling wind speed compared with the Weibull pdf. Zhou et al. [37] evaluated 9 wind speed pdf models to describe the data of wind speed in several sites within the North Dakota region. Based on the goodness-of-fit assessment, they found that different sites have different suitable pdf models for wind speed data. Carta et al. [38] provided a comprehensive review of the wind speed pdf in wind energy applications. Overall, they have determined that 11 of the wind speed pdf models have been used by most of the researchers all over the world for modeling and assessing wind energy potential. Masseran et al. [39] used 9 different wind speed pdf models to describe the variability of the wind regime in the Malaysian regions by working with 67 wind station sites. Based on their analysis, they found that the Gamma, Weibull and Inverse Gamma pdfs provide a good, fitted model to the data. Chang [40] estimates the wind energy potential in Taiwan using six different probability density functions, namely; Weibull, mixture GammaWeibull, mixture Normal, mixture Normal-Weibull, Mixture Weibull and MEP(maximum entropy principle distribution ). The results of the study indicate that the unimodal distribution of wind speed data does not provide a significant differential between the fitted of each pdfs. However, if the wind speed data is bimodal, the mixture pdfs and MEP pdf can provide better characterizations of wind speed than the Weibull pdf. Usta & Kantar [41] analyze two flexible families of pdfs for estimation of the wind speed distributions. The families of pdfs being proposed include the STD (skewed t-distribution) and skewed generalized error distribution (SGED). Their results found that the STD and SGED can provide better results than the Weibull with a two- or three-parameter distribution. Apart from that, there are many research studies that have performed analysis regarding wind energy by considering several probability densities simultaneously to provide more accurate results for wind energy calculation and estimation.
2. Deriving the wind power density using the transformation method for random variables As mentioned above, the wind power estimation can be described by Equations (1)e(3). Thus, to determine the theoretical wind power density function, a method of transformation for random variables is needed. The transformation method is a technique that is commonly used in statistical analysis to derive the pdf for the function of a random variable,h(X). As is mentioned above, X is a random variable for wind speed data. Then, let P ¼ h(X) be a function of the random variable X, while the pdf for X isfX(x). Because the value for the wind speed data is always greater than zero, the function of h(x) is always a monotonic function. Then, by using the transformation method, the pdf for P can be derived by
8 1 > < f h1 ðpÞ d h ðpÞ ; X dp fp ðpÞ ¼ > : 0 ;
p2P
(4)
otherwise
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where h1(p) is an inverse function and has a continuous derivative on P [11,42]. To facilitate the application of this method, let u ¼ 12 Ark . Then, from Equation (2), we have P ¼ uX3 as a monotonic function for each x > 0. Thus the inverse function for P ¼ uX3 can be 13 1 obtained as h1 ðpÞ ¼ up and d½h dpðpÞ ¼ 11 2 . Thus, by using 3u3 p3
Equation (4), the pdf for the wind power,fp(p) can be flexibly derived from various wind speed density functions, fX(x). Next, from the pdf of the wind power, fp(p), the theoretical properties of the wind power can be computed by using the concept of a raw moment for the wind speed pdf or the method of Monte Carlo integration. 3. Statistical properties of the wind power density function The wind power density model is useful for describing the distributions of wind energy at various wind speed values. As discussed above, wind power density is obtained by considering a suitable wind speed density function. Thus, the utilization of a more accurate wind speed pdf will minimize the uncertainty in the wind resource estimates, and consequently, it will improve the result in the site assessment phase of planning. According to the study by Masseran et al. [39], the Weibull, Gamma and Inverse Gamma distributions can provide a good fitted model for the wind speed data in Malaysia. In addition, these statistical distributions imply a more precise estimate for the mean power. Thus, in this study, we expend the results by exploring the properties of the wind power density that have been derived from the Weibull, Gamma and Inverse Gamma pdf. 3.1. Wind power density from the Weibull pdf The Weibull density function was first introduced by Weibull [43]. In the wind energy application, the Weibull pdf has been used extensively by a large number of researchers [14e25]. The Weibull pdf is a generalization of the Rayleigh pdf. In fact, the Weibull pdf has been determined to be capable of providing a better result in wind speed modeling compared with the Rayleigh pdf. The Weibull pdf for the wind speed variable is given by:
fX ðxÞ ¼
b b x b1 x exp a a a
(5)
where b is a shape parameter that explains the variation of the hourly mean speed around the overall mean (or annual mean). The variation of the hourly mean speed around the overall mean is
g2 ¼
overall wind speed is higher [44]. Next, to derive the wind power density from the Weibull pdf, a transformation method in Equation (4) will be applied, which can be written as:
dh1 ðpÞ fp ðpÞ ¼ fX h1 ðpÞ dp 2 1 0p13 1b 3 2 p 3 3b1 u b4 6 @ u A 7 1 5 ¼ exp4 5 1 2 3u3 p3 a a a
small when the parameter b is higher. Thus, a higher value of b corresponds to a lower dispersion of the wind speed, which implies a greater concentration toward the overall mean of the wind speed in specific areas. In addition, a is a scale parameter that determines how “windy” a location is or how high the overall (annual) wind speed is in a specific area. If the parameter a is larger, then the
b12
2
"
b p p p exp 1 2 3 ua3 ua3 3u3 p3 a ua " b
b # b p 31 p 3 exp fp ðpÞ ¼ ua3 3ua3 ua3 3
¼
3
3
3
(6)
b # 3
0
Let b0 ¼ 3b and a ¼ua3, thus this pdf is also known as a Weibull 0 0 pdf with the shape parameter b and the scale parameter.a With this approach, the properties of the power density function can be obtained easily by matching up and performing some modifications to the properties of the Weibull pdf for wind speed. Here, the mean of the power density is obtained, which is given by:
mp ¼
Ar 3 3 a G 1þ 2 b
(7)
The standard deviation of the power density is given by:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u
2 # u Ar 6 3 t 3 a G 1þ G 1þ sp ¼ 2 b b
(8)
The power density skewness is given by:
g1 ¼
3 Ar 3 G 1 þ 9b a 3EðPÞVarðPÞ ½EðPÞ3 2 (9)
3
½VarðPÞ2
Next, to compute the kurtosis, a new parameter must be defined as a function of the mean and variance of the power density [45], which is given by:
"
#1:086 EðPÞ b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðPÞ
(10)
Then, the kurtosis for the power density can be written as:
G 1 þ 4b 4G 1 þ 1b G 1 þ 3b þ 6G2 1 þ 1b G 1 þ 2b 3G4 1 þ 1b
2 G 1 þ 2b G2 1 þ 1b
3
3
(11)
In addition, the corresponding cumulative power density function can be written as:
" Fp ðpÞ ¼ PrðP pÞ ¼ 1 exp
p 3 ua
b # 3
(12)
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3.2. Wind power density from the gamma pdf
Based on the transformation method, the wind power density derived from the Inverse Gamma density function can be written as:
The Gamma pdf is among the earliest statistical distributions that were used in describing the wind behaviors [46]. The Gamma pdf is also known as a Pearson type III pdf [9]. A study by Kiss & Janosi [47] found that the Gamma pdf could explain almost all of the wind speed distribution in the European regions. In addition, it should be mentioned that other researchers who used a Gamma pdf to model the wind speed distribution are Li & Shi [6]; Morgan et al. [9]; Carta et al. [10]; Brano et al. [35]; Safari [36]; Zhou et al. [37]; Carta et al. [38] and Masseran et al. [48]. The Gamma pdf for the wind speed variable is given by:
2 3 1 p1 1 b p p 3 b 4 5 fP ðpÞ ¼ exp 1 1 2 GðpÞ u p 3 3u3 p3 u " # 1 pþ1 i h p b p 3 bu3 ¼ exp 1 2 1 3GðpÞu3 p3 u p3 " # 1 hpi3p1323 hpi23 bp bu3 ¼ exp 1 1 2 u 3GðpÞu3 p3 u p3 " # 1 p i h bp p 31 bu3 ¼ exp 1 3GðpÞu u p3 " # 1 p b bu3 3p1 fP ðpÞ ¼ exp 1 p ½p 3GðpÞu3 p3
fX ðxÞ ¼
1 x a1 exp ax GðaÞb b
(13)
where G(.) is the gamma function. Then, by using the transformation method, the wind power density can be derived as follows:
fP ðpÞ ¼
¼
¼
1 GðaÞab
1 a1 p 3 u
1 3u3 p3 GðaÞab 1
2
1
As with the Gamma density function, this type of power density is also difficult to match with any existing statistical model. Thus, the computation for the statistical properties such as the mean, variance, skewness and kurtosis of the power density in Equation (16) is difficult to solve by using an analytical approach.
p13 3 1 u 5 4 exp b 3u13 p23 2
1 a1 p 3 u
exp4 u 5 b
pa31323 p23
1 2 3GðaÞab u3 p3 u
p13 3
2
u
" exp
# " 1 pa31 1 p3 exp 1 3GðaÞab u u u3 b # " 1 3 a 1 p 1 3 exp 1 fP ðpÞ ¼ a ðpÞ 3GðaÞab u3 u3 b
1
p3
(16)
3.4. Monte Carlo integration
#
(14)
1
u3 b
To overcome the difficulties that are presented in Equations (14) and (16), Monte Carlo integration will be used. Monte Carlo integration provides a solution that addresses the problem of the evaluation of integrals of the type:
Z
¼
E½gðXÞ ¼
gðxÞfX ðxÞdx
(17)
x
The power density function in Equation (14) cannot be matched by any commonly used statistical distribution. Thus, the computation for the mean, variance, skewness and kurtosis of the power density in Equation (14) will be difficult to solve by using an analytical approach.
where the function gðXÞ ¼ 12 Ark X 3 explains the function of the wind power in Equations (1) and (2), and fX(x) is the Gamma or Inverse Gamma density function. Then, by generating T random
3.3. Wind power density from the Inverse Gamma pdf The inverse Gamma pdf is also known as a Pearson type IV pdf. The inverse Gamma pdf is obtained from the reciprocal of the random variable from the Gamma pdf. Other researchers who used the Gamma pdf to model the wind speed distribution are Li & Shi [6]; Zhou et al. [37]; and Masseran et al. [48]. The Inverse Gamma pdf for the wind speed variable is given by:
fX ðxÞ ¼
bp p1 b x exp x GðpÞ
(15)
Table 1 Geographical coordinates for each station. Station
Latitude
Longitude
Mersing Cameron Highlands Malacca Kudat Putrajaya Sandakan
2 4 2 6 2 5
105 101 102 116 101 118
270 N 280 N 160 N 550 N 550 N 540 N
500 E 220 E 150 E 500 E 400 E 040 E
Fig. 1. The 010C wind speed sensor [50].
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N. Masseran / Energy xxx (2015) 1e9 Table 2 010C wind speed sensor specifications [50]. Performance characteristics Maximum operating range Starting speed Calibrated range Accuracy Temperature range Distance constant Electrical characteristics Power requirements
Table 4 Results of goodness-of-fit based on the KolmogoroveSmirnov statistics and the selected distribution (in bold) for each station. Station
0e125 mph (0e60 m/s) 0.5 mph (0.22 m/s) 0e100 mph (0e50 m/s) ±1% (0.15 mph) (0.07 m/s) 50 C to þ65 C (58 F to þ149 F) Less than 5 ft (1.5 m) of flow (meets EPA specifications)
Output impedance Physical characteristics Weight Finish
PT
Gamma
Inverse gamma
0.0867 0.0722 0.0899 0.0568 0.0663 0.0513
0.0582 0.0761 0.0866 0.0619 0.2480 0.0343
0.0822 0.0980 0.0945 0.0938 0.0844 0.0824
Z g1 ¼ skew½gðXÞ ¼ x
Cable assembly; specify length in feet or meters PN 191 Crossarm assembly
samples fromfX(x), the unbiased estimator for E[g(X)] can be written as:
g T ðXÞ ¼
Weibull
iii) Skewness of the wind power density:
1.5 lbs (0.68 kg) Clear anodized aluminum; Lexan cup assembly
Cable and mounting PN 1953 Mounting
KolmogoroveSmirnov statistic
Mersing C. Highlands Malacca Kudat Putrajaya Sandakan
12 VDC at 10 mA, 12 VDC at 350 mA for interal heater 11 V (pulse frequency equivalent to speed) 100 U maximum
Output signal
5
gðxi Þ T
i¼1
(18)
gðxÞ g m ðXÞ Var½gðXÞ
PT
i¼1
¼
fX ðxÞdx
gðxi Þ E½gðXÞ Var½gðXÞ T
3
(21)
iv) Kurtosis of the wind power density:
g2 ¼ kurtðgðXÞÞ ¼
Z gðxÞ g m ðXÞ 4 fX ðxÞdx Var½gðXÞ x
which is known as a sample mean for the g(X) function [39,49]. In fact, because the function of g(X) explains the function of the wind power, the mean power for the power density, E(P) is equivalent to the value of E[g(X)]. Then, by using the same approach, the solutions for each of the wind power density properties can be easily computed. The equations below summarize the wind power density properties, which are determined using Monte Carlo integration.
3
PT ¼
i¼1
gðxi Þ E½gðXÞ 4 Var½gðXÞ T
(22)
v) Cumulative power density function:
b gðxÞ 3 Fp ðpÞ ¼ Pr½gðXÞ gðxÞ ¼ 1 exp ua3
(23)
i) Mean of the wind power density:
PT
Z mp ¼ E½gðXÞ ¼
gðxÞ fX ðxÞdp ¼ x
gðxi Þ T
i¼1
(19)
ii) Standard deviation of the wind power density:
sp ¼
In this study, the statistical properties of the wind power that are derived from the Gamma pdf for the wind speed variable can be obtained using Monte Carlo integration. However, this method can also be applied to the other wind speed pdfs, especially for a more complex wind speed pdf such as the Wakeby pdf, Kappa pdf or Mixture Weibull pdf.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½gðXÞ 4. Case study in Peninsular Malaysia
where:
Z
2
Var½gðXÞ ¼
ðgðxÞ E½gðXÞÞ fX ðxÞdx x
(20)
PT ¼
2 i¼1 ðgðxi Þ E½gðXÞÞ T
Table 3 Parameter estimations using the maximum likelihood method. Station
Mersing C. Highlands Malacca Kudat Putrajaya Sandakan
Weibull
Gamma
Inverse gamma
a
b
a
b
p
b
3.166 2.285 2.367 2.874 1.399 2.668
2.187 1.746 1.690 1.673 1.720 1.958
4.317 2.763 2.511 2.473 1.399 3.418
0.648 0.734 0.838 1.033 0.581 0.690
3.578 2.331 2.129 2.154 2.468 2.868
7.667 3.100 2.815 3.458 2.018 4.826
To provide a practical application regarding the methodologies that have been discussed above, a case study that involves a real data set will be presented in this section. The data are involved with six selected stations in Peninsular Malaysia, namely, the Mersing, Cameron Highlands, Malacca, Kudat, Putrajaya and Sandakan stations (shown in Table 1). The 010C Wind Speed Sensor has been used by the Department of Environment to collect the hourly wind speed data in Malaysia. The wind speed sensor of type 010C can provide accurate information on the horizontal wind speed. The lightweight three-cup anemometer has been used in virtually all of the applications in which a fast response and a low starting threshold are of paramount importance. The instrument was provided by Met One Instruments. Fig. 1 shows the 010C Wind Speed Sensor model, while Table 2 describes the specifications of the 010C Wind Speed Sensor, including its characteristics, its accuracy and its range value (for more details, please refer to [50]). In this study, the hourly wind speed data was collected from January 1, 2007 to November 30,
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Fig. 2. Fitting a gamma, Weibull and inverse gamma pdf to the wind speed data.
2009. The missing data has been estimated by using the method of single imputation [51]. The parameter estimates for each density function have been performed using the maximum likelihood method. In addition, the goodness-of-fit procedures are determined by the KolmogoroveSmirnov statistic (KS) method. A distribution with a minimum value of KS will be selected to be the best model for the wind speed distribution at each station. Table 3 shows the results of the maximum likelihood estimation, while Table 4 shows the results of the goodness-of-fit for each station. In addition, Fig. 2 shows the fitted pdf model for the observed wind speed data for each station. From Table 4 and Fig. 2, a comparison between the three distributions reveals that the Weibull distribution is the best model for the wind speed at the Cameron Highlands, Kudat and Putrajaya stations, while for the stations of Mersing, Malacca and Sandakan, the Gamma distribution is better than the Weibull and Inverse Gamma distributions. In fact, the fitted pdf model in Fig. 2 shows that the Weibull and Gamma pdf is able to provide a good approximation of the observed wind speed data in each station. Thus, these pdf models are reliable for estimating wind energy, particularly the energy density function. However, because this study does not involve real wind turbines, the value of the airstream area for wind turbines as shown in Equation (2) is assumed to be equal to 1 m2,A ¼ 1. Thus, the computed power will be measured in terms of the power per unit area. However, those who are involved with real wind turbines can easily change the A value according to their study. In addition to this, the value of constant air density is rk ¼ 1.16 kg m3 [52]. Then, by using the results from Table 3, the statistical characteristics of wind power
density for each station can be computed using Equations (7)e(12) and (17e23). Fig. 3 shows the density plots that have information on the statistical properties of wind power for each station. Clearly, theoretical wind power density fits well with the histogram of empirical wind power data. Thus, it can be concluded that theoretical wind power density derived by the method of transformation provides good results. From Fig. 3, the mean power per unit area is in the range of 2e24 W/m2. The Mersing and Kudat stations have a greater mean power per unit area compared with the other stations. In addition, the standard deviation for all of the stations ranges between 4 and 45 W/m2. The Mersing and Kudat stations have a greater standard deviation compared with other stations. This can be seen clearly from the power density plot for the Mersing and Kudat stations. The maximum power per unit area for the Mersing and Kudat stations is more than 800 W/m2 and 600 W/m2, respectively. In addition, all of the density plots are found to be skewed to the right with each of the coefficients of skewness g1 > 4, which indicates a lack of symmetric properties for the power density. The kurtosis measures the peakness of the wind power density for each station. A higher value for the kurtosis indicates a sharper peak in the density function. This finding is clearly in agreement with the results shown in Fig. 3. The value of the kurtosis coefficient is greater than 35 for all of the stations,g2 > 4. Finally, the cumulative power density function for each station is shown in Fig. 4. Thus, based on Figs. 3 and 4, more information regarding the wind energy potential can be extracted from the wind speed data. This information provides a better understanding of the characteristics and properties of the power density for each station.
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Fig. 3. Wind power density and its statistical properties for each station.
Fig. 4. Cumulative power density functions for each station.
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5. Conclusions This study discusses the statistical properties of the wind power density function, particularly the mean power, standard deviation, skewness and kurtosis. The transformation method has been proposed for deriving a theoretical density function of wind power based on the wind speed pdf, such as the Gamma, Weibull and Inverse Gamma pdfs. The power density function derived from the Weibull pdf is also found to follow a Weilbull pdf with some modification of the parameters. Thus, its statistical properties of power density, such as the mean power, standard deviation, skewness and the kurtosis can be obtained by using the knowledge of the Weibull properties. However, for power density derived from the Gamma and Inverse Gamma pdfs, the equations for the power density function are difficult to match with any existing statistical model. Thus, the statistical properties of power density cannot be easily obtained. In fact, the solution using the method of integration will also be difficult. To overcome this problem, this study proposed the method of Monte Carlo integration. The method of Monte Carlo integration is efficient and reliable in providing a flexible solution for determining the statistical properties of wind power density. Although this study only provides a solution for the Gamma and Inverse Gamma pdfs, the proposed method also can be applied to any other statistical model, particularly for researchers that involve more complex pdf functions, such as the Kappa, Wakeby, Mixture Weibull, etc. Apart from that, the application of the proposed method has been demonstrated by a case study that involves wind speed data from several stations in Malaysia. Acknowledgments The authors are indebted to the staff of the Department of Environment and the Malaysian Meteorology Department for providing the hourly mean wind speed data that made this paper possible. This research would also not have been possible without the sponsorship from Universiti Kebangsaan Malaysia and the Ministry of Higher Education, Malaysia (grant number GGPM-2014056 and FRGS/1/2014/SG04/UKM/03/1) References [1] Akpinar EK, Akpinar S. A statistical analysis of wind speed data used in installation of wind energy conversion systems. Energy Convers Manag 2005;46:515e32. [2] Masseran N, Razali AM, Ibrahim K, Wan Zin WZ. Evaluating the wind speed persistence for several wind stations in Peninsular Malaysia. Energy 2012;37: 649e56. [3] Yaniktepe B, Savrun MM, Koroglu T. Current status of wind energy and wind energy policy in Turkey. Energy Convers Manag 2013;72:103e10. [4] Abbasi Tabassum, Premalatha M, Abbasi T, Abbasi SA. Wind energy: Increasing deployment, rising environmental concerns. Renew Sustain Energy Rev 2014;31:270e88. [5] Jung J, Broadwater RP. Current status and future advances for wind speed and power forecasting. Renew Sustain Energy Rev 2014;31:762e77. [6] Li G, Shi J. Application of Bayesian model averaging in modeling long-term wind speed distributions. Renew Energy 2010;6:1192e201. [7] Masseran N, Razali AM, Ibrahim K, Latif MT. Fitting a mixture of von Mises distributions in order to model data on wind direction in Peninsular Malaysia. Energy Convers Manag 2013;72:94e102. [8] Carrillo C, Montano Obando AF, Cidras J, Diaz-Dorado E. Review of power curve modeling for wind turbines. Renew Sustain Energy Rev 2013;21: 571e81. [9] Morgan EC, Lackner M, Vodel RM, Baise LG. Probability distribution for offshore wind speeds. Energy Convers Manag 2011;52:15e26. [10] Carta JA, Ramirez P, Velazquez S. Influence of the level of fit of a density probability function to wind-speed data on the WECS mean power output estimation. Energy Convers Manag 2008;49:2647e55. [11] Villanueva D, Feijoo A. Wind power distributions: a review of their applications. Renew Sustain Energy Rev 2010;14:1490e5. [12] Carta JA, Mentado D. A continuous bivariate model for wind power density and wind turbine energy output estimations. Energy Convers Manag 2007;48: 420e32.
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Wind resource estimation and wind park design in El-Kef region, Tunisia. Energy 2012;40:348e57. [26] Abed KA, El-Mallah AA. Capacity factor of wind turbines. Energy 1997;22: 487e91. [27] Celik AN. Weibull representative compressed wind speed data for energy and performance calculations of wind energy systems. Energy Convers Manag 2003;44:3057e72. [28] Celik AN. Energy output estimation for small-scale wind power generators using Weibull-representative wind data. J Wind Eng Industrial Aerodynamics 2003;91:693e707. [29] Chang TJ, Tu YL. Evaluation of monthly capacity factor of WECS using chronological and probabilistic wind speed data: a case study of Taiwan. Renew Energy 2007;32:1999e2010. [30] Pallabazzer R. Previsional estimation of the energy output of wind generators. Renew Energy 2004;29:413e20. [31] Celik AN. A simplified model for estimating the monthly performance of autonomous wind energy systems with battery storage. Renew Energy 2003;28:561e72. [32] Celik AN. A simplified model for estimating yearly wind fraction in hybrid wind energy systems. Renew Energy 2006;31:105e18. [33] Andrawus JA, Watson J, Kishk M. Wind turbine maintenance optimization principles of quantitative maintenance optimisation. Wind Eng 2007;31: 101e10. [34] Jaramillo OA, Borja MA. Bimodal versus Weibull wind speed distributions: an analysis of wind energy potential in La Venta, Mexico. Wind Eng 2004;28: 225e34. [35] Brano VL, Orioli A, Ciulla G, Culotta S. Quality of wind speed fitting distribution for urban area of Palermo, Italy. Renew Energy 2011;36:1026e39. [36] Safari B. Modeling wind speed and wind power distributions in Rwanda. Renew Sustain Energy Rev 2011;15:925e35. [37] Zhou L, Erdem E, Li G, Shi J. Comprehensive evaluation of wind speed distribution model: a case study for North Dakota sites. Energy Convers Manag 2010;51:1449e58. [38] Carta JA, Ramirez P, Velazquez S. A review of wind speed probability distributions used in wind energy analysis: case studies in the Canary Islands. Renew Sustain Energy Rev 2009;13:933e55. [39] Masseran N, Razali AM, Ibrahim K. An analysis of wind power density derived from several wind speed density functions: the regional assessment on wind power in Malaysia. Renew Sustain Energy Rev 2012;16:6476e87. [40] Chang TP. Estimation of wind energy potential using different probability density functions. Appl Energy 2011;88:1848e56. [41] Usta I, Kantar YM. Analysis of some flexible families of distributions for estimation of wind speed distributions. Appl Energy 2012;89:355e67. [42] Casella G, Berger RL. Statistical inference. 2nd ed. Pacific Grove: Thomson Learning Inc; 2002. [43] Weibull W. A statistical distribution function of wind applicability. J Appl Mech 1951;18:293e7. [44] Munteanu I, Bratcu AI, Cutululis N-A, Ceanga E. Optimal control of wind energy systems: towards a global approach. 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[50] Met One Instrument. Wind energy. 2010. http://metone.com/windpower.php. 010C-020C. [51] Masseran N, Razali AM, Ibrahim K, Zaharim A, Sopian K. Application of the single imputation method to estimate missing wind speed data in Malaysia. Res J Appl Sci Eng Technol 2013;6:1780e4. [52] Sopian K, Othman MY, Wirsat HA. The wind energy potential of Malaysia. Renew Energy 1995;6:1005e16.
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Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Evaluating wind power density models and their statistical properties Nurulkamal Masseran a, b, * a b
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia Centre for Modeling and Data Analysis (DELTA), Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 September 2014 Received in revised form 29 December 2014 Accepted 5 March 2015 Available online xxx
Information about the wind power df(density function ) is very important when measuring the wind energy potential for a specific area. Usually, the wind power df provides knowledge about the mean power, which is an indicator of the energy potential. However, the mean power does not describe well the characteristics of power density. Thus, by knowing information about other statistical properties, such as standard deviation, skewness and kurtosis, better insight about the characteristics and properties of power density can be obtained. This study proposes a method to derive a wind power density model and its statistical properties particularly from well-known dfs, namely, the Weibull, Gamma and Inverse Gamma dfs. Applying the method of transformation and Monte Carlo integration has been discussed to address the difficulty of finding the different statistical properties of power density. In addition, an application of the proposed method is demonstrated by a case study that involves wind speed data from several stations in Malaysia. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Statistical modeling Wind energy Wind power density Wind speed probability distribution
1. Introduction Currently, wind energy has become an important alternative source of energy because it is cost-effective and readily available. In addition to its low cost and its availability everywhere in the world, wind energy also provides advantages with its low, local and manageable impact on the environment. These advantages exist specifically because the process involved in generating the electricity from wind turbines does not release carbon dioxide. These advantages have allowed wind energy to be among the potential alternatives for renewable clean energy; thus, wind energy could substitute for fossil-fuel-based energy sources, which contaminate the lower layers of the troposphere [1]. In addition, wind energy does not pose a transportation problem, and its utilization does not require advanced technology [2]. In fact, investment into wind energy will provide benefits in terms of employment, research, economic activity and energy independence in the electricity sector [3]. In developed countries such as Germany, USA and Spain, the use of wind energy has gained recognition, and currently, it is
* School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia. Tel.: þ60 3 8921 3424; fax: þ60 3 8925 4519. E-mail address: [email protected].
gaining more and more advanced implementation. In addition, Asia, China and India have also actively enhanced and expanded the use of wind energy applications [4]. Thus, to take advantage of this freely available energy source, a large amount of research has been conducted toward the development of an accurate and reliable wind energy assessment model using many different approaches [5]. An effective utilization of wind energy requires detailed knowledge of wind characteristics in a specific area. The characteristics of wind speed can be explained by using a wind speed density function. The wind speed density function is important in determining the selection of suitable sites for a wind generator, designing the wind farm, designing the power generator, determining the dominant direction of the wind and evaluating the management operations of the wind power conversion system [6,7]. Thus, it can be concluded that the information regarding the wind speed density function is very important for assessing the capacity and the potential performance of wind energy in a specific area. A wind speed density function is usually described by its pdf (probability density function). Then, the energy density E in term of W/m2 for a specific wind site and a wind turbine can be obtained by using the power curve and the probability density function of wind speed [8]. Morgan et al. [9] and Carta et al. [10] stated that engineering practice has identified the mean of the wind energy that
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can be produced by a wind turbine associated with the probability distribution function of wind speed which can be calculated as follows:
Z∞ Pw ¼
Pw ðxÞf ðxÞdx
(1)
0
where P w is the mean power that is produced, x is the value for the wind speed random variable X, f(x) is a pdf for the wind speed, andPw(x) is a turbine power curve. The uncertainties in the wind power estimation when using Equation (1) depend on the pdf for wind speed, f(x). This relationship occurs because the value of the turbine power curve can usually be determined accurately. Thus, the selection of the ‘good’ wind speed pdf can provide a more precise estimation and a better result for the analysis of wind energy potential. In fact, the mean of the wind power P w, described by Equation (1) can also be known as a first-moment of the wind speed density function corresponding to the turbine power curvePw(x). The turbine power curve mentioned in Equation (1) defined wind power as a proportion of the cube of wind speed X. Instead of the wind speed, the power curve of a wind turbine also depends on the value of constant air density, rk and the area of the airstream that has been measured at a perpendicular plane to wind speed direction, A [11,12]. Thus, the wind power equation, based on the criteria of the power curve, can be written as follows:
Pw ðXÞ ¼
1 Ar X 3 2 k
(2)
Equation (2) provides the power of the wind that flows at the speed of X through a blade sweep area A. This equation has been used to compute the availability of the wind power in a specific area. To be more precise, we must consider Betz's law in the power coefficientCp, which is used in windmills after considering several factors in the process of generating power [11]. Thus, the complete wind power equation is:
PCp ðXÞ ¼
1 Ar X 3 Cp ðl; bÞ 2 k
(3)
where Cp(l,b) is the power coefficient value (which complies with Betz’ law and the type of wind turbine that is used). However, the calculation involved in Equation (3) must be performed by involving several engineering aspects, such as the parameter of the rotational speed of a wind turbine, the turbine blade's angle of attack, the pitch angle etc. [13]. Thus, in this study, I attempt to provide a theoretical approach to the estimation of wind power using Equations (1) and (2). The researchers who are involved with the real application of wind power generators (using Equation (3)) can easily make modifications to our approach in order to meet the requirements of their study. Based on the wind power equations discussed above, it can be concluded that the probability density function of the wind speed is very important in determining and evaluating wind energy potential. In fact, the Weibull pdf is among the most popular statistical distributions in the field of wind energy applications. For example, the Weibull pdf has been used widely in modeling and the assessment of wind energy potential for particular areas for examples,see [14e25]. The information from the Weibull pdf can be used to estimate the wind power corresponding to the wind turbine capacity factor see [26e30]. In addition to this, the Weibull pdf has also been used as an estimation model to evaluate the wind power performance system [31,32] and the failure model for the wind turbine [33]. However, not all of the wind regimes can be
modeled using the Weibull distribution. For example, Jaramillo & Borja [34] have shown that the mixture in the Weibull pdf can provide a better result in modeling the wind regime than using a bimodal distribution. Brano et al. [35] have found that the Burr distribution is among the best pdf models for wind regime in the urban area of Palermo, Italy. Safari [36] has found that the Gamma distribution can also be used to model the wind-regime data in Rwanda. Morgan et al. [9] compare 14 different wind speed pdfs to determine the best model for offshore wind speed calculations in North America. The results of their study conclude that the Lognormal, Kappa, Bimodal Weibull and Wakeby pdfs can perform better for modeling wind speed compared with the Weibull pdf. Zhou et al. [37] evaluated 9 wind speed pdf models to describe the data of wind speed in several sites within the North Dakota region. Based on the goodness-of-fit assessment, they found that different sites have different suitable pdf models for wind speed data. Carta et al. [38] provided a comprehensive review of the wind speed pdf in wind energy applications. Overall, they have determined that 11 of the wind speed pdf models have been used by most of the researchers all over the world for modeling and assessing wind energy potential. Masseran et al. [39] used 9 different wind speed pdf models to describe the variability of the wind regime in the Malaysian regions by working with 67 wind station sites. Based on their analysis, they found that the Gamma, Weibull and Inverse Gamma pdfs provide a good, fitted model to the data. Chang [40] estimates the wind energy potential in Taiwan using six different probability density functions, namely; Weibull, mixture GammaWeibull, mixture Normal, mixture Normal-Weibull, Mixture Weibull and MEP(maximum entropy principle distribution ). The results of the study indicate that the unimodal distribution of wind speed data does not provide a significant differential between the fitted of each pdfs. However, if the wind speed data is bimodal, the mixture pdfs and MEP pdf can provide better characterizations of wind speed than the Weibull pdf. Usta & Kantar [41] analyze two flexible families of pdfs for estimation of the wind speed distributions. The families of pdfs being proposed include the STD (skewed t-distribution) and skewed generalized error distribution (SGED). Their results found that the STD and SGED can provide better results than the Weibull with a two- or three-parameter distribution. Apart from that, there are many research studies that have performed analysis regarding wind energy by considering several probability densities simultaneously to provide more accurate results for wind energy calculation and estimation.
2. Deriving the wind power density using the transformation method for random variables As mentioned above, the wind power estimation can be described by Equations (1)e(3). Thus, to determine the theoretical wind power density function, a method of transformation for random variables is needed. The transformation method is a technique that is commonly used in statistical analysis to derive the pdf for the function of a random variable,h(X). As is mentioned above, X is a random variable for wind speed data. Then, let P ¼ h(X) be a function of the random variable X, while the pdf for X isfX(x). Because the value for the wind speed data is always greater than zero, the function of h(x) is always a monotonic function. Then, by using the transformation method, the pdf for P can be derived by
8 1 > < f h1 ðpÞ d h ðpÞ ; X dp fp ðpÞ ¼ > : 0 ;
p2P
(4)
otherwise
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N. Masseran / Energy xxx (2015) 1e9
where h1(p) is an inverse function and has a continuous derivative on P [11,42]. To facilitate the application of this method, let u ¼ 12 Ark . Then, from Equation (2), we have P ¼ uX3 as a monotonic function for each x > 0. Thus the inverse function for P ¼ uX3 can be 13 1 obtained as h1 ðpÞ ¼ up and d½h dpðpÞ ¼ 11 2 . Thus, by using 3u3 p3
Equation (4), the pdf for the wind power,fp(p) can be flexibly derived from various wind speed density functions, fX(x). Next, from the pdf of the wind power, fp(p), the theoretical properties of the wind power can be computed by using the concept of a raw moment for the wind speed pdf or the method of Monte Carlo integration. 3. Statistical properties of the wind power density function The wind power density model is useful for describing the distributions of wind energy at various wind speed values. As discussed above, wind power density is obtained by considering a suitable wind speed density function. Thus, the utilization of a more accurate wind speed pdf will minimize the uncertainty in the wind resource estimates, and consequently, it will improve the result in the site assessment phase of planning. According to the study by Masseran et al. [39], the Weibull, Gamma and Inverse Gamma distributions can provide a good fitted model for the wind speed data in Malaysia. In addition, these statistical distributions imply a more precise estimate for the mean power. Thus, in this study, we expend the results by exploring the properties of the wind power density that have been derived from the Weibull, Gamma and Inverse Gamma pdf. 3.1. Wind power density from the Weibull pdf The Weibull density function was first introduced by Weibull [43]. In the wind energy application, the Weibull pdf has been used extensively by a large number of researchers [14e25]. The Weibull pdf is a generalization of the Rayleigh pdf. In fact, the Weibull pdf has been determined to be capable of providing a better result in wind speed modeling compared with the Rayleigh pdf. The Weibull pdf for the wind speed variable is given by:
fX ðxÞ ¼
b b x b1 x exp a a a
(5)
where b is a shape parameter that explains the variation of the hourly mean speed around the overall mean (or annual mean). The variation of the hourly mean speed around the overall mean is
g2 ¼
overall wind speed is higher [44]. Next, to derive the wind power density from the Weibull pdf, a transformation method in Equation (4) will be applied, which can be written as:
dh1 ðpÞ fp ðpÞ ¼ fX h1 ðpÞ dp 2 1 0p13 1b 3 2 p 3 3b1 u b4 6 @ u A 7 1 5 ¼ exp4 5 1 2 3u3 p3 a a a
small when the parameter b is higher. Thus, a higher value of b corresponds to a lower dispersion of the wind speed, which implies a greater concentration toward the overall mean of the wind speed in specific areas. In addition, a is a scale parameter that determines how “windy” a location is or how high the overall (annual) wind speed is in a specific area. If the parameter a is larger, then the
b12
2
"
b p p p exp 1 2 3 ua3 ua3 3u3 p3 a ua " b
b # b p 31 p 3 exp fp ðpÞ ¼ ua3 3ua3 ua3 3
¼
3
3
3
(6)
b # 3
0
Let b0 ¼ 3b and a ¼ua3, thus this pdf is also known as a Weibull 0 0 pdf with the shape parameter b and the scale parameter.a With this approach, the properties of the power density function can be obtained easily by matching up and performing some modifications to the properties of the Weibull pdf for wind speed. Here, the mean of the power density is obtained, which is given by:
mp ¼
Ar 3 3 a G 1þ 2 b
(7)
The standard deviation of the power density is given by:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u
2 # u Ar 6 3 t 3 a G 1þ G 1þ sp ¼ 2 b b
(8)
The power density skewness is given by:
g1 ¼
3 Ar 3 G 1 þ 9b a 3EðPÞVarðPÞ ½EðPÞ3 2 (9)
3
½VarðPÞ2
Next, to compute the kurtosis, a new parameter must be defined as a function of the mean and variance of the power density [45], which is given by:
"
#1:086 EðPÞ b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðPÞ
(10)
Then, the kurtosis for the power density can be written as:
G 1 þ 4b 4G 1 þ 1b G 1 þ 3b þ 6G2 1 þ 1b G 1 þ 2b 3G4 1 þ 1b
2 G 1 þ 2b G2 1 þ 1b
3
3
(11)
In addition, the corresponding cumulative power density function can be written as:
" Fp ðpÞ ¼ PrðP pÞ ¼ 1 exp
p 3 ua
b # 3
(12)
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N. Masseran / Energy xxx (2015) 1e9
3.2. Wind power density from the gamma pdf
Based on the transformation method, the wind power density derived from the Inverse Gamma density function can be written as:
The Gamma pdf is among the earliest statistical distributions that were used in describing the wind behaviors [46]. The Gamma pdf is also known as a Pearson type III pdf [9]. A study by Kiss & Janosi [47] found that the Gamma pdf could explain almost all of the wind speed distribution in the European regions. In addition, it should be mentioned that other researchers who used a Gamma pdf to model the wind speed distribution are Li & Shi [6]; Morgan et al. [9]; Carta et al. [10]; Brano et al. [35]; Safari [36]; Zhou et al. [37]; Carta et al. [38] and Masseran et al. [48]. The Gamma pdf for the wind speed variable is given by:
2 3 1 p1 1 b p p 3 b 4 5 fP ðpÞ ¼ exp 1 1 2 GðpÞ u p 3 3u3 p3 u " # 1 pþ1 i h p b p 3 bu3 ¼ exp 1 2 1 3GðpÞu3 p3 u p3 " # 1 hpi3p1323 hpi23 bp bu3 ¼ exp 1 1 2 u 3GðpÞu3 p3 u p3 " # 1 p i h bp p 31 bu3 ¼ exp 1 3GðpÞu u p3 " # 1 p b bu3 3p1 fP ðpÞ ¼ exp 1 p ½p 3GðpÞu3 p3
fX ðxÞ ¼
1 x a1 exp ax GðaÞb b
(13)
where G(.) is the gamma function. Then, by using the transformation method, the wind power density can be derived as follows:
fP ðpÞ ¼
¼
¼
1 GðaÞab
1 a1 p 3 u
1 3u3 p3 GðaÞab 1
2
1
As with the Gamma density function, this type of power density is also difficult to match with any existing statistical model. Thus, the computation for the statistical properties such as the mean, variance, skewness and kurtosis of the power density in Equation (16) is difficult to solve by using an analytical approach.
p13 3 1 u 5 4 exp b 3u13 p23 2
1 a1 p 3 u
exp4 u 5 b
pa31323 p23
1 2 3GðaÞab u3 p3 u
p13 3
2
u
" exp
# " 1 pa31 1 p3 exp 1 3GðaÞab u u u3 b # " 1 3 a 1 p 1 3 exp 1 fP ðpÞ ¼ a ðpÞ 3GðaÞab u3 u3 b
1
p3
(16)
3.4. Monte Carlo integration
#
(14)
1
u3 b
To overcome the difficulties that are presented in Equations (14) and (16), Monte Carlo integration will be used. Monte Carlo integration provides a solution that addresses the problem of the evaluation of integrals of the type:
Z
¼
E½gðXÞ ¼
gðxÞfX ðxÞdx
(17)
x
The power density function in Equation (14) cannot be matched by any commonly used statistical distribution. Thus, the computation for the mean, variance, skewness and kurtosis of the power density in Equation (14) will be difficult to solve by using an analytical approach.
where the function gðXÞ ¼ 12 Ark X 3 explains the function of the wind power in Equations (1) and (2), and fX(x) is the Gamma or Inverse Gamma density function. Then, by generating T random
3.3. Wind power density from the Inverse Gamma pdf The inverse Gamma pdf is also known as a Pearson type IV pdf. The inverse Gamma pdf is obtained from the reciprocal of the random variable from the Gamma pdf. Other researchers who used the Gamma pdf to model the wind speed distribution are Li & Shi [6]; Zhou et al. [37]; and Masseran et al. [48]. The Inverse Gamma pdf for the wind speed variable is given by:
fX ðxÞ ¼
bp p1 b x exp x GðpÞ
(15)
Table 1 Geographical coordinates for each station. Station
Latitude
Longitude
Mersing Cameron Highlands Malacca Kudat Putrajaya Sandakan
2 4 2 6 2 5
105 101 102 116 101 118
270 N 280 N 160 N 550 N 550 N 540 N
500 E 220 E 150 E 500 E 400 E 040 E
Fig. 1. The 010C wind speed sensor [50].
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N. Masseran / Energy xxx (2015) 1e9 Table 2 010C wind speed sensor specifications [50]. Performance characteristics Maximum operating range Starting speed Calibrated range Accuracy Temperature range Distance constant Electrical characteristics Power requirements
Table 4 Results of goodness-of-fit based on the KolmogoroveSmirnov statistics and the selected distribution (in bold) for each station. Station
0e125 mph (0e60 m/s) 0.5 mph (0.22 m/s) 0e100 mph (0e50 m/s) ±1% (0.15 mph) (0.07 m/s) 50 C to þ65 C (58 F to þ149 F) Less than 5 ft (1.5 m) of flow (meets EPA specifications)
Output impedance Physical characteristics Weight Finish
PT
Gamma
Inverse gamma
0.0867 0.0722 0.0899 0.0568 0.0663 0.0513
0.0582 0.0761 0.0866 0.0619 0.2480 0.0343
0.0822 0.0980 0.0945 0.0938 0.0844 0.0824
Z g1 ¼ skew½gðXÞ ¼ x
Cable assembly; specify length in feet or meters PN 191 Crossarm assembly
samples fromfX(x), the unbiased estimator for E[g(X)] can be written as:
g T ðXÞ ¼
Weibull
iii) Skewness of the wind power density:
1.5 lbs (0.68 kg) Clear anodized aluminum; Lexan cup assembly
Cable and mounting PN 1953 Mounting
KolmogoroveSmirnov statistic
Mersing C. Highlands Malacca Kudat Putrajaya Sandakan
12 VDC at 10 mA, 12 VDC at 350 mA for interal heater 11 V (pulse frequency equivalent to speed) 100 U maximum
Output signal
5
gðxi Þ T
i¼1
(18)
gðxÞ g m ðXÞ Var½gðXÞ
PT
i¼1
¼
fX ðxÞdx
gðxi Þ E½gðXÞ Var½gðXÞ T
3
(21)
iv) Kurtosis of the wind power density:
g2 ¼ kurtðgðXÞÞ ¼
Z gðxÞ g m ðXÞ 4 fX ðxÞdx Var½gðXÞ x
which is known as a sample mean for the g(X) function [39,49]. In fact, because the function of g(X) explains the function of the wind power, the mean power for the power density, E(P) is equivalent to the value of E[g(X)]. Then, by using the same approach, the solutions for each of the wind power density properties can be easily computed. The equations below summarize the wind power density properties, which are determined using Monte Carlo integration.
3
PT ¼
i¼1
gðxi Þ E½gðXÞ 4 Var½gðXÞ T
(22)
v) Cumulative power density function:
b gðxÞ 3 Fp ðpÞ ¼ Pr½gðXÞ gðxÞ ¼ 1 exp ua3
(23)
i) Mean of the wind power density:
PT
Z mp ¼ E½gðXÞ ¼
gðxÞ fX ðxÞdp ¼ x
gðxi Þ T
i¼1
(19)
ii) Standard deviation of the wind power density:
sp ¼
In this study, the statistical properties of the wind power that are derived from the Gamma pdf for the wind speed variable can be obtained using Monte Carlo integration. However, this method can also be applied to the other wind speed pdfs, especially for a more complex wind speed pdf such as the Wakeby pdf, Kappa pdf or Mixture Weibull pdf.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½gðXÞ 4. Case study in Peninsular Malaysia
where:
Z
2
Var½gðXÞ ¼
ðgðxÞ E½gðXÞÞ fX ðxÞdx x
(20)
PT ¼
2 i¼1 ðgðxi Þ E½gðXÞÞ T
Table 3 Parameter estimations using the maximum likelihood method. Station
Mersing C. Highlands Malacca Kudat Putrajaya Sandakan
Weibull
Gamma
Inverse gamma
a
b
a
b
p
b
3.166 2.285 2.367 2.874 1.399 2.668
2.187 1.746 1.690 1.673 1.720 1.958
4.317 2.763 2.511 2.473 1.399 3.418
0.648 0.734 0.838 1.033 0.581 0.690
3.578 2.331 2.129 2.154 2.468 2.868
7.667 3.100 2.815 3.458 2.018 4.826
To provide a practical application regarding the methodologies that have been discussed above, a case study that involves a real data set will be presented in this section. The data are involved with six selected stations in Peninsular Malaysia, namely, the Mersing, Cameron Highlands, Malacca, Kudat, Putrajaya and Sandakan stations (shown in Table 1). The 010C Wind Speed Sensor has been used by the Department of Environment to collect the hourly wind speed data in Malaysia. The wind speed sensor of type 010C can provide accurate information on the horizontal wind speed. The lightweight three-cup anemometer has been used in virtually all of the applications in which a fast response and a low starting threshold are of paramount importance. The instrument was provided by Met One Instruments. Fig. 1 shows the 010C Wind Speed Sensor model, while Table 2 describes the specifications of the 010C Wind Speed Sensor, including its characteristics, its accuracy and its range value (for more details, please refer to [50]). In this study, the hourly wind speed data was collected from January 1, 2007 to November 30,
Please cite this article in press as: Masseran N, Evaluating wind power density models and their statistical properties, Energy (2015), http:// dx.doi.org/10.1016/j.energy.2015.03.018
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N. Masseran / Energy xxx (2015) 1e9
Fig. 2. Fitting a gamma, Weibull and inverse gamma pdf to the wind speed data.
2009. The missing data has been estimated by using the method of single imputation [51]. The parameter estimates for each density function have been performed using the maximum likelihood method. In addition, the goodness-of-fit procedures are determined by the KolmogoroveSmirnov statistic (KS) method. A distribution with a minimum value of KS will be selected to be the best model for the wind speed distribution at each station. Table 3 shows the results of the maximum likelihood estimation, while Table 4 shows the results of the goodness-of-fit for each station. In addition, Fig. 2 shows the fitted pdf model for the observed wind speed data for each station. From Table 4 and Fig. 2, a comparison between the three distributions reveals that the Weibull distribution is the best model for the wind speed at the Cameron Highlands, Kudat and Putrajaya stations, while for the stations of Mersing, Malacca and Sandakan, the Gamma distribution is better than the Weibull and Inverse Gamma distributions. In fact, the fitted pdf model in Fig. 2 shows that the Weibull and Gamma pdf is able to provide a good approximation of the observed wind speed data in each station. Thus, these pdf models are reliable for estimating wind energy, particularly the energy density function. However, because this study does not involve real wind turbines, the value of the airstream area for wind turbines as shown in Equation (2) is assumed to be equal to 1 m2,A ¼ 1. Thus, the computed power will be measured in terms of the power per unit area. However, those who are involved with real wind turbines can easily change the A value according to their study. In addition to this, the value of constant air density is rk ¼ 1.16 kg m3 [52]. Then, by using the results from Table 3, the statistical characteristics of wind power
density for each station can be computed using Equations (7)e(12) and (17e23). Fig. 3 shows the density plots that have information on the statistical properties of wind power for each station. Clearly, theoretical wind power density fits well with the histogram of empirical wind power data. Thus, it can be concluded that theoretical wind power density derived by the method of transformation provides good results. From Fig. 3, the mean power per unit area is in the range of 2e24 W/m2. The Mersing and Kudat stations have a greater mean power per unit area compared with the other stations. In addition, the standard deviation for all of the stations ranges between 4 and 45 W/m2. The Mersing and Kudat stations have a greater standard deviation compared with other stations. This can be seen clearly from the power density plot for the Mersing and Kudat stations. The maximum power per unit area for the Mersing and Kudat stations is more than 800 W/m2 and 600 W/m2, respectively. In addition, all of the density plots are found to be skewed to the right with each of the coefficients of skewness g1 > 4, which indicates a lack of symmetric properties for the power density. The kurtosis measures the peakness of the wind power density for each station. A higher value for the kurtosis indicates a sharper peak in the density function. This finding is clearly in agreement with the results shown in Fig. 3. The value of the kurtosis coefficient is greater than 35 for all of the stations,g2 > 4. Finally, the cumulative power density function for each station is shown in Fig. 4. Thus, based on Figs. 3 and 4, more information regarding the wind energy potential can be extracted from the wind speed data. This information provides a better understanding of the characteristics and properties of the power density for each station.
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N. Masseran / Energy xxx (2015) 1e9
Fig. 3. Wind power density and its statistical properties for each station.
Fig. 4. Cumulative power density functions for each station.
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N. Masseran / Energy xxx (2015) 1e9
5. Conclusions This study discusses the statistical properties of the wind power density function, particularly the mean power, standard deviation, skewness and kurtosis. The transformation method has been proposed for deriving a theoretical density function of wind power based on the wind speed pdf, such as the Gamma, Weibull and Inverse Gamma pdfs. The power density function derived from the Weibull pdf is also found to follow a Weilbull pdf with some modification of the parameters. Thus, its statistical properties of power density, such as the mean power, standard deviation, skewness and the kurtosis can be obtained by using the knowledge of the Weibull properties. However, for power density derived from the Gamma and Inverse Gamma pdfs, the equations for the power density function are difficult to match with any existing statistical model. Thus, the statistical properties of power density cannot be easily obtained. In fact, the solution using the method of integration will also be difficult. To overcome this problem, this study proposed the method of Monte Carlo integration. The method of Monte Carlo integration is efficient and reliable in providing a flexible solution for determining the statistical properties of wind power density. Although this study only provides a solution for the Gamma and Inverse Gamma pdfs, the proposed method also can be applied to any other statistical model, particularly for researchers that involve more complex pdf functions, such as the Kappa, Wakeby, Mixture Weibull, etc. Apart from that, the application of the proposed method has been demonstrated by a case study that involves wind speed data from several stations in Malaysia. Acknowledgments The authors are indebted to the staff of the Department of Environment and the Malaysian Meteorology Department for providing the hourly mean wind speed data that made this paper possible. This research would also not have been possible without the sponsorship from Universiti Kebangsaan Malaysia and the Ministry of Higher Education, Malaysia (grant number GGPM-2014056 and FRGS/1/2014/SG04/UKM/03/1) References [1] Akpinar EK, Akpinar S. A statistical analysis of wind speed data used in installation of wind energy conversion systems. Energy Convers Manag 2005;46:515e32. [2] Masseran N, Razali AM, Ibrahim K, Wan Zin WZ. Evaluating the wind speed persistence for several wind stations in Peninsular Malaysia. Energy 2012;37: 649e56. [3] Yaniktepe B, Savrun MM, Koroglu T. Current status of wind energy and wind energy policy in Turkey. Energy Convers Manag 2013;72:103e10. [4] Abbasi Tabassum, Premalatha M, Abbasi T, Abbasi SA. Wind energy: Increasing deployment, rising environmental concerns. Renew Sustain Energy Rev 2014;31:270e88. [5] Jung J, Broadwater RP. Current status and future advances for wind speed and power forecasting. Renew Sustain Energy Rev 2014;31:762e77. [6] Li G, Shi J. Application of Bayesian model averaging in modeling long-term wind speed distributions. Renew Energy 2010;6:1192e201. [7] Masseran N, Razali AM, Ibrahim K, Latif MT. Fitting a mixture of von Mises distributions in order to model data on wind direction in Peninsular Malaysia. Energy Convers Manag 2013;72:94e102. [8] Carrillo C, Montano Obando AF, Cidras J, Diaz-Dorado E. Review of power curve modeling for wind turbines. Renew Sustain Energy Rev 2013;21: 571e81. [9] Morgan EC, Lackner M, Vodel RM, Baise LG. Probability distribution for offshore wind speeds. Energy Convers Manag 2011;52:15e26. [10] Carta JA, Ramirez P, Velazquez S. Influence of the level of fit of a density probability function to wind-speed data on the WECS mean power output estimation. Energy Convers Manag 2008;49:2647e55. [11] Villanueva D, Feijoo A. Wind power distributions: a review of their applications. Renew Sustain Energy Rev 2010;14:1490e5. [12] Carta JA, Mentado D. A continuous bivariate model for wind power density and wind turbine energy output estimations. Energy Convers Manag 2007;48: 420e32.
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