Analysis of the upper-truncated Weibull distribution for wind speed

Energy Conversion and Management 96 (2015) 81–88

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Analysis of the upper-truncated Weibull distribution for wind speed Yeliz Mert Kantar ⇑, Ilhan Usta Department of Statistics, Faculty of Science, Anadolu University, Eskisehir 26470, Turkey

a r t i c l e

i n f o

Article history: Received 18 November 2014 Accepted 20 February 2015 Available online 9 March 2015 Keywords: Wind speed Weibull distribution Upper-truncated Weibull distribution Truncation point Wind power Model selection criteria

a b s t r a c t Accurately modeling wind speed is critical in estimating the wind energy potential of a certain region. In order to model wind speed data smoothly, several statistical distributions have been studied. Truncated distributions are defined as a conditional distribution that results from restricting the domain of statistical distribution and they also cover base distribution. This paper proposes, for the first time, the use of upper-truncated Weibull distribution, in modeling wind speed data and also in estimating wind power density. In addition, a comparison is made between upper-truncated Weibull distribution and well known Weibull distribution using wind speed data measured in various regions of Turkey. The obtained results indicate that upper-truncated Weibull distribution shows better performance than Weibull distribution in estimating wind speed distribution and wind power. Therefore, upper-truncated Weibull distribution can be an alternative for use in the assessment of wind energy potential. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction In the literature, it is known that Weibull is the most widely-used distribution to estimate wind energy potential because of its flexibility and easy computation [1–13]. However, it should be noted that the Weibull distribution is unable to model all the wind structures encountered in nature. For this reason, in recent years, in order to model wind speed data more smoothly, the use of a variety of statistical distributions has been proposed in a large number of studies. For instance, [14] considers normal-mixture distribution and mixture Weibull distribution versus the classical Weibull distribution. [15–17,14,18] introduce two mixture Weibull distribution to model wind speed data. [19–22] propose the distributions derived from the maximum entropy principle. Similarly, [23] and [24] respectively introduce distributions derived from the maximum entropy principle and minimum cross entropy principle. [25] presents a comparison of log-normal, gamma, Weibull and Rayleigh models. [26] provides a review and literature on the usage of various statistical distributions in modeling wind speed data. As well as the mentioned statistical distributions, Erlang, inverse normal and gumbel-maximum distributions are presented as wind speed distributions in [27], while a generalized extreme value distribution is used in [28]. On the other hand, [29] firstly introduces mixture Gamma–Weibull and mixture truncated normal distributions, while [30] proposes certain flexible families of distributions as an alternative to the Weibull ⇑ Corresponding author. Tel.: +90 222 335 05 80x4685. E-mail address: [email protected] (Y.M. Kantar). http://dx.doi.org/10.1016/j.enconman.2015.02.063 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.

distribution in estimating wind speed distribution. Similar to [30,31] evaluates the performance of the Johnson SB distribution in comparison to the Weibull distribution. Moreover, different statistical distributions have been proposed as an alternative to the Weibull distribution, and certain comparisons and reviews concerning these distributions have been conducted in the literature [12,27,32]. For example, [12] compares various distributions and shows that the Gamma distribution could be an efficient alternative to the Weibull distribution. [27] concludes that multiple distributions are found to produce good fit for wind speed data compared to the Weibull distribution. Consequently, the mentioned studies [15–17,14,18–31] emphasize that Weibull distribution (WD) does not present good performance in the modeling of wind speed data in comparison with a distribution with more parameters, for all wind types encountered in nature, such as low or high, skewed or kurtotic, or skewed and kurtotic wind speed. Thus, in order to minimize errors in wind power estimation, it is necessary to select the most appropriate distribution for the description of wind speed measured for a specific area. In statistics, a truncated distribution is defined as a conditional distribution that results from restricting the domain of the statistical distribution. Hence, truncated distributions are used in cases where occurrences are limited to values which lie above or below a given threshold or within a specified range. If occurrences are limited to values which lie below a given threshold, the lower (left) truncated distribution is obtained. Similarly, if occurrences are limited to values which lie above a given threshold, the upper (right) truncated distribution arises [33,34].

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Nomenclature A AIC cdf fWD (v) fTWD (v) k KS lnL LS m ML MM MSE MSEk MSEp n N p pdf PREF

wind turbine blade sweep area (m2) akaike information criterion cumulative distribution function Weibull pdf with two parameters upper-truncated Weibull pdf scale parameter of both WD and TWD (m/s) Kolmogorov–Smirnov distance negative of the natural logarithm of the likelihood function least squares number of parameters of the distribution maximum likelihood method of moments mean square error mean square error for k parameter mean square error for p parameter number of all observed wind speed data number of the wind speed class shape parameter of both WD and TWD probability density function mean power density based on time series wind data (Ws/m)

Empirical wind speed data can be represented by upper-truncated distributions, since observed wind speed data is limited to a maximum value of wind speed value. For this reason, we firstly propose an upper-truncated Weibull distribution (TWD) to model wind speed data. The main objective of this paper is to introduce, for the first time, TWD for a description of wind speed distribution and also to evaluate the capability of TWD in modeling wind speed. Therefore, the present study might contribute to the related literature by adopting a new perspective of modeling wind speed data and showing the superiority of TWD over WD. Additionally, the present study might provide different and useful insights to engineers and scientists dealing with wind energy by firstly introducing a truncation parameter into WD. The remainder of this paper is organized as follows: Section 2 introduces TWD and provides a discussion on the estimation of parameters of TWD. Section 3 provides the wind speed data used in the study. Section 4 presents the results of analyses concerning TWD and WD. Section 5 concludes the study with the obtained results and certain suggestions for further research. 2. Upper-truncated Weibull distribution for wind speed distribution

FðxÞ ; FðTÞ

06x6T

vi WD xj yj zj

ð1Þ

where F(x) is the cdf of the X random variable. By using Eq. (1), an upper-truncated distribution can be obtained from the non-truncated distribution and suitable truncation point.

mean power density based on TWD (Ws/m) mean power density based on WD (Ws/m) coefficient of determination root mean square error truncation point in the TWD upper-truncated Weibull distribution wind speed (m/s) ith observed wind speed data (m/s) Weibull distribution jth observed probability j = 1, . . ., n jth predicted probability calculated from a special distribution jth computed value from the correlation equation for the same value of xj

Greek letters U gamma function q air density (kg/m3) 2 rTWD variance of TWD lTWD mean of TWD

   p , cdf and pdf of If F(x) is taken as Weibull cdf, 1  exp  kx upper-truncated Weibull random variable can be given respectively as Eqs. (2) and (3):

    p 1  exp  kx     ; GðxÞ ¼ p 1  exp  Tk

06x6T

     p1 p ðp=kÞ kx exp  kx     gðxÞ ¼ ; p 1  exp  Tk

06x6T

ð2Þ

ð3Þ

where k is the scale parameter and p is the shape parameter based on the characteristics of the usual WD [34] and the truncation point T is generally assumed to be known. 2.1. Mean and variance of upper-truncated Weibull distribution The mean and variance of TWD with a known truncation point are respectively given as follows [33–35]:

lTWD

Upper-truncated distribution is applicable to a situation where the range of random variable is bounded from above by an unknown cut-off point, called a truncation point. In other words, if the values of random variables are observed in the interval [0, T], then upper-truncated distribution can be used. Wind speed measurements are generally observed in the range of [0, V], and therefore, upper-truncated distribution can be applied to model wind speed data. Upper truncated cumulative distribution function (cdf) is defined as follows:

GðxÞ ¼

PTWD PWD R2 RMSE T TWD v

  kC 1 þ 1p    ¼ p 1  exp  Tk    p  2 T 1  exp  p k   ,   p  2 1 T 1  exp  C2 1 þ p k

ð4Þ

 

r2TWD ¼ k2 C 1 þ

ð5Þ

Depending on the Eqs. (4) and (5), moment estimation procedure can be applied to obtain the parameters of TWD. 2.2. Discussion on parameter estimation methods In this section, we provide a description of the maximum likelihood (ML), least squares (LS) methods and the method of moments (MM) for estimating the unknown parameters of TWD with a known truncation point T, which is taken to be the largest sample observation, as in the literature. In addition, we carry out

Y.M. Kantar, I. Usta / Energy Conversion and Management 96 (2015) 81–88

a study of the Monte Carlo simulation to decide which of the considered methods is best for estimation of the parameters of TWD. 2.2.1. Maximum likelihood method ML estimates are obtained by maximizing the log-likelihood function of a random sample from TWD. The log-likelihood function corresponding to TWD is given as follows:

X xi  X xi p log L ¼ n log p  n log k  ðp  1Þ  ln k k    p  T  nl n  exp  k

ð6Þ

where xi 2 ½0; T; i ¼ 1; . . . ; n. If the truncation point (T) is taken as the largest sample observation (T ? xmax), the log-likelihood function reaches the maximum value. By maximizing the log-likelihood function given Eq. (6) with respect to unknown parameters, k and p, the estimates are obtained with simultaneous solution of the following equations:

       p p exp  Tk ln Tk Tk @logL n X xi  X xi xi p   ¼  n þ ln ln ¼0  p @p p k k k 1  exp  T k

ð7Þ       p p T p exp  Tk k k @logL n n p X  xi  p     ¼0 ¼  þ ðp  1Þ þ n p @k k k k k 1  exp  Tk

It should be noted that there is no explicit solution for Eqs. (7) and (8), and thus iterative methods should be used [33,35,36]. In this paper, we apply the well-known Newton-Rapson to obtain parameters of TWD. See [37] for detailed information about the method. 2.2.2. Least squares method The LS estimates of the parameters of TWD can be derived by minimizing the functions given in Eqs. (9) or (10) with respect to parameters k and p.

ð9Þ

i¼1

or

    0 12 p n 1  exp  xki X i  0:4 @ A      p n þ 0:3 1  exp  Tk i¼1

ð10Þ

b i Þ, which is the estimate of G(xi), is generally taken as where Gðx i0:4 : In order to estimate parameters (k, p), we have to use a numernþ0:3 ical method [36]. 2.2.3. Method of moments The method of moments involves equating sample moments with population moments. Thus, moment estimates are obtained with simultaneous solution of Eqs. (11) and (12) for the known truncation point.

  n kC 1 þ 1p X xi    ¼ p T n 1  exp  k i¼1

ð11Þ

83

ð12Þ

2.3. Performance comparison of parameter estimation methods To analyze the performance of the three aforementioned methods in estimating the parameters of TWD, a simulation is carried out for various combinations of values of shape, scale and truncation parameters. It can be observed in wind speed studies that shape parameters change typically in the 1.5–2.5 range, and the range of scale parameter is generally 3.5-6. For this reason, we consider the shape parameter as 1.5, 2, 2.5 and the scale as 4, 4.5. In addition, the truncation point T is chosen so as to be the quantile of 95% for WD. Random data from TWD is generated by the quantile function of the upper-truncated Weibull random variable, given as follows:

     p   1=p T QðuÞ ¼ k ln 1  u 1  exp  k

ð13Þ

where U as a random variable is distributed as uniform distribution (U  Uniform(0, 1)) in Eq. (13). 10,000/n replications are generated for each simulation combination. Parameters of TWD are estimated by ML, LS methods and MM for each sample. In order to compare these methods, MSE criterion is used [38,39]. MSE formulae for k and p are given in Eqs. (14) and (15):

MSEk ¼ ð8Þ

n  2 X b iÞ Gðxi Þ  Gðx

 h    i p 2 n k C 1 þ 2p 1  exp  Tk X x2i ¼ h     i2 p n i¼1 1  exp  Tk

10;000=n 2 X  1 ^ kk 10; 000=n i¼1

ð14Þ

10;000=n X 1 ^ Þ2 ðp  p 10; 000=n i¼1

ð15Þ

and

MSEp ¼

All computations for the simulation are performed using the Matlab program. From the simulation results presented in Table 1 for scale parameter k, we conclude that while ML apparently shows the best performance, MM yields the worst performance for all the considered parameters and sample sizes. In particular, the ML estimation method provides a substantial improvement over LS and MM for large sample sizes. As can be seen in Table 2, for shape parameter p, ML gives more satisfactory results than other considered methods. Furthermore, ML is the best method for large sample sizes, n = 500 and 750. On the other hand, we compare the methods for parameters p = 2.5, 3 and k = 3.5, 5, 5.5. However, since the considered method produces similar results, it is not reported here. Consequently, the ML method is used to find the parameters of the TWD as well as WD in this study. 3. Wind speed data In order to evaluate the suitability of the TWD distribution, monthly and yearly wind speed data, measured hourly at 10 m above ground level in three different regions of Turkey, with different geographical structures and climates, are used. The considered regions are Florya, Yunak and Ulukısßla denoted by Station A, Station B and Station C, respectively. Wind speed (m/s) data, used for the considered regions, is taken from the Turkish State Meteorological Service. Information regarding these regions is given in Table 3. Descriptive statistics of the monthly and yearly wind speed data of these regions are given in Table 4. As seen in Table 4, Stations A, B and C exhibit different statistical characteristics of wind speed,

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Table 1 MSE values for k parameter of the TWD.

Table 3 Details of the regions for which the wind speed data were analyzed.

Scale

Shape

Methods

Sample size, n 50

100

250

500

750

4

1.5

ML LS MM

0.2201 0.2261 0.3180

0.1660 0.1679 0.2813

0.0813 0.0901 0.2407

0.0351 0.0413 0.2273

0.0223 0.0276 0.2252

4

2

ML LS MM

0.1730 0.1850 0.1962

0.1168 0.1227 0.1481

0.0409 0.0467 0.1276

0.0219 0.0246 0.1180

0.0112 0.0124 0.1131

4.5

1.5

ML LS MM

0.2818 0.2926 0.4140

0.2175 0.2122 0.3505

0.1075 0.1158 0.3116

0.0534 0.0581 0.2874

0.0319 0.0375 0.2794

4.5

2

ML LS MM

0.2185 0.2160 0.2338

0.1349 0.1347 0.1813

0.0539 0.0598 0.1457

0.0282 0.0315 0.1313

0.0126 0.0158 0.1282

Table 2 MSE values for p parameter of TWD. Scale

4

4

4.5

4.5

Shape

1.5

2

1.5

2

Methods

Sample size, n

Stations

Latitude

Longitude

Height of ground above sea level (m)

Florya Yunak Ulukısßla

40°970 38°820 37°540

28°780 31°720 34°480

37 1148 1453

Table 4 Descriptive statistics for monthly and yearly wind speed data (m/s) for 2006. Mean

Variance

Skewness

Kurtosis

n

January

Station A Station B Station C

2.8200 2.8719 2.8032

2.5379 6.1973 3.8633

0.5392 1.9693 1.0360

2.8208 8.5355 3.4990

744 743 744

February

Station A Station B Station C

2.2350 3.6637 3.1140

2.0287 10.6338 3.7883

0.6079 1.0912 0.8048

2.3520 3.7198 3.1801

666 669 672

March

Station A Station B Station C

2.3770 4.9007 4.0083

2.0030 11.8172 6.3268

0.4275 0.8045 0.7548

2.2487 3.2887 3.5171

732 744 743

April

Station A Station B Station C

2.1356 4.2006 3.1612

1.7201 6.9411 4.0567

0.7328 0.9539 0.7629

3.2950 4.0633 3.1472

716 719 719

50

100

250

500

750

ML LS MM

0.0299 0.0337 0.0305

0.0176 0.0220 0.0304

0.0103 0.0144 0.0298

0.0057 0.0077 0.0267

0.0033 0.0048 0.0249

May

ML LS MM

0.0505 0.0578 0.0366

0.0345 0.0420 0.0239

0.0164 0.0218 0.0159

0.0083 0.0097 0.0086

0.0059 0.0073 0.0077

Station A Station B Station C

1.8662 3.3430 2.5312

1.2823 3.5205 2.0861

1.0040 0.6306 0.6451

4.2461 3.1285 3.0161

737 744 744

June

ML LS MM

0.0290 0.0330 0.0304

0.0213 0.0240 0.0306

0.0109 0.0135 0.0285

0.0056 0.0070 0.0269

0.0029 0.0039 0.0233

Station A Station B Station C

2.0397 3.7960 2.8038

1.1063 3.8974 2.4375

0.1353 0.6797 0.4139

2.2038 3.2800 2.6984

711 720 719

July

ML LS MM

0.0523 0.0606 0.0382

0.0354 0.0412 0.0239

0.0157 0.0215 0.0150

0.0084 0.0109 0.0089

0.0067 0.0080 0.0076

Station A Station B Station C

2.9067 3.5648 2.6942

2.3713 2.7747 2.1501

0.3674 0.4806 0.2441

2.8063 2.7156 2.3944

733 744 744

August

Station A Station B Station C

2.1096 3.9113 2.4763

1.6841 3.8796 2.2246

0.3592 0.4442 0.3017

2.3203 2.5747 2.1282

726 744 743

September

Station A Station B Station C

2.2902 3.2286 2.3974

1.9580 4.1185 2.1965

0.6577 0.9141 0.5445

3.0568 3.4955 2.5538

711 718 720

October

Station A Station B Station C

2.3007 2.6817 2.2755

2.1332 2.7896 2.2250

0.6793 1.2028 0.7777

2.7463 6.0894 2.8756

729 743 744

November

Station A Station B Station C

2.4281 2.7619 2.1704

1.6709 5.6693 2.0680

0.3915 1.7507 1.2070

2.8252 5.9644 4.3516

704 720 720

December

Station A Station B Station C

1.8041 1.9755 1.9922

1.2884 2.7192 1.8604

0.4750 1.5195 1.2531

2.1548 4.8736 4.5327

608 744 744

Whole year

Station A Station B Station C

2.2843 3.4052 2.6995

1.9240 5.9278 3.2048

0.6419 1.2867 1.0527

3.0161 5.3517 4.5123

8517 8752 8756

since they have different degrees of positive skewness and kurtosis. It is clear that Station B has a higher mean, variance, skewness and kurtosis than Station A or C for most of the considered months. Also, if we look at the amount of measured hourly wind speed data, there is an almost 1% loss in wind speed data. 4. Results and discussion Various analyses have been made based on monthly and yearly wind speed data measured in Stations A, B and C to evaluate the suitability of TWD relative to WD. For this purpose, we consider different criteria; R2, RMSE, KS and AIC, as shown in Table 5. The best distributional model can be determined according to the highest value of R2 and the lowest values of RMSE, KS and AIC. It can be seen from Table 5 that AIC has two factors; while lnL presents the goodness-of-fit of the ML estimates, k penalizes AIC for each added parameter. Moreover, we take into account errors in calculating the mean power densities using TWD and WD compared to those using measured wind speed distribution. It should be noted that different criteria may provide inconsistent ranking orders of fit performance among the candidate distributions. Estimates of the parameters of WD and TWD and the criteria for Station A are given in Table 6. It can be concluded from Table 6 that TWD shows good performance relative to WD in terms of all the considered criteria. For example, in the case of wind speed data measured in December, TWD provides a good fit to wind data in terms of AIC, since the AIC value of TWD is less than the value of WD. Similarly, MSE and KS values show the superiority of TWD

Table 5 The formulas of criteria for model evaluation. Criteria R2

Formulas 1

PN  j¼1

yj  xj

2 P

N j¼1

 2 yj  zj

RMSE



KS

max16i6n ðFðv i Þ  ði  1Þ=n; i=n  Fðv i ÞÞ Qn  2 ln i¼1 f ðv i Þ þ 2m

AIC

2 1=2 PN  N j¼1 yj  xj

over WD. On the other hand, this result in favor of TWD is supported by the highest R2. Considering Table 6 for all months of Station A, the following conclusions can be derived.

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 According to the AIC criterion, TWD shows the best performance for all considered months and for the whole year.  For all months except July and October, TWD provides comparable efficiency in terms of R2 and RMSE.  KS generally favors TWD compared with WD for almost all months. The calculated results for Station B are given in Table 7. The derived results can be presented as follows:

 In terms of AIC, TWD generally shows better performance than WD for all months, except January and October.  With regard to RMSE, KS and R2, TWD provides a good fit compared to WD for all months, except July, September and December.  While TWD presents good performance for February, March, April, August and October, according to all the criteria, TWD provides a comparable performance with WD for the other months.

Table 6 Estimates of parameters of WD, TWD and criteria for Station A. Months

Station A January February March April May June July August September October November December Whole year

Parameters

WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD

k

p

3.1716 3.1916 2.4899 2.6164 2.6583 2.8483 2.3857 2.3883 2.0914 2.0943 2.2877 2.4281 3.2544 3.3420 2.3337 2.4763 2.5561 2.5716 2.5644 2.6854 2.7228 2.7634 2.0080 2.2183 2.5501 2.5547

1.8259 1.7982 1.5885 1.4898 1.6959 1.5548 1.6547 1.6492 1.7014 1.6949 1.9689 1.7958 1.8987 1.8072 1.5705 1.4506 1.6530 1.6306 1.5977 1.5034 1.9043 1.8446 1.5921 1.4339 1.6612 1.6528

AIC

RMSE

KS

R2

2710.67 2708.75 2225.07 2204.24 2487.54 2454.24 2293.59 2292.00 2130.97 2130.22 2077.89 2046.47 2689.33 2675.09 1178.36 1164.35 1187.92 1186.31 2472.24 2450.70 1160.49 1156.00 1772.91 1736.81 28379.58 28370.54

0.02003 0.01938 0.03272 0.03274 0.02203 0.01546 0.01053 0.01062 0.02770 0.02782 0.03053 0.02581 0.02557 0.02603 0.03022 0.02616 0.02047 0.02008 0.00958 0.01092 0.03155 0.03100 0.03614 0.02682 0.01107 0.01097

0.05582 0.05462 0.05779 0.05791 0.05055 0.03142 0.03212 0.03212 0.04342 0.04478 0.06234 0.04810 0.06231 0.06230 0.06229 0.05157 0.04782 0.04501 0.05899 0.05213 0.07161 0.07129 0.06415 0.05592 0.03972 0.03884

0.94918 0.95096 0.89843 0.89745 0.95208 0.97025 0.99131 0.99109 0.96077 0.95972 0.92758 0.93924 0.91096 0.90502 0.91713 0.91977 0.96127 0.96200 0.99306 0.98618 0.90911 0.90509 0.90733 0.92979 0.98940 0.98941

AIC

RMSE

KS

R2

3003.38 3004.83 3070.73 3061.74 3737.16 3733.13 3274.62 3274.17 2957.94 2957.95 2939.39 2939.76 2821.15 2813.06 3049.58 3048.55 2887.11 2885.86 2694.39 2696.38 2840.49 2840.86 2428.52 2427.75 37267.36 37268.48

0.00816 0.00819 0.02326 0.02177 0.01121 0.01049 0.01339 0.01338 0.00812 0.00782 0.00972 0.00972 0.01731 0.01809 0.01463 0.01406 0.01101 0.01115 0.01530 0.01530 0.03361 0.03393 0.03515 0.03564 0.00514 0.00514

0.04850 0.04813 0.09118 0.08670 0.05076 0.04615 0.03839 0.03671 0.02554 0.02688 0.03240 0.03223 0.04167 0.04704 0.04032 0.03763 0.04318 0.04457 0.06326 0.06326 0.09722 0.09722 0.09140 0.09274 0.02891 0.02879

0.98939 0.98934 0.88346 0.89786 0.94483 0.95014 0.94851 0.94853 0.98934 0.98963 0.98205 0.98207 0.95489 0.95232 0.95666 0.95871 0.97883 0.97795 0.97158 0.97158 0.87469 0.87243 0.92240 0.92049 0.99412 0.99411

T

8.5 6.3 6.1 8.3 6.9 4.7 7.4 5.9 7.7 6.5 6.5 4.7 8.5

Table 7 Estimates of parameters of WD and TWD and criteria for Station B. Months

Station B January February March April May June July August September October November December Whole year

Parameters

WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD

k

p

3.0918 3.0961 3.7948 3.9199 5.3807 5.4479 4.7006 4.7222 3.7569 3.7692 4.2892 4.2936 4.0266 4.0792 4.4230 4.4417 3.6177 3.6395 3.0082 3.0082 2.9964 3.0084 2.1501 2.1638 3.7645 3.7651

1.2405 1.2371 1.0959 1.0577 1.4148 1.3846 1.6458 1.6291 1.8326 1.8166 2.0247 2.0179 2.2729 2.2060 2.1047 2.0793 1.6523 1.6316 1.6770 1.6769 1.2716 1.2635 1.2932 1.2806 1.4497 1.4489

T

16.4 16.5 18.3 14.6 10.6 11.8 8.3 10.6 10.9 13.9 13.6 9 18.3

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Table 8 Estimates of parameters of WD and TWD and criteria for Station C. Months

Parameters

Station C January

WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD WD TWD

February March April May June July August September October November December Whole year

k

p

3.1190 3.1777 3.4905 3.5535 4.4809 4.4968 3.5322 3.5860 2.8560 2.8611 3.1566 3.1834 3.0360 3.0546 2.7716 2.7995 2.6864 2.6940 2.5438 2.5799 2.4392 2.4425 2.2317 2.2345 3.0157 3.0158

1.4964 1.4557 1.6582 1.6088 1.6314 1.6174 1.6110 1.5685 1.8380 1.8282 1.8572 1.8202 1.9053 1.8741 1.6807 1.6438 1.6618 1.6494 1.5828 1.5446 1.6176 1.6121 1.5648 1.5601 1.5719 1.5717

9.4 9.4 14.4 10 8.4 8.1 7.8 7.8 8.6 7.4 8.6 8.3 14.4

 TWD apparently shows the best performance according to the AIC criterion for all months except May, November and December.  TWD exhibits a significantly better fit to wind data when taking into account the KS criterion for all months and the whole year.  TWD clearly provides a good fit compared to WD for all months, except December, and for the whole year in terms of RMSE and R2.  Taking into account all the criteria, TWD provides a convenient representation of the wind speed data for Station C. On the other hand, it is known that the power density based on the probability density function f(v) is calculated as follows:

1 qA 2

Z

1

v 3 f ðv Þdv

ð16Þ

0

where v shows wind speed, q is air density, which is assumed to be equal to 1.225 kg/m3 in this study, and A is the wind turbine blade sweep area. If f(v) is WD, the mean wind power density is provided by Eq. (17).

PWD

  1 3 3 ¼ qk C 1 þ 2 p

ð17Þ

If f(v) is taken as TWD in Eq. (16), we calculate the mean power density (PTWD) using a basic numerical integration technique. The mean power density of time series wind data, denoted as PREF is estimated as follows:

PREF ¼

n v 3i 1 X qA 2 n i¼1

In Eq. (18), PREF is known as ‘reference mean power density’.

RMSE

KS

R2

1427.05 1422.23 1324.73 1320.14 3326.61 3326.58 2886.51 2880.41 2536.16 2537.06 2599.97 2596.75 2612.41 2610.41 2595.86 2592.06 2470.51 2470.99 2508.91 2503.21 2330.04 2331.35 2308.23 2309.61 32504.77 32506.62

0.02202 0.02180 0.00857 0.00692 0.01962 0.01907 0.01980 0.01928 0.02932 0.02915 0.09264 0.08942 0.06480 0.06397 0.06900 0.06751 0.04261 0.04207 0.07892 0.07572 0.02905 0.02900 0.04179 0.04189 0.02089 0.02089

0.07134 0.07043 0.05345 0.04863 0.05731 0.05574 0.06264 0.05843 0.08483 0.08389 0.04634 0.04551 0.12255 0.12007 0.13565 0.13139 0.09473 0.09358 0.02007 0.01782 0.10708 0.10669 0.13364 0.13340 0.07799 0.07797

0.93221 0.93389 0.98877 0.99117 0.89821 0.90001 0.92776 0.92939 0.91787 0.91858 0.75842 0.76272 0.60496 0.60959 0.56888 0.57902 0.82483 0.82788 0.96202 0.96817 0.93601 0.93625 0.89389 0.89344 0.94167 0.94169

T

The following conclusions can be drawn from the results of the criteria for Station C given in Table 8:

Pf ¼

AIC

ð18Þ

The other performance criterion is the power density error to compare plausible distributional models for wind speed. Therefore, in order to assess the capability of TWD in estimating wind power, we calculate the power density error given in the following equation [40–43]:

  PREF  PD    100 error ¼   P

ð19Þ

REF

where PD is the calculated mean wind power based on TWD and WD. In Eq. (19), errors for WD and TWD are provided as percentages. Table 9 provides the wind power density error, calculated from Eq. (19), for WD and TWD. Taking into account the error values of TWD and WD for Station A, it can be seen that TWD can provide better power estimation results than WD for the all considered months except May. Moreover, TWD provides substantial improvement over WD in estimating wind power for February and December. On the other hand, it can be deduced from Table 9 that for Station B, TWD yields less errors than WD for most of the

Table 9 Wind power density error (%) for WD and TWD. Station A

January February March April May June July August September October November December Whole year

Station B

Station C

PWD

PTWD

PWD

PTWD

PWD

PTWD

2.3538 5.3907 6.1681 2.7705 1.6480 5.0888 4.2990 12.5827 4.0711 4.2285 3.7300 7.8766 4.1963

1.4618 0.0074 1.3106 2.3644 2.0799 3.1753 2.7185 7.2068 2.7915 0.8520 2.5913 0.3877 3.6516

16.2689 16.9582 7.8293 2.8469 1.5525 0.2142 0.9997 0.5635 4.4092 4.3947 16.1247 11.0285 3.9949

17.4324 2.5274 4.1902 3.9453 0.9888 0.0476 0.3961 0.1896 5.7144 4.3929 18.3369 13.9997 4.1603

2.2003 0.0153 2.3252 2.2040 0.4476 3.1818 4.1774 7.8370 4.2264 1.4535 6.4749 7.0352 2.2557

6.1309 0.0117 1.3587 0.4265 0.0590 2.2006 3.5125 6.2121 3.4517 1.2909 7.0025 7.5836 2.2287

87

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considered months. Also, the results of Station C show that TWD provides a smaller error than WD in calculating wind power density for the considered months, except for January, November and December. If the results for Station A, B and C are evaluated together, it can be derived that TWD can provide better results than WD in terms of power error criterion for the considered wind speed cases, except data, which has a noticeably high positive skewness and kurtosis. In addition, we provide a few examples of graphs of the pdfs corresponding to the TWD and WD to observe the suitability of fitting these distributions to wind speed data. Station A

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Station A

0.45

WD: k=2.6583, p=1.6959 TWD: k=2.8483, p=1.5548, T=6.1

Probability density function

Probability density function

Fig. 1. provides a comparison of the histogram of the observed monthly wind speed, the estimated WD and TWD for two months of Station A. It can be seen from Fig. 1(a) that the probability of wind speed of 2–3 m/s is overestimated, whereas it is underestimated for a speed of 4–7 m/s while using WD. On the other hand, TWD accurately describes an empirical distribution of wind speed data, particularly the peak and tail of wind speed data. Similarly, good performance of TWD is observed in Fig 1(b). Fig. 2(a) and (b) demonstrate the histogram, and the estimated WD and TWD for Station B. These figures illustrate that WD and TWD provide similar graphs. However, TWD exhibits good fit to the peak of wind speed data.

WD: k=2.2877, p=1.9689 TWD: k=2.4281, p=1.7958, T=4.7

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0

1

2

3

4

5

6

0

7

0

1

2

3

4

Wind speed (m/s)

Wind speed (m/s)

(a)

(b)

5

Fig. 1. Histogram and pdf graphs of WD and TWD for monthly wind speed data measured at Station A.

Station B

0.2

0.15

0.1

0.05

0

0

2

4

6

8

Station B

0.3

WD: k=3.7948, p=1.0959 TWD: k=3.9199, p=1.0577, T=16.5

Probability density function

Probability density function

0.25

10

12

14

16

WD: k=4.0266, p=2.2729 TWD: k=4.0792, p=2.2060, T=8.3

0.25 0.2 0.15 0.1 0.05 0

18

0

1

2

3

4

5

6

Wind speed (m/s)

Wind speed (m/s)

(a)

(b)

7

8

9

Fig. 2. Histogram and pdf graphs of WD and TWD for monthly wind speed data measured at Station B. Station C

0.2

0.15

0.1

0.05

0

0

2

4

Station C

0.35

WD: k=3.4905, p=1.6582 TWD: k=3.5535, p=1.6088, T=9.4

6

8

10

Probability density function

Probability density function

0.25

WD: k=2.5438, p=1.5828 TWD: k=2.5799, p=1.5446, T=7.4

0.3 0.25 0.2 0.15 0.1 0.05 0

0

1

2

3

4

5

Wind speed (m/s)

Wind speed (m/s)

(a)

(b)

6

Fig. 3. Histogram and pdf graphs of WD and TWD for monthly wind speed data measured at Station C.

7

8

88

Y.M. Kantar, I. Usta / Energy Conversion and Management 96 (2015) 81–88

Lastly, we provide two examples of graphs of the pdfs corresponding to the TWD and WD for Station C in Fig. 3(a) and (b). Similar to Station A and B, TWD shows good fit to wind data and is particularly suitable for a peak of wind data. 5. Conclusions The main results obtained from the presented study can be listed as follows: 1. TWD, which includes usual WD as a special case, is used to model wind speed data for the first time. 2. A Monte Carlo simulation is carried out to find the best estimation method for TWD. The simulation results show that ML among the considered methods seems to be a more efficient estimator for the parameters of TWD in terms of MSE. 3. The fitting accuracy of the proposed TWD is judged from different model selection criteria commonly used in wind energy literature. It is found that TWD outperforms WD for most of the wind speed data measured in the considered regions of Turkey. 4. Moreover, a comparison has also been made between TWD and WD in terms of ability to describe reference mean wind power. It is observed that TWD yields less mean power error than WD for most of the considered wind speed data. 5. Consequently, TWD can be alternatively used to evaluate the wind energy potential of a specified region.

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