Validity analysis of maximum entropy distribution based on different moment constraints for wind energy assessment

Energy 36 (2011) 1820e1826

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Energy journal homepage: www.elsevier.com/locate/energy

Validity analysis of maximum entropy distribution based on different moment constraints for wind energy assessment Feng Jiao Liu a, Tian Pau Chang b, * a b

Department of Electrical Engineering, Nankai University of Technology, Nantou 542, Taiwan Department of Computer Science and Information Engineering, Nankai University of Technology, Nantou 542, Taiwan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 July 2010 Received in revised form 18 November 2010 Accepted 20 November 2010 Available online 4 February 2011

Knowing about wind speed distribution for a specific site is very essential step in wind resource utilizations. In this paper, a probability density function with the maximum entropy principle is derived using different algorithm from previous studies. Its validity considering various numbers of moment constraints is tested and compared with the conventional Weibull function in terms of computation accuracy. Judgment criterions include the Chi-square error, root mean square error, maximum error in cumulative distribution function as well as the relative error of wind power density between theoretical function and observation data. Wind sample data are observed at four wind farms having different weather conditions in Taiwan. The results show that the entropy quantities reveal a negative correlation with the number of constraints used, regardless of station considered. For a specific site experiencing more stable weather condition where wind regimes are not too dispersive, the conventional Weibull function may accurately describe the distribution. While for wind regimes having two humps on it, the maximum entropy distributions proposed outperform a lot the Weibull function, irrespective of wind speed or power density analyzed. For the consideration of computation burden, using four moment constraints in calculating maximum entropy parameters is recommended in wind analysis. Ó 2010 Elsevier Ltd. All rights reserved.

Keywords: Probability density function Maximum entropy distribution Weibull distribution Wind speed Wind power density

1. Introduction The utilization of wind resource becomes more and more popular around the world due to shortage of fossil fuels. Knowing about wind characteristics for a specific site is pretty important before utilizing wind energy. Two-parameter Weibull probability density function (pdf) has widely been adopted to describe the wind speed distribution in specialized literature over the last decades [1e5]. It even became a reference distribution in commercial wind energy software such as Wind Atlas Analysis and Application Program [6]. Celik [7] analyzed the wind data measured at the southern region of Turkey and summarized that the Weibull function is better than Rayleigh function in fitting wind speed distributions. Akpinar and Akpinar [8] applied the Weibull function to evaluate the wind energy potential in Turkey using hourly wind data from 1998 to 2002 and found that the annual shape and scale parameters of Weibull function varied a lot among the five years. Shape parameters range from 1.51 to 1.70 while scale parameters range from 5.54 to 6.12. Sulaiman et al. [9] described the wind

* Corresponding author. Tel.: þ886 49 2563489; fax: þ886 49 2315030. E-mail address: [email protected] (T.P. Chang). 0360-5442/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2010.11.033

observation data in Oman using Weibull function; the KolmogoroveSmirnov test was adopted to examine the goodness of fit considering 1% and 5% of confident level. Chang [10] analyzed the wind speed data observed in Taiwan and concluded that the maximum likelihood method outperforms other numerical methods in estimating Weibull parameters. However many researches have shown that the conventional Weibull function does not fit very well with wind distribution measured in some cases. For example, there are two or more humps appeared in the distribution resulting maybe from special climatic conditions; in this context, a bimodal mixture Weibull function (so called WeibulleWeibull pdf) has been verified that it is feasible in describing this kind of data [11e13]. Therein five parameters need to be resolved, i.e. two shape parameters and two scale parameters and one weighting value. On the other hand, if the appearance opportunity of null winds for a particular location is significant, then the applicability of both the Weibull and WeibulleWeibull pdfs may face a new challenge as stated in Refs [14e16]. The maximum entropy principle (MEP) has been applied to various engineering fields to predict the most likely probability distribution constrained by a given set of physical conditions if the information available is limited. Li and Li [14,17,18] adopted the concept of maximum entropy principle and proposed a kind of

F.J. Liu, T.P. Chang / Energy 36 (2011) 1820e1826

Nomenclature f () G H n O() P T() v v3

Probability density function Gradient matrix of Lagrange multiplier Entropy Sample size, dimensionless Observed cumulative distribution function Wind power density, W/m2 Theoretical cumulative distribution function Wind speed, m/s Mean of wind speed cubes, m3/s3

Greek letters a Shape parameter of Weibull function, dimensionless b scale parameter of Weibull function, m/s r air density, kg/m3 ln Lagrange multiplier, dimensionless mn moment constraint d residual vector of Lagrange multiplier q residual vector of moment constraint

probability density function to analyze wind speed distribution, in which a term of r-powers of wind speed is added before an exponential function. Where, r is a non-negative number ranging from 0 to 5. They concluded that the MEP type distribution describes both wind speed and power density more accurately than the Weibull function. Similar results were obtained by other researches using different wind data [19,20]. Shamilov et al. [21] applied firstly the MinMaxEnt distribution, a kind of MEP distributions, to wind energy field to model wind data; they stated that the MinMaxEnt distribution shows better flexibility than the known Weibull distribution in estimating wind speed distribution and power density. Akpinar and Akpinar [16] compared the Weibull, WeibulleWeibull, MEP and Singly Truncated Normal Weibull mixture (TNW) pdfs in assessing wind energy potential and concluded that the TNW pdf is superior to others in terms of the analysis of root mean square error, Chi-square error and R2 coefficient. However, in the researches concerning MEP distribution aforementioned, only three low-order physical constraints were imposed in solving maximum entropy distribution, i.e. the conservation of mass, momentum and energy for air stream. A study using more physical constraints to calculate the distribution of wind speed is rarely found in literature. In this paper, the MEP distributions with three, four and five constraints are compared; a new algorithm used to calculate its relevant parameters is presented. Hourly mean wind speed observed in Taiwan from 2006 to 2008 at four wind farms having different weather conditions are selected as sample data to test their performance. The first station Dayuan locates at the northwestern plain of Taiwan; the northeastern monsoon is active in winter months; the height of anemometer is 64.7 m above ground level. The second station Changhua locates near shoreline in central Taiwan having strong wind in winter but the southwestern monsoon becomes prevailing in summer and autumn, with the anemometer height of 67 m. The third one Hengchun is at the southern peninsula experiencing more stable weather conditions throughout the year, with the same anemometer height as Dayuan. The fourth one Chungtun is at a small island in Taiwan Strait experiencing the highest wind in winter and spring, anemometer height is 46 m above ground level.

1821

The analyses of the present study include the wind speed and wind power density considering the Chi-square error, root mean square error and maximum error of cumulative distribution function in the KolmogoroveSmirnov test. Additionally the relative error between the power density calculated from theoretical function and that from observed time-series data is considered as a judgment too. The performance of conventional Weibull function will be compared as well. A powerful computer program written in MATLAB languages would be available through email request via [email protected]. 2. Weibull function The probability density function of the Weibull distribution can be expressed as:

f ðvÞ ¼


a1

a v b b


a  v exp 

(1)

b

where v is the wind speed, a is the shape parameter (dimensionless), b is the scale parameter having the same unit as speed. Weibull shape parameter reflects the width of data distribution, the larger the shape parameter the narrower the distribution and the higher its peak value. Scale parameter influences the abscissa scale of a plot of data distribution. Note that the Weibull distribution is

Define the Integration Range of Wind Speed

Read in Moment Constraints

μn

Read in Initial Estimate of Lagrange Multipliers λ o

Calculate the Nonlinear Equations and Gradient Matrix g nk

Gn (λ )

Find the Residual Values of Lagrange Multipliers δ

No

λ = λ +δ o

Are the δ Acceptable ?

Obtain the Lagrange Multipliers

λ

Fig. 1. Flow chart of computation procedures of the Lagrange multipliers.

1822

F.J. Liu, T.P. Chang / Energy 36 (2011) 1820e1826

Table 1 Moment constraints used in calculating maximum entropy distribution. Stations

First order (m/s)

Second order (m2/s2)

Third order (m3/s3)

Fourth order (m4/s4)

Fifth order (m5/s5)

1 2 3 4

8.090021 9.632943 7.553532 9.996750

84.032042 119.489121 70.880152 137.660739

985.888332 1724.087144 772.333733 2255.617916

12,381.508722 27,417.346737 9388.757834 40,623.908925

162,705.337226 466,598.947361 123,925.481780 774,308.007434

(Dayuan) (Changhua) (Hengchun) (Chungtun)

expectation data. The classic solution of the maximum entropy problem can be written as:

related to a number of other probability distributions; in particular, it becomes the exponential distribution (if shape parameter a ¼ 1), and the Rayleigh distribution (if a ¼ 2). The Weibull shape parameters are usually greater than 1 in real wind speed distributions [3e6,10]. Several numerical methods can be applied to calculate the Weibull parameters in literature such as the moment method, empirical method, graphical method and maximum likelihood method [10]. In the present study the maximum likelihood method is selected for its accuracy given as:

f ðxÞ ¼ exp 

ln fn ðxÞ

(6)

"

Z

fn ðxÞexp 

N X

#

ln fn ðxÞ dx ¼ mn for n

n¼0

a ¼ 6 6

n 1X va n i¼1 i

#

where ln are the Lagrange multipliers that can be obtained by solving the following (N þ 1) nonlinear equations:

Gn ðlÞ ¼

6Pn va lnðv Þ Pn lnðv Þ7 i i¼1 i i 7  i¼1 Pn 7 a 5 4 n i ¼ 1 vi

N X n¼0

31

2

b¼

"

¼ 0; 1; .; N

(2)

(7)

These equations can be solved by the standard Newton method by expanding Gn ðlÞ in Taylor’s series around trial Lambda values o ðl Þ. Neglect the quadratic and higher order terms, then the resulted linear system can be solved iteratively as shown below.

!1=a (3)

 o  o t Gn ðlÞyGn l þ l  l ½VGn ðlÞðl ¼ lo Þ ¼ mn

where vi is the wind speed in time step i and n is the number of data points.

for n

¼ 0; 1; .; N

(8)

It can be simplified as a matrix form: 3. Probability density function with maximum entropy principle

Gd ¼ q

(9)

where G is the gradient matrix given by: The entropy of a probability density function f(x) is defined as [22]:

 G ¼

Z H ¼ 

f ðxÞln f ðxÞ dx

(4)

"

Z gnk ¼ 

(10)

fn ðxÞfk ðxÞexp 

#

N X

ln fn ðxÞ dx

(11)

n¼0

d and q are the residual vectors given by:

Z

fn ðxÞf ðxÞ dx ¼ mn for n ¼ 0; 1; .; N

for n; k ¼ 0; 1; .; N

where

Maximizing the entropy subject to some constraints enables one to find the most likely probability density function when the information available is limited. The (N þ 1) constraints for a physical system can generally be expressed as:

Effn ðxÞg ¼

 vGn ðlÞ ¼ ½gnk  vlk ðl ¼ lo Þ

d ¼ l  lo

(5)

(12)



where fn ðxÞ, n ¼ 0; 1; .; N with f0 ðxÞ ¼ 1, are the known functions for the system; mn , n ¼ 0; 1; .; N with m0 ¼ 1, are the

q ¼ m0  G0 ðlo Þ; .; mN  GN ðlo Þ

t

(13)

Table 2 Lagrange multipliers and Weibull parameters computed for different stations. Stations

Distributions

Lagrange multipliers

l0

l1

l2

l3

1

ME3 ME4 ME5 ME3 ME4 ME5 ME3 ME4 ME5 ME3 ME4 ME5

3.05843065176 4.46372824333 5.10002289108 3.83028168141 4.84898165530 5.35448667575 4.26322794118 5.10085163642 5.12837920416 3.31694130691 5.02276188718 5.62126059453

0.00750677342 1.37253895949 2.13471466196 0.29045255452 0.97727103034 1.42853447305 0.64582861768 1.27133026717 1.30074950535 0.13803966626 1.27182598872 1.77818316463

0.02179343430 0.31984164542 0.58527785419 0.01810670293 0.14152843762 0.25972547087 0.05730910740 0.18858305870 0.19795029384 0.00941624156 0.20579344310 0.33379422961

0.00155530914 0.02864190374 0.06624839930 0.00010355369 0.00812330517 0.02077288543 0.00096999466 0.01110984833 0.01234138986 0.00003749938 0.01212952915 0.02531770941

2

3

4

l4

Weibull parameters

2.82118 2.76477 2.75181 3.01740 2.99267 2.98876 2.69788 2.68681 2.68680 3.14866 3.06209 3.05712

1.9441

9.1154

1.9521

10.8748

2.1501

8.5388

1.6891

11.2314

l5 0.00005152025

0.00086949579 0.00318504790 0.00000978693 0.00016966715 0.00075651654 0.00000145060 0.00025591017 0.00032638369 0.00000939957 0.00024082626 0.00082824931

Entropy

a

b (m/s)

F.J. Liu, T.P. Chang / Energy 36 (2011) 1820e1826

observed wind speed ME3 pdf ME4 pdf ME5 pdf Weibull pdf ME3 cdf ME4 cdf ME5 cdf Weibull cdf observed cdf

0.08

0.12

1.6 1.4 1.2 1

0.06

0.8 0.6

0.04

0.1

0.08

1.8 1.6 1.4 1.2 1

0.06

0.8 0.6

0.04

0.4

0.4 0.02

0.02

0.2

0.2 0 0

2

4

6

8

0

0 10 12 14 16 18 20 22 24 26 28 30

0

2

4

6

8

10

12 14 16 18 20 Wind speed (m/s)

Wind speed (m/s) Fig. 2. Wind speed frequency and its cumulative distribution function for Station 1.

fn ðvÞ ¼ vn for n ¼ 0; 1; .; N

(14)

Then mn, n ¼ 0; 1; .; N with m0 ¼ 1, are the moments of the distribution representing the mean values of n powers of wind speed, which can be calculated from wind observation data. Consequently the corresponding expressions are given by:

" f ðvÞ ¼ exp 

N X

#

lm vm

(15)

m¼0

0.12

Wind speed frequency

0.1

0.08

0.06

0.8 0.6

0.04

lm vm dv ¼ mn for n ¼ 0; 1; .; N

"

Z gnk ¼ 

n k

v v exp 

N X

# m

lm v

dv

for n; k ¼ 0; 1; .; N

m¼0

(17) In the present study, three kinds of probability density functions of maximum entropy type will be considered, i.e. the ones with N ¼ 3, 4 and 5, that will be named ME3, ME4 and ME5 hereafter in this study. Flow chart illustrating the computation procedures of Lagrange multipliers is shown in Fig. 1. The integration range in Eqs. (16) and (17) is from minimum to maximum wind speed measured;

2

0.14 observed wind speed ME3 pdf ME4 pdf ME5 pdf Weibull pdf ME3 cdf ME4 cdf ME5 cdf Weibull cdf observed cdf

Station 4

1.6

1

0 30

#

N X

vn exp 

0.12

1.2

28

(16)

1.8

1.4

26

m¼0

0.1

Wind speed frequency

Station 2

Cumulative distribution function

observed wind speed ME3 pdf ME4 pdf ME5 pdf Weibull pdf ME3 cdf ME4 cdf ME5 cdf Weibull cdf observed cdf

"

Z Gn ðlÞ ¼

2

0.14

24

Fig. 4. Wind speed frequency and its cumulative distribution function for Station 3.

o

This system is solved for d, from which we let l ¼ l þ d, which o becomes a new initial vector l , the iteration procedure is continued until the d approaches a predefined tolerance. As for the purpose of wind speed distribution analysis, we let fn ðxÞ be the powers of wind speed (v) such that:

22

0.08

1.8 1.6 1.4 1.2 1

0.06

0.8 0.6

0.04

0.4 0.02

Cumulative distribution function

Wind speed frequency

0.1

observed wind speed ME3 pdf ME4 pdf ME5 pdf Weibull pdf ME3 cdf ME4 cdf ME5 cdf Weibull cdf observed cdf

Station 3

1.8

Wind speed frequency

0.12

Cumulative distribution function

Station 1

2

0.14

Cumulative distribution function

2

0.14

1823

0.4 0.02

0.2 0 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

0 30

0.2 0

0 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

Wind speed (m/s)

Wind speed (m/s)

Fig. 3. Wind speed frequency and its cumulative distribution function for Station 2.

Fig. 5. Wind speed frequency and its cumulative distribution function for Station 4.

F.J. Liu, T.P. Chang / Energy 36 (2011) 1820e1826 2

Station 1

0.18

0.16

Wind power frequency

0.14

0.12

0.2

1.8

0.18

1.6

0.16

1.4

1.2

0.1

1

0.08

0.8

0.06

0.6

0.04

0.02

0 2

4

6

8

10

12

14

16

18

20

22

24

26

28

Station 3

0.12

1.8

1.6

1.4

1.2

0.1

1

0.08

0.8

0.06

0.6

0.4

0.04

0.4

0.2

0.02

0.2

0 0

2 observed ME3 pdf ME4 pdf ME5 pdf Weibull pdf ME3 cdf ME4 cdf ME5 cdf Weibull cdf observed cdf

0.14

Wind power frequency

observed ME3 pdf ME4 pdf ME5 pdf Weibull pdf ME3 cdf ME4 cdf ME5 cdf Weibull cdf observed cdf

Cumulative distribution function

0.2

0

30

Cumulative distribution function

1824

0 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

Wind speed (m/s)

Wind speed (m/s)

Fig. 6. Wind power frequency and its cumulative distribution function for Station 1.

Fig. 8. Wind power frequency and its cumulative distribution function for Station 3.

acceptable residual values of Lagrange multipliers is set to be 1e6. All the procedures are implemented using computer program written in MATLAB languages.

The second one is the root mean square error (RMSE) defined as:

RMSE ¼

4. Fitness judgments

i¼1

1 ðy  yic Þ2 yic i

 (18)

where yi is the actual value at time stage i, yic is the value computed from correlation expression for the same stage, n is the number of data. 0.2

Q ¼ maxjTðvÞ  OðvÞj

observed ME3 pdf ME4 pdf ME5 pdf Weibull pdf ME3 cdf ME4 cdf ME5 cdf Weibull cdf observed cdf

0.2

0.16

0.14

0.12

1.2

0.1

1

0.04

0.4

0.2

0.02

0.2

0.02

0 8

10

12

14

16

18

20

22

24

26

28

1

0.4

0.04

6

0.1

1.2

0.6

0.6

4

0.12

1.4

0.06

0.06

2

0.14

1.6

0.8

0.8

0

1.8

observed ME3 pdf ME4 pdf ME5 pdf Weibull pdf ME3 cdf ME4 cdf ME5 cdf Weibull cdf observed cdf

0.08

0.08

0

Station 4

0.16

1.6

1.4

2

0.18

1.8

Cumulative distribution function

Station 2

(20)

where T(v) and O(v) are the cumulative distribution functions (cdf), for wind speed not exceeding v, calculated from theoretical function and from observation data, respectively. The relative error of wind power density is proposed by the present paper and is considered as a useful judgment, because it is most accurate and meaningful for wind energy application. Wind power density calculated from actual time-series data is expressed by:

2

0.18

Wind power frequency

The third judgment of accuracy is the max error in the KolmogoroveSmirnov test given by:

Wind power frequency

c2 ¼

n  X

(19)

30

0

Cumulative distribution function

To examine how well a theoretical probability density function fits with observation data, four judgments are considered in this paper, which shown as follows: The first one is the Chi-square error given as:

#1=2 " n 1X ðyi  yic Þ2 n i¼1

0 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

Wind speed (m/s)

Wind speed (m/s)

Fig. 7. Wind power frequency and its cumulative distribution function for Station 2.

Fig. 9. Wind power frequency and its cumulative distribution function for Station 4.

F.J. Liu, T.P. Chang / Energy 36 (2011) 1820e1826

1825

Table 3 Statistical analysis parameters for Station 1. Distributions

Weibull ME3 ME4 ME5

Wind speed

Power density

Power density

c2

RMSE

Max error in cdf

c2

RMSE

Max error in cdf

Relative error (%)

0.194212 0.157531 0.060950 0.022140

0.018064 0.016409 0.008202 0.005607

0.105412 0.064868 0.022457 0.014744

0.406750 0.083832 0.173930 0.067117

0.027295 0.012829 0.010646 0.008782

0.232550 0.068505 0.038869 0.032167

5.192725 6.4538e007 6.3593e006 0.010913

Table 4 Statistical analysis parameters for Station 2. Distributions

Weibull ME3 ME4 ME5

Wind speed

Power density RMSE

Max error in cdf

c2

RMSE

Max error in cdf

Relative error (%)

0.036883 0.064249 0.023932 0.015646

0.006238 0.008566 0.003508 0.003281

0.058718 0.038503 0.011819 0.008080

0.123883 0.085706 0.066775 0.033291

0.012539 0.010599 0.007833 0.005110

0.094171 0.055493 0.032352 0.019101

0.434152 0.014216 1.0897e006 0.001695

1 P1 ¼ rv3 2

(21)

where r is the air density; v3 is the mean of wind speed cubes. Similar power density based on theoretical probability density function can be obtained by the integration:

P2 ¼

1 r 2

Power density

c2

Z v3 f ðvÞdv

(22)

5. Results and discussion Table 1 lists the moment constraints in calculating maximum entropy distribution for the four stations studied. These values are given with six digits behind decimal point. It needs the former 3, 4 and 5 orders of constraints, respectively, while calculating ME3, ME4 and ME5 probability distributions. Table 2 summarizes the computed Lagrange multipliers for different stations. Eleven digits behind decimal point are available in the list; that is required to guarantee the accuracy of computation since their tiny changes may cause an extreme variation in its exponential probability function. Clearly, for a particular maximum entropy distribution, the Lagrange multiplier for higher power of wind speed is smaller than that for lower one. The max iteration numbers in searching acceptable Lagrange multipliers never exceed 20 for the four stations in the present study. The entropy quantity depending on the number of constraints used has the smallest value for ME5 and the largest value for ME3, regardless of station considered. Both the shape and scale parameters computed for the Weibull probability density function are also provided in the table. Figs. 2e5 show the wind speed frequency histograms for the four stations; Figs. 6e9 show the corresponding power density histograms. Maximum entropy distributions computed with 3, 4 and 5 moment constraints (i.e. ME3, ME4 and ME5) are plotted

together with the Weibull distribution. Corresponding cumulative distribution functions are referred to the right ordinate. Tables 3e6 illustrate relevant statistical parameters for the stations. As for Stations 1 and 4 at which two significant speed humps are found on it, the Weibull distribution describes worse the observation data getting larger errors as listed in Tables 3 and 6. For example, its cdf max error in Station 1 exceeds 0.105 and 0.232 for wind speed and power density respectively; and they exceed 0.107 and 0.140 in Station 4. All these values are close to or greater than the critical value of 95% confident level in the KolmogoroveSmirnov test. Meanwhile the relative percent error of power density between the Weibull function and observation data reaches 5.19% and 5.40% for Stations 1 and 4, respectively. On the other hand, ME3 distribution fits with the observation data slightly better than the Weibull distribution, while ME4 and ME5 present better than the Weibull one having pretty lower errors for both wind speed and power density data. The relative errors of power density between the maximum entropy distributions and observation data are far less than those between the Weibull function and observation data, even the highest error occurred in Station 1 obtained from ME5 is only 0.010%. While for Stations 2 and 3, wind observations are more concentrated and not too dispersive as compared with those in Stations 1 and 4, therefore the Weibull distribution, in addition to the ME-series distributions, can characterize the observation data properly. Note that the maximum entropy distribution describe effectively the appearance of null winds that might happen in some cases, so its curve may start at a non-zero value; however the Weibull function does not. This is one of the reasons why the maximum entropy distributions outperform the conventional Weibull function. Another reason is that the maximum entropy distributions are able to describe bimodal distributions that usually happen in some measurements due to special wind structures. Another thing worth to note is that the c2 , RMSE and cdf max error calculated from wind power density data are generally larger

Table 5 Statistical analysis parameters for Station 3. Distributions

Weibull ME3 ME4 ME5

Wind speed

Power density

Power density

c2

RMSE

Max error in cdf

c2

RMSE

Max error in cdf

Relative error (%)

0.016998 0.041669 0.022035 0.021978

0.006281 0.005849 0.005848 0.005840

0.054187 0.021762 0.023462 0.023215

0.070221 0.115101 0.027691 0.028596

0.009769 0.011252 0.007019 0.007178

0.042610 0.056393 0.026668 0.027642

0.178057 0.006018 1.6937e005 0.000235

1826

F.J. Liu, T.P. Chang / Energy 36 (2011) 1820e1826

Table 6 Statistical analysis parameters for Station 4. Distributions

Weibull ME3 ME4 ME5

Wind speed

Power density

Power density

c2

RMSE

Max error in cdf

c2

RMSE

Max error in cdf

Relative error (%)

0.146833 0.195836 0.038291 0.026325

0.013426 0.016205 0.006457 0.006031

0.107162 0.060583 0.019987 0.012058

0.321641 0.267628 0.079938 0.043327

0.021264 0.019296 0.007852 0.005743

0.140072 0.125811 0.032372 0.012958

5.408826 0.078518 1.5773e006 0.000546

than those from wind speed data, independent of distribution functions and independent of stations considered. That is, a probability function showing a good fit to speed data does not absolutely show a good fit to power density data. This is because power density distribution is heavily weighted towards high wind speed range. Therefore the fitness judgment for a given probability function based on the analysis of wind power density would be of importance. Meanwhile the main disadvantage concerning the maximum entropy distribution is that the computation procedures of parameters are more complex when compared with conventional Weibull distribution. The performance of the ME4 or ME5 distribution proposed is superior to those probability functions adopted by other researches [14,18,19] that reveal unstable accuracy when facing different wind regimes having calms. For the consideration of computation burden, four-constraints maximum entropy distribution (ME4) is recommend therefore. 6. Conclusions The validity of a probability density function with maximum entropy principle has been analyzed considering different numbers of moment constraints. Relevant algorithm procedures used to calculate the distribution parameters different from previous researches are presented. Statistical parameters including the Chisquare error, root mean square error and max error in KolmogoroveSmirnov test as well as the relative error of power density between theoretical and observed data are considered as judgment criterions. The conclusions are drawn as follows: (a) Maximum entropy distribution outperforms a lot the conventional Weibull function irrespective of wind speed or power density analyzed, especially for wind regimes having two humps on it. (b) Statistical errors calculated from wind power density are commonly larger than those from wind speed data, independent of distribution functions and stations considered. As a result, the judgment criterion in choosing a probability function based on the analysis of power density would be of importance. (c) For the consideration of computation burden using four moment constraints to calculate maximum entropy distribution (ME4) is recommended for wind energy application. Acknowledgments The author would deeply appreciate the Central Weather Bureau and Ministry of Economic Affairs for providing observation data and deeply thank Dr. Wu CF and Dr. Huang MW, researchers of the Institute of Earth Sciences, Academia Sinica, Taiwan, for their

precious comments. This study was partly supported by the National Science Council under contract NSC99-2221-E-252-011.

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